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+# Black-Box Dissector: Towards Erasing-based Hard-Label Model Stealing Attack
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+Previous studies have verified that the functionality of black-box models can be stolen with full probability outputs. However, under the more practical hard-label setting, we observe that existing methods suffer from catastrophic performance degradation. We argue this is due to the lack of rich information in the probability prediction and the overfitting caused by hard labels. To this end, we propose a novel hard-label model stealing method termed black-box dissector, which consists of two erasing-based modules. One is a CAM-driven erasing strategy that is designed to increase the information capacity hidden in hard labels from the victim model. The other is a random-erasing-based self-knowledge distillation module that utilizes soft labels from the substitute model to mitigate overfitting. Extensive experiments on four widely-used datasets consistently demonstrate that our method outperforms state-of-the-art methods, with an improvement of at most $8 . 2 7 \%$ . We also validate the effectiveness and practical potential of our method on real-world APIs and defense methods. Furthermore, our method promotes other downstream tasks, i.e., transfer adversarial attacks.
+
+# 16 1 Introduction
+
+17 Machine learning models deployed on the cloud can serve users through the application program   
+18 interfaces (APIs) to improve productivity. Since developing these cloud models is a product of   
+19 intensive labor and monetary effort, these models are valuable intellectual property and AI companies   
+20 try to keep them private. However, the exposure of the model’s predictions represents a significant   
+21 risk as an adversary can leverage this information to steal the model’s functionality, a.k.a. model   
+22 stealing attack [22, 20, 21]. With such an attack, adversaries are able to not only use the stolen model   
+23 to make a profit, but also mount further adversarial attacks [34, 29]. Besides, the model stealing   
+24 attacks is a kind of black-box knowledge distillation which is a hot research topic. Studying various   
+25 mechanisms of model stealing attack is of great interest both to AI companies and researchers.   
+26 Previous methods [20, 34, 21] mainly assume the complete probability predictions of the victim   
+27 model available, while the real-world APIs usually only return partial probability values (top- $k$   
+28 predictions) or even the top-1 prediction (i.e., hard label). In this paper, we focus on the more   
+29 challenging and realistic scenario, i.e., the victim model only outputs the hard labels. However, under   
+30 this setting, existing methods suffer from a significant performance degradation, even by $3 0 . 5 0 \%$ (as   
+31 shown in the Fig. 1 (a) and the appendix Tab. I).   
+32 To investigate the reason for the degradation, we evaluate the performance of attack methods with   
+33 different numbers of prediction probability categories available and hard labels as in Fig. 1 (b). With   
+34 the observation that the performance degrades when the top- $k$ information missing, we conclude   
+35 that the top- $k$ predictions are informative as it indicates the similarity of different categories or   
+36 multiple objects in the picture, and previous attack methods suffer from such information obscured   
+37 by the top-1 prediction under the hard-label setting. It motivates us to re-mine this information by   
+38 eliminating the top-1 prediction. Particularly, we design a novel CAM-based erasing method, which   
+39 erases the important area on the pictures based on the substitute model’s top-1 class activation maps   
+40 (CAM) [24, 33] and queries the victim model for a new prediction. Note that we can dig out other   
+41 class information in this sample if the new prediction changes. Otherwise, it proves that the substitute   
+42 model pays attention to the wrong area. Then we can align the attention of the substitute and the   
+43 victim model by learning clean samples and the corresponding erased samples simultaneously.   
+4 Besides, previous works on the   
+45 self-Knowledge Distillation (self  
+46 KD) [15], calibration [8], and noisy   
+47 label [31] have pointed out the   
+48 hard and noisy labels will introduce   
+49 overfitting and miscalibration. More   
+50 specifically, the attack algorithms   
+51 cannot access the training data, and   
+52 thus can only use the synthetic data or   
+53 other datasets as a substitute, which   
+54 is noisy. Therefore, the hard-label   
+55 setting will suffer from overfitting,   
+56 which leads to worse performance,   
+57 and we verify it by plotting the loss   
+58 curves in Fig. 1 (c). To mitigate   
+59 this problem, we introduce a simple   
+60 self-knowledge distillation module   
+61 with random erasing $( R E )$ to utilize   
+62 soft labels for generalization. Partic  
+63 ularly, we randomly erase one sample   
+64 a certain number of times, query   
+65 the substitute model for soft-label   
+66 outputs, and take the average value   
+67 of these outputs as the pseudo-label.   
+68 After that, we use both hard labels
+
+![](images/57d39e95c20569e8951b065bee0a280df2a9efb4bec0e20510253a11377359f2.jpg)  
+Figure 1: (a) The test accuracies of previous methods with hard labels are much lower than the ones with soft labels. (KN: KnockoffNets, ‘AT’: ActiveThief, $\mathbf { \hat { E } } ^ { \prime }$ : entropy, ‘K’: $\mathbf { k }$ -Center, ‘D’: DFAL) (b) The performance decreases as the number of available classes decreases (dotted line : hardlabel setting). (c) & (d) Loss curves for training/test set during model training without and with self-KD. All results are on the CIFAR10 dataset.
+
+from the victim model and pseudo labels from the previous substitute model to train a new substitute model. Therefore, we can also consider the ensemble of the two models as the teacher in knowledge distillation. As in Fig. 1 (d), such a module helps generalization and better performance.
+
+72 In summary, we propose a novel model stealing framework termed black-box dissector, which   
+73 includes a CAM-driven erasing strategy and a RE-based self-KD module. Our method is orthogonal   
+74 to previous approaches [20, 21] and can be integrated with them. The experiments on four widely  
+75 used datasets demonstrate our method achieves $4 3 . 0 4 - 9 0 . 5 7 \%$ test accuracy $( 4 7 . 6 0 - 9 1 . 3 7 \%$   
+76 agreement) to the victim model, which is at most $8 . 2 7 \%$ higher than the state of the art method.   
+77 We also proved that our method can defeat popular defense methods and is effective for real-world   
+78 APIs like services provided by Amazon Web Services (AWS). Furthermore, our method promotes   
+79 downstream tasks, i.e., transfer adversarial attack, with $4 . 9 1 \% - 1 6 . 2 0 \%$ improvement.
+
+# 80 2 Background and Notions
+
+81 Model stealing attack is aim to find a substitute model $\hat { f } \colon [ 0 , 1 ] ^ { d } \mapsto \mathbb { R } ^ { N }$ that performs as similarly   
+82 as possible to the black-box victim model $f \colon [ 0 , 1 ] ^ { d } \mapsto \mathbb { R } ^ { N }$ (with only outputs accessed). Papernot   
+83 et al. [22] first observed that online models could be stolen through multiple queries. After that, due   
+84 to the practical threat to real-world APIs, several studies paid attention to this problem and proposed   
+85 many attack algorithms.   
+86 These algorithms consist of two stages: 1) constructing a transfer dataset $D _ { T }$ (step 1 in Fig. 2) and   
+87 2) training a substitute model. The transfer dataset is constructed based on data synthesis or data   
+88 selection and then feed into the victim model for labels. Methods based on data synthesis [34, 14, 2]   
+89 adopt the GAN-based models to generate a virtual dataset. And the substitute model and the GAN   
+90 model are trained alternatively on this virtual dataset by querying the victim model iteratively. The   
+91 data selection methods prepare an attack dataset as the data pool, and then sample the most informative   
+92 data via machine learning algorithms, e.g., reinforcement learning [20] or active learning strategy [21],   
+93 uncertainty-based strategy [17], k-Center strategy [25], and DFAL strategy [5]. Considering that   
+94 querying the victim model will be costly, the attacker usually sets a budget on the number of the   
+95 queries, so the size of the transfer dataset should be limited as well. Previous methods assume the   
+96 victim model returns a complete probability prediction $f ( x )$ , which is less practical.   
+97 In this paper, we focus on a more practical scenario that is about hard-label $\phi ( f ( x ) )$ setting, where $\phi$   
+98 is the truncation function used to truncate the information contained in the victim’s output and return   
+99 the corresponding one-hot vector:
+
+![](images/6580de836653139a0778be4cd12a5c4ed0c61b4f161900ec361366e026646038.jpg)  
+Figure 2: Details of our proposed black-box dissector with a CAM-driven erasing strategy (step 2.1) and a RE-based self-KD module (step 2.2). In step 2.1, the images in transfer set $D _ { T }$ are erased according to the Grad-CAM, and we selected the erased images with the largest difference from the original images according to the substitute model’s outputs. In step 2.2, we randomly erase the unlabeled image $N$ times, and then average the outputs of the $N$ erased images by the substitute model as the pseudo-label.
+
+$$
+\phi ( f ( x ) ) _ { i } : = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ } } i = \arg \operatorname* { m a x } _ { n } f ( x ) _ { n } ; } \\ { 0 } & { { \mathrm { o t h e r w i s e } } . } \end{array} \right. }
+$$
+
+100 With the transfer dataset, the substitute model is optimized by minimizing a loss function $\mathcal { L }$ (e.g.,   
+101 cross-entropy loss function):
+
+$$
+\left\{ \begin{array} { l l } { \mathbb { E } _ { x \sim \mathcal { D } _ { T } } \left[ \mathcal { L } \big ( f ( x ) , \hat { f } ( x ) \big ) \right] , } & { \mathrm { f o r ~ s o f t ~ l a b e l s ; } } \\ { \mathbb { E } _ { x \sim \mathcal { D } _ { T } } \left[ \mathcal { L } \big ( \phi ( f ( x ) ) , \hat { f } ( x ) \big ) \right] , } & { \mathrm { f o r ~ h a r d ~ l a b e l s . } } \end{array} \right.
+$$
+
+102 Knowledge distillation (KD) has been widely studied in machine learning [10, 1, 6], which transfers   
+103 the knowledge from a teacher model to a student model. Model stealing attacks can be regarded as a   
+104 black-box KD problem where the victim model is the teacher with only outputs accessible and the   
+105 substitute model is the student. The main reason for the success of KD is the valuable information   
+106 that defines a rich similarity structure over the data in the probability prediction [10]. However,   
+107 for the hard-label setting discussed in this paper, this valuable information is lost. Inspired by KD,   
+108 our method tries to dig out the hidden information in the data and models, and then transfers more   
+109 knowledge to the substitute model.   
+110 The erasing-based method, e.g., random erasing (RE) [32, 3], is currently one of the widely used   
+111 data augmentation methods, which generates training images with various levels of occlusion, thereby   
+112 reducing the risk of over-fitting and improving the robustness of the model. Our work is inspired   
+113 by RE and designs a prior-driven erasing operation, which erases the area corresponding to the hard   
+114 label to re-mine missing information.   
+116 The overview of our proposed black-box dissector is shown in Fig. 2. In addition to the conventional   
+117 process (i.e., the transfer dataset $D _ { T }$ constructing in step 1 and the substitute model training in the   
+118 right), we introduce two key modules: a CAM-driven erasing strategy (step 2.1) and a RE-based   
+119 self-KD module (step 2.2).
+
+# 20 3.1 A CAM-driven erasing strategy
+
+121 Since the lack of class similarity infor  
+122 mation degrades the performance of   
+123 previous methods under the hard-label   
+124 setting, we try to re-dig out such hid  
+125 den information. Taking an example   
+126 from the ILSVRC-2012 dataset for il  
+127 lustration as in Fig. 3. Querying the   
+128 CUBS200 trained victim model with   
+129 this image, we get two classes with   
+130 the highest confidence score: “Anna   
+131 hummingbird" (0.1364) and “Com  
+132 mon yellowthroat" (0.1165), and show   
+133 their corresponding attention map in   
+134 the first column of Fig. 3. It is easy to   
+135 conclude that two different attention   
+136 regions response for different classes   
+137 according to the attention map. When   
+138 training the substitute model with the   
+139 hard label “Anna hummingbird" and   
+140 without the class similarity informa
+
+![](images/96fc3450a533eeb97226572a27ca1d168a7cefcf39bb07ad10603539c78f7c73.jpg)  
+Figure 3: An example from the ILSVRC-2012 dataset and its attention map corresponding to two most likely class “Anna humming bird" and “Common yellow throat" on the CUBS200 trained model. The attention areas share similar visual apparent with images of “Anna humming bird" and “Common yellow throat", respectively.
+
+tion, the model can not learn from the area related to the “Common yellowthroat" class, which means this area is wasted. To re-dig out the information about the “Common yellowthroat" class, we need to erase the impact of the “Anna hummingbird" class.
+
+144 To this end, a natural idea is to erase the response area corresponding to the hard label. Since the   
+145 victim model is a black-box model, we use the substitute model to approximately calculate the   
+146 attention map instead. If the attention map calculated by the substitute model is inaccurate and the   
+147 victim model’s prediction on the erased image does not change, we can also align the attention map of   
+148 two models by letting the substitute model learn the original image and the erased one simultaneously.   
+149 The attention map is also a kind of supervision signal pushing two models to be similar [30]. To   
+150 get the attention map, we utilize the Grad-CAM [24] in this paper. With the input image $x \in [ 0 , 1 ] ^ { d }$   
+151 and the trained DNN $\mathcal { F } \colon [ 0 , 1 ] ^ { d } \mapsto \mathbb { R } ^ { N }$ , we let $\alpha _ { k } ^ { c }$ denote the weight of class $c$ corresponding to the   
+152 k-th feature map, and calculate it as αck = 1Z Pi Pj , where $Z$ is the number of pixels in the   
+153 feature map, $\mathcal { F } ( x ) ^ { c }$ is the score of class $c$ and $A _ { i j } ^ { k }$ is the value of pixel at $( i , j )$ in the $k$ -th feature   
+154 map. After obtaining the weights corresponding to all feature maps, the final attention map can be   
+155 obtained as $\begin{array} { r } { S _ { \mathrm { G r a d - C A M } } ^ { c } = \tilde { \mathrm { R e L U } } ( \sum _ { k } \hat { \alpha _ { k } ^ { c } } A ^ { k } ) } \end{array}$ via weighted summation.   
+156 To erase the corresponding area, inspired by [32], we define a prior-driven erasing operation as   
+157 $\psi ( I , P )$ , shown in Alg. 1, which randomly erases a rectangle region in the image $I$ with random   
+158 values while the central position of the rectangle region is randomly selected following the prior   
+159 probability $P$ . The prior probability $P$ is of the same size as the input image and is used to determine   
+160 161 the prior. Let the probability of different pixels being erased. Here, we use the attention map from Grad-CAM as $x \in [ 0 , 1 ] ^ { d }$ denote the input image from the transfer set and $S _ { \mathrm { G r a d - C A M } } ^ { \mathrm { a r g m a x } \hat { f } ( x ) } ( x , \hat { f } )$ denote   
+162 the attention map of the substitute model $\hat { f }$ . This CAM-driven erasing operation can be represented:
+
+$$
+\psi \left( x , S _ { \mathrm { G r a d - C A M } } ^ { \mathrm { a r g m a x } \hat { f } ( x ) } ( x , \hat { f } ) \right) .
+$$
+
+163 We abbreviate it as $\psi ( x , S ( x , { \hat { f } } ) )$ . To alleviate the impact of inaccurate CAM caused by the difference   
+164 between the substitute model and the victim one, for each image, we perform this operation $N$ times   
+165 $\psi _ { i }$ means the $i$ -th erasing) and select the one with the largest difference from the original label.
+
+# Algorithm 1: Prior-driven Erasing Operation $\psi ( I , P )$
+
+Input: Input image $I$ , prior probability $P$ , image size $W$ and $H$ , area of image $S$ , erasing area ratio range $s _ { l }$ and $s _ { h }$ , erasing aspect ratio range $r _ { 1 }$ and $r _ { 2 }$ .
+
+Output: Erased image $I ^ { \prime }$
+
+$S _ { e } \overset { \bar { \mathbf { \sigma } } } {  } \operatorname { R a n d } ( s _ { l } , s _ { h } ) \overset { \bar { \mathbf { \sigma } } } { \times } S$ , $r _ { e } \gets \mathrm { R a n d } ( r _ { 1 } , r _ { 2 } ) ^ { 1 }$
+
+3 $x _ { e } , y _ { e }$ sampled randomly according to $P$   
+4 $I _ { e } \gets ( x _ { e } - W _ { e } , y _ { e } - H _ { e } , x _ { e } + W _ { e } , y _ { e } + H _ { e } )$   
+5 $I ( I _ { e } ) \gets \mathrm { R a n d } ( 0 , 2 5 5 )$   
+6 $I ^ { \prime }  I$   
+166 Such a data augment operation helps the erasing process to be more robust. We use the cross-entropy   
+167 to calculate the difference between the new label and the original label, and we want to select the   
+168 sample with the biggest difference. Formally, we define $\Pi ( x )$ as the function to select the most   
+169 different variation of image $x$ :
+
+$$
+\begin{array} { r l r } {  { \Pi ( x ) : = \psi _ { k } ( x , S ( x , \hat { f } ) ) , } } \\ & { } & { \quad \mathrm { w h e r e } \ k : = \underset { i \in [ N ] } { \arg \operatorname* { m a x } } - \sum _ { j } \phi ( f ( x ) ) _ { j } \cdot \log \Big ( \hat { f } \big ( \psi _ { i } ( x , S ( x , \hat { f } ) ) \big ) _ { j } \Big ) } \\ & { } & { = \underset { i \in [ N ] } { \arg \operatorname* { m a x } } - \log \bigg ( \hat { f } \big ( \psi _ { i } ( x , S ( x , \hat { f } ) ) \big ) _ { \arg \operatorname* { m a x } \phi \big ( f ( x ) \big ) } \bigg ) } \\ & { } & { = \underset { i \in [ N ] } { \arg \operatorname* { m i n } } \hat { f } ( \psi _ { i } ( x , S ( x , \hat { f } ) ) ) _ { \arg \operatorname* { m a x } \phi \big ( f ( x ) \big ) } . } \end{array}
+$$
+
+170 Due to the limitation of the number of queries, we cannot query the victim model for each erased   
+171 image to obtain a new label. We continuously choose the erased image with the highest substitute’s   
+172 confidence until reaching the budget. To measure the confidence of the model, we adopt the Maximum   
+173 Softmax Probability (MSP) for its simplicity:
+
+$$
+\begin{array} { r l } & { \underset { x \sim \mathcal { D } _ { T } } { \arg \operatorname* { m a x } } M S P \left( \hat { f } \left( \Pi \left( x \right) \right) \right) } \\ & { = \underset { x \sim \mathcal { D } _ { T } } { \arg \operatorname* { m a x } } \hat { f } \left( \Pi \left( x \right) \right) _ { \mathrm { a r g m a x } } \hat { f } ( \Pi ( x ) ) , } \end{array}
+$$
+
+174 where $D _ { T }$ is the transfer set. The erased images selected in this way are most likely to change the   
+175 prediction class. Then, we query the victim model to get these erased images’ labels and construct   
+176 an erased sample set $D _ { E }$ . Note that when the victim model’s predictions on the erased images   
+177 change, it means our erasing method does dig out other related class information in the sample. With   
+178 the unchanged predictions, it points out the attentions of the substitute model and the victim are   
+179 inconsistent. Though wrong attention areas erased, training with these samples benefits aligning the   
+180 attentions of two models. As [30] stated, the attention alignment can help more powerful KD.
+
+# 181 3.2 A random-erasing-based self-KD module
+
+182 We also find that in training with limited hard-label OOD samples, the substitute model is likely   
+183 to overfit the training set, which damages its generalization ability [15, 31]. Therefore, based on   
+184 the above erasing operation, we further design a simple RE-based self-KD method to improve the   
+185 generalization ability of the substitute model.   
+186 Formally, let $x \in [ 0 , 1 ] ^ { d }$ denote the unlabeled input image. We perform the erasing operation with a   
+187 uniform prior $U$ on it $N$ times, and then average the substitute’s outputs on these erased images as   
+188 the pseudo-label of the original image:
+
+$$
+y _ { p } ( x , \hat { f } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \hat { f } \big ( \psi _ { i } ( x , U ) \big ) .
+$$
+
+${ } ^ { 1 } \mathrm { R a n d } ( a , b )$ returns an evenly distributed random real number in the range of $a$ to $b$ .
+
+# Algorithm 2: Black-box Dissector
+
+Input: Unlabeled pool $D _ { U }$ , victim model $f$ , maximum number of queries $Q$ .   
+Output: Substitute model $\hat { f }$ .   
+1 Initialize $q  0 , D _ { T }  \emptyset , D _ { E }  \emptyset$   
+2 while $q < Q$ do   
+3 // Step 1   
+4 Select samples from $D _ { U }$ according to budget and query $f$ to updata $D _ { T }$   
+5 $q = q +$ budget   
+6 $\begin{array} { r } { \mathcal { L } = \sum _ { x \in D _ { T } } \mathcal { L } ^ { \prime } \big ( \phi ( f ( x ) ) , \hat { f } ( x ) \big ) } \end{array}$   
+7 $\hat { f } \gets u p d a t e ( \hat { f } , \mathcal { L } )$   
+8 // A CAM-driven erasing strategy (step 2.1)   
+9 Erase samples in $D _ { T }$ according to Eq. 4   
+10 Choose samples from erased samples according to Eq. 5 and budget   
+11 Query $f$ to get labels and updata $D _ { E }$   
+12 $\begin{array} { r } { \mathcal { L } = \sum _ { x \in D _ { T } \cup D _ { E } } \mathcal { L } ^ { \prime } \big ( \phi ( f ( x ) ) , \hat { f } ( x ) \big ) } \end{array}$   
+13 $\hat { f } \gets u p d a t e ( \hat { f } , \mathcal { L } )$   
+14 $q = q +$ budget   
+15 // A random-erasing-based self-KD (step 2.2)   
+16 Select samples from $D _ { U }$   
+17 Get pseudo-labels according to Eq. 6 and construct a pseudo-label set $D _ { P }$   
+18 $\begin{array} { r l } & { \mathcal { L } = \sum _ { x \in D _ { T } \cup D _ { E } } \mathcal { L } ^ { \prime } \big ( \phi ( f ( x ) ) , \hat { f } ( x ) \big ) + \sum _ { x \in D _ { P } } \mathcal { L } ^ { \prime } \big ( y _ { p } ( x , \hat { f } ) , \hat { f } ( x ) \big ) } \\ & { \hat { f }  u p d a t e ( \hat { f } , \mathcal { L } ) } \end{array}$   
+19   
+20 end   
+189 This is a type of consistency regularization, which enforces the model to have the same predictions   
+190 for the perturbed images and enhances the generalization ability. With Eq.6, we construct a new soft   
+191 pseudo label set $D _ { P } = \{ { \bigl ( } x , y _ { p } ( x , { \hat { f } } ) { \bigr ) } , \ldots \}$ .   
+192 With the transfer set $D _ { T }$ , the erased sample set $D _ { E }$ , and the pseudo-label set $D _ { P }$ , we train a new   
+193 substitute model using the ensemble of the victim model and the previous substitute model as the   
+194 teacher. Our final objective function is:
+
+$$
+\operatorname* { m i n } \mathcal { L } = \operatorname* { m i n } \big [ \sum _ { x \in D _ { T } \cup D _ { E } } \mathcal { L } ^ { \prime } \big ( \phi ( f ( x ) ) , \hat { f } ( x ) \big ) + \sum _ { x \in D _ { P } } \mathcal { L } ^ { \prime } \big ( y _ { p } ( x , \hat { f } ) , \hat { f } ( x ) \big ) \big ] .
+$$
+
+where 195 $\mathcal { L } ^ { \prime }$ can be commonly used loss functions, e.g., cross-entropy loss function.
+
+196 To sum up, we built our method on the conventional process of the model stealing attack (step   
+197 1), and proposed a CAM-driven erasing strategy (step 2.1) and a RE-based self-KD module (step   
+198 2.2) unified by a novel erasing method. The former strategy digs out missing information between   
+199 classes and aligns the attention while the latter module helps to mitigate overfitting and enhance the   
+200 generalization. We name the whole framework as black-box dissector and present the algorithm   
+201 detail of it in Alg. 2.
+
+# 4 Experiment
+
+# 4.1 Experiment settings
+
+Victim model. The victim models we used (ResNet-34 [9]) are trained on four datasets, namely, CIFAR10 [16], SVHN [19], Caltech256 [7], and CUBS200 [28], and their test accuracy are $9 1 . 5 6 \%$ , $9 6 . 4 5 \%$ , $7 8 . 4 0 \%$ , and $7 7 . 1 0 \%$ , respectively. All models are trained using the SGD optimizer with momentum (of 0.5) for 200 epochs with a base learning rate of 0.1 decayed by a factor of 0.1 every 30 epochs. Following [20, 21, 34], we use the same architecture for the substitute model and will analyze the impact of different architectures in the supplementary.
+
+210 Attack dataset. We use $1 . 2 M$ images without labels from the ILSVRC-2012 challenge [23] as the   
+211 attack dataset. In a real attack scenario, the attacker may use pictures collected from the Internet, and   
+212 the ILSVRC-2012 dataset can simulate this scenario well. Note that we resize all images in the attack   
+213 dataset to fit the size of the target datasets, which is similar to the existing setting [20, 21, 34].
+
+Table 1: The agreement and test accuracy (in $\%$ ) of each method under $3 0 \mathrm { k }$ queries. For our model, we report the average accuracy as well as the standard deviation computed over 5 runs. (Boldface: the best value, italics: the second best value.)   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR10</td><td colspan="2">SVHN</td><td colspan="2">Caltech256</td><td colspan="2">CUBS200</td></tr><tr><td>Agreement</td><td>Acc</td><td>Agreement</td><td>Acc</td><td>Agreement</td><td>Acc</td><td>Agreement</td><td>Acc</td></tr><tr><td>KnockoffNets</td><td>75.32</td><td>74.44</td><td>85.00</td><td>84.50</td><td>57.64</td><td>55.28</td><td>30.01</td><td>28.03</td></tr><tr><td>ActiveThief(Entropy)</td><td>75.26</td><td>74.21</td><td>90.47</td><td>89.85</td><td>56.28</td><td>54.14</td><td>32.05</td><td>29.43</td></tr><tr><td>ActiveThief(k-Center)</td><td>75.71</td><td>74.24</td><td>81.45</td><td>80.79</td><td>61.19</td><td>58.84</td><td>37.68</td><td>34.64</td></tr><tr><td>ActiveThief(DFAL)</td><td>76.72</td><td>75.62</td><td>84.79</td><td>84.17</td><td>46.92</td><td>44.91</td><td>20.31</td><td>18.69</td></tr><tr><td>ActiveThief(DFAL+k-Center)</td><td>74.97</td><td>73.98</td><td>81.40</td><td>80.86</td><td>55.70</td><td>53.69</td><td>26.60</td><td>24.42</td></tr><tr><td>Ours+Random</td><td>82.14±0.16</td><td>80.47±0.02</td><td>92.33±0.47</td><td>91.57±0.29</td><td>62.15±0.52</td><td>59.91±0.58</td><td>38.28±0.31</td><td>35.24±0.49</td></tr><tr><td>Ours+k-Center</td><td>80.84±0.21</td><td>79.27±0.15</td><td>91.47±0.09</td><td>90.68±0.14</td><td>65.12±0.56</td><td>62.72±0.57</td><td>46.69±0.87</td><td>42.91±0.46</td></tr></table>
+
+![](images/2bac624872db9b5c764a32c26a8ace18d0db089797185f4c817042cd0158bb85.jpg)  
+Figure 4: Curves of the test accuracy versus the number of queries.
+
+Training process. We use the SGD optimizer with momentum (of 0.9) for 200 epochs and a base learning rate of $\begin{array} { r } { 0 . 0 2 \times \frac { b a t c h s i z e } { 1 2 8 } } \end{array}$ decayed by a factor of 0.1 every 60 epochs. The weight decay is set to $5 \times 1 0 ^ { - 4 }$ for small datasets (CIFAR10 [16] and SVHN [19]) and 0 for others. We set up a query sequence {0.1K, 0.2K, 0.5K, 0.8K, 1K, 2K, 5K, 10K, 20K, $\mathrm { 3 0 K } \}$ as the iterative maximum query budget, and stop the sampling stage whenever reaching the budget at each iteration.
+
+Baselines and evaluation metric. We mainly compare our method with KnockoffNets [20] and ActiveThief [21]. Follow Jagielski et al. [12], we mainly report the test accuracy (Acc) as the evaluation metric. We also report the Agreement metric proposed by Pal et al. [21] which counts how often the prediction of the substitute model is the same as the victim’s as a supplement.
+
+# 4.2 Experiment results
+
+We first report the performance of our method compared with previous methods. After that, we conduct ablation experiments to analyze the contribution of each module. Finally, we also analyze the performance of our method when encountering defense methods and real-world online APIs. More experiments (e.g., adversarial attack and overfitting analysis) can be found in our supplementary.
+
+Effectiveness of our method. As in Tab. 1, the test accuracy and agreement of our method are all better than the previous methods. We also plot the curves of the test accuracy versus the number of queries in Fig. 4. The performance of our method consistently outperforms other methods throughout the process. Since our method does not conflict with the previous sample selection strategy, they can be used simultaneously to further improve the performance of these attacks. Here, we take the $\mathbf { k }$ -Center algorithm as an example. Note that, with or without the sample selection strategy, our method beats the previous methods by a large margin. Particularly, the test accuracies of our method are $4 . 8 5 \%$ , $1 . 7 2 \%$ , $3 . 8 8 \%$ , and $8 . 2 7 \%$ higher than the previous best method, respectively. And the agreement metric shares similar results. It is also interesting that it is less necessary to use the $\mathbf { k }$ -Center algorithm on datasets with a small number of classes (i.e., CIFAR10 and SVHN). While for the datasets with a large number of classes, the k-Center algorithm can make the selected samples better cover each class and improve the effectiveness of the method.
+
+240 Ability to evade the SOTA defense method. The SOTA perturbation-based defense method, adap  
+241 tive misinformation [13], introduces an Out-Of-Distribution (OOD) detection module based on the   
+242 maximum predicted value and punishes the OOD samples with a perturbed model $f ^ { \prime } ( \cdot ; \theta ^ { \prime } )$ . The   
+243 model $f ^ { \prime } ( \cdot ; \bar { \theta } ^ { \prime } )$ is trained with arg minθ0 $\mathbb { E } _ { ( x , y ) } [ - \log ( 1 - f ^ { \prime } ( x ; \theta ^ { \prime } ) _ { y } ) ]$ to minimize the probability of
+
+Table 2: Ability to evade the state-of-the-art defense method (adaptive misinformation) on CIFAR10 dataset. The larger the threshold, the better the defence effect while the low victim model’s accuracy (threshold 0 means no defence). Our method evades the defense best, and the self-KD part makes a great difference.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Threshold</td></tr><tr><td>0</td><td>0.5</td><td>0.7</td><td>0.9</td></tr><tr><td>KnockoffNets</td><td>74.44%</td><td>74.13%</td><td>73.61%</td><td>54.98%</td></tr><tr><td>ActiveThief(k-Center)</td><td>74.24%</td><td>69.14%</td><td>59.78%</td><td>50.19%</td></tr><tr><td>ActiveThief(Entropy)</td><td>74.21%</td><td>71.61%</td><td>64.84%</td><td>51.07%</td></tr><tr><td>Ours</td><td>80.47 %</td><td>79.95%</td><td>78.25%</td><td>74.40 %</td></tr><tr><td>Ours w/o self-KD</td><td>79.02%</td><td>78.66%</td><td>73.61%</td><td>61.81%</td></tr><tr><td>victimmodel</td><td>91.56%</td><td>91.23%</td><td>89.10%</td><td>85.14%</td></tr></table>
+
+244 the correct class. Finally, the output will be:
+
+$$
+y ^ { \prime } = ( 1 - \alpha ) f ( x ; \theta ) + ( \alpha ) f ^ { \prime } ( x ; \theta ^ { \prime } ) ,
+$$
+
+245 where $\alpha = 1 / ( 1 + e ^ { \nu ( \operatorname* { m a x } f ( x ; \theta ) - \tau ) } )$ with a hyper-parameter $\nu$ is the coefficient to control how   
+246 much correct results will be returned, and $\tau$ is the threshold used for OOD detection. The model   
+247 returns incorrect predictions for the OOD samples without having much impact on the in-distribution   
+248 samples.   
+49 We choose four values of the threshold $\tau$ to compare the effects of our method with the previous   
+50 methods. The threshold value of 0 means no defence. The result is shown in Tab. 2. Compared   
+51 with other methods, adaptive misinformation is almost invalid to our method. Furthermore, we find   
+52 that if we remove the self-KD in our method, the performance is greatly reduced. We conclude that   
+53 this is because adaptive misinformation adds noise labels to the substitute model’s training dataset,   
+54 and self-KD can alleviate the overfitting of the substitute model to the training dataset, making this   
+55 defence method not effective enough.
+
+Ablation study. To evaluate the contribution of different modules in our method, we conduct the ablation study on CUBS200 dataset and plot the results in Fig. 5. If the CAM-driven erasing strategy is removed, the performance of our method will be greatly reduced, showing that it has an indispensable position in our method. We also give some visual examples in Fig. 7 to demonstrate that this strategy can help align the attention of two models. As depicted in the Fig. 7, at the beginning time, the substitute model learns the wrong attention map. Along with the iterative training stages, the attention area of the substitute model tends to fit the victim model’s, which conforms to our intention. We further remove the self-KD module to evaluate its performance. It can be found from Fig. 1 and Fig. 5 that the self-KD can improve the generalization of our method and further improve the performance.
+
+![](images/422f0249cb08516bac5147a376079b7cae97c37d1c6e95548f1260ea1f225012.jpg)  
+Figure 5: Ablation study on CUBS200 dataset for the contribution of the CAM-driven erasing and the self-KD in our method.
+
+Stealing functionality of a real-world API. We validate our method is applicable to real-world APIs. The AWS Marketplace is an online store that provides a variety of trained ML models for users. It can only be used in the form of a black-box setting. We choose a popular model (waste classifier 2) as the victim model. We use ILSVRC-2012 dataset as the attack dataset and choose another small public waste classifier dataset 3, containing 2, 527 images as the test dataset. As in Fig. 6, the substitute model obtained by our method achieves $1 2 . 6 3 \%$ and $7 . 3 2 \%$ improvements in test accuracy compared with two previous methods, which show our method has stronger practicality in the real world.
+
+![](images/f68d411ac8b069ce7d038c7675b82dab953585e2c8df673559e86e1af0b345de.jpg)  
+Figure 6: The experiment on AWS online API.
+
+Table 3: Transferability of adversarial samples generated with PGD attack on the substitute models.   
+Substitute’s CAM of different stages   
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Substitute&#x27;sarchitecture</td></tr><tr><td>ResNet-34</td><td>ResNet-18</td><td>ResNet-50</td><td>VGG-16</td><td>DenseNet</td></tr><tr><td>KnockoffNets</td><td>57.85%</td><td>63.33%</td><td>52.04%</td><td>42.88%</td><td>60.77%</td></tr><tr><td>ActiveThief(k-Center)</td><td>57.44%</td><td>57.90%</td><td>57.01%</td><td>16.49%</td><td>60.72%</td></tr><tr><td>ActiveThief(Entropy)</td><td>63.56%</td><td>66.76%</td><td>58.19%</td><td>55.43%</td><td>62.05%</td></tr><tr><td>Ours</td><td>76.63%</td><td>74.10%</td><td>74.28%</td><td>67.03%</td><td>66.96%</td></tr></table>
+
+Victim’s CAM
+
+![](images/73e4a57a663fe6f033f3426daadadb7feab4fd28c4410ec52a22855617ac7911.jpg)  
+Figure 7: The visualized attention maps of the victim model and different stages substitute models using the Grad-CAM. Along with the training stages, the attention map of the substitute model tends to fit the victim model’s.
+
+Transferability of adversarial samples. Though with the dominant performance on a wide range of tasks, deep neural networks are shown to be vulnerable to imperceptible perturbations, i.e., adversarial examples [27]. Since the model stealing attack can obtain a functionally similar substitute model, some previous works (e.g., JBDA [22], DaST [34] and ActiveThief [21]) used this substitute model to generate adversarial samples and then performed the transferable adversarial attack on the victim model. We argue that a more similar substitute model leads to a more successful adversarial attacks. We test the transferability of adversarial samples on the test set of the CIFAR10 dataset. Keeping the architecture of the victim model as the ResNet-34, we evaluate the attack success rate of adversarial samples generated from different substitute models (i.e., ResNet-34, ResNet-18, ResNet-50 [9], VGG16 [26], DenseNet [11]). All adversarial samples are generated using Projected Gradient Descent (PGD) attack [18] with maximum $L _ { \infty }$ -norm of perturbations as 8/255. As shown in Tab. 3, the adversarial samples generated by our substitute models have stronger transferability in all substitute’s architectures. This again proves that our method is more practical in real-world scenarios.
+
+# 296 5 Conclusion
+
+297 We investigated the problem of model stealing attacks under the hard-label setting and pointed out   
+298 why previous methods are not effective enough. We presented a new method, termed black-box   
+299 dissector, which contains a CAM-driven erasing strategy and a RE-based self-KD module. We   
+300 showed its superiority on four widely-used datasets and verified the effectiveness of our method   
+301 with defense methods, real-world APIs, and the downstream adversarial attack. Though focusing   
+302 on image data in this paper, our method is general for other tasks as long as the CAM and similar   
+303 erasing method work, e.g., synonym saliency words replacement for NLP tasks [4]. We believe our   
+304 method can be easily extended to other fields and inspire future researchers. Model stealing attack   
+305 poses a threat to the deployed machine learning models. We hope this work will draw attention to   
+306 the protection of deployed models and furthermore shed more light on the attack mechanisms and   
+307 prevention methods.
+
+# 308 References
+
+frey E Hinton. Large scale distributed neural network training through online distillation. arXiv preprint arXiv:1804.03235, 2018.   
+[2] Antonio Barbalau, Adrian Cosma, Radu Tudor Ionescu, and Marius Popescu. Black-box ripper: Copying black-box models using generative evolutionary algorithms. In NeurIPS, 2020.   
+[3] Terrance DeVries and Graham W Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.   
+[4] Xinshuai Dong, Anh Tuan Luu, Rongrong Ji, and Hong Liu. Towards robustness against natural language word substitutions. In ICLR, 2021.   
+[5] Melanie Ducoffe and Frederic Precioso. Adversarial active learning for deep networks: a margin based approach. In ICML, 2018.   
+[6] Tommaso Furlanello, Zachary C Lipton, Michael Tschannen, Laurent Itti, and Anima Anandkumar. Born again neural networks. In ICML, 2018.   
+[7] Gregory Griffin, Alex Holub, and Pietro Perona. Caltech-256 object category dataset. 2007.   
+[8] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In ICML, 2017.   
+[9] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016.   
+[10] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2015.   
+[11] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In CVPR, 2017.   
+[12] Matthew Jagielski, Nicholas Carlini, David Berthelot, Alex Kurakin, and Nicolas Papernot. High accuracy and high fidelity extraction of neural networks. In 29th Usenix Security, 2020.   
+[13] Sanjay Kariyappa and Moinuddin K Qureshi. Defending against model stealing attacks with adaptive misinformation. In CVPR, 2020.   
+[14] Sanjay Kariyappa, Atul Prakash, and Moinuddin Qureshi. Maze: Data-free model stealing attack using zeroth-order gradient estimation. arXiv preprint arXiv:2005.03161, 2020.   
+[15] Kyungyul Kim, ByeongMoon Ji, Doyoung Yoon, and Sangheum Hwang. Self-knowledge distillation: A simple way for better generalization. arXiv preprint arXiv:2006.12000, 2020.   
+[16] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.   
+[17] David D Lewis and William A Gale. A sequential algorithm for training text classifiers. In SIGIR, 1994.   
+[18] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In ICLR, 2018.   
+[19] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.   
+[20] Tribhuvanesh Orekondy, Bernt Schiele, and Mario Fritz. Knockoff nets: Stealing functionality of black-box models. In CVPR, 2019.   
+[21] Soham Pal, Yash Gupta, Aditya Shukla, Aditya Kanade, Shirish Shevade, and Vinod Ganapathy. Activethief: Model extraction using active learning and unannotated public data. In AAAI, 2020.   
+[22] Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against machine learning. In ACM AsiACCS, 2017.   
+[23] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. IJCV, 2015.   
+[24] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In ICCV, 2017.   
+[25] Ozan Sener and Silvio Savarese. Active learning for convolutional neural networks: A core-set approach. In ICLR, 2018.   
+[26] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.   
+[27] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In ICLR, 2014.   
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See section 3   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See section 5   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See the supplemental material   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See section 4.1   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Tab. 1   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See section 4.1 (b) Did you mention the license of the assets? [No] (c) Did you include any new assets either in the supplemental material or as a URL? [No]
+
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "Black-Box Dissector: Towards Erasing-based Hard-Label Model Stealing Attack ",
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+        "text": "Previous studies have verified that the functionality of black-box models can be stolen with full probability outputs. However, under the more practical hard-label setting, we observe that existing methods suffer from catastrophic performance degradation. We argue this is due to the lack of rich information in the probability prediction and the overfitting caused by hard labels. To this end, we propose a novel hard-label model stealing method termed black-box dissector, which consists of two erasing-based modules. One is a CAM-driven erasing strategy that is designed to increase the information capacity hidden in hard labels from the victim model. The other is a random-erasing-based self-knowledge distillation module that utilizes soft labels from the substitute model to mitigate overfitting. Extensive experiments on four widely-used datasets consistently demonstrate that our method outperforms state-of-the-art methods, with an improvement of at most $8 . 2 7 \\%$ . We also validate the effectiveness and practical potential of our method on real-world APIs and defense methods. Furthermore, our method promotes other downstream tasks, i.e., transfer adversarial attacks. ",
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+        "type": "text",
+        "text": "16 1 Introduction ",
+        "text_level": 1,
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+        "text": "17 Machine learning models deployed on the cloud can serve users through the application program   \n18 interfaces (APIs) to improve productivity. Since developing these cloud models is a product of   \n19 intensive labor and monetary effort, these models are valuable intellectual property and AI companies   \n20 try to keep them private. However, the exposure of the model’s predictions represents a significant   \n21 risk as an adversary can leverage this information to steal the model’s functionality, a.k.a. model   \n22 stealing attack [22, 20, 21]. With such an attack, adversaries are able to not only use the stolen model   \n23 to make a profit, but also mount further adversarial attacks [34, 29]. Besides, the model stealing   \n24 attacks is a kind of black-box knowledge distillation which is a hot research topic. Studying various   \n25 mechanisms of model stealing attack is of great interest both to AI companies and researchers.   \n26 Previous methods [20, 34, 21] mainly assume the complete probability predictions of the victim   \n27 model available, while the real-world APIs usually only return partial probability values (top- $k$   \n28 predictions) or even the top-1 prediction (i.e., hard label). In this paper, we focus on the more   \n29 challenging and realistic scenario, i.e., the victim model only outputs the hard labels. However, under   \n30 this setting, existing methods suffer from a significant performance degradation, even by $3 0 . 5 0 \\%$ (as   \n31 shown in the Fig. 1 (a) and the appendix Tab. I).   \n32 To investigate the reason for the degradation, we evaluate the performance of attack methods with   \n33 different numbers of prediction probability categories available and hard labels as in Fig. 1 (b). With   \n34 the observation that the performance degrades when the top- $k$ information missing, we conclude   \n35 that the top- $k$ predictions are informative as it indicates the similarity of different categories or   \n36 multiple objects in the picture, and previous attack methods suffer from such information obscured   \n37 by the top-1 prediction under the hard-label setting. It motivates us to re-mine this information by   \n38 eliminating the top-1 prediction. Particularly, we design a novel CAM-based erasing method, which   \n39 erases the important area on the pictures based on the substitute model’s top-1 class activation maps   \n40 (CAM) [24, 33] and queries the victim model for a new prediction. Note that we can dig out other   \n41 class information in this sample if the new prediction changes. Otherwise, it proves that the substitute   \n42 model pays attention to the wrong area. Then we can align the attention of the substitute and the   \n43 victim model by learning clean samples and the corresponding erased samples simultaneously.   \n4 Besides, previous works on the   \n45 self-Knowledge Distillation (self  \n46 KD) [15], calibration [8], and noisy   \n47 label [31] have pointed out the   \n48 hard and noisy labels will introduce   \n49 overfitting and miscalibration. More   \n50 specifically, the attack algorithms   \n51 cannot access the training data, and   \n52 thus can only use the synthetic data or   \n53 other datasets as a substitute, which   \n54 is noisy. Therefore, the hard-label   \n55 setting will suffer from overfitting,   \n56 which leads to worse performance,   \n57 and we verify it by plotting the loss   \n58 curves in Fig. 1 (c). To mitigate   \n59 this problem, we introduce a simple   \n60 self-knowledge distillation module   \n61 with random erasing $( R E )$ to utilize   \n62 soft labels for generalization. Partic  \n63 ularly, we randomly erase one sample   \n64 a certain number of times, query   \n65 the substitute model for soft-label   \n66 outputs, and take the average value   \n67 of these outputs as the pseudo-label.   \n68 After that, we use both hard labels ",
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+        "img_path": "images/57d39e95c20569e8951b065bee0a280df2a9efb4bec0e20510253a11377359f2.jpg",
+        "image_caption": [
+            "Figure 1: (a) The test accuracies of previous methods with hard labels are much lower than the ones with soft labels. (KN: KnockoffNets, ‘AT’: ActiveThief, $\\mathbf { \\hat { E } } ^ { \\prime }$ : entropy, ‘K’: $\\mathbf { k }$ -Center, ‘D’: DFAL) (b) The performance decreases as the number of available classes decreases (dotted line : hardlabel setting). (c) & (d) Loss curves for training/test set during model training without and with self-KD. All results are on the CIFAR10 dataset. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "from the victim model and pseudo labels from the previous substitute model to train a new substitute model. Therefore, we can also consider the ensemble of the two models as the teacher in knowledge distillation. As in Fig. 1 (d), such a module helps generalization and better performance. ",
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+        "text": "72 In summary, we propose a novel model stealing framework termed black-box dissector, which   \n73 includes a CAM-driven erasing strategy and a RE-based self-KD module. Our method is orthogonal   \n74 to previous approaches [20, 21] and can be integrated with them. The experiments on four widely  \n75 used datasets demonstrate our method achieves $4 3 . 0 4 - 9 0 . 5 7 \\%$ test accuracy $( 4 7 . 6 0 - 9 1 . 3 7 \\%$   \n76 agreement) to the victim model, which is at most $8 . 2 7 \\%$ higher than the state of the art method.   \n77 We also proved that our method can defeat popular defense methods and is effective for real-world   \n78 APIs like services provided by Amazon Web Services (AWS). Furthermore, our method promotes   \n79 downstream tasks, i.e., transfer adversarial attack, with $4 . 9 1 \\% - 1 6 . 2 0 \\%$ improvement. ",
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+    {
+        "type": "text",
+        "text": "80 2 Background and Notions ",
+        "text_level": 1,
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+        "text": "81 Model stealing attack is aim to find a substitute model $\\hat { f } \\colon [ 0 , 1 ] ^ { d } \\mapsto \\mathbb { R } ^ { N }$ that performs as similarly   \n82 as possible to the black-box victim model $f \\colon [ 0 , 1 ] ^ { d } \\mapsto \\mathbb { R } ^ { N }$ (with only outputs accessed). Papernot   \n83 et al. [22] first observed that online models could be stolen through multiple queries. After that, due   \n84 to the practical threat to real-world APIs, several studies paid attention to this problem and proposed   \n85 many attack algorithms.   \n86 These algorithms consist of two stages: 1) constructing a transfer dataset $D _ { T }$ (step 1 in Fig. 2) and   \n87 2) training a substitute model. The transfer dataset is constructed based on data synthesis or data   \n88 selection and then feed into the victim model for labels. Methods based on data synthesis [34, 14, 2]   \n89 adopt the GAN-based models to generate a virtual dataset. And the substitute model and the GAN   \n90 model are trained alternatively on this virtual dataset by querying the victim model iteratively. The   \n91 data selection methods prepare an attack dataset as the data pool, and then sample the most informative   \n92 data via machine learning algorithms, e.g., reinforcement learning [20] or active learning strategy [21],   \n93 uncertainty-based strategy [17], k-Center strategy [25], and DFAL strategy [5]. Considering that   \n94 querying the victim model will be costly, the attacker usually sets a budget on the number of the   \n95 queries, so the size of the transfer dataset should be limited as well. Previous methods assume the   \n96 victim model returns a complete probability prediction $f ( x )$ , which is less practical.   \n97 In this paper, we focus on a more practical scenario that is about hard-label $\\phi ( f ( x ) )$ setting, where $\\phi$   \n98 is the truncation function used to truncate the information contained in the victim’s output and return   \n99 the corresponding one-hot vector: ",
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+        "image_caption": [
+            "Figure 2: Details of our proposed black-box dissector with a CAM-driven erasing strategy (step 2.1) and a RE-based self-KD module (step 2.2). In step 2.1, the images in transfer set $D _ { T }$ are erased according to the Grad-CAM, and we selected the erased images with the largest difference from the original images according to the substitute model’s outputs. In step 2.2, we randomly erase the unlabeled image $N$ times, and then average the outputs of the $N$ erased images by the substitute model as the pseudo-label. "
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+        "img_path": "images/94419c59aa7eee1b4ad57cb7881e93be9486270b6418c5b86710651be2a43d7e.jpg",
+        "text": "$$\n\\phi ( f ( x ) ) _ { i } : = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } i = \\arg \\operatorname* { m a x } _ { n } f ( x ) _ { n } ; } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } . } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
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+        "text": "100 With the transfer dataset, the substitute model is optimized by minimizing a loss function $\\mathcal { L }$ (e.g.,   \n101 cross-entropy loss function): ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/4294df8d7bff68671c89428c1a267accd16d754af7e5a1e89b6c6f60e2666cfd.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { \\mathbb { E } _ { x \\sim \\mathcal { D } _ { T } } \\left[ \\mathcal { L } \\big ( f ( x ) , \\hat { f } ( x ) \\big ) \\right] , } & { \\mathrm { f o r ~ s o f t ~ l a b e l s ; } } \\\\ { \\mathbb { E } _ { x \\sim \\mathcal { D } _ { T } } \\left[ \\mathcal { L } \\big ( \\phi ( f ( x ) ) , \\hat { f } ( x ) \\big ) \\right] , } & { \\mathrm { f o r ~ h a r d ~ l a b e l s . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "102 Knowledge distillation (KD) has been widely studied in machine learning [10, 1, 6], which transfers   \n103 the knowledge from a teacher model to a student model. Model stealing attacks can be regarded as a   \n104 black-box KD problem where the victim model is the teacher with only outputs accessible and the   \n105 substitute model is the student. The main reason for the success of KD is the valuable information   \n106 that defines a rich similarity structure over the data in the probability prediction [10]. However,   \n107 for the hard-label setting discussed in this paper, this valuable information is lost. Inspired by KD,   \n108 our method tries to dig out the hidden information in the data and models, and then transfers more   \n109 knowledge to the substitute model.   \n110 The erasing-based method, e.g., random erasing (RE) [32, 3], is currently one of the widely used   \n111 data augmentation methods, which generates training images with various levels of occlusion, thereby   \n112 reducing the risk of over-fitting and improving the robustness of the model. Our work is inspired   \n113 by RE and designs a prior-driven erasing operation, which erases the area corresponding to the hard   \n114 label to re-mine missing information.   \n116 The overview of our proposed black-box dissector is shown in Fig. 2. In addition to the conventional   \n117 process (i.e., the transfer dataset $D _ { T }$ constructing in step 1 and the substitute model training in the   \n118 right), we introduce two key modules: a CAM-driven erasing strategy (step 2.1) and a RE-based   \n119 self-KD module (step 2.2). ",
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+    {
+        "type": "text",
+        "text": "20 3.1 A CAM-driven erasing strategy ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "121 Since the lack of class similarity infor  \n122 mation degrades the performance of   \n123 previous methods under the hard-label   \n124 setting, we try to re-dig out such hid  \n125 den information. Taking an example   \n126 from the ILSVRC-2012 dataset for il  \n127 lustration as in Fig. 3. Querying the   \n128 CUBS200 trained victim model with   \n129 this image, we get two classes with   \n130 the highest confidence score: “Anna   \n131 hummingbird\" (0.1364) and “Com  \n132 mon yellowthroat\" (0.1165), and show   \n133 their corresponding attention map in   \n134 the first column of Fig. 3. It is easy to   \n135 conclude that two different attention   \n136 regions response for different classes   \n137 according to the attention map. When   \n138 training the substitute model with the   \n139 hard label “Anna hummingbird\" and   \n140 without the class similarity informa",
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+        "img_path": "images/96fc3450a533eeb97226572a27ca1d168a7cefcf39bb07ad10603539c78f7c73.jpg",
+        "image_caption": [
+            "Figure 3: An example from the ILSVRC-2012 dataset and its attention map corresponding to two most likely class “Anna humming bird\" and “Common yellow throat\" on the CUBS200 trained model. The attention areas share similar visual apparent with images of “Anna humming bird\" and “Common yellow throat\", respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "tion, the model can not learn from the area related to the “Common yellowthroat\" class, which means this area is wasted. To re-dig out the information about the “Common yellowthroat\" class, we need to erase the impact of the “Anna hummingbird\" class. ",
+        "bbox": [
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+        "type": "text",
+        "text": "144 To this end, a natural idea is to erase the response area corresponding to the hard label. Since the   \n145 victim model is a black-box model, we use the substitute model to approximately calculate the   \n146 attention map instead. If the attention map calculated by the substitute model is inaccurate and the   \n147 victim model’s prediction on the erased image does not change, we can also align the attention map of   \n148 two models by letting the substitute model learn the original image and the erased one simultaneously.   \n149 The attention map is also a kind of supervision signal pushing two models to be similar [30]. To   \n150 get the attention map, we utilize the Grad-CAM [24] in this paper. With the input image $x \\in [ 0 , 1 ] ^ { d }$   \n151 and the trained DNN $\\mathcal { F } \\colon [ 0 , 1 ] ^ { d } \\mapsto \\mathbb { R } ^ { N }$ , we let $\\alpha _ { k } ^ { c }$ denote the weight of class $c$ corresponding to the   \n152 k-th feature map, and calculate it as αck = 1Z Pi Pj , where $Z$ is the number of pixels in the   \n153 feature map, $\\mathcal { F } ( x ) ^ { c }$ is the score of class $c$ and $A _ { i j } ^ { k }$ is the value of pixel at $( i , j )$ in the $k$ -th feature   \n154 map. After obtaining the weights corresponding to all feature maps, the final attention map can be   \n155 obtained as $\\begin{array} { r } { S _ { \\mathrm { G r a d - C A M } } ^ { c } = \\tilde { \\mathrm { R e L U } } ( \\sum _ { k } \\hat { \\alpha _ { k } ^ { c } } A ^ { k } ) } \\end{array}$ via weighted summation.   \n156 To erase the corresponding area, inspired by [32], we define a prior-driven erasing operation as   \n157 $\\psi ( I , P )$ , shown in Alg. 1, which randomly erases a rectangle region in the image $I$ with random   \n158 values while the central position of the rectangle region is randomly selected following the prior   \n159 probability $P$ . The prior probability $P$ is of the same size as the input image and is used to determine   \n160 161 the prior. Let the probability of different pixels being erased. Here, we use the attention map from Grad-CAM as $x \\in [ 0 , 1 ] ^ { d }$ denote the input image from the transfer set and $S _ { \\mathrm { G r a d - C A M } } ^ { \\mathrm { a r g m a x } \\hat { f } ( x ) } ( x , \\hat { f } )$ denote   \n162 the attention map of the substitute model $\\hat { f }$ . This CAM-driven erasing operation can be represented: ",
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+        "img_path": "images/be277cedaccdadd712cfd798348a61819d3d90de99006679c8b5c0ecb5c0f2cb.jpg",
+        "text": "$$\n\\psi \\left( x , S _ { \\mathrm { G r a d - C A M } } ^ { \\mathrm { a r g m a x } \\hat { f } ( x ) } ( x , \\hat { f } ) \\right) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "163 We abbreviate it as $\\psi ( x , S ( x , { \\hat { f } } ) )$ . To alleviate the impact of inaccurate CAM caused by the difference   \n164 between the substitute model and the victim one, for each image, we perform this operation $N$ times   \n165 $\\psi _ { i }$ means the $i$ -th erasing) and select the one with the largest difference from the original label. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Prior-driven Erasing Operation $\\psi ( I , P )$ ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Input: Input image $I$ , prior probability $P$ , image size $W$ and $H$ , area of image $S$ , erasing area ratio range $s _ { l }$ and $s _ { h }$ , erasing aspect ratio range $r _ { 1 }$ and $r _ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Output: Erased image $I ^ { \\prime }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$S _ { e } \\overset { \\bar { \\mathbf { \\sigma } } } {  } \\operatorname { R a n d } ( s _ { l } , s _ { h } ) \\overset { \\bar { \\mathbf { \\sigma } } } { \\times } S$ , $r _ { e } \\gets \\mathrm { R a n d } ( r _ { 1 } , r _ { 2 } ) ^ { 1 }$ ",
+        "bbox": [
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+        "type": "text",
+        "text": "3 $x _ { e } , y _ { e }$ sampled randomly according to $P$   \n4 $I _ { e } \\gets ( x _ { e } - W _ { e } , y _ { e } - H _ { e } , x _ { e } + W _ { e } , y _ { e } + H _ { e } )$   \n5 $I ( I _ { e } ) \\gets \\mathrm { R a n d } ( 0 , 2 5 5 )$   \n6 $I ^ { \\prime }  I$   \n166 Such a data augment operation helps the erasing process to be more robust. We use the cross-entropy   \n167 to calculate the difference between the new label and the original label, and we want to select the   \n168 sample with the biggest difference. Formally, we define $\\Pi ( x )$ as the function to select the most   \n169 different variation of image $x$ : ",
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+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/435cc187c297b4487cce8377718fe6601780d2d55ca46ebb9b4ced8dfcf524f4.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\Pi ( x ) : = \\psi _ { k } ( x , S ( x , \\hat { f } ) ) , } } \\\\ & { } & { \\quad \\mathrm { w h e r e } \\ k : = \\underset { i \\in [ N ] } { \\arg \\operatorname* { m a x } } - \\sum _ { j } \\phi ( f ( x ) ) _ { j } \\cdot \\log \\Big ( \\hat { f } \\big ( \\psi _ { i } ( x , S ( x , \\hat { f } ) ) \\big ) _ { j } \\Big ) } \\\\ & { } & { = \\underset { i \\in [ N ] } { \\arg \\operatorname* { m a x } } - \\log \\bigg ( \\hat { f } \\big ( \\psi _ { i } ( x , S ( x , \\hat { f } ) ) \\big ) _ { \\arg \\operatorname* { m a x } \\phi \\big ( f ( x ) \\big ) } \\bigg ) } \\\\ & { } & { = \\underset { i \\in [ N ] } { \\arg \\operatorname* { m i n } } \\hat { f } ( \\psi _ { i } ( x , S ( x , \\hat { f } ) ) ) _ { \\arg \\operatorname* { m a x } \\phi \\big ( f ( x ) \\big ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "170 Due to the limitation of the number of queries, we cannot query the victim model for each erased   \n171 image to obtain a new label. We continuously choose the erased image with the highest substitute’s   \n172 confidence until reaching the budget. To measure the confidence of the model, we adopt the Maximum   \n173 Softmax Probability (MSP) for its simplicity: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6165f18ecdb4a741cf976d58d79b4de4a3f3e4246af1bf230173d5bf39c7217c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { x \\sim \\mathcal { D } _ { T } } { \\arg \\operatorname* { m a x } } M S P \\left( \\hat { f } \\left( \\Pi \\left( x \\right) \\right) \\right) } \\\\ & { = \\underset { x \\sim \\mathcal { D } _ { T } } { \\arg \\operatorname* { m a x } } \\hat { f } \\left( \\Pi \\left( x \\right) \\right) _ { \\mathrm { a r g m a x } } \\hat { f } ( \\Pi ( x ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "174 where $D _ { T }$ is the transfer set. The erased images selected in this way are most likely to change the   \n175 prediction class. Then, we query the victim model to get these erased images’ labels and construct   \n176 an erased sample set $D _ { E }$ . Note that when the victim model’s predictions on the erased images   \n177 change, it means our erasing method does dig out other related class information in the sample. With   \n178 the unchanged predictions, it points out the attentions of the substitute model and the victim are   \n179 inconsistent. Though wrong attention areas erased, training with these samples benefits aligning the   \n180 attentions of two models. As [30] stated, the attention alignment can help more powerful KD. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "181 3.2 A random-erasing-based self-KD module ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "182 We also find that in training with limited hard-label OOD samples, the substitute model is likely   \n183 to overfit the training set, which damages its generalization ability [15, 31]. Therefore, based on   \n184 the above erasing operation, we further design a simple RE-based self-KD method to improve the   \n185 generalization ability of the substitute model.   \n186 Formally, let $x \\in [ 0 , 1 ] ^ { d }$ denote the unlabeled input image. We perform the erasing operation with a   \n187 uniform prior $U$ on it $N$ times, and then average the substitute’s outputs on these erased images as   \n188 the pseudo-label of the original image: ",
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+        "img_path": "images/d1af20ee9193d1f640bf24fdf9c6bf9b936914d45c17f79720f9ee1c66d1f7f6.jpg",
+        "text": "$$\ny _ { p } ( x , \\hat { f } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\hat { f } \\big ( \\psi _ { i } ( x , U ) \\big ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "${ } ^ { 1 } \\mathrm { R a n d } ( a , b )$ returns an evenly distributed random real number in the range of $a$ to $b$ . ",
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+    {
+        "type": "text",
+        "text": "Algorithm 2: Black-box Dissector ",
+        "text_level": 1,
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Input: Unlabeled pool $D _ { U }$ , victim model $f$ , maximum number of queries $Q$ .   \nOutput: Substitute model $\\hat { f }$ .   \n1 Initialize $q  0 , D _ { T }  \\emptyset , D _ { E }  \\emptyset$   \n2 while $q < Q$ do   \n3 // Step 1   \n4 Select samples from $D _ { U }$ according to budget and query $f$ to updata $D _ { T }$   \n5 $q = q +$ budget   \n6 $\\begin{array} { r } { \\mathcal { L } = \\sum _ { x \\in D _ { T } } \\mathcal { L } ^ { \\prime } \\big ( \\phi ( f ( x ) ) , \\hat { f } ( x ) \\big ) } \\end{array}$   \n7 $\\hat { f } \\gets u p d a t e ( \\hat { f } , \\mathcal { L } )$   \n8 // A CAM-driven erasing strategy (step 2.1)   \n9 Erase samples in $D _ { T }$ according to Eq. 4   \n10 Choose samples from erased samples according to Eq. 5 and budget   \n11 Query $f$ to get labels and updata $D _ { E }$   \n12 $\\begin{array} { r } { \\mathcal { L } = \\sum _ { x \\in D _ { T } \\cup D _ { E } } \\mathcal { L } ^ { \\prime } \\big ( \\phi ( f ( x ) ) , \\hat { f } ( x ) \\big ) } \\end{array}$   \n13 $\\hat { f } \\gets u p d a t e ( \\hat { f } , \\mathcal { L } )$   \n14 $q = q +$ budget   \n15 // A random-erasing-based self-KD (step 2.2)   \n16 Select samples from $D _ { U }$   \n17 Get pseudo-labels according to Eq. 6 and construct a pseudo-label set $D _ { P }$   \n18 $\\begin{array} { r l } & { \\mathcal { L } = \\sum _ { x \\in D _ { T } \\cup D _ { E } } \\mathcal { L } ^ { \\prime } \\big ( \\phi ( f ( x ) ) , \\hat { f } ( x ) \\big ) + \\sum _ { x \\in D _ { P } } \\mathcal { L } ^ { \\prime } \\big ( y _ { p } ( x , \\hat { f } ) , \\hat { f } ( x ) \\big ) } \\\\ & { \\hat { f }  u p d a t e ( \\hat { f } , \\mathcal { L } ) } \\end{array}$   \n19   \n20 end   \n189 This is a type of consistency regularization, which enforces the model to have the same predictions   \n190 for the perturbed images and enhances the generalization ability. With Eq.6, we construct a new soft   \n191 pseudo label set $D _ { P } = \\{ { \\bigl ( } x , y _ { p } ( x , { \\hat { f } } ) { \\bigr ) } , \\ldots \\}$ .   \n192 With the transfer set $D _ { T }$ , the erased sample set $D _ { E }$ , and the pseudo-label set $D _ { P }$ , we train a new   \n193 substitute model using the ensemble of the victim model and the previous substitute model as the   \n194 teacher. Our final objective function is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/24b267c08daba11dc8b407898c1653a455f2259a166732ee3770a77cc325a3f3.jpg",
+        "text": "$$\n\\operatorname* { m i n } \\mathcal { L } = \\operatorname* { m i n } \\big [ \\sum _ { x \\in D _ { T } \\cup D _ { E } } \\mathcal { L } ^ { \\prime } \\big ( \\phi ( f ( x ) ) , \\hat { f } ( x ) \\big ) + \\sum _ { x \\in D _ { P } } \\mathcal { L } ^ { \\prime } \\big ( y _ { p } ( x , \\hat { f } ) , \\hat { f } ( x ) \\big ) \\big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where 195 $\\mathcal { L } ^ { \\prime }$ can be commonly used loss functions, e.g., cross-entropy loss function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "196 To sum up, we built our method on the conventional process of the model stealing attack (step   \n197 1), and proposed a CAM-driven erasing strategy (step 2.1) and a RE-based self-KD module (step   \n198 2.2) unified by a novel erasing method. The former strategy digs out missing information between   \n199 classes and aligns the attention while the latter module helps to mitigate overfitting and enhance the   \n200 generalization. We name the whole framework as black-box dissector and present the algorithm   \n201 detail of it in Alg. 2. ",
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+    {
+        "type": "text",
+        "text": "4 Experiment ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 Experiment settings ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Victim model. The victim models we used (ResNet-34 [9]) are trained on four datasets, namely, CIFAR10 [16], SVHN [19], Caltech256 [7], and CUBS200 [28], and their test accuracy are $9 1 . 5 6 \\%$ , $9 6 . 4 5 \\%$ , $7 8 . 4 0 \\%$ , and $7 7 . 1 0 \\%$ , respectively. All models are trained using the SGD optimizer with momentum (of 0.5) for 200 epochs with a base learning rate of 0.1 decayed by a factor of 0.1 every 30 epochs. Following [20, 21, 34], we use the same architecture for the substitute model and will analyze the impact of different architectures in the supplementary. ",
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+    {
+        "type": "text",
+        "text": "210 Attack dataset. We use $1 . 2 M$ images without labels from the ILSVRC-2012 challenge [23] as the   \n211 attack dataset. In a real attack scenario, the attacker may use pictures collected from the Internet, and   \n212 the ILSVRC-2012 dataset can simulate this scenario well. Note that we resize all images in the attack   \n213 dataset to fit the size of the target datasets, which is similar to the existing setting [20, 21, 34]. ",
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+        "type": "table",
+        "img_path": "images/5331fcb83fd2695d6c809e4825055cc0d60bc89743405f3a8572a782f09219b5.jpg",
+        "table_caption": [
+            "Table 1: The agreement and test accuracy (in $\\%$ ) of each method under $3 0 \\mathrm { k }$ queries. For our model, we report the average accuracy as well as the standard deviation computed over 5 runs. (Boldface: the best value, italics: the second best value.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">CIFAR10</td><td colspan=\"2\">SVHN</td><td colspan=\"2\">Caltech256</td><td colspan=\"2\">CUBS200</td></tr><tr><td>Agreement</td><td>Acc</td><td>Agreement</td><td>Acc</td><td>Agreement</td><td>Acc</td><td>Agreement</td><td>Acc</td></tr><tr><td>KnockoffNets</td><td>75.32</td><td>74.44</td><td>85.00</td><td>84.50</td><td>57.64</td><td>55.28</td><td>30.01</td><td>28.03</td></tr><tr><td>ActiveThief(Entropy)</td><td>75.26</td><td>74.21</td><td>90.47</td><td>89.85</td><td>56.28</td><td>54.14</td><td>32.05</td><td>29.43</td></tr><tr><td>ActiveThief(k-Center)</td><td>75.71</td><td>74.24</td><td>81.45</td><td>80.79</td><td>61.19</td><td>58.84</td><td>37.68</td><td>34.64</td></tr><tr><td>ActiveThief(DFAL)</td><td>76.72</td><td>75.62</td><td>84.79</td><td>84.17</td><td>46.92</td><td>44.91</td><td>20.31</td><td>18.69</td></tr><tr><td>ActiveThief(DFAL+k-Center)</td><td>74.97</td><td>73.98</td><td>81.40</td><td>80.86</td><td>55.70</td><td>53.69</td><td>26.60</td><td>24.42</td></tr><tr><td>Ours+Random</td><td>82.14±0.16</td><td>80.47±0.02</td><td>92.33±0.47</td><td>91.57±0.29</td><td>62.15±0.52</td><td>59.91±0.58</td><td>38.28±0.31</td><td>35.24±0.49</td></tr><tr><td>Ours+k-Center</td><td>80.84±0.21</td><td>79.27±0.15</td><td>91.47±0.09</td><td>90.68±0.14</td><td>65.12±0.56</td><td>62.72±0.57</td><td>46.69±0.87</td><td>42.91±0.46</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2bac624872db9b5c764a32c26a8ace18d0db089797185f4c817042cd0158bb85.jpg",
+        "image_caption": [
+            "Figure 4: Curves of the test accuracy versus the number of queries. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "Training process. We use the SGD optimizer with momentum (of 0.9) for 200 epochs and a base learning rate of $\\begin{array} { r } { 0 . 0 2 \\times \\frac { b a t c h s i z e } { 1 2 8 } } \\end{array}$ decayed by a factor of 0.1 every 60 epochs. The weight decay is set to $5 \\times 1 0 ^ { - 4 }$ for small datasets (CIFAR10 [16] and SVHN [19]) and 0 for others. We set up a query sequence {0.1K, 0.2K, 0.5K, 0.8K, 1K, 2K, 5K, 10K, 20K, $\\mathrm { 3 0 K } \\}$ as the iterative maximum query budget, and stop the sampling stage whenever reaching the budget at each iteration. ",
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+    {
+        "type": "text",
+        "text": "Baselines and evaluation metric. We mainly compare our method with KnockoffNets [20] and ActiveThief [21]. Follow Jagielski et al. [12], we mainly report the test accuracy (Acc) as the evaluation metric. We also report the Agreement metric proposed by Pal et al. [21] which counts how often the prediction of the substitute model is the same as the victim’s as a supplement. ",
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+    {
+        "type": "text",
+        "text": "4.2 Experiment results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first report the performance of our method compared with previous methods. After that, we conduct ablation experiments to analyze the contribution of each module. Finally, we also analyze the performance of our method when encountering defense methods and real-world online APIs. More experiments (e.g., adversarial attack and overfitting analysis) can be found in our supplementary. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effectiveness of our method. As in Tab. 1, the test accuracy and agreement of our method are all better than the previous methods. We also plot the curves of the test accuracy versus the number of queries in Fig. 4. The performance of our method consistently outperforms other methods throughout the process. Since our method does not conflict with the previous sample selection strategy, they can be used simultaneously to further improve the performance of these attacks. Here, we take the $\\mathbf { k }$ -Center algorithm as an example. Note that, with or without the sample selection strategy, our method beats the previous methods by a large margin. Particularly, the test accuracies of our method are $4 . 8 5 \\%$ , $1 . 7 2 \\%$ , $3 . 8 8 \\%$ , and $8 . 2 7 \\%$ higher than the previous best method, respectively. And the agreement metric shares similar results. It is also interesting that it is less necessary to use the $\\mathbf { k }$ -Center algorithm on datasets with a small number of classes (i.e., CIFAR10 and SVHN). While for the datasets with a large number of classes, the k-Center algorithm can make the selected samples better cover each class and improve the effectiveness of the method. ",
+        "bbox": [
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+        "type": "text",
+        "text": "240 Ability to evade the SOTA defense method. The SOTA perturbation-based defense method, adap  \n241 tive misinformation [13], introduces an Out-Of-Distribution (OOD) detection module based on the   \n242 maximum predicted value and punishes the OOD samples with a perturbed model $f ^ { \\prime } ( \\cdot ; \\theta ^ { \\prime } )$ . The   \n243 model $f ^ { \\prime } ( \\cdot ; \\bar { \\theta } ^ { \\prime } )$ is trained with arg minθ0 $\\mathbb { E } _ { ( x , y ) } [ - \\log ( 1 - f ^ { \\prime } ( x ; \\theta ^ { \\prime } ) _ { y } ) ]$ to minimize the probability of ",
+        "bbox": [
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+        "img_path": "images/1ef16364636bc1485746807703f06bf97d0bc89a5d73d50173a2a9c5e39a7833.jpg",
+        "table_caption": [
+            "Table 2: Ability to evade the state-of-the-art defense method (adaptive misinformation) on CIFAR10 dataset. The larger the threshold, the better the defence effect while the low victim model’s accuracy (threshold 0 means no defence). Our method evades the defense best, and the self-KD part makes a great difference. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">Threshold</td></tr><tr><td>0</td><td>0.5</td><td>0.7</td><td>0.9</td></tr><tr><td>KnockoffNets</td><td>74.44%</td><td>74.13%</td><td>73.61%</td><td>54.98%</td></tr><tr><td>ActiveThief(k-Center)</td><td>74.24%</td><td>69.14%</td><td>59.78%</td><td>50.19%</td></tr><tr><td>ActiveThief(Entropy)</td><td>74.21%</td><td>71.61%</td><td>64.84%</td><td>51.07%</td></tr><tr><td>Ours</td><td>80.47 %</td><td>79.95%</td><td>78.25%</td><td>74.40 %</td></tr><tr><td>Ours w/o self-KD</td><td>79.02%</td><td>78.66%</td><td>73.61%</td><td>61.81%</td></tr><tr><td>victimmodel</td><td>91.56%</td><td>91.23%</td><td>89.10%</td><td>85.14%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "244 the correct class. Finally, the output will be: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edd5b018104223f06f1c44ef7dac9cb0afd04ad99085e998b082c682529b80e2.jpg",
+        "text": "$$\ny ^ { \\prime } = ( 1 - \\alpha ) f ( x ; \\theta ) + ( \\alpha ) f ^ { \\prime } ( x ; \\theta ^ { \\prime } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "245 where $\\alpha = 1 / ( 1 + e ^ { \\nu ( \\operatorname* { m a x } f ( x ; \\theta ) - \\tau ) } )$ with a hyper-parameter $\\nu$ is the coefficient to control how   \n246 much correct results will be returned, and $\\tau$ is the threshold used for OOD detection. The model   \n247 returns incorrect predictions for the OOD samples without having much impact on the in-distribution   \n248 samples.   \n49 We choose four values of the threshold $\\tau$ to compare the effects of our method with the previous   \n50 methods. The threshold value of 0 means no defence. The result is shown in Tab. 2. Compared   \n51 with other methods, adaptive misinformation is almost invalid to our method. Furthermore, we find   \n52 that if we remove the self-KD in our method, the performance is greatly reduced. We conclude that   \n53 this is because adaptive misinformation adds noise labels to the substitute model’s training dataset,   \n54 and self-KD can alleviate the overfitting of the substitute model to the training dataset, making this   \n55 defence method not effective enough. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Ablation study. To evaluate the contribution of different modules in our method, we conduct the ablation study on CUBS200 dataset and plot the results in Fig. 5. If the CAM-driven erasing strategy is removed, the performance of our method will be greatly reduced, showing that it has an indispensable position in our method. We also give some visual examples in Fig. 7 to demonstrate that this strategy can help align the attention of two models. As depicted in the Fig. 7, at the beginning time, the substitute model learns the wrong attention map. Along with the iterative training stages, the attention area of the substitute model tends to fit the victim model’s, which conforms to our intention. We further remove the self-KD module to evaluate its performance. It can be found from Fig. 1 and Fig. 5 that the self-KD can improve the generalization of our method and further improve the performance. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/422f0249cb08516bac5147a376079b7cae97c37d1c6e95548f1260ea1f225012.jpg",
+        "image_caption": [
+            "Figure 5: Ablation study on CUBS200 dataset for the contribution of the CAM-driven erasing and the self-KD in our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Stealing functionality of a real-world API. We validate our method is applicable to real-world APIs. The AWS Marketplace is an online store that provides a variety of trained ML models for users. It can only be used in the form of a black-box setting. We choose a popular model (waste classifier 2) as the victim model. We use ILSVRC-2012 dataset as the attack dataset and choose another small public waste classifier dataset 3, containing 2, 527 images as the test dataset. As in Fig. 6, the substitute model obtained by our method achieves $1 2 . 6 3 \\%$ and $7 . 3 2 \\%$ improvements in test accuracy compared with two previous methods, which show our method has stronger practicality in the real world. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f68d411ac8b069ce7d038c7675b82dab953585e2c8df673559e86e1af0b345de.jpg",
+        "image_caption": [
+            "Figure 6: The experiment on AWS online API. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/af5354d5bd906182a3d6f917c8efb1c156f59c545c60b338fc938a757c7a3318.jpg",
+        "table_caption": [
+            "Table 3: Transferability of adversarial samples generated with PGD attack on the substitute models. ",
+            "Substitute’s CAM of different stages "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"5\">Substitute&#x27;sarchitecture</td></tr><tr><td>ResNet-34</td><td>ResNet-18</td><td>ResNet-50</td><td>VGG-16</td><td>DenseNet</td></tr><tr><td>KnockoffNets</td><td>57.85%</td><td>63.33%</td><td>52.04%</td><td>42.88%</td><td>60.77%</td></tr><tr><td>ActiveThief(k-Center)</td><td>57.44%</td><td>57.90%</td><td>57.01%</td><td>16.49%</td><td>60.72%</td></tr><tr><td>ActiveThief(Entropy)</td><td>63.56%</td><td>66.76%</td><td>58.19%</td><td>55.43%</td><td>62.05%</td></tr><tr><td>Ours</td><td>76.63%</td><td>74.10%</td><td>74.28%</td><td>67.03%</td><td>66.96%</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Victim’s CAM ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/73e4a57a663fe6f033f3426daadadb7feab4fd28c4410ec52a22855617ac7911.jpg",
+        "image_caption": [
+            "Figure 7: The visualized attention maps of the victim model and different stages substitute models using the Grad-CAM. Along with the training stages, the attention map of the substitute model tends to fit the victim model’s. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Transferability of adversarial samples. Though with the dominant performance on a wide range of tasks, deep neural networks are shown to be vulnerable to imperceptible perturbations, i.e., adversarial examples [27]. Since the model stealing attack can obtain a functionally similar substitute model, some previous works (e.g., JBDA [22], DaST [34] and ActiveThief [21]) used this substitute model to generate adversarial samples and then performed the transferable adversarial attack on the victim model. We argue that a more similar substitute model leads to a more successful adversarial attacks. We test the transferability of adversarial samples on the test set of the CIFAR10 dataset. Keeping the architecture of the victim model as the ResNet-34, we evaluate the attack success rate of adversarial samples generated from different substitute models (i.e., ResNet-34, ResNet-18, ResNet-50 [9], VGG16 [26], DenseNet [11]). All adversarial samples are generated using Projected Gradient Descent (PGD) attack [18] with maximum $L _ { \\infty }$ -norm of perturbations as 8/255. As shown in Tab. 3, the adversarial samples generated by our substitute models have stronger transferability in all substitute’s architectures. This again proves that our method is more practical in real-world scenarios. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "296 5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "297 We investigated the problem of model stealing attacks under the hard-label setting and pointed out   \n298 why previous methods are not effective enough. We presented a new method, termed black-box   \n299 dissector, which contains a CAM-driven erasing strategy and a RE-based self-KD module. We   \n300 showed its superiority on four widely-used datasets and verified the effectiveness of our method   \n301 with defense methods, real-world APIs, and the downstream adversarial attack. Though focusing   \n302 on image data in this paper, our method is general for other tasks as long as the CAM and similar   \n303 erasing method work, e.g., synonym saliency words replacement for NLP tasks [4]. We believe our   \n304 method can be easily extended to other fields and inspire future researchers. Model stealing attack   \n305 poses a threat to the deployed machine learning models. We hope this work will draw attention to   \n306 the protection of deployed models and furthermore shed more light on the attack mechanisms and   \n307 prevention methods. ",
+        "bbox": [
+            140,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "308 References ",
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+    {
+        "type": "text",
+        "text": "frey E Hinton. Large scale distributed neural network training through online distillation. arXiv preprint arXiv:1804.03235, 2018.   \n[2] Antonio Barbalau, Adrian Cosma, Radu Tudor Ionescu, and Marius Popescu. Black-box ripper: Copying black-box models using generative evolutionary algorithms. In NeurIPS, 2020.   \n[3] Terrance DeVries and Graham W Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.   \n[4] Xinshuai Dong, Anh Tuan Luu, Rongrong Ji, and Hong Liu. Towards robustness against natural language word substitutions. In ICLR, 2021.   \n[5] Melanie Ducoffe and Frederic Precioso. Adversarial active learning for deep networks: a margin based approach. In ICML, 2018.   \n[6] Tommaso Furlanello, Zachary C Lipton, Michael Tschannen, Laurent Itti, and Anima Anandkumar. Born again neural networks. In ICML, 2018.   \n[7] Gregory Griffin, Alex Holub, and Pietro Perona. Caltech-256 object category dataset. 2007.   \n[8] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In ICML, 2017.   \n[9] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016.   \n[10] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2015.   \n[11] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In CVPR, 2017.   \n[12] Matthew Jagielski, Nicholas Carlini, David Berthelot, Alex Kurakin, and Nicolas Papernot. High accuracy and high fidelity extraction of neural networks. In 29th Usenix Security, 2020.   \n[13] Sanjay Kariyappa and Moinuddin K Qureshi. Defending against model stealing attacks with adaptive misinformation. In CVPR, 2020.   \n[14] Sanjay Kariyappa, Atul Prakash, and Moinuddin Qureshi. Maze: Data-free model stealing attack using zeroth-order gradient estimation. arXiv preprint arXiv:2005.03161, 2020.   \n[15] Kyungyul Kim, ByeongMoon Ji, Doyoung Yoon, and Sangheum Hwang. Self-knowledge distillation: A simple way for better generalization. arXiv preprint arXiv:2006.12000, 2020.   \n[16] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.   \n[17] David D Lewis and William A Gale. A sequential algorithm for training text classifiers. In SIGIR, 1994.   \n[18] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In ICLR, 2018.   \n[19] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.   \n[20] Tribhuvanesh Orekondy, Bernt Schiele, and Mario Fritz. Knockoff nets: Stealing functionality of black-box models. In CVPR, 2019.   \n[21] Soham Pal, Yash Gupta, Aditya Shukla, Aditya Kanade, Shirish Shevade, and Vinod Ganapathy. Activethief: Model extraction using active learning and unannotated public data. In AAAI, 2020.   \n[22] Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against machine learning. In ACM AsiACCS, 2017.   \n[23] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. IJCV, 2015.   \n[24] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In ICCV, 2017.   \n[25] Ozan Sener and Silvio Savarese. Active learning for convolutional neural networks: A core-set approach. In ICLR, 2018.   \n[26] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.   \n[27] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In ICLR, 2014.   \n[28] Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011.   \n[29] Jiancheng Yang, Yangzhou Jiang, Xiaoyang Huang, Bingbing Ni, and Chenglong Zhao. Learning black-box attackers with transferable priors and query feedback. In NeurIPS, 2020.   \n[30] Sergey Zagoruyko and Nikos Komodakis. Paying more attention to attention: Improving the performance of convolutional neural networks via attention transfer. In ICLR, 2017.   \n[31] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. In ICLR, 2017.   \n[32] Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random erasing data augmentation. In AAAI, 2020.   \n[33] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. In CVPR, 2016.   \n[34] Mingyi Zhou, Jing Wu, Yipeng Liu, Shuaicheng Liu, and Ce Zhu. Dast: Data-free substitute training for adversarial attacks. In CVPR, 2020. ",
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parse/train/BklhAj09K7/BklhAj09K7.md

@@ -0,0 +1,431 @@
+# UNSUPERVISED DOMAIN ADAPTATION FOR DISTANCE METRIC LEARNING
+
+Kihyuk Sohn1 Wenling Shang2 Xiang $\mathbf { Y u } ^ { 1 }$ Manmohan Chandraker1,3 1NEC Labs America 2University of Amsterdam 3UC San Diego
+
+# ABSTRACT
+
+Unsupervised domain adaptation is a promising avenue to enhance the performance of deep neural networks on a target domain, using labels only from a source domain. However, the two predominant methods, domain discrepancy reduction learning and semi-supervised learning, are not readily applicable when source and target domains do not share a common label space. This paper addresses the above scenario by learning a representation space that retains discriminative power on both the (labeled) source and (unlabeled) target domains while keeping representations for the two domains well-separated. Inspired by a theoretical analysis, we first reformulate the disjoint classification task, where the source and target domains correspond to non-overlapping class labels, to a verification one. To handle both within and cross domain verifications, we propose a Feature Transfer Network (FTN) to separate the target feature space from the original source space while aligned with a transformed source space. Moreover, we present a non-parametric multi-class entropy minimization loss to further boost the discriminative power of FTNs on the target domain. In experiments, we first illustrate how FTN works in a controlled setting of adapting from MNIST-M to MNIST with disjoint digit classes between the two domains and then demonstrate the effectiveness of FTNs through state-of-the-art performances on a cross-ethnicity face recognition problem.
+
+# 1 INTRODUCTION
+
+Despite strong performances on facial analysis using deep neural networks (Taigman et al., 2014; Sun et al., 2014; Schroff et al., 2015; Parkhi et al., 2015), learning a model that generalizes across variations in attributes like ethnicity, gender or age remains a challenge. For example, it is reported by Buolamwini & Gebru (2018) that commercial engines tend to make mistakes at detecting gender for images of darker-skinned females. Such biases have enormous social consequences, such as conscious or unconscious discrimination in law enforcement, surveillance or security (WIRED, 2018a;b; NYTimes, 2018; GIZMODO, 2018). A typical solution is to collect and annotate more data along the underrepresented dimension, but such efforts are laborious and time consuming. This paper proposes a novel deep unsupervised domain adaptation approach to overcome such biases in face verification and identification.
+
+Deep domain adaptation (Long et al., 2013; 2015; 2016; Tzeng et al., 2015; Ganin et al., 2016; Sohn et al., 2017; Haeusser et al., 2017; Luo et al., 2017) allows porting a deep neural network to a target domain without extensive labeling efforts. Currently, there are two predominant approaches to deep domain adaptation. The first approach, domain divergence reduction learning, is motivated by the works of (Ben-David et al., 2007; 2010). It aims to reduce the source-target domain divergence using domain adversarial training (Ganin et al., 2016; Sohn et al., 2017; Tran et al., 2018) or maximum mean discrepancy minimization (Tzeng et al., 2015; Long et al., 2015; 2016), while leveraging supervised loss from labeled source examples to maintain feature space discriminative power. Since the theoretical basis of this approach (Ben-David et al., 2007) assumes a common task between domains, it is usually applied to a classification problem where the source and target domains share the same label space and task definition. The second approach considers domain adaptation as a semi-supervised learning problem and applies techniques such as entropy minimization (Grandvalet & Bengio, 2005) or self-ensembling (Laine & Aila, 2017; Tarvainen & Valpola, 2017; French et al., 2018) on target examples to encourage decisive and consistent predictions.
+
+However, neither of those are applicable if the label spaces of source and target domains do not align. As a motivating example, consider a cross-ethnicity generalization of face recognition problem, where the source ethnicity (e.g., Caucasian) contains labeled examples and the target ethnicity (e.g., African-American) contains only unlabeled examples. When it is cast as a classification problem, the tasks of the two domains are different due to disjoint label spaces. Moreover, examples from different ethnicity domains almost certainly belong to different identity classes. To satisfy such additional label constraints, representations of examples from different domains should ideally be distant from each other in the embedding space, which conflicts with the requirements of domain divergence reduction learning as well as entropy minimization on target examples with source domain class labels.
+
+In this work, we aim at learning a shared representation space between a source and target domain with disjoint label spaces that not only remains discriminative over both domains but also keep representations of examples from different domains well-separated, when provided with additional label constraints. Firstly, to overcome the limitation of domain adversarial neural network (DANN) (Ganin et al., 2016), we propose to convert disjoint classification tasks (i.e., the source and target domains correspond to non-overlapping class labels) into a unified binary verification task. We term adaptation across such source and target domains as cross-domain distance metric adaptation (CD2MA). We demonstrate a generalization of the theory of domain adaptation (Ben-David et al., 2007) to our setup, which bounds the empirical risk for within-domain verification of two examples drawn from the unlabeled target domain. While the theory does not guarantee verification between examples from different domains, we propose approaches that also address such cross-domain verification tasks.
+
+To this end, we introduce a Feature Transfer Network (FTN) that separates the target features from the source features while simultaneously aligning them with an auxiliary domain of transformed source features. Specifically, we learn a shared feature extractor that maps examples from different domains to representations far apart. Simultaneously, we learn a feature transfer module that transforms the source representation space to another space used to align with the target representation space through a domain adversarial loss. By forging this alignment, the discriminative power from the augmented source representation space would ideally be transferred to the target representation space. The verification setup also allows us to introduce a novel entropy minimization loss in the form of $N$ -pair metric loss (Sohn, 2016), termed multi-class entropy minimization (MCEM), to further leverage unlabeled target examples whose label structure is not known. MCEM samples pairs of examples from a discovered label structure within the target domain using an offline hierarchical clustering algorithm such as HDBSCAN (Campello et al., 2013), computes the $N$ -pair metric loss among these examples (Sohn, 2016), and backpropagates the resulting error derivatives.
+
+In experiments, we first perform on a controlled setting by adapting between disjoint sets of digit classes. Specifically, we adapt from 0–4 of MNIST-M (Ganin et al., 2016) dataset to 5–9 of MNIST dataset and demonstrate the effectiveness of FTN in learning to align and separate domains. Then, we assess the impact of our proposed unsupervised CD2MA method on a challenging cross-ethnicity face recognition task, whose source domain contains face images of Caucasian identities and the target domain of non-Caucasian identities, such as African-American or East-Asian. This is an important problem since existing face recognition datasets show significant label biases towards Caucasian ethnicity, leading to sub-optimal recognition performance for other ethnicities. The proposed method demonstrates significant improvement in face verification and identification compared to a source-only baseline model and a standard DANN. Our proposed method also closely matches the performance upper bounds obtained by training with fully labeled source and target domains.
+
+# 2 RELATED WORK
+
+Research efforts in deep domain adaptation have explored a proper metric to measure the variational distance between two domains and subsequently regularize neural networks to minimize this distance. For example, maximum mean discrepancy (Long et al., 2013; 2016; Tzeng et al., 2014; Fernando et al., 2015; Tzeng et al., 2015; Sun & Saenko, 2016) estimates the domain difference based on kernels. As another example, domain adversarial neural networks (Ganin et al., 2016; Bousmalis et al., 2016; 2017; Sohn et al., 2017; Luo et al., 2017; Tran et al., 2018), measuring the distance using a trainable and flexible discriminator often parameterized by an MLP, have been successfully adopted for several computer vision applications, such as semantic segmentation (Hoffman et al., 2016; Tsai et al., 2018; Zhang et al., 2018) and object detection (Chen et al., 2018). Most of those works assume a common classification task between two domains, whereas we tackle a cross-domain distance metric adaptation problem where label spaces of source and target domains are different.
+
+Moreover, our problem setting, an adaptation from labeled source to unlabeled target with disjoint label spaces, contains flavors from both domain adaptation (DA) and transfer learning (TL), following the nomenclature of (Pan et al., 2010). The difference in input distribution between source and target domains and the lack of labels in the target domain are similar to that of DA or transductive TL (Pan et al., 2010), while the difference in label distribution and task definitions between two domains is akin to inductive TL (Pan et al., 2010; Daumé III, 2007). In our work, we formalize this problem in domain adaptation framework using verification as a common task. This is a key contribution that allows theoretical analysis on the generalization bound as presented in Section 3 and Appendix A, while allowing novel applications like cross-ethnicity face recognition.
+
+In terms of task objective, (Hu et al., 2015; Ganin et al., 2016; Sohn et al., 2017) also deal with domain adaptation in distance metric learning, but neither learns a representation space capable of separating the source and target domains. Resembling CD2MA, Luo et al. (2017) considers domain adaptation with disjoint label spaces, but the problem is still cast as classification with an assumption that the target label space is known and a few labeled target examples are provided for training.
+
+In terms of network design, residual transfer network (Long et al., 2016), which learns two classifiers differ by a residual function for the source and the target domain, is closely related. However, it only tackles the scenario where source and target domains share a common label space for classification.
+
+# 3 REVISITING THE THEORY OF DOMAIN ADAPTATION FOR VERIFICATION
+
+Under the domain adaptation assumption, Ben-David et al. (2007) show that the empirical risk on the target domain $\mathcal { X } _ { T }$ is bounded by the empirical risk on the source domain $\chi _ { S }$ and the variational distance between the two domains, provided that the source and the target domains share the classifiers. Therefore, this bound is not applicable to our CD2MA setup where the label spaces of two domains are often different. To generalize those theoretical results to our setting, we reformulate the verification task as a binary classification task shared across two domains. This new binary classification task takes a pair of images as an input and predicts the label of 1 if the pair of images shares the same identity and 0 otherwise. Furthermore, if we now define the new source domain to be pairs of source images and the new target domain to be pairs of target images, then Theorem 1 and 2 from (Ben-David et al., 2007) can be directly carried over to bound the new target domain binary classification error in the same manner. That is, the empirical with-in target domain verification loss is bounded by with-in source domain verification loss and the variational distance between $\mathcal { X } _ { S } \times \mathcal { X } _ { S }$ and $\mathscr { X } _ { T } \times \mathscr { X } _ { T }$ .1 Note that inputs to the binary classifier are pairs of images from the same domain. Thus, this setup only addresses adaptation of within-domain verification to unlabeled target domains.
+
+There are two implications from the theoretical insights on domain adaptation using verification as a shared classification task. Firstly, domain adversarial training, reducing the discrepancy between the source and the target product spaces, coupled with supervised source domain binary classification loss (i.e., verification loss using source domain labels) can yield target representations with high discriminative power when performing within-domain verification. Note that in practice we approximately reduce the product space discrepancy by generic adversarial learning as done in (Ganin et al., 2016; Sohn et al., 2017). Secondly, there is no guarantee that the aligned source and target feature spaces possess any discriminative power for cross-domain verification task. Thus, additional actions in the form of a feature transfer module and domain separation objective are required to address this issue. These two consequences together motivate the design of our proposed framework, which is introduced in the next section.
+
+# 4 FEATURE TRANSFER NET: LEARNING TO ALIGN AND SEPARATE DOMAINS
+
+In this section, we first define the CD2MA problem setup and motivate our proposed feature transfer network (FTN). Then we elaborate on the training objectives that help our model achieve its desired properties. Lastly, we provide practical considerations to implement our proposed algorithm.
+
+# 4.1 PROBLEM STATEMENT AND ALGORITHM OVERVIEW
+
+Recall the description of CD2MA, given source and target domain data distributions $\chi _ { S }$ and $\mathcal { X } _ { T }$ , our goal is to verify whether two random samples $x , x ^ { \prime }$ drawn from either of the two distributions (and we do not know which distribution $x$ or $x ^ { \prime }$ come from a priori) belong to the same class.
+
+There are 3 scenarios of constructing a pair: $x , x ^ { \prime } \in \mathcal { X } _ { S }$ , $x , x ^ { \prime } \in \mathcal { X } _ { T }$ , or $x \in \mathcal { X } _ { S } , x ^ { \prime } \in \mathcal { X } _ { T }$ . We refer the task of the first two cases as within-domain verification and the last as cross-domain verification.
+
+![](images/8cfeb16e958c5f11cf1c6ec4dcf6ae64daf6322ef950f6bc112c510328987eef.jpg)  
+Figure 1: Training of Feature Transfer Network (FTN) for verification, composed of feature generation module (Gen; $f )$ , feature transfer module $( \operatorname { T x } ; g )$ , and two domain discriminators $D _ { 1 }$ and $D _ { 2 }$ . Verification objective ${ \mathcal { L } } _ { \mathrm { v r f } }$ ’s are applied to source $( f _ { s } )$ pairs and transformed source $( g ( f _ { s } ) ) ,$ ) pairs. Our FTN applies domain adversarial objective ${ \mathcal { L } } _ { \mathrm { a d v } }$ for domain alignment between transformed source and target domains by $D _ { 1 }$ and applies $\mathcal { L } _ { \mathrm { s e p } }$ to distinguish source domain from both target and transformed source domains by $D _ { 2 }$ .
+
+If $x , x ^ { \prime } \in \mathcal { X } _ { S }$ (or $\chi _ { T } .$ ), we need a source (or target) domain classifier2. For the source domain, we are provided with adequate labeled training examples to learn a competent classifier. For the target domain, we are only given unlabeled examples. However, with our extension of Theorem 1 and 2 from (Ben-David et al., 2007), discriminative power of the classifier can be transferred to the target domain by adapting the representation spaces of $\mathcal { X } _ { T } \times \mathcal { X } _ { T }$ and $\mathcal { X } _ { S } \times \mathcal { X } _ { S }$ , that is, we can utilize the same competent classifier from the source domain to verify target domain pairs if two domains are well-aligned. For the third scenario where $x \in X _ { S }$ but $x ^ { \prime } \in X _ { T }$ , we assume that the two examples cannot be of the same class, which is true for problems such as cross-ethnicity face verification.
+
+Our proposed framework, Feature Transfer Network (FTN), is designed to solve all these verification scenarios in an unified framework. FTN is composed of multiple modules as illustrated in Figure 1. First, a feature generation module $f : \mathcal { X } \to \mathcal { Z }$ denoted as “Gen” in Figure 1 ideally maps $\chi _ { S }$ and $\mathcal { X } _ { T }$ to distinguishable representation spaces, that is, $f ( \mathcal { X } _ { S } )$ and $f ( \mathcal { X } _ { T } )$ are far apart. To achieve this, we introduce a domain separation objective.3 Next, the feature transfer module $g : { \mathcal { Z } } \to { \mathcal { Z } }$ denoted as “Tx” in Figure 1 transforms $f ( \mathcal { X } _ { S } )$ to $g ( f ( \mathcal { X } _ { S } ) )$ for it to be aligned with $f ( \mathcal { X } _ { T } )$ . To achieve this, we introduce a domain adversarial objective. Finally, we apply verification losses on $f ( \mathcal { X } _ { S } )$ and $g ( f ( \mathcal { X } _ { S } ) )$ using classifiers $h _ { f } , h _ { g } : \mathcal { Z } \times \mathcal { Z }  \{ 0 , 1 \}$ . During testing, we compare the metric distance between ${ \dot { f } } ( x )$ and $f ( x ^ { \prime } )$ . Overall, we achieve the following desired capabilities:
+
+• If $x , x ^ { \prime }$ are from different domains, $f ( x )$ and $f ( x ^ { \prime } )$ will be far away due to the functionality of the feature generation module. • If $x , x ^ { \prime } \in \mathcal { X } _ { S }$ , then $f ( x )$ and $f ( x ^ { \prime } )$ will be close if they belong to the same class and far away otherwise, due to the discriminative power acquired from optimizing $h _ { f }$ . • If $x , x ^ { \prime } \in \mathcal { X } _ { T }$ , then $f ( x )$ and $f ( x ^ { \prime } )$ will be close if they belong to the same class and far otherwise, due to the discriminative power acquired by optimizing $h _ { g }$ with domain adversarial training.
+
+# 4.2 TRAINING OBJECTIVES
+
+We first define individual learning objectives of the proposed Feature Transfer Network and then present overall training objectives of FTN. For ease of exposition, all objectives are to be maximized.
+
+Verification Objective. For a pair of source examples, we evaluate the verification losses at two representations spaces $f ( \mathcal { X } _ { S } )$ and $g ( f ( \mathcal { X } _ { S } ) )$ using classifiers $h _ { f }$ and $h _ { g }$ as follows:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { v r f } } ( f ) = \mathbb { E } _ { ( x _ { 1 } , x _ { 2 } ) \in \mathcal { X } _ { S } \times \mathcal { X } _ { S } } \left[ y _ { 1 2 } \log h _ { f } ( f _ { 1 } , f _ { 2 } ) + ( 1 \mathrm { - } y _ { 1 2 } ) \log ( 1 - h _ { f } ( f _ { 1 } , f _ { 2 } ) ) \right] } \\ & { \mathcal { L } _ { \mathrm { v r f } } ( g ) = \mathbb { E } _ { ( x _ { 1 } , x _ { 2 } ) \in \mathcal { X } _ { S } \times \mathcal { X } _ { S } } \left[ y _ { 1 2 } \log h _ { g } ( g _ { 1 } , g _ { 2 } ) + ( 1 \mathrm { - } y _ { 1 2 } ) \log ( 1 - h _ { g } ( g _ { 1 } , g _ { 2 } ) ) \right] } \end{array}
+$$
+
+where $g _ { i } = g ( f ( x _ { i } ) ) , f _ { i } = f ( x _ { i } )$ and $y _ { 1 2 } = 1$ if $x _ { 1 }$ and $x _ { 2 }$ are from the same class and 0 otherwise. While classifiers $h _ { f } , h _ { g }$ can be parameterized by neural networks, we aim to learn a generator $f$ and $g$ whose embeddings can be directly used as a distance metric. Therefore, we use non-parameteric classifiers hf = σ(f >1 f2), hg = σ(g>1 g2) where σ(a) = 11+exp(−a) .
+
+Domain Adversarial Objective. Let $D _ { 1 } : \mathcal { Z } \to ( 0 , 1 )$ be a domain discriminator. As mentioned earlier, $D _ { 1 }$ is trained to discriminate distributions $f ( \mathcal { X } _ { T } )$ and $g ( f ( \mathcal { X } _ { S } ) )$ and then produces gradient for them to be indistinguishable. The learning objectives are written as follows:
+
+$$
+\mathcal { L } _ { D _ { 1 } } = \mathbb { E } _ { x \in \mathcal { X } _ { S } } \log D _ { 1 } ( g ) + \mathbb { E } _ { x \in \mathcal { X } _ { T } } \log \left( 1 - D _ { 1 } ( f ) \right) , \ \mathcal { L } _ { \mathrm { a d v } } = \mathbb { E } _ { x \in \mathcal { X } _ { T } } \log D _ { 1 } ( f )
+$$
+
+Note that when feature transform module is an identity mapping, i.e., $g ( f ( x ) ) = f ( x )$ , Equation (3) defines the training objective of standard DANN.
+
+Domain Separation Objective. The goal of this objective is to distinguish between source and target at representation spaces of generation module. To this end, we formulate the objective using another domain discriminator $D _ { 2 } : \mathcal { Z } \to ( 0 , 1 )$ :
+
+$$
+\mathcal { L } _ { \mathrm { s e p } } = \mathbb { E } _ { x \in \mathcal { X } _ { S } } \log D _ { 2 } ( f ) + \frac { 1 } { 2 } \big [ \mathbb { E } _ { x \in \mathcal { X } _ { S } } \log ( 1 - D _ { 2 } ( g ) ) + \mathbb { E } _ { x \in \mathcal { X } _ { T } } \log ( 1 - D _ { 2 } ( f ) ) \big ]
+$$
+
+Note that, in $\mathcal { L } _ { \mathrm { s e p } }$ , the source space $f ( \mathcal { X } _ { S } )$ is not only pushed apart from the target space $f ( \mathcal { X } _ { T } )$ but also from the augmented source space $g ( f ( \mathcal { X } _ { S } ) )$ to ensure that $g$ learns meaningful transformation of source domain representation beyond identity transformation.
+
+Training FTN. Now we are ready to present the overall training objectives $\mathcal { L } _ { f }$ and $\mathcal { L } _ { g }$ :
+
+$$
+\mathcal { L } _ { f } = \frac { 1 } { 2 } \big [ \mathcal { L } _ { \mathrm { v r f } } ( g ) + \mathcal { L } _ { \mathrm { v r f } } ( f ) \big ] + \lambda _ { 1 } \mathcal { L } _ { \mathrm { a d v } } + \lambda _ { 2 } \mathcal { L } _ { \mathrm { s e p } } , \mathcal { L } _ { g } = \mathcal { L } _ { \mathrm { v r f } } ( g ) + \lambda _ { 2 } \mathbb { E } _ { \mathcal { X } _ { S } } \log ( 1 - D _ { 2 } ( g ) )
+$$
+
+with $\lambda _ { 1 }$ for domain adversarial objective and $\lambda _ { 2 }$ for domain separation objective. We use $\mathcal { L } _ { D _ { 1 } }$ in Equation (3) for $D _ { 1 }$ and $\mathcal { L } _ { D _ { 2 } } = \mathcal { L } _ { \mathrm { s e p } }$ for $D _ { 2 }$ . We alternate updating between $D _ { 1 }$ and $( f , g , D _ { 2 } )$ .
+
+# 4.3 PRACTICAL CONSIDERATIONS
+
+Preventing Mode Collapse via Feature Reconstruction Loss. The mode collapsing phenomenon with generative adversarial networks (GANs) (Goodfellow et al., 2014) has received much attention (Salimans et al., 2016). In the context of domain adaptation, we also find it critical to treat the domain adversarial objective with care to avoid similar optimization instability.
+
+In this work, we prevent the mode collapse issue for domain adversarial learning with an additional regularization method similar to (Sohn et al., 2017). Assuming the representation of the source domain is already close to optimal, we regularize the features of source examples to be similar to those from the reference network $f _ { \mathrm { r e f } } : \mathcal { X } \xrightarrow { } \mathcal { Z }$ , which is pretrained on labeled source data and fixed during the training of $f$ . Furthermore, we add a similar but less emphasized $( \lambda _ { 4 } < \lambda _ { 3 } )$ regularization to target examples, simultaneously avoiding collapsing and allowing more room for target features to diverge from the original representations. Finally, the feature reconstruction loss is written as follows:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { r e c o n } } = - \big [ \lambda _ { 3 } \mathbb { E } _ { x \in \mathcal { X } _ { S } } \| f ( x ) - f _ { \mathrm { r e f } } ( x ) \| _ { 2 } ^ { 2 } + \lambda _ { 4 } \mathbb { E } _ { x \in \mathcal { X } _ { T } } \| f ( x ) - f _ { \mathrm { r e f } } ( x ) \| _ { 2 } ^ { 2 } \big ] } \end{array}
+$$
+
+We empirically find that without the feature reconstruction loss, the training would become unstable, reach an early local optimum and lead to suboptimal performance (see Section 6 and Appendix C). Thus, we always include the feature reconstruction loss to train DANN or FTN models unless stated otherwise.
+
+Replacing Verification Loss with $N$ -pair Loss. Our theoretical analysis in Section 3 (and Appendix A) suggests to use a verification loss that compares similarity between a pair of images. In practice, however, the pairwise verification loss is too weak to learn a good deep distance metric. Following (Sohn, 2016), we propose to replace the verification loss with an $N$ -pair loss, defined as follows:
+
+$$
+\mathcal { L } _ { N } ( f ) = \mathbb { E } _ { \{ x _ { n } , x _ { n } ^ { + } \} _ { n = 1 } ^ { N } , x _ { n } , x _ { n } ^ { + } \in \mathcal { X } _ { S } } \Big [ \sum _ { n = 1 } ^ { N } \log p _ { n } ( f ) \Big ] , \ p _ { n } ( f ) = \frac { \exp ( f ( x _ { n } ) ^ { \top } f ( x _ { n } ^ { + } ) ) } { \sum _ { k = 1 } ^ { N } \exp ( f ( x _ { n } ) ^ { \top } f ( x _ { k } ^ { + } ) ) }
+$$
+
+where $x _ { n }$ and $x _ { n } ^ { + }$ are from the same class and $x _ { n }$ and $x _ { k } ^ { + }$ , $n \neq k$ , are from different classes. Replacing ${ \mathcal { L } } _ { \mathrm { v r f } }$ into $\mathcal { L } _ { N }$ , the training objective of FTN with $N$ -pair loss is written as follows:
+
+$$
+\mathcal { L } _ { f } = \frac { 1 } { 2 } \big [ \mathcal { L } _ { N } ( g ) + \mathcal { L } _ { N } ( f ) \big ] + \lambda _ { 1 } \mathcal { L } _ { \mathrm { a d v } } + \lambda _ { 2 } \mathcal { L } _ { \mathrm { s e p } } + \mathcal { L } _ { \mathrm { r e c o n } } , \ \mathcal { L } _ { g } = \mathcal { L } _ { N } ( g ) + \lambda _ { 2 } \mathbb { E } _ { \mathcal { X } _ { S } } \log ( 1 - D _ { 2 } ( g ) )
+$$
+
+![](images/615d4cd79fd19165dc5663fa4acc13f4c916387923ea7b2e30633301e743bb85.jpg)  
+Figure 2: t-SNE visualizations of source (0–4 from MNIST-M) and target (5–9 from MNIST) representations by different learning methods: (a) deep neural network without adaptation, (b) domain adversarial neural network (DANN) and (c) our feature transfer network (FTN). While domain adversarial learning results in significant confusion of digits classes between source and target domains (e.g., 3/5, 2/8, 4/9, or 0/6 in (b)), the proposed FTN transfers discriminative power to target domain while successfully separating them from the source domain.
+
+# 5 ENTROPY MINIMIZATION VIA HIERARCHICAL CLUSTERING
+
+Entropy minimization (Grandvalet & Bengio, 2005) is a popular training objective in unsupervised domain adaptation: unlabeled data is trained to minimize entropy of a class prediction distribution so as to form features that convey confident decision rules. However, it is less straightforward how to apply entropy minimization when label spaces for source and target are disjoint. Motivated from Section 3, we extend entropy minimization for distance metric adaptation using verification as a common task for both domains:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { v r f } } ^ { \mathrm { e n t } } ( f ) = \mathbb { E } _ { x _ { i } , x _ { j } \in \mathcal { X } _ { T } } \left[ p _ { i j } \log p _ { i j } + ( 1 - p _ { i j } ) \log ( 1 - p _ { i j } ) \right] } \end{array}
+$$
+
+where $p _ { i j } \triangleq p _ { i j } ( f ) = \sigma ( f ( x _ { i } ) ^ { \top } f ( x _ { j } ) )$ . This formulation encourages a more confident prediction for verifying two unlabeled images, whether or not coming from the same class.
+
+However, recall that for the source domain, we use $N$ -pair loss instead of pair-wise verification loss for better representation learning. Therefore, we would like to similarly incorporate the concept of $N$ - pair loss on the target domain by forging a multi-class entropy minimization (MCEM) objective. This demands $N$ pair examples to be sampled from the target domain. As the target domain is unlabeled, we ought to first discover a plausible label structure, which is done off-line via HDBSCAN (Campello et al., 2013; McInnes et al., 2017), a fast and scalable density-based hierarchical clustering algorithm. The returned clusters provide pseudo-labels to individual examples of the target domain, allowing us to sample $N$ pair examples to evaluate the following MCEM objective:
+
+$$
+\mathcal { L } _ { N } ^ { \mathrm { e n t } } ( f ) = \mathbb { E } _ { \{ x _ { n } , x _ { n } ^ { + } \} _ { n = 1 } ^ { N } , x _ { n } , x _ { n } ^ { + } \in \mathcal { X } _ { T } } \Big [ \sum _ { n = 1 } ^ { N } \big \{ \sum _ { m = 1 } ^ { N } p _ { n m } \log p _ { n m } \big \} \Big ] , \ p _ { n m } ( f ) = \frac { \exp \bigl ( f _ { n } ^ { \top } f _ { m } ^ { + } \bigr ) } { \sum _ { k = 1 } ^ { N } \exp \bigl ( f _ { n } ^ { \top } f _ { k } ^ { + } \bigr ) }
+$$
+
+where $x _ { n }$ and $x _ { n } ^ { + }$ are from the same cluster and $x _ { n }$ and $x _ { k } ^ { + }$ , $n \neq k$ are from different clusters. The objective can be combined with $\mathcal { L } _ { f }$ in Equation (8) to optimize $f$ .
+
+# 6 EXPERIMENTS
+
+In this section, we first experiment on digit datasets as a proof of concept and compare our proposed FTN to DANN. Then, we tackle the problem of cross-ethnicity generalization in the context of face recognition to demonstrate the effectiveness of FTN. In all experiments, we use $N$ -pair loss as defined in Equation (8) to update $f$ and $g$ for better convergence and improved performance. We also use the same learning objectives for DANN while fixing $g$ to the identity mapping and $\lambda _ { 2 } = 0$ .
+
+# 6.1 PROOF OF CONCEPT: MNIST-M (0–4) TO MNIST (5–9)
+
+To provide insights on the functionality of FTN, we conduct an experiment adapting the digits 0–4 from MNIST-M (Ganin et al., 2016) to 5–9 from MNIST. In other words, the two domains in our setting not only differ in foreground and background patterns but also contain non-overlapping digit classes, contrasting the usual adaptation setup with a shared label space. Our goal is to learn a feature space that separates the digit classes not only within each domain, but also across the two.
+
+We construct a feature generator $f$ composed of a CNN encoder followed by two fully-connected (FC) layers and a feature transfer module $g$ composed of MLP with residual connections. Outputs of $f$ and $g$ are then fed to discriminators $D _ { 1 }$ and $D _ { 2 }$ parameterized by MLPs to induce domain adversarial and domain separation losses respectively. We provide more architecture details in Appendix B.1.
+
+We visualize t-SNE plots of generator features in Figure 2. Without an adaptation (Figure 2(a)), features of digits from the target domain are heavily mixed with those from the source domain as well as one another. The model reaches $1 . 3 \%$ verification error in the source domain but as high as $2 7 . 3 \%$ in the target domain. Though DANN in Figure 2(b) shows better separation with a reduced target verification error of $2 . 2 \%$ , there still exists significant overlap between digit classes across two domains, such as 3/5, 4/9, 0/6 and 2/8. As a result, a domain classifier trained to distinguish source and target on top of generator features can only attain $1 1 . 5 \%$ classification error. In contrast, the proposed FTN in Figure 2(c) shows 10 clean clusters without any visual overlap among 10 digits classes from either source or target domain, implying that it not only separates digits within the target domain ( $2 . 1 \%$ verification error), but also differentiates them across domains $( 0 . 3 \%$ domain classification error).
+
+Table 1: Verification and identification accuracy on the Cross Ethnicity Faces (CEF) dataset. For supervised models, we report results trained on labeled CAU $( \mathrm { S u p } ^ { \mathrm { C } } )$ or on labeled CAU, AA, EA domains $( \mathbf { S } \mathbf { u p } ^ { \mathbf { C } , \mathrm { A } , \mathrm { E } } )$ ; for adaptation, we evaluate DANN and FTN, without and with multi-class entropy minimization (MCEM).   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Verification</td><td colspan="4">Identification</td></tr><tr><td>CAU</td><td>AA</td><td>EA</td><td>ALL</td><td>CAU</td><td>AA</td><td>EA</td><td>ALL</td></tr><tr><td>SupC</td><td>98.39</td><td>92.24</td><td>93.41</td><td>95.58</td><td>90.07</td><td>69.64</td><td>76.37</td><td>77.97</td></tr><tr><td>SupCA.E</td><td>98.43</td><td>97.16</td><td>97.05</td><td>98.15</td><td>90.16</td><td>84.02</td><td>84.38</td><td>85.75</td></tr><tr><td>DANN\Lrecon</td><td>98.36</td><td>94.54</td><td>95.02</td><td>96.84</td><td>90.01</td><td>73.05</td><td>74.94</td><td>77.99</td></tr><tr><td>DANN</td><td>98.36</td><td>95.37</td><td>96.36</td><td>97.34</td><td>90.34</td><td>74.88</td><td>79.39</td><td>79.83</td></tr><tr><td>FTN</td><td>98.36</td><td>95.62</td><td>96.64</td><td>97.68</td><td>90.54</td><td>75.35</td><td>80.69</td><td>81.28</td></tr><tr><td>DANN+MCEM</td><td>98.39</td><td>96.36</td><td>97.34</td><td>97.88</td><td>90.77</td><td>80.30</td><td>83.07</td><td>82.69</td></tr><tr><td>FTN+MCEM</td><td>98.37</td><td>96.76</td><td>97.40</td><td>98.08</td><td>90.95</td><td>80.75</td><td>83.71</td><td>84.16</td></tr></table>
+
+Table 2: Cross domain identification accuracy on CEF, with CAU evaluated against AA $^ +$ EA combined, AA against CAU and EA against CAU.   
+
+<table><tr><td>Model</td><td>CAU vs. AA, EA</td><td>AA vs. CAU</td><td>EA vs. CAU</td></tr><tr><td>SupC</td><td>91.67</td><td>95.42</td><td>94.87</td></tr><tr><td>DANN</td><td>89.91</td><td>84.78</td><td>91.47</td></tr><tr><td>FTN</td><td>92.29</td><td>88.09</td><td>92.07</td></tr></table>
+
+# 6.2 CROSS ETHNICITY FACE VERIFICATION AND RECOGNITION
+
+The performances of face recognition engines have significantly improved thanks to recent advances in deep learning for image recognition (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; Szegedy et al., 2015; He et al., 2016) and publicly available large-scale face recognition datasets (Yi et al., 2014; Guo et al., 2016). However, most public datasets are collected from the web by querying celebrities, with significant label bias towards Caucasian ethnicity. For example, more than $\bar { 8 } 5 \%$ of identities are Caucasian for CASIA Web face dataset (Yi et al., 2014). Similarly, $8 2 \%$ are Caucasian (CAU) for MS-Celeb-1M (MS-1M) dataset (Guo et al., 2016), while there are only $9 . 7 \%$ African-American (AA), $6 . 4 \%$ East-Asian (EA) and less than $2 \%$ Latino and South-Asian combined.4
+
+Such imbalance across ethnicity in labeled training data can result in significant drop in identification performance on data-scarce minorities: the second row of Table 1 shows a model trained on Caucasian dominated dataset performs poorly on the other ethnicities. As expected, if the training data is composed of only Caucasian identities as source domain, the performance over the target domains consisting of the other ethnicities further deteriorates (see row 1 of Table 1). Provided the available labeled source domain contains only Caucasian identities, we subsequently demonstrate that our method can effectively leverage unlabeled data from the non-Caucasian target ethnicity to substantially improve their face verification performances.
+
+Experimental Setup. We perform an adaptation from CAU to a mixture of AA and EA. Our experiments use the MS-1M dataset. We first remove identities that both appear in the training and testing sets. The resulting training set consists of $4 . 0 4 M$ images from $6 0 K$ CAU identities, $3 9 8 K$ images from $7 K$ AA identities, and $3 0 8 K$ images from $4 . 6 K$ EA identities. For domain adaptation experiments, we use labeled CAU images and unlabeled AA, EA images for training. For supervised experiments to obtain performance lower and upper bound, we use labeled CAU images to train $\mathrm { S u p ^ { C } }$ and labeled CAU, AA, EA images to train $\mathrm { S u p } ^ { \mathrm { \hat { C } , A , E } }$ .
+
+We adopt a 38-layer ResNet (He et al., 2016) for the feature generation module. Feature transfer module and discriminators are parameterized with MLPs similarly to Section 6.1. We use 4096-pair loss for training, including for the supervised CNNs. It is worth mentioning that our network architecture and training scheme result in strongly competitive face recognition performance, comparing to other state-of-the-art methods such as FaceNet (Schroff et al., 2015) on YouTube Faces (Wolf et al., 2011) $( 9 7 . 3 2 \%$ (ours) vs $9 5 . 1 2 \%$ ) and Neural Aggregation Network (Yang et al., 2017) on IJB-A (see row 2 of Table 3). The complete network architecture and training details are provided in Appendix B.2.
+
+Evaluation. We report the performance of the baseline and our proposed models on two standard face recognition benchmarks LFW (Huang et al., 2007) and IJB-A (Klare et al., 2015). Note that these datasets also exhibit significant ethnicity bias.5
+
+To highlight the effectiveness of the proposed adaptation approach, we construct individual test set for CAU, AA, EA, each of which contains 10 face images from 200 identities. We refer to our testing set as the Cross-Ethnicity Faces (CEF) dataset. We apply two evaluation metrics on CEF dataset, verification accuracy and identification accuracy. For verification, following the standard protocol (Huang et al., 2007), we construct 10 splits, each containing 900 positive and 900 negative pairs, and compute the accuracy on each split using the threshold found from the other 9 splits. For identification, a pair composed of the reference and the query images from the same identity is considered correct if there is no image from different identity that has higher similarity to the reference image than the query image. We evaluate identification accuracy per ethnicity (200-way) as well as across all ethnicities (600-way).
+
+Results. The results on CEF are summarized in Table 1. Cross domain identification accuracy is reported in Table 2, where we use AA and EA as negative classes when evaluating accuracy on CAU and vice versa, as a measure to indicate domain discrepancy. Among adaptation models, DANN without feature reconstruction loss $\left( \mathrm { D A N N } \backslash \mathcal { L } _ { \mathrm { r e c o n } } \right)$ shows unstable training and easily degenerate, which leads to only marginal improvement upon $\mathrm { S u p ^ { C } }$ . Similar trend is observed while training FTN. Therefore, to ensure training stability, we impose ${ \mathcal { L } } _ { \mathrm { r e c o n } }$ as a regularization term for all adaptation models. More analysis on the effectiveness of $\scriptstyle { \mathcal { L } } _ { \mathrm { r e c o n } }$ is provided in Appendix C.
+
+When testing on AA and EA with model trained on only the labeled source CAU domain $( \mathrm { S u p } ^ { \mathrm { C } } )$ , we observe significant performance drops in Table 1. Meanwhile, in Table 2, cross domain identification accuracy is much higher than within domain identification accuracy, i.e., $9 6 . 1 4 \%$ of AA vs. CAU is much higher than $7 1 . 9 2 \%$ of AA identification in Table 1, indicating 1) significant discrepancy between the feature spaces of the source and target domains and 2) lack of discriminative power for within domain verification task on target ethnicity.
+
+Comparing to $\mathrm { S u p ^ { C } }$ , both DANN and FTN show moderate improvement when testing on AA and EA from CEF (Table 1), demonstrating the effectiveness of domain adversarial learning in transferring within domain verification capability from labeled source domain to unlabeled target domain. Despite the improvement, DANN suffers a notable drawback from adversarial objective which attempts to align identities from different domains, resulting a poor cross domain identification accuracy as shown in Table 2. In contrast, the proposed FTN achieves much higher cross domain identification accuracy, demonstrating both within and cross domain discriminative power.
+
+Additionally, in combination with the multi-class entropy minimization $\mathbf { ( F T N + M C E M ) }$ ), we further boost the verification and identification accuracy over FTN on AA and EA as well as approach the accuracy of $\operatorname { S u p } ^ { \mathrm { C , A , E } }$ , the performance upper bound. This indicates that the HDBSCAN-based hierarchical clustering provides high quality pseudo-class labels for MCEM to be effective. Indeed, the clustering algorithm achieves F-score as high as $9 6 . 3 1 \%$ and $9 6 . 3 4 \%$ on AA and EA. We provide more in-depth analysis on the clustering strategy in Appendix D.
+
+Finally, Table 3 reports the performance of face recognition models on standard verification and recognition benchmarks. We observe similar improvements with our proposed distance metric adaptation when only using labeled CAU, i.e., source domain, as training data. Once the task becomes more challenging thus demands more discriminative power, the advantage of our method becomes more evident, such as in the case of open-set recognition and verification at low FAR.
+
+# 7 CONCLUSION
+
+We address the challenge of unsupervised domain adaptation when the source and the target domains have disjoint label spaces by formulating the classification problem into a verification task. We propose a Feature Transfer Network, allowing simultaneous optimization of domain adversarial loss and domain separation loss, as well as a variant of $N$ -pair metric loss for entropy minimization on the target domain where the ground-truth label structure is unknown, to further improve the adaptation quality. Our proposed framework excels at both within-domain and cross-domain verification tasks. As an application, we demonstrate cross-ethnicity face verification that overcomes label biases in training data, achieving high accuracy even for unlabeled ethnicity domains, which we believe is a result with vital social significance.
+
+Table 3: Face verification and recognition performance on LFW and IJB-A. From left to right, verification (VRF), closed-set (CLS) and open-set recognition at $\mathrm { F A R } = 0 . 0 1$ and 0.001 (Best-Rowden et al., 2014) on LFW, and verification at different FAR and identification (id.) at rank- $k$ on IJB-A are reported.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">LFW</td><td colspan="3">IJB-A (verification)</td><td colspan="2">IJB-A (id.)</td></tr><tr><td>VRF</td><td>CLS</td><td>0.01</td><td>0.001</td><td>0.01</td><td>0.001</td><td>0.0001</td><td>rank-1</td><td>rank-5</td></tr><tr><td>SupC</td><td>99.57</td><td>98.95</td><td>86.07</td><td>66.61</td><td>92.67</td><td>76.65</td><td>50.32</td><td>94.31</td><td>97.25</td></tr><tr><td>SupCA.E</td><td>99.72</td><td>98.79</td><td>96.81</td><td>91.11</td><td>95.57</td><td>87.45</td><td>76.45</td><td>94.73</td><td>97.19</td></tr><tr><td>DANN\Lrecon</td><td>99.43</td><td>98.98</td><td>96.81</td><td>91.44</td><td>94.23</td><td>86.87</td><td>73.80</td><td>94.27</td><td>97.03</td></tr><tr><td>DANN</td><td>99.63</td><td>98.95</td><td>97.15</td><td>93.46</td><td>95.54</td><td>88.64</td><td>77.13</td><td>94.59</td><td>97.31</td></tr><tr><td>FTN</td><td>99.63</td><td>99.11</td><td>97.15</td><td>92.95</td><td>95.07</td><td>88.45</td><td>77.70</td><td>94.48</td><td>97.19</td></tr><tr><td>DANN+MCEM</td><td>99.63</td><td>99.08</td><td>97.65</td><td>94.97</td><td>95.28</td><td>88.78</td><td>77.30</td><td>94.75</td><td>97.30</td></tr><tr><td>FTN+MCEM</td><td>99.65</td><td>99.14</td><td>96.98</td><td>93.46</td><td>94.63</td><td>88.28</td><td>77.98</td><td>94.79</td><td>97.00</td></tr></table>
+
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+# Appendix
+
+# A DERIVATION FOR GENERALIZATION BOUND OF TARGET DOMAIN VERIFICATION LOSS
+
+Let $( \mathcal { X } , \mathcal { F } )$ and $( { \mathcal { Z } } , { \mathcal { G } } )$ be measurable input and feature spaces respectively and a feature extractor $R : { \mathcal { X } } \to { \mathcal { Z } }$ be a measurable function. Let $\mu$ be a probability measure on $\mathcal { X }$ corresponding to the data distribution. Let $( \mathcal { X } _ { 1 } , \mathcal { F } , \mu _ { 1 } ) = ( \mathcal { X } _ { 2 } , \mathcal { F } , \mu _ { 2 } ) = ( \mathcal { X } , \mathcal { F } , \mu )$ and $\mu _ { 1 2 } = \mu _ { 1 } \times \mu _ { 2 }$ on $\mathcal { X } _ { 1 } \times \mathcal { X } _ { 2 }$ be the unique product measure (Forrest). Similarly, we construct $\mathcal { Z } _ { 1 } \times \mathcal { Z } _ { 2 }$ where $( \mathcal { Z } _ { 1 } , \mathcal { G } ) = ( \mathcal { Z } _ { 2 } , \mathcal { G } ) = ( \mathcal { Z } , \mathcal { G } )$ . Since $R$ is measurable, $R ^ { 2 } : \mathcal { X } _ { 1 } \times \mathcal { X } _ { 2 } \to \mathcal { Z } _ { 1 } \times \mathcal { Z } _ { 2 }$ where $R ^ { 2 } ( x _ { 1 } , x _ { 2 } ) = ( R ( x _ { 1 } ) , R ( x _ { 2 } ) )$ is also measurable (see Lemma 1 for the proof). Then we can obtain an induced probability measure for $\mathcal { Z } _ { 1 } \times \mathcal { Z } _ { 2 }$ from $R ^ { 2 }$ , denoted as ${ \tilde { \mu } _ { 1 2 } } = \mu _ { 1 2 } \circ ( R ^ { 2 } ) ^ { - 1 }$ (Proposition 1.34 from (Lalley, 2017)).
+
+Let $Y : \mathcal { X } _ { 1 } \times \mathcal { X } _ { 2 }  \{ 0 , 1 \}$ , where 1 represents the pair from the same identity and 0 otherwise,6 be the stochastic target function for ground truth labeling, $\phi ( x _ { 1 } , x _ { 2 } ) = \mathbb { E } \left[ Y ( \dot { x _ { 1 } } , x _ { 2 } ) \right]$ be the expectation of the label at $( x _ { 1 } , x _ { 2 } )$ , and $\tilde { \phi } ( z _ { 1 } , z _ { 2 } ) = \mathbb { E } \left[ \phi ( x _ { 1 } , x _ { 2 } ) | R ( x _ { 1 } ) { = } z _ { 1 } , R ( x _ { 2 } ) { = } z _ { 2 } \right]$ be the conditional expectation of $\phi$ given the value of $R ^ { 2 } ( x _ { 1 } , x _ { 2 } ) = ( z _ { 1 } , z _ { 2 } )$ . Now, consider two domains, namely the source domain with probability measure $\mu ^ { S }$ over $\mathcal { X }$ and induced probability measure $\tilde { \mu } ^ { S }$ over $\mathcal { Z } _ { 1 } \times \mathcal { Z } _ { 2 }$ , as well as target domain counterparts $\mu ^ { T }$ and $\tilde { \mu } ^ { T }$ . Provided with a deterministic hypothesis class $\mathcal { H } \subseteq \left\{ g : \mathcal { Z } _ { 1 } \times \mathcal { Z } _ { 2 } \overset { \cdot } { \to } \left\{ 0 , 1 \right\} \right\}$ of VC-dimension $d$ , suppose there exists a function $h \in \mathcal H$ that can predict both source and target domains reasonably well. Then, we can quantify $\tilde { \phi }$ to be $\lambda$ -close to $\mathcal { H }$ :
+
+$$
+\operatorname* { i n f } _ { h \in \mathcal { H } } \epsilon _ { S } ( h ) + \epsilon _ { T } ( h ) \leq \lambda , \mathrm { w h e r e } \epsilon _ { i } ( h ) = \int | \tilde { \phi } ( z _ { 1 } , z _ { 2 } ) - h ( z _ { 1 } , z _ { 2 } ) | d \tilde { \mu } ^ { i } .
+$$
+
+We are ready to define the variational distance between the two domains with respect to $\mathcal { H }$ :
+
+$$
+\begin{array} { r } { d _ { \mathcal { H } } ( \tilde { \mu } ^ { S } , \tilde { \mu } ^ { T } ) = 2 \underset { A \in \mathcal { A } } { \operatorname* { s u p } } | \tilde { \mu } ^ { S } ( A ) - \tilde { \mu } ^ { T } ( A ) | , \ A = \{ A _ { h } = \{ ( z _ { 1 } , z _ { 2 } ) \in \mathcal { Z } _ { 1 } \times \mathcal { Z } _ { 2 } : h ( z _ { 1 } , z _ { 2 } ) = 1 \} , h \in \mathcal { H } \} . } \end{array}
+$$
+
+So far, we have successfully prepared the components in our verification setup to meet the assumptions and the format required by Theorem 1 from (Ben-David et al., 2007). We may now directly apply the theorem:
+
+Theorem 1. Randomly sample a labeled set of size m by applying $R ^ { 2 }$ to samples from $\mathcal { X } _ { 1 } \times \mathcal { X } _ { 2 }$ with labels defined according to $Y$ , with probability at least $1 - \delta$ , $\forall h \in { \mathcal { H } }$
+
+$$
+\epsilon _ { T } ( h ) \leq \hat { \epsilon } _ { S } ( h ) + \sqrt { \frac { 4 } { m } ( d \log \frac { 2 m } { d } + d + \log \frac { 4 } { \delta } } ) + d _ { \mathcal { H } } ( \tilde { \mu } ^ { S } , \tilde { \mu } ^ { T } ) + \lambda .
+$$
+
+Furthermore, $d _ { \mathcal { H } } ( \tilde { \mu } ^ { S } , \tilde { \mu } ^ { T } )$ can be empirically approximated by finite samples from both domain (Kifer et al., 2004), using the binary classifier from $\mathcal { H }$ that can best distinguishes pairs of samples between two domains. Following Theorem 2 from (Ben-David et al., 2007), let $\tilde { U } _ { S }$ and $\tilde { U } _ { T }$ consist of $n$ random pairs of samples from source and target each, with probability at least $1 - \delta$ , we have :
+
+$$
+\begin{array} { r l } & { d _ { \mathcal { H } } ( \tilde { \mu } ^ { S } , \tilde { \mu } ^ { T } ) \leq d _ { \mathcal { H } } ( \tilde { U } _ { S } , \tilde { U } _ { T } ) + \sqrt { \frac { d \log ( 2 n ) + \log \frac { 4 } { \delta } } { n } } , } \\ & { \cdot d _ { \mathcal { H } } ( \tilde { U } _ { S } , \tilde { U } _ { T } ) = 2 \left( 1 - 2 \operatorname* { m i n } _ { h \in \mathcal { H } } \frac { 1 } { 2 n } \sum _ { i = 1 } ^ { 2 n } \big | h \big ( z _ { 1 , i } , z _ { 2 , i } \big ) - \mathbf { 1 } \big \{ \big ( z _ { 1 , i } , z _ { 2 , i } \big ) \in \tilde { U } _ { S } \big \} \big | \right) . } \end{array}
+$$
+
+For completeness of our analysis, we formalize and prove in Lemma 1 that $R ^ { 2 }$ is measurable.
+
+Lemma 1. Let $( \mathcal { X } , \mathcal { F } , \mu )$ and $( \mathcal { Z } , \mathcal { G } , \tilde { \mu } )$ be measurable spaces and let $( \boldsymbol { \mathcal { X } } \times \boldsymbol { \mathcal { X } } , \sigma ( \boldsymbol { \mathcal { F } } \times \boldsymbol { \mathcal { F } } ) , \mu \times \mu )$ , $( \mathcal { Z } \times \mathcal { Z } , \sigma ( \mathcal { G } \times \mathcal { G } ) , \tilde { \mu } \times \tilde { \mu } )$ be their product spaces with the product measures. Let $R : { \mathcal { X } } \to { \mathcal { Z } }$ be $a$ measurable function, then $R ^ { 2 } : \mathcal { X } \times \mathcal { X } \to \mathcal { Z } \times \mathcal { Z }$ where $R ^ { 2 } \bar { ( x _ { 1 } , x _ { 2 } ) } = ( R ( x _ { 1 } ) , R ( x _ { 2 } ) )$ is also measurable.
+
+Proof. As the $\sigma$ -algebra of $\mathcal { Z } \times \mathcal { Z }$ is generated by $\mathcal { G } \times \mathcal { G }$ , we only need to show that the pre-image of any generator is measurable. Let $G _ { 1 } \times G _ { 2 } \in \mathcal { G } \times \mathcal { G }$ , then it is easy to see that $\hat { ( } R ^ { 2 } ) ^ { - 1 } \bar { ( } G _ { 1 } \times G _ { 2 } \bar { ) } \stackrel { - } { = } R ^ { - 1 } ( G _ { 1 } ) \times R ^ { - 1 } ( G _ { 2 } )$ . Since $R$ is a measurable function, hence $\overline { { R } } ^ { - 1 } ( G _ { 1 } )$ and $R ^ { - 1 } ( G _ { 2 } )$ are measurable and so is $R ^ { - 1 } ( G _ { 1 } ) \times R ^ { - 1 } ( G _ { 2 } )$ measurable. □
+
+![](images/feabfc061fb298eb630fbc66b69a6ffa2ac433469c82c5673bdc6aba18120b73.jpg)  
+Figure S1: Network architecture of feature transfer module and domain discriminators.
+
+![](images/c0a4d56878dd80fee9680f7f001096706ad75341901b73053839ce0bfae6f53a.jpg)  
+(b) domain discriminator
+
+(a) feature transfer module
+
+Table S1: Network architecture for digit experiments.   
+
+<table><tr><td rowspan=1 colspan=1>operation</td><td rowspan=1 colspan=1>kernel</td><td rowspan=1 colspan=1>output size</td></tr><tr><td rowspan=1 colspan=1>Conv1-1+ReLUConv1-2 +ReLU max pooling</td><td rowspan=1 colspan=1>3×33×32×2</td><td rowspan=1 colspan=1>32×32×3232×32×3216×16×32</td></tr><tr><td rowspan=1 colspan=1>Conv2-1 +ReLUConv2-2 +ReLUmax pooling</td><td rowspan=1 colspan=1>3×33×32×2</td><td rowspan=1 colspan=1>16×16×6416×16×648×8×64</td></tr><tr><td rowspan=1 colspan=1>Conv3-1+ReLUConv3-2+ReLUmax pooling</td><td rowspan=1 colspan=1>3×33×32×2</td><td rowspan=1 colspan=1>8×8×1288×8×1284×4×128</td></tr><tr><td rowspan=1 colspan=1>FC1+ReLUFC2Normalize and Scale (2)</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>128128128</td></tr></table>
+
+# B NETWORK ARCHITECTURE AND TRAINING DETAILS
+
+B.1 TOY EXPERIMENTS: MNIST-M $( 0 - 4 )$ TO MNIST $( 5 - 9 )$
+
+Following (Haeusser et al., 2017), we preprocess the data by subtracting a channel-wise pixel mean and dividing by channel-wise standard deviation of pixel values. For MNIST examples, we also apply color-intensity inversion. All images are resized into $3 2 \times 3 2$ with 3 channels.
+
+Our feature generator module is composed of 6 convolution layers and 3 max-pooling layers followed by 2 fully-connected layers. We use ReLU (Nair & Hinton, 2010) after convolution layers. The output dimension of the feature generator module is 128 and is normalized to have L2-norm of 2. The full description of the generator module is in Table S1.
+
+The feature transfer module maps 128 dimensional vector into the same dimensional vector using two fully-connected layers $( 1 2 8 - 2 5 6 - 2 5 6 - 1 2 8 )$ and residual connection as in Figure 1(a). Discriminator architectures are similar to that in Figure 1(b) but with fully-connected layers whose output dimensions are 128 instead of 256.
+
+We use Adam stochastic optimizer with learning rate of 0.0003, $\lambda _ { 1 } = 0 . 3$ and $\lambda _ { 2 } = 0 . 0 3$ to train FTN.
+
+# B.2 CROSS ETHNICITY FACE VERIFICATION AND RECOGNITION
+
+Our experimental protocols, such as data preprocessing and network architecture, closely follow those of (Sohn et al., 2017). We preprocess face images by detecting (Yang et al., 2016), aligning (Yu et al., 2016), and cropping to provide face images of size $1 1 0 \times 1 1 0$ . The data is prepared for network training by random cropping into $1 0 0 \times 1 0 0$ with horizontal flip with a $5 0 \%$ chance and converting into gray-scale.
+
+Our feature generation module contains 38 layers of convolution with several residual blocks and max pooling layers. We use ReLU (Nair & Hinton, 2010) for most of the layers in combination with maxout nonlinearities (Goodfellow et al., 2013). We add $7 \times 7$ average pooling layer on top of the last convolution layer. The output of the feature generation module is 320 dimensional vector and is normalized to have L2-norm of size 12. The full description of the model is in Table S2.
+
+The feature transfer module maps 320 dimensional output vector from feature generation module into the same dimensional vector using two fully-connected layers and residual connection. The architecture of feature transfer module is described in Figure 1(a). Discriminators have similar network architecture besides different numbers of neurons and omitted residual connection.
+
+All models, including supervised CNNs $\mathrm { \langle S u p ^ { C } }$ , $\operatorname { S u p } ^ { \mathrm { C , A , E } }$ ), are trained with 4096-pair loss. For SupC and $\operatorname { S u p } ^ { \mathrm { C , A , E } }$ , we use Adam stochastic optimizer (Kingma & Ba, 2015) with the learning rate of 0.0003 for the first $1 2 K$ updates and 0.0001 and 0.00003 for the next two subsequent $3 K$ updates.
+
+Our feature generation module is initialized with the $\mathrm { S u p ^ { C } }$ model, which is also used as a reference network for feature reconstruction loss as described in Section 4.3. Other modules of our model, such as feature generation module and discriminators, are initialized randomly. All modules are then updated with the learning rate of 0.00003. Hyperparameters of different models are summarized in Table S3.
+
+Table S2: Network architecture for face experiments.   
+
+<table><tr><td rowspan=1 colspan=1>operation</td><td rowspan=1 colspan=1>kernel</td><td rowspan=1 colspan=1>output size</td></tr><tr><td rowspan=1 colspan=1>Conv1-1+ReLUConv1-2 + Maxout (2)max pooling</td><td rowspan=1 colspan=1>3×33×32×2</td><td rowspan=1 colspan=1>100×100×32100×100×6450×50×64</td></tr><tr><td rowspan=1 colspan=1>ResBlock+ReLU×2Conv2 + Maxout (2)max pooling</td><td rowspan=1 colspan=1>3×3,64-64-643×32×2</td><td rowspan=1 colspan=1>50×50×6450×50×12825×25×128</td></tr><tr><td rowspan=1 colspan=1>ResBlock+ReLU×4Conv3 + Maxout (2) max pooling</td><td rowspan=1 colspan=1>3×3,128-96-1283×32×2</td><td rowspan=1 colspan=1>25×25×12825×25×19213×13×192</td></tr><tr><td rowspan=1 colspan=1>ResBlock + ReLU ×8Conv4 + Maxout (2)max pooling</td><td rowspan=1 colspan=1>3×3,192-128－1923×32×2</td><td rowspan=1 colspan=1>13×13×19213×13×2567×7×256</td></tr><tr><td rowspan=1 colspan=1>ResBlock + ReLU ×2Conv5 + Maxout (2)avg poolingNormalize and Scale (12)</td><td rowspan=1 colspan=1>3×3,256-160－2563×37×71</td><td rowspan=1 colspan=1>7×7×2567×7×3201×1×320320</td></tr></table>
+
+Table S3: Optimal hyperparameter settings of different adaptation models.   
+
+<table><tr><td></td><td>入1</td><td>入2</td><td>入3</td><td>入4</td></tr><tr><td>DANN</td><td>0.1</td><td>1</td><td>0.1</td><td>0.01</td></tr><tr><td>FTN</td><td>0.03</td><td>0.1</td><td>0.03</td><td>0.01</td></tr><tr><td>FTN+MCEM</td><td>0.03</td><td>0.1</td><td>0.03</td><td>0.003</td></tr></table>
+
+# C IMPACT OF FEATURE RECONSTRUCTION LOSS ON DOMAIN ADVERSARIALTRAINING
+
+We demonstrate the effectiveness of feature reconstruction loss in stabilizing the domain adversarial training in DANN framework. We train four different DANN models with different configurations of $\lambda _ { 3 }$ and $\lambda _ { 4 }$ . We visualize in Figure S2 the performance curves of identification accuracy evaluated on the AA, EA, and CAU ethnicities of CEF dataset. Note that we stop training early on when the performance start to degrade significantly. Therefore, $x$ -axis, the number of training epoch, of different curves are different. $y$ -axis represents the identification accuracy.
+
+As we see in Figure S2, the performance of all models on the target ethnicities start to improve in the beginning of training from those of the pretrained reference network. Soon after, however, the accuracy starts to drop when values of either $\lambda _ { 3 }$ or $\lambda _ { 4 }$ are set to 0. Note that even in that situation the performance on the CAU set still remains high, which implies the failure of discriminative information transfer. On the other hand, our proposed feature reconstruction loss with non-zero values of $\lambda _ { 3 }$ and $\lambda _ { 4 }$ (Figure 2(d)) shows much more stable performance curve. Nonetheless, values of $\lambda _ { 3 }$ and $\lambda _ { 4 }$ should be carefully selected since the feature generation module of DANNs or FTNs will remain almost the same to the reference network when they are set too strong and the effectiveness of the domain adversarial loss will be reduced. In our experiment, we use $\lambda _ { 3 } = 0 . 1$ and $\lambda _ { 4 } = 0 . 0 1$ for DANN, $\lambda _ { 3 } = 0 . 0 3$ and $\lambda _ { 4 } = 0 . 0 1$ for FTN. For FTN with entropy minimization we further reduce $\lambda _ { 4 } = 0 . 0 0 3$ to give more flexibility in updating model parameters based on entropy loss.
+
+![](images/9f2925f537752b9feccf5b37c8a6a65271f7e1ef3ba71f846ae7406be68620aa.jpg)  
+Figure S2: Performance curves of identification accuracy per ethnicity subset on the CEF datasets. The accuracy of DANNs with different values of $\lambda _ { 3 }$ for $\lambda _ { 4 }$ are visualized.
+
+# D PERFORMANCE OF UNSUPERVISED HIERARCHICAL CLUSTERING
+
+In this section, we provide analysis on the performance of our clustering strategy by measuring the clustering accuracy. Specifically, we measure the verification precision and recall as follows:
+
+$$
+\mathrm { P r e c i s i o n } = \frac { \sum _ { x _ { 1 } , x _ { 2 } \in \mathcal { X } _ { T } } 1 \big \{ y _ { 1 } = y _ { 2 } , \hat { y } _ { 1 } = \hat { y } _ { 2 } \big \} } { \sum _ { x _ { 1 } , x _ { 2 } \in \mathcal { X } _ { T } } 1 \big \{ \hat { y } _ { 1 } = \hat { y } _ { 2 } \big \} } , \ \mathrm { R e c a l l } = \frac { \sum _ { x _ { 1 } , x _ { 2 } \in \mathcal { X } _ { T } } 1 \big \{ y _ { 1 } = y _ { 2 } , \hat { y } _ { 1 } = \hat { y } _ { 2 } \big \} } { \sum _ { x _ { 1 } , x _ { 2 } \in \mathcal { X } _ { T } } 1 \big \{ y _ { 1 } = y _ { 2 } \big \} }
+$$
+
+where $y _ { i }$ is the ground-truth class label of an example $x _ { i }$ , and $\hat { y } _ { i }$ is an index of an assigned cluster. Precision computes the proportion of positive pairs among pairs assigned to the same cluster, i.e., purity of returned clusters, and recall computes the proportion of positive pairs assigned to the same cluster. Ideally, we expect high precision and high recall, i.e., high F-score, to ensure examples with the same class labels are assigned to the same cluster. Note that we only use clusters of size 5 or larger as new target classes and discard examples assigned to a cluster whose size is less than 5.
+
+Here, in addition to our proposed clustering strategy, we also evaluate the clustering performance that clusters target examples by finding a nearest classes or examples from the source domain, which are shown to be effective for zero-shot learning (Vinyals et al., 2016) or semi-supervised domain adaptation with disjoint source and target classes (Luo et al., 2017). In this case, we call two examples from the target domain are assigned to the same cluster if the nearest source examples are the same. We also measure the clustering performance by matching the nearest source classes.
+
+The summary result is provided in Table S4. Firstly, we observe extremely low precision when using source domain examples or clusters as a proxy to relate target examples. We believe that this idea of “clustering by finding the nearest source classes” works under a cross-category similarity assumption between disjoint classes of source and target domains. In other words, it assumes that there exists a certain source class closer to examples from certain target class, so that those examples from the same target class can be clustered around that source class, even though those matching source and target classes are indeed different (e.g., 3/5, 2/8, 4/9, and 0/6 in Section 6.1). Unfortunately, such an assumption does not hold for our problem, maybe due to the huge number of identity classes $( 6 0 K )$ in the source domain.
+
+On the other hand, using hierarchical clustering on target features achieves significantly higher precision and recall. Especially, when using embedding vectors of $\mathrm { S u p ^ { C } }$ , we achieve $1 0 0 \%$ precision, which means that all clusters are pure even though some ground-truth classes might be separated into multiple clusters. We observe slightly lower precision using FTN features but much higher recall, achieving higher F-score overall. Further, the number of examples returned with FTN feature $2 5 3 K$ and $1 9 5 K$ for AA and EA, respectively) is higher than with $\mathrm { S u p ^ { C } }$ feature $2 1 7 K$ and $1 6 5 K _ { , }$ ). Repeating the process using feature of FTN+MCEM model improves the F-score while returning more target examples that are with cluster assignment ( $2 7 6 K$ and $2 1 4 K$ ). This not only shows the
+
+<table><tr><td rowspan="3"></td><td colspan="2">source example</td><td colspan="2">source center</td><td colspan="2">HDBSCAN</td><td colspan="2">FTN</td><td colspan="2">FTN+MCEM</td></tr><tr><td>AA</td><td>EA</td><td>AA</td><td>EA</td><td>AA</td><td>EA</td><td>AA</td><td>EA</td><td>AA</td><td>EA</td></tr><tr><td>Precision</td><td>0.11</td><td>0.12</td><td>0.30</td><td>0.08</td><td>100</td><td>100</td><td>95.36</td><td>96.23</td><td>95.79</td><td>96.10</td></tr><tr><td>Recall</td><td>25.64</td><td>31.11</td><td>22.54</td><td>48.60</td><td>88.66</td><td>81.74</td><td>97.29</td><td>96.46</td><td>97.81</td><td>96.89</td></tr><tr><td>F-score</td><td>0.22</td><td>0.25</td><td>0.58</td><td>0.16</td><td>93.99</td><td>89.95</td><td>96.31</td><td>96.34</td><td>96.79</td><td>96.49</td></tr></table>
+
+Table S4: Verification precision and recall of clustering methods, such as projection to source example or source class center, or hierarchical clustering using embeddings of $\mathrm { { S u p } ^ { C } }$ (HDBSCAN) or our proposed FTN model. Furthermore, we repeat the clustering using the FTN with multi-class entropy minimization model $\left( \mathrm { F T N + M C E M } \right)$ and report the clustering accuracy.
+
+improved discriminative quality of features by FTNs, but also suggests a potential tool for automatic labeling of unlabeled data by iterative training of FTN model and hierarchical clustering.
+
+# E VISUALIZATION OF ETHNICITY ANNOTATED IMAGE SAMPLES
+
+![](images/13fd2f997473f27a70c6ad3585a72aa49d7970f3d27566f7b5b1accdcef89cd0.jpg)  
+Figure S3: Face images of Caucasian, African-American, and East-Asian sampled from MS-1M dataset.

parse/train/ByeWogStDS/ByeWogStDS.md

@@ -0,0 +1,378 @@
+# SUB-POLICY ADAPTATION FOR HIERARCHICALREINFORCEMENT LEARNING
+
+Alexander C. Li∗, Carlos Florensa∗, Ignasi Clavera, Pieter Abbeel University of California, Berkeley {alexli1, florensa, iclavera, pabbeel}@berkeley.edu
+
+# ABSTRACT
+
+Hierarchical reinforcement learning is a promising approach to tackle long-horizon decision-making problems with sparse rewards. Unfortunately, most methods still decouple the lower-level skill acquisition process and the training of a higher level that controls the skills in a new task. Leaving the skills fixed can lead to significant sub-optimality in the transfer setting. In this work, we propose a novel algorithm to discover a set of skills and continuously adapt them along with the higher level even when training on a new task. Our main contributions are two-fold. First, we derive a new hierarchical policy gradient with an unbiased latent-dependent baseline, and we introduce Hierarchical Proximal Policy Optimization (HiPPO), an on-policy method to efficiently train all levels of the hierarchy jointly. Second, we propose a method of training time-abstractions that improves the robustness of the obtained skills to environment changes. Code and videos are available. 1.
+
+# 1 INTRODUCTION
+
+Reinforcement learning (RL) has made great progress in a variety of domains, from playing games such as Pong and Go (Mnih et al., 2015; Silver et al., 2017) to automating robotic locomotion (Schulman et al., 2015; Heess et al., 2017), dexterous manipulation (Florensa et al., 2017b; OpenAI et al., 2018), and perception (Nair et al., 2018; Florensa et al., 2018). Yet, most work in RL is still learning from scratch when faced with a new problem. This is particularly inefficient when tackling multiple related tasks that are hard to solve due to sparse rewards or long horizons.
+
+A promising technique to overcome this limitation is hierarchical reinforcement learning (HRL) (Sutton et al., 1999). In this paradigm, policies have several modules of abstraction, allowing to reuse subsets of the modules. The most common case consists of temporal hierarchies (Precup, 2000; Dayan & Hinton, 1993), where a higher-level policy (manager) takes actions at a lower frequency, and its actions condition the behavior of some lower level skills or sub-policies. When transferring knowledge to a new task, most prior works fix the skills and train a new manager on top. Despite having a clear benefit in kick-starting the learning in the new task, having fixed skills can considerably cap the final performance on the new task (Florensa et al., 2017a). Little work has been done on adapting pre-trained sub-policies to be optimal for a new task.
+
+In this paper, we develop a new framework for simultaneously adapting all levels of temporal hierarchies. First, we derive an efficient approximated hierarchical policy gradient. The key insight is that, despite the decisions of the manager being unobserved latent variables from the point of view of the Markovian environment, from the perspective of the sub-policies they can be considered as part of the observation. We show that this provides a decoupling of the manager and sub-policy gradients, which greatly simplifies the computation in a principled way. It also theoretically justifies a technique used in other prior works (Frans et al., 2018). Second, we introduce a sub-policy specific baseline for our hierarchical policy gradient. We prove that this baseline is unbiased, and our experiments reveal faster convergence, suggesting efficient gradient variance reduction. Then, we introduce a more stable way of using this gradient, Hierarchical Proximal Policy Optimization (HiPPO). This method helps us take more conservative steps in our policy space (Schulman et al., 2017), critical in hierarchies because of the interdependence of each layer. Results show that HiPPO is highly efficient both when learning from scratch, i.e. adapting randomly initialized skills, and when adapting pretrained skills on a new task. Finally, we evaluate the benefit of randomizing the time-commitment of the sub-policies, and show it helps both in terms of final performance and zero-shot adaptation on similar tasks.
+
+# 2 PRELIMINARIES
+
+We define a discrete-time finitehorizon discounted Markov decision process (MDP) by a tuple $\begin{array} { r l } { M } & { { } = } \end{array}$ $( \boldsymbol { \bar { s } } , \boldsymbol { \mathcal { A } } , \mathcal { P } , \boldsymbol { r } , \rho _ { 0 } , \gamma , \boldsymbol { \dot { H } } )$ , where $s$ is a state set, $\mathcal { A }$ is an action set, $\mathcal { P } :$ $S \times \mathcal { A } \times \mathcal { S } \to \mathbb { R } _ { + }$ is the transition probability distribution, $\gamma ~ \in ~ [ 0 , 1 ]$ is a discount factor, and $H$ the horizon. Our objective is to find a stochastic policy $\pi _ { \theta }$ that maximizes the expected discounted return within the MDP, $\begin{array} { r } { \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { H } \gamma ^ { t } r ( s _ { t } , a _ { t } ) ] } \end{array}$ We use ${ \boldsymbol \tau } = ( s _ { 0 } , a _ { 0 } , . . . , )$ to denote the entire state-action trajectory, where $s _ { 0 } ~ \sim ~ \rho _ { 0 } ( s _ { 0 } )$ , $a _ { t } \sim \dot { \pi } _ { \theta } ( a _ { t } | s _ { t } )$ , and $s _ { t + 1 } \sim \mathcal { P } ( s _ { t + 1 } | s _ { t } , a _ { t } )$ .
+
+![](images/7ba3bb08e3131bf7c0453118b8ce8cd1f7c6946bcc17a481702825cd49f02921.jpg)  
+Figure 1: Temporal hierarchy studied in this paper. A latent code $z _ { t }$ is sampled from the manager policy $\bar { \pi _ { \boldsymbol { \theta } _ { h } } ( \boldsymbol { z } _ { t } | \boldsymbol { s } _ { t } ) }$ every $p$ time-steps, using the current observation $s _ { k p }$ . The actions $a _ { t }$ are sampled from the sub-policy $\pi _ { \boldsymbol { \theta } _ { l } } ( a _ { t } | \bar { s } _ { t } , z _ { k p } )$ conditioned on the same latent code from $t = k p$ to $( k \bar { + } 1 ) p - 1$
+
+In this work, we propose a method to learn a hierarchical policy and efficiently adapt all the levels in the hierarchy to perform a new task. We study hierarchical policies composed of a higher level, or manager $\pi _ { \boldsymbol { \theta } _ { h } } \big ( \boldsymbol { z } _ { t } | \boldsymbol { s } _ { t } \big )$ , and a lower level, or sub-policy $\pi _ { \theta _ { l } } ( a _ { t ^ { \prime } } | z _ { t } , s _ { t ^ { \prime } } )$ . The higher level does not take actions in the environment directly, but rather outputs a command, or latent variable $z _ { t } \in \mathcal { Z }$ , that conditions the behavior of the lower level. We focus on the common case where ${ \mathcal { Z } } = \mathbb { Z } _ { n }$ making the manager choose among $n$ sub-policies, or skills, to execute. The manager typically operates at a lower frequency than the sub-policies, only observing the environment every $p$ time-steps. When the manager receives a new observation, it decides which low level policy to commit to for $p$ environment steps by the means of a latent code $z$ . Figure 1 depicts this framework where the high level frequency $p$ is a random variable, which is one of the contribution of this paper as described in Section 4.4. Note that the class of hierarchical policies we work with is more restrictive than others like the options framework, where the time-commitment is also decided by the policy. Nevertheless, we show that this loss in policy expressivity acts as a regularizer and does not prevent our algorithm from surpassing other state-of-the art methods.
+
+# 3 RELATED WORK
+
+There has been growing interest in HRL for the past few decades (Sutton et al., 1999; Precup, 2000), but only recently has it been applied to high-dimensional continuous domains as we do in this work (Kulkarni et al., 2016; Daniel et al., 2016). To obtain the lower level policies, or skills, most methods exploit some additional assumptions, like access to demonstrations (Le et al., 2018; Merel et al., 2019; Ranchod et al., 2015; Sharma et al., 2018), policy sketches (Andreas et al., 2017), or task decomposition into sub-tasks (Ghavamzadeh & Mahadevan, 2003; Sohn et al., 2018). Other methods use a different reward for the lower level, often constraining it to be a “goal reacher” policy, where the signal from the higher level is the goal to reach (Nachum et al., 2018; Levy et al., 2019; Vezhnevets et al., 2017). These methods are very promising for state-reaching tasks, but might require access to goal-reaching reward systems not defined in the original MDP, and are more limited when training on tasks beyond state-reaching. Our method does not require any additional supervision, and the obtained skills are not constrained to be goal-reaching.
+
+When transferring skills to a new environment, most HRL methods keep them fixed and simply train a new higher-level on top (Hausman et al., 2018; Heess et al., 2016). Other work allows for building on previous skills by constantly supplementing the set of skills with new ones (Shu et al., 2018), but they require a hand-defined curriculum of tasks, and the previous skills are never fine-tuned.
+
+Our algorithm allows for seamless adaptation of the skills, showing no trade-off between leveraging the power of the hierarchy and the final performance in a new task. Other methods use invertible functions as skills (Haarnoja et al., 2018), and therefore a fixed skill can be fully overwritten when a new layer of hierarchy is added on top. This kind of “fine-tuning” is promising, although similar to other works (Peng et al., 2019), they do not apply it to temporally extended skills as we do here.
+
+One of the most general frameworks to define temporally extended hierarchies is the options framework (Sutton et al., 1999), and it has recently been applied to continuous state spaces (Bacon et al., 2017). One of the most delicate parts of this formulation is the termination policy, and it requires several regularizers to avoid skill collapse (Harb et al., 2017; Vezhnevets et al., 2016). This modification of the objective may be difficult to tune and affects the final performance. Instead of adding such penalties, we propose to have skills of a random length, not controlled by the agent during training of the skills. The benefit is two-fold: no termination policy to train, and more stable skills that transfer better. Furthermore, these works only used discrete action MDPs. We lift this assumption, and show good performance of our algorithm in complex locomotion tasks. There are other algorithms recently proposed that go in the same direction, but we found them more complex, less principled (their per-action marginalization cannot capture well the temporal correlation within each option), and without available code or evidence of outperforming non-hierarchical methods (Smith et al., 2018).
+
+The closest work to ours in terms of final algorithm structure is the one proposed by Frans et al. (2018). Their method can be included in our framework, and hence benefits from our new theoretical insights. We introduce a modification that is shown to be highly beneficial: the random timecommitment mentioned above, and find that our method can learn in difficult environments without their complicated training scheme.
+
+# 4 EFFICIENT HIERARCHICAL POLICY GRADIENTS
+
+When using a hierarchical policy, the intermediate decision taken by the higher level is not directly applied in the environment. Therefore, technically it should not be incorporated into the trajectory description as an observed variable, like the actions. This makes the policy gradient considerably harder to compute. In this section we first prove that, under mild assumptions, the hierarchical policy gradient can be accurately approximated without needing to marginalize over this latent variable. Then, we derive an unbiased baseline for the policy gradient that can reduce the variance of its estimate. Finally, with these findings, we present our method, Hierarchical Proximal Policy Optimization (HiPPO), an on-policy algorithm for hierarchical policies, allowing learning at all levels of the policy jointly and preventing sub-policy collapse.
+
+# 4.1 APPROXIMATE HIERARCHICAL POLICY GRADIENT
+
+Policy gradient algorithms are based on the likelihood ratio trick (Williams, 1992) to estimate the gradient of returns with respect to the policy parameters as
+
+$$
+\begin{array} { c l } { \displaystyle \nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb E _ { \tau } \big [ \nabla _ { \theta } \log P ( \tau ) R ( \tau ) \big ] \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { n } \nabla _ { \theta } \log P ( \tau _ { i } ) R ( \tau _ { i } ) } \\ { \displaystyle = \frac { 1 } { N } \sum _ { i = 1 } ^ { n } \frac { 1 } { H } \sum _ { t = 1 } ^ { H } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) R ( \tau _ { i } ) } \end{array}
+$$
+
+In a temporal hierarchy, a hierarchical policy with a manager $\pi _ { \boldsymbol { \theta } _ { h } } \big ( \boldsymbol { z } _ { t } | \boldsymbol { s } _ { t } \big )$ selects every $p$ time-steps one of $n$ sub-policies to execute. These sub-policies, indexed by $z \in \mathbb { Z } _ { n }$ , can be represented as a single conditional probability distribution over actions $\pi _ { \boldsymbol { \theta } _ { l } } ( a _ { t } | \boldsymbol { z } _ { t } , \boldsymbol { s } _ { t } )$ . This allows us to not only use a given set of sub-policies, but also leverage skills learned with Stochastic Neural Networks (SNNs) (Florensa et al., 2017a). Under this framework, the probability of a trajectory $\tau = ( s _ { 0 } , a _ { 0 } , s _ { 1 } , \dots , s _ { H } )$ can be written as
+
+$$
+P ( \tau ) = \Bigg ( \prod _ { k = 0 } ^ { H / p } \Big [ \sum _ { j = 1 } ^ { n } \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \Big ] \Bigg ) \Bigg [ P ( s _ { 0 } ) \prod _ { t = 1 } ^ { H } P ( s _ { t + 1 } | s _ { t } , a _ { t } ) \Bigg ] .
+$$
+
+The mixture action distribution, which presents itself as an additional summation over skills, prevents additive factorization when taking the logarithm, as from Eq. 1 to 2. This can yield numerical instabilities due to the product of the $p$ sub-policy probabilities. For instance, in the case where all the skills are distinguishable all the sub-policies’ probabilities but one will have small values, resulting in an exponentially small value. In the following Lemma, we derive an approximation of the policy gradient, whose error tends to zero as the skills become more diverse, and draw insights on the interplay of the manager actions.
+
+Lemma 1. If the skills are sufficiently differentiated, then the latent variable can be treated as part of the observation to compute the gradient of the trajectory probability. Let $\pi _ { \boldsymbol { \theta } _ { h } } ( z | s )$ and $\pi _ { \boldsymbol { \theta } _ { l } } ( a | s , z )$ be Lipschitz functions w.r.t. their parameters, and assume that $0 < \pi _ { \theta _ { l } } ( a | s , z _ { j } ) < \epsilon \forall j \neq k p$ , then
+
+$$
+\nabla _ { \theta } \log P ( \tau ) = \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log \pi _ { \theta _ { h } } ( z _ { k p } | s _ { k p } ) + \sum _ { t = 0 } ^ { H } \nabla _ { \theta } \log \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) + \mathcal { O } ( n H \epsilon ^ { p - 1 } )
+$$
+
+Proof. See Appendix.
+
+Our assumption can be seen as having diverse skills. Namely, for each action there is just one sub-policy that gives it high probability. In this case, the latent variable can be treated as part of the observation to compute the gradient of the trajectory probability. Many algorithms to extract lowerlevel skills are based on promoting diversity among the skills (Florensa et al., 2017a; Eysenbach et al., 2019), therefore usually satisfying our assumption. We further analyze how well this assumption holds in our experiments section and Table 2.
+
+# 4.2 UNBIASED SUB-POLICY BASELINE
+
+The policy gradient estimate obtained when applying the log-likelihood ratio trick as derived above is known to have large variance. A very common approach to mitigate this issue without biasing the estimate is to subtract a baseline from the returns (Peters & Schaal, 2008). It is well known that such baselines can be made state-dependent without incurring any bias. However, it is still unclear how to formulate a baseline for all the levels in a hierarchical policy, since an action dependent baseline does introduce bias in the gradient (Tucker et al., 2018). It has been recently proposed to use latent-conditioned baselines (Weber et al., 2019). Here we go further and prove that, under the assumptions of Lemma 1, we can formulate an unbiased latent dependent baseline for the approximate gradient (Eq. 5).
+
+Lemma 2. For any functions $b _ { h } : S  \mathbb { R }$ and $b _ { l } : \mathcal { S } \times \mathcal { Z } \to \mathbb { R }$ we have:
+
+$$
+\mathbb { E } _ { \tau } [ \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log P ( z _ { k p } | s _ { k p } ) b _ { h } ( s _ { k p } ) ] = 0 \quad a n d \quad \mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { H } \nabla _ { \theta } \log \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) b _ { l } ( s _ { t } , z _ { k p } ) ] = 0
+$$
+
+Proof. See Appendix.
+
+Now we apply Lemma 1 and Lemma 2 to Eq. 1. By using the corresponding value functions as the function baseline, the return can be replaced by the Advantage function $A ( s _ { k p } , z _ { k p } )$ (see details in Schulman et al. (2016)), and we obtain the following approximate policy gradient expression:
+
+$$
+\hat { g } = \mathbb { E } _ { \tau } \Big [ \big ( \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log \pi _ { \theta _ { h } } ( z _ { k p } | s _ { k p } ) A ( s _ { k p } , z _ { k p } ) \big ) + \big ( \sum _ { t = 0 } ^ { H } \nabla _ { \theta } \log \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) A ( s _ { t } , a _ { t } , z _ { k p } ) \big ) \Big ]
+$$
+
+This hierarchical policy gradient estimate can have lower variance than without baselines, but using it for policy optimization through stochastic gradient descent still yields an unstable algorithm. In the next section, we further improve the stability and sample efficiency of the policy optimization by incorporating techniques from Proximal Policy Optimization (Schulman et al., 2017).
+
+# 4.3 HIERARCHICAL PROXIMAL POLICY OPTIMIZATION
+
+Using an appropriate step size in policy space is critical for stable policy learning. Modifying the policy parameters in some directions may have a minimal impact on the distribution over actions, whereas small changes in other directions might change its behavior drastically and hurt training
+
+# Algorithm 1 HiPPO Rollout
+
+# Algorithm 2 HiPPO
+
+1: Input: skills $\pi _ { \boldsymbol { \theta } _ { l } } ( a | s , z )$ , manager $\pi _ { \boldsymbol { \theta } _ { h } } ( z | s )$ , time   
+commitment bounds $P _ { \mathrm { m i n } }$ and $P _ { \mathrm { m a x } }$ , horizon $H$   
+2: Reset environment: $s _ { 0 } \sim \rho _ { 0 }$ , $t = 0$ .   
+3: while $t < H$ do   
+4: Sample time-commitment $p \sim \mathsf { C a t } ( [ P _ { \operatorname* { m i n } } , P _ { \operatorname* { m a x } } ] )$   
+5: Sample skill $z _ { t } \sim \pi _ { \theta _ { h } } ( \cdot | s _ { t } )$   
+6: for $t ^ { \prime } = t \ldots ( t + p )$ do   
+7: Sample action $a _ { t ^ { \prime } } \sim \pi _ { \theta _ { l } } ( \cdot | s _ { t ^ { \prime } } , z _ { t } )$   
+8: Observe new state $s _ { t ^ { \prime } + 1 }$ and reward $\boldsymbol { r } _ { t ^ { \prime } }$   
+9: end for   
+10: $t \gets t + p$   
+11: end while   
+12: Output: $\left( s _ { 0 } , z _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , \ldots , s _ { H } , z _ { H } , a _ { H } , s _ { H + 1 } \right)$
+
+1: Input: skills $\pi _ { \boldsymbol { \theta } _ { l } } ( a | s , z )$ , manager $\pi _ { \boldsymbol { \theta } _ { h } } ( z | s )$ , horizon $H$ , learning rate $\alpha$   
+2: while not done do   
+3: for actor $= 1$ , 2, ..., N do   
+4: Obtain trajectory with HiPPO Rollout   
+5: Estimate advantages $\hat { A } ( a _ { t ^ { \prime } } , s _ { t ^ { \prime } } , z _ { t } )$ and $\bar { A } ( z _ { t } , s _ { t } )$   
+6: 7: $\theta  \theta + \alpha \nabla _ { \theta } L _ { H i P P O } ^ { C L I P } ( \theta )$   
+8: end while
+
+efficiency (Kakade, 2002). Trust region policy optimization (TRPO) uses a constraint on the KLdivergence between the old policy and the new policy to prevent this issue (Schulman et al., 2015). Unfortunately, hierarchical policies are generally represented by complex distributions without closed form expressions for the KL-divergence. Therefore, to improve the stability of our hierarchical policy gradient we turn towards Proximal Policy Optimization (PPO) (Schulman et al., 2017). PPO is a more flexible and compute-efficient algorithm. In a nutshell, it replaces the KL-divergence constraint with a cost function that achieves the same trust region benefits, but only requires the computation of the likelihood. Letting $\begin{array} { r } { w _ { t } ( \theta ) = \frac { \pi _ { \theta } \left( a _ { t } | s _ { t } \right) } { \pi _ { \theta _ { o l d } } \left( a _ { t } | s _ { t } \right) } } \end{array}$ , the PPO objective is:
+
+$$
+L ^ { C L I P } ( \theta ) = \mathbb { E } _ { t } \operatorname* { m i n } \left\{ w _ { t } ( \theta ) A _ { t } , \mathrm { ~ c ~ l ~ i p } ( w _ { t } ( \theta ) , 1 - \epsilon , 1 + \epsilon ) A _ { t } \right\}
+$$
+
+We can adapt our approximated hierarchical policy gradient with the same approach by letting $\begin{array} { r } { w _ { h , k p } ( \theta ) \ = \ \frac { \pi _ { \theta _ { h } } ( z _ { k p } | s _ { k p } ) } { \pi _ { \theta _ { h , o l d } } ( z _ { k p } | s _ { k p } ) } } \end{array}$ and $\begin{array} { r } { w _ { l , t } ( \theta ) = \frac { \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) } { \pi _ { \theta _ { l , o l d } } ( a _ { t } | s _ { t } , z _ { k p } ) } } \end{array}$ , and using the super-index clip to denote the clipped objective version, we obtain the new surrogate objective:
+
+$$
+\begin{array} { c } { { { \displaystyle { \cal L } _ { H i P P O } ^ { C L I P } } ( \theta ) = \mathbb { E } _ { \tau } \Big [ \displaystyle { \sum _ { k = 0 } ^ { H / p } \operatorname* { m i n } \left\{ w _ { h , k p } ( \theta ) A ( s _ { k p } , z _ { k p } ) , w _ { h , k p } ^ { \mathrm { c l i p } } ( \theta ) A ( s _ { k p } , z _ { k p } ) \right\} } \qquad } } \\ { { + \displaystyle { \sum _ { t = 0 } ^ { H } \operatorname* { m i n } \left\{ w _ { l , t } ( \theta ) A ( s _ { t } , a _ { t } , z _ { k p } ) , w _ { l , t } ^ { \mathrm { c l i p } } ( \theta ) A ( s _ { t } , a _ { t } , z _ { k p } ) \right\} } \Big ] } } \end{array}
+$$
+
+We call this algorithm Hierarchical Proximal Policy Optimization (HiPPO). Next, we introduce a critical additions: a switching of the time-commitment between skills.
+
+# 4.4 VARYING TIME-COMMITMENT
+
+Most hierarchical methods either consider a fixed time-commitment to the lower level skills (Florensa et al., 2017a; Frans et al., 2018), or implement the complex options framework (Precup, 2000; Bacon et al., 2017). In this work we propose an in-between, where the time-commitment to the skills is a random variable sampled from a fixed distribution Categorical $( T _ { \mathrm { m i n } } , T _ { \mathrm { m a x } } )$ just before the manager takes a decision. This modification does not hinder final performance, and we show it improves zero-shot adaptation to a new task. This approach to sampling rollouts is detailed in Algorithm 1. The full algorithm is detailed in Algorithm 2.
+
+# 5 EXPERIMENTS
+
+We designed our experiments to answer the following questions: 1) How does HiPPO compare against a flat policy when learning from scratch? 2) Does it lead to policies more robust to environment changes? 3) How well does it adapt already learned skills? and 4) Does our skill diversity assumption hold in practice?
+
+![](images/698cac3c55e1a99afc5dbc3e8a69b575139080c5784a5c1282ae08d45458591a.jpg)  
+Figure 2: Environments used to evaluate the performance of our method. Every episode has a different configuration: wall heights for (a)-(b), ball positions for (c)-(d)
+
+![](images/6630bd4ec4adf24f9991b32a9c529630605655bc9ee9e48e1f0804e13fee17ac.jpg)  
+Figure 3: Analysis of different time-commitment strategies on learning from scratch.
+
+# 5.1 TASKS
+
+We evaluate our approach on a variety of robotic locomotion and navigation tasks. The Block environments, depicted in Fig. 2a-2b, have walls of random heights at regular intervals, and the objective is to learn a gait for the Hopper and Half-Cheetah robots to jump over them. The agents observe the height of the wall ahead and their proprioceptive information (joint positions and velocities), receiving a reward of $+ 1$ for each wall cleared. The Gather environments, described by Duan et al. (2016), require agents to collect apples (green balls, $+ 1$ reward) while avoiding bombs (red balls, -1 reward). The only available perception beyond proprioception is through a LIDAR-type sensor indicating at what distance are the objects in different directions, and their type, as depicted in the bottom left corner of Fig. 2c-2d. This is challenging hierarchical task with sparse rewards that requires simultaneously learning perception, locomotion, and higher-level planning capabilities. We use the Snake and Ant robots in Gather. Details for all robotic agents are provided in Appendix B.
+
+# 5.2 LEARNING FROM SCRATCH AND TIME-COMMITMENT
+
+In this section, we study the benefit of using our HiPPO algorithm instead of standard PPO on a flat policy (Schulman et al., 2017). The results, reported in Figure 3, demonstrate that training from scratch with HiPPO leads to faster learning and better performance than flat PPO. Furthermore, we show that the benefit of HiPPO does not just come from having temporally correlated exploration: PPO with action repeat converges at a lower performance than our method. HiPPO leverages the time-commitment more efficiently, as suggested by the poor performance of the ablation where we set $p = 1$ , when the manager takes an action every environment step as well. Finally, Figure 4 shows the effectiveness of using the presented skill-dependent baseline.
+
+# 5.3 COMPARISON TO OTHER METHODS
+
+We compare HiPPO to current state-of-the-art hierarchical methods. First, we evaluate HIRO (Nachum et al., 2018), an off-policy RL method based on training a goal-reaching lower level policy. Fig. 5 shows that HIRO achieves poor performance on our tasks. As further detailed in Appendix D, this algorithm is sensitive to access to ground-truth information, like the exact $( x , y )$ position of the robot in Gather. In contrast, our method is able to perform well directly from the raw sensory inputs described in Section 5.1. We evaluate Option-Critic (Bacon et al., 2017), a variant of the options framework (Sutton et al., 1999) that can be used for continuous action-spaces. It fails to learn, and we hypothesize that their algorithm provides less time-correlated exploration and learns less diverse skills. We also compare against MLSH (Frans et al., 2018), which repeatedly samples new environment configurations to learn primitive skills. We take these hyperparameters from their Ant Twowalk experiment: resetting the environment configuration every 60 iterations, a warmup period of 20 during which only the manager is trained, and a joint training period of 40 during which both manager and skills are trained. Our results show that such a training scheme does not provide any benefits. Finally, we provide a comparison to a direct application of our Hierarchical Vanilla Policy Gradient (HierVPG) algorithm, and we see that the algorithm is unstable without PPO’s trust-region-like technique.
+
+![](images/104a04a5797b71d11aafffde40d1191b91b356cc7b5c8611ac4e6b31f18c96b0.jpg)  
+Figure 4: Using a skill-conditioned baseline, as defined in Section 4.2, generally improves performance of HiPPO when learning from scratch.
+
+![](images/09aff31033112d9cd52a822e7e3b242507e89a86d92aef55e0b80822e72c1caa.jpg)  
+Figure 5: Comparison of HiPPO and HierVPG to prior hierarchical methods on learning from scratch.
+
+# 5.4 ROBUSTNESS TO DYNAMICS PERTURBATIONS
+
+We investigate the robustness of HiPPO to changes in the dynamics of the environment. We perform several modifications to the base Snake Gather and Ant Gather environments. One at a time, we change the body mass, dampening of the joints, body inertia, and friction characteristics of both robots. The results, presented in Table 1, show that HiPPO with randomized period Categorical $( [ T _ { \operatorname* { m i n } } , T _ { \operatorname* { m a x } } ] )$ is able to better handle these dynamics changes. In terms of the drop in policy performance between the training environment and test environment, it outperforms HiPPO with fixed period on 6 out of 8 related tasks. These results suggest that the randomized period exposes the policy to a wide range of scenarios, which makes it easier to adapt when the environment changes.
+
+<table><tr><td>Gather</td><td>Algorithm</td><td>Initial</td><td>Mass</td><td>Dampening</td><td>Inertia</td><td>Friction</td></tr><tr><td rowspan="3">Snake</td><td>Flat PPO</td><td>2.72</td><td>3.16 (+16%)</td><td>2.75 (+1%)</td><td>2.11 (-22%)</td><td>2.75 (+1%)</td></tr><tr><td>HiPPO,p = 10</td><td>4.38</td><td>3.28 (-25%)</td><td>3.27 (-25%)</td><td>3.03 (-31%)</td><td>3.27 (-25%)</td></tr><tr><td>HiPPO random p</td><td>5.11</td><td>4.09 (-20%)</td><td>4.03 (-21%)</td><td>3.21 (-37%)</td><td>4.03 (-21%)</td></tr><tr><td rowspan="3">Ant</td><td>Flat PPO</td><td>2.25</td><td>2.53 (+12%)</td><td>2.13 (-5%)</td><td>2.36 (+5%)</td><td>1.96 (-13%)</td></tr><tr><td>HiPPO,p = 10</td><td>3.84</td><td>3.31 (-14%)</td><td>3.37 (-12%)</td><td>2.88 (-25%)</td><td>3.07 (-20%)</td></tr><tr><td>HiPPO random p</td><td>3.22</td><td>3.37 (+5%)</td><td>2.57 (-20%)</td><td>3.36 (+4%)</td><td>2.84 (-12%)</td></tr></table>
+
+Table 1: Zero-shot transfer performance. The final return in the initial environment is shown, as well as the average return over 25 rollouts in each new modified environment.
+
+# 5.5 ADAPTATION OF PRE-TRAINED SKILLS
+
+For the Block task, we use DIAYN (Eysenbach et al., 2019) to train 6 differentiated subpolicies in an environment without any walls. Here, we see if these diverse skills can improve performance on a downstream task that’s out of the training distribution. For Gather, we take 6 pretrained subpolicies encoded by a Stochastic Neural Network (Tang & Salakhutdinov, 2013) that was trained in a diversity-promoting environment (Florensa et al., 2017a). We fine-tune them with HiPPO on the Gather environment, but with an extra penalty on the velocity of the Center of Mass. This can be understood as a preference for cautious behavior. This requires adjustment of the sub-policies, which were trained with a proxy reward encouraging them to move as far as possible (and hence quickly). Fig. 6 shows that using HiPPO to simultaneously train a manager and fine-tune the skills achieves higher final performance than fixing the sub-policies and only training a manager with PPO. The two initially learn at the same rate, but HiPPO’s ability to adjust to the new dynamics allows it to reach a higher final performance. Fig. 6 also shows that HiPPO can fine-tune the same given skills better than Option-Critic (Bacon et al., 2017), MLSH (Frans et al., 2018), and HIRO (Nachum et al., 2018).
+
+![](images/70943baaa443c6192a1e00d29a7bad11943570b86e84a71b5f3b01009b4b31d7.jpg)  
+Figure 6: Benefit of adapting some given skills when the preferences of the environment are different from those of the environment where the skills were originally trained. Adapting skills with HiPPO has better learning performance than leaving the skills fixed or learning from scratch.
+
+# 5.6 SKILL DIVERSITY ASSUMPTION
+
+In Lemma 1, we derived a more efficient and numerically stable gradient by assuming that the sub-policies are diverse. In this section, we empirically test the validity of our assumption and the quality of our approximation. We run the HiPPO algorithm on Ant Gather and Snake Gather both from scratch and with given pretrained skills, as done in the previous section. In Table 2, we report the average maximum probability under other sub-policies, corresponding to $\epsilon$ from the assumption. In all settings, this is on the order of magnitude of 0.1. Therefore, under the $p \approx 1 0$ that we use in our experiments, the term we neglect has a factor $\epsilon ^ { p - 1 } = 1 0 ^ { - 1 0 }$ . It is not surprising then that the average cosine similarity between the full gradient and our approximation is almost 1, as reported in Table 2.
+
+<table><tr><td>Gather</td><td>Algorithm</td><td>Cosine Sim.</td><td>maxz&#x27;+zkp T0(（at|st,2&#x27;）</td><td>π0(at|st,2kp)</td></tr><tr><td rowspan="2">Snake</td><td>HiPPO on given skills</td><td>0.98±0.01</td><td>0.09± 0.04</td><td>0.44 ± 0.03</td></tr><tr><td>HiPPO on random skills</td><td>0.97 ± 0.03</td><td>0.12 ± 0.03</td><td>0.32 ± 0.04</td></tr><tr><td rowspan="2">Ant</td><td>HiPPO on given skills</td><td>0.96±0.04</td><td>0.11 ± 0.05</td><td>0.40±0.08</td></tr><tr><td>HiPPO on random skills</td><td>0.94 ± 0.03</td><td>0.13 ± 0.05</td><td>0.31 ± 0.09</td></tr></table>
+
+Table 2: Empirical evaluation of Lemma 1. In the middle and right columns, we evaluate the quality of our assumption by computing the largest probability of a certain action under other skills (), and the action probability under the actual latent. We also report the cosine similarity between our approximate gradient and the exact gradient from Eq. 3. The mean and standard deviation of these values are computed over the full batch collected at iteration 10.
+
+# 6 CONCLUSIONS AND FUTURE WORK
+
+In this paper, we examined how to effectively adapt temporal hierarchies. We began by deriving a hierarchical policy gradient and its approximation. We then proposed a new method, HiPPO, that can stably train multiple layers of a hierarchy jointly. The adaptation experiments suggest that we can optimize pretrained skills for downstream environments, and learn emergent skills without any unsupervised pre-training. We also demonstrate that HiPPO with randomized period can learn from scratch on sparse-reward and long time horizon tasks, while outperforming non-hierarchical methods on zero-shot transfer.
+
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+# A HYPERPARAMETERS AND ARCHITECTURES
+
+The Block environments used a horizon of 1000 and a batch size of 50,000, while Gather used a batch size of 100,000. Ant Gather has a horizon of 5000, while Snake Gather has a horizon of 8000 due to its larger size. For all experiments, both PPO and HiPPO used learning rate $3 \times 1 0 ^ { - 3 }$ , clipping parameter $\epsilon = 0 . 1$ , 10 gradient updates per iteration, and discount $\gamma = 0 . 9 9 9$ . The learning rate, clipping parameter, and number of gradient updates come from the OpenAI Baselines implementation.
+
+HiPPO used $n = 6$ sub-policies. HiPPO uses a manager network with 2 hidden layers of 32 units, and a skill network with 2 hidden layers of 64 units. In order to have roughly the same number of parameters for each algorithm, flat PPO uses a network with 2 hidden layers with 256 and 64 units respectively. For HiPPO with randomized period, we resample $p \sim \mathrm { U n i f o r m } \{ 5 , 1 5 \}$ every time the manager network outputs a latent, and provide the number of timesteps until the next latent selection as an input into both the manager and skill networks. The single baselines and skill-dependent baselines used a MLP with 2 hidden layers of 32 units to fit the value function. The skill-dependent baseline receives, in addition to the full observation, the active latent code and the time remaining until the next skill sampling. All runs used five random seeds.
+
+# B ROBOT AGENT DESCRIPTION
+
+Hopper is a 3-link robot with a 14-dimensional observation space and a 3-dimensional action space. Half-Cheetah has a 20-dimensional observation space and a 6-dimensional action space. We evaluate both of these agents on a sparse block hopping task. In addition to observing their own joint angles and positions, they observe the height and length of the next wall, the $\mathbf { X }$ -position of the next wall, and the distance to the wall from the agent. We also provide the same wall observations for the previous wall, which the agent can still interact with.
+
+Snake is a 5-link robot with a 17-dimensional observation space and a 4-dimensional action space. Ant is a quadrupedal robot with a 27-dimensional observation space and a 8-dimensional action space. Both Ant and Snake can move and rotate in all directions, and Ant faces the added challenge of avoiding falling over irrecoverably. In the Gather environment, agents also receive 2 sets of 10-dimensional lidar observations, whcih correspond to separate apple and bomb observations. The observation displays the distance to the nearest apple or bomb in each $3 6 ^ { \circ }$ bin, respectively. All environments are simulated with the physics engine MuJoCo (Todorov et al., 2012).
+
+# C PROOFS
+
+Lemma 1. If the skills are sufficiently differentiated, then the latent variable can be treated as part of the observation to compute the gradient of the trajectory probability. Concretely, if $\pi _ { \boldsymbol { \theta } _ { h } } ( z | s )$ and $\pi _ { \boldsymbol { \theta } _ { l } } ( a | s , z )$ are Lipschitz in their parameters, and $0 \overset { \cdot } { < } \pi _ { \theta _ { l } } ( \bar { a } _ { t } \vert s _ { t } , z _ { j } ) < \epsilon \forall j \neq k p$ , then
+
+$$
+\nabla _ { \theta } \log P ( \tau ) = \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log \pi _ { \theta _ { h } } ( z _ { k p } | s _ { k p } ) + \sum _ { t = 1 } ^ { p } \nabla _ { \theta } \log \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) + \mathcal { O } ( n H \epsilon ^ { p - 1 } )
+$$
+
+Proof. From the point of view of the MDP, a trajectory is a sequence $\begin{array} { r l } { \tau } & { { } = } \end{array}$ $( s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , \ldots , a _ { H - 1 } , s _ { H } )$ . Let’s assume we use the hierarchical policy introduced above, with a higher-level policy modeled as a parameterized discrete distribution with $n$ possible outcomes $\pi _ { \boldsymbol { \theta } _ { h } } ( \bar { z } | s ) = C \bar { a } t e g o r i c a l _ { \boldsymbol { \theta } _ { h } } ( n )$ . We can expand $P ( \tau )$ into the product of policy and environment dynamics terms, with $z _ { j }$ denoting the $j$ th possible value out of the $n$ choices,
+
+$$
+P ( \tau ) = \Bigg ( \prod _ { k = 0 } ^ { H / p } \Big [ \sum _ { j = 1 } ^ { n } \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \Big ] \Bigg ) \Bigg [ P ( s _ { 0 } ) \prod _ { t = 1 } ^ { H } P ( s _ { t + 1 } | s _ { t } , a _ { t } ) \Bigg ]
+$$
+
+Taking the gradient of $\log { P ( \tau ) }$ with respect to the policy parameters $\theta = [ \theta _ { h } , \theta _ { l } ]$ , the dynamics terms disappear, leaving:
+
+$$
+\begin{array} { l } { { \displaystyle 7 _ { \theta } \log P ( \tau ) = \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log \Big ( \sum _ { j = 1 } ^ { n } \pi _ { \theta _ { l } } ( z _ { j } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { s , \theta } ( a _ { t } | s _ { t } , z _ { j } ) \Big ) } } \\ { { \displaystyle \qquad = \sum _ { k = 0 } ^ { H / p } \frac { 1 } { \sum _ { j = 1 } ^ { n } \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) } \sum _ { j = 1 } ^ { n } \nabla _ { \theta } \Big ( \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \Big ) } } \end{array}
+$$
+
+The sum over possible values of $z$ prevents the logarithm from splitting the product over the $p$ -step sub-trajectories. This term is problematic, as this product quickly approaches 0 as $p$ increases, and suffers from considerable numerical instabilities. Instead, we want to approximate this sum of products by a single one of the terms, which can then be decomposed into a sum of logs. For this we study each of the terms in the sum: the gradient of a sub-trajectory probability under a specific latent $\begin{array} { r } { \nabla _ { \theta } \Big ( \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , \bar { z } _ { j } ) \Big ) } \end{array}$ . Now we can use the assumption that the skills are easy to distinguish, $0 < \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) < \epsilon \forall j \neq k p$ . Therefore, the probability of the sub-trajectory under a latent different than the one that was originally sampled $z _ { j } \neq z _ { k p }$ , is upper bounded by $\epsilon ^ { p }$ . Taking the gradient, applying the product rule, and the Lipschitz continuity of the policies, we obtain that for all $z _ { j } \neq z _ { k p }$ ,
+
+$$
+\begin{array} { l l } { { \displaystyle 7 _ { \theta } \Big ( \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) } } & { { \displaystyle \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \Big ) = \nabla _ { \theta } \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) } } & { { \displaystyle \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) } + } \\ { { } } & { { \displaystyle \sum _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { h } } ( z _ { j } | s _ { k p } ) \big ( \nabla _ { \theta } \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \big ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { l } } ( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } ) \Big \} } } \\ { { } } & { { \displaystyle = \mathcal { O } ( p \epsilon ^ { p - 1 } ) } } \end{array}
+$$
+
+Thus, we can across the board replace the summation over latents by the single term corresponding to the latent that was sampled at that time.
+
+$$
+\begin{array} { l } { { \displaystyle \zeta _ { \theta } \log P ( \tau ) = \sum _ { k = 0 } ^ { H / p } \frac { 1 } { \pi _ { \theta _ { h } } ( z _ { k p } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) } \nabla _ { \theta } \Big ( P ( z _ { k p } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) \Big ) } } \\ { { \displaystyle \qquad = \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log \Big ( \pi _ { \theta _ { h } } ( z _ { k p } | s _ { k p } ) \prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \pi _ { \theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) \Big ) + \mathcal { O } ( n H \epsilon ^ { p - 1 } ) } } \\ { { \displaystyle \qquad = \mathbb { E } _ { \tau } \bigg [ \Big ( \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log \pi _ { \theta _ { h } } \big ( z _ { k p } | s _ { k p } ) + \sum _ { t = 1 } ^ { H } \nabla _ { \theta } \log \pi _ { \theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) \Big ) \bigg ] + \mathcal { O } ( n H \epsilon ^ { p - 1 } ) } } \end{array}
+$$
+
+Interestingly, this is exactly $\nabla _ { \theta } P ( s _ { 0 } , z _ { 0 } , a _ { 0 } , s _ { 1 } , \dots )$ . In other words, it’s the gradient of the probability of that trajectory, where the trajectory now includes the variables $z$ as if they were observed.
+
+Lemma 2. For any functions $b _ { h } : S  \mathbb { R }$ and $b _ { l } : \mathcal { S } \times \mathcal { Z } \to \mathbb { R }$ we have:
+
+$$
+\mathbb { E } _ { \tau } [ \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] = 0
+$$
+
+$$
+\mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { H } \nabla _ { \theta } \log \pi _ { \theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) b ( s _ { t } , z _ { k p } ) ] = 0
+$$
+
+Proof. We can use the tower property as well as the fact that the interior expression only depends on $s _ { k p }$ and $z _ { k p }$ :
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { \tau } [ \displaystyle \sum _ { k = 0 } ^ { H / p } \nabla _ { \theta } \log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] = \displaystyle \sum _ { k = 0 } ^ { H / p } \mathbb { E } _ { s _ { k p } , z _ { k p } } [ \mathbb { E } _ { \tau \setminus s _ { k p } , z _ { k p } } [ \nabla _ { \theta } \log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] } & { } \\ { \displaystyle } & { ~ = \displaystyle \sum _ { k = 0 } ^ { H / p } \mathbb { E } _ { s _ { k p } , z _ { k p } } [ \nabla _ { \theta } \log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] } \end{array}
+$$
+
+Then, we can write out the definition of the expectation and undo the gradient-log trick to prove that the baseline is unbiased.
+
+$$
+\begin{array} { r l } { \nabla _ { \theta } \log \mathfrak { A } _ { \theta _ { 1 } } ( \mathfrak { L } _ { \mathcal { F } _ { 1 } } | \mathfrak { L } _ { \mathcal { F } _ { 1 } } ) \hat { \mathfrak { H } } ( \mathfrak { L } _ { \mathcal { F } _ { 1 } } ) \rVert ( \mathfrak { L } _ { \mathcal { F } _ { 1 } } ) } & { = \displaystyle \operatorname* { l i m } _ { \le t \le j } \int _ { ( s _ { \phi _ { 1 } , \phi _ { 2 } } , \mathfrak { L } _ { \mathcal { F } _ { 2 } } ) \in \mathcal { L } _ { \mathcal { F } _ { 1 } } } \nabla _ { ( s _ { \phi _ { 2 } } , \mathfrak { L } _ { \mathcal { F } _ { 2 } } ) \in \mathcal { L } _ { \mathcal { F } _ { 1 } } } \left. \mathfrak { L } _ { \mathcal { F } _ { 1 } } ( \mathcal { L } _ { \mathcal { F } _ { 2 } } | \mathfrak { L } _ { \mathcal { F } _ { 1 } } ) \hat { \mathfrak { H } } ( \mathfrak { L } _ { \mathcal { F } _ { 2 } } ) \hat { \mathfrak { H } } ( \mathfrak { L } _ { \mathcal { F } _ { 2 } } ) \hat { \mathfrak { H } } ( \mathfrak { L } _ { \mathcal { F } _ { 2 } } ) \hat { \mathfrak { H } } ( \mathfrak { L } _ { \mathcal { F } _ { 2 } } ) \right. } \\ &  = \displaystyle \sum _ { s = 0 } ^ { \mathcal { N } _ { 0 } } \int _ { s _ { \phi _ { 2 } } \in \mathcal { L } _ { \mathcal { F } _ { 2 } } } P ( s _ { \phi _ { 3 } , \phi _ { 3 } } ) \hat { \mathfrak { H } } ( s _ { \phi _ { 3 } } ) \int _ { s _ { \phi _ { 2 } } \le s _ { \phi _ { 1 } } } \left( \mathfrak { L } _ { \mathcal { F } _ { 2 } } | \mathfrak { L } _ { \mathcal { F } _ { 3 } } \right) \nabla _ { \phi } \log \mathfrak { L } _ { \mathcal { F } _ { 3 } } ( \mathcal { L } _ { \mathcal { F } _ { 1 } } | \mathfrak { L } _ { \mathcal { F } _ { 3 } } ) \hat { \mathfrak { H } } ( \mathfrak { L } _ { \mathcal { F } _ { 2 } } | \hat { \mathfrak { L } } _ { \mathcal { F } _ { 3 } } ) \hat { \mathfrak { H } } ( \mathcal { L } _ { \mathcal { F } _ { 1 } } | \mathfrak { L } _ { \mathcal { F } _ { 1 } } ) \prod _ { s = 0 }  \end{array}
+$$
+
+Subtracting a state- and subpolicy- dependent baseline from the second term is also unbiased, i.e.
+
+$$
+\mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { H } \nabla _ { \theta } \log \pi _ { s , \theta } ( a _ { t } | s _ { t } , z _ { k p } ) b ( s _ { t } , z _ { k p } ) ] = 0
+$$
+
+We’ll follow the same strategy to prove the second equality: apply the tower property, express the expectation as an integral, and undo the gradient-log trick.
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { x } | \displaystyle \sum _ { t = 1 } ^ { M } \nabla \cdot \nabla ^ { \theta } \log \pi _ { \theta } \{ \alpha _ { t } | s _ { t } , z _ { t , p } \} b ( s _ { t } , z _ { t , p } ) \Big | } \\ &  \begin{array} { r l } &  \displaystyle = \sum _ { t = 0 } ^ { H } \sum _ { \alpha _ { t } , \alpha _ { t } , z _ { t , p } \} \mathbb { E } _ { \Gamma ( s ) \cup \mathcal { L } _ { q } \cup \mathcal { L } _ { q } } [ \nabla _ { s } \log \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) ] \Big | } \\ &  \displaystyle = \sum _ { t = 0 } ^ { H } \sum _ { \alpha _ { t } , \alpha _ { t } , z _ { t , p } \} \big [ \nabla _ { s } \log \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \big ] \big ( \nabla _ { s } \log \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \big ) \big | } \\ &  \displaystyle = \sum _ { t = 0 } ^ { H } \mathbb { E } _ { \kappa _ { t } , \alpha _ { t } , z _ { t , p } \} \big [ \nabla _ { s } \log \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \big ] \big ( \nabla _ { s } \log \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) \big ) } \\ & { \displaystyle = \sum _ { t = 0 } ^ { H } \int _ { ( a _ { t } , z _ { t , p } ) } P ( s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \int _ { a _ { t } } \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( a _ { t } | s _ { t } , z _ { t , p } ) d a _ { t } d a _ { t } d a _ { t } d a _ { t } } \\ &  \displaystyle = \sum _ { t = 0 } ^ { H } \int _ { ( a _ { t } , z _ { t , p } ) } P ( s _ { t } , z _ { t , p } \end{array} \end{array}
+$$
+
+![](images/ee0ef8b7d618f7c26e204fd303b0f1de7942fc7e9d8b5b2ff23a13be001b23f1.jpg)  
+Figure 7: HIRO performance on Ant Gather with and without access to the ground truth $( x , y )$ , which it needs to communicate useful goals.
+
+# D HIRO SENSITIVITY TO OBSERVATION-SPACE
+
+In this section we provide a more detailed explanation of why HIRO (Nachum et al., 2018) performs poorly under our environments. As explained in our related work section, HIRO belongs to the general category of algorithms that train goal-reaching policies as lower levels of the hierarchy (Vezhnevets et al., 2017; Levy et al., 2017). These methods rely on having a goal-space that is meaningful for the task at hand. For example, in navigation tasks they require having access to the $( x , y )$ position of the agent such that deltas in that space can be given as meaningful goals to move in the environment. Unfortunately, in many cases the only readily available information (if there’s no GPS signal or other positioning system installed) are raw sensory inputs, like cameras or the LIDAR sensors we mimic in our environments. In such cases, our method still performs well because it doesn’t rely on the goal-reaching extra supervision that is leveraged (and detrimental in this case) in HIRO and similar methods. In Figure 7, we show that knowing the ground truth location is critical for its success. We have reproduced the HIRO results in Fig. 7 using the published codebase, so we are convinced that our results showcase a failure mode of HIRO.
+
+# E HYPERPARAMETER SENSITIVITY PLOTS
+
+![](images/bf71f0947b79c1dacc6adb6a4abac4033bdc6c446907c6564290d12f3e9db82b.jpg)  
+Figure 8: Sensitivity of HiPPO to variation in the time-commitment.
+
+![](images/6e231db06c188eafb086f33848ba035a6839f3368bff333cca0e31a26f509a6d.jpg)  
+Figure 9: Sensitivity of HiPPO to variation in the number of skills.

parse/train/ByeWogStDS/ByeWogStDS_content_list.json

@@ -0,0 +1,1879 @@
+[
+    {
+        "type": "text",
+        "text": "SUB-POLICY ADAPTATION FOR HIERARCHICALREINFORCEMENT LEARNING",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            745,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alexander C. Li∗, Carlos Florensa∗, Ignasi Clavera, Pieter Abbeel University of California, Berkeley {alexli1, florensa, iclavera, pabbeel}@berkeley.edu ",
+        "bbox": [
+            184,
+            169,
+            684,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hierarchical reinforcement learning is a promising approach to tackle long-horizon decision-making problems with sparse rewards. Unfortunately, most methods still decouple the lower-level skill acquisition process and the training of a higher level that controls the skills in a new task. Leaving the skills fixed can lead to significant sub-optimality in the transfer setting. In this work, we propose a novel algorithm to discover a set of skills and continuously adapt them along with the higher level even when training on a new task. Our main contributions are two-fold. First, we derive a new hierarchical policy gradient with an unbiased latent-dependent baseline, and we introduce Hierarchical Proximal Policy Optimization (HiPPO), an on-policy method to efficiently train all levels of the hierarchy jointly. Second, we propose a method of training time-abstractions that improves the robustness of the obtained skills to environment changes. Code and videos are available. 1. ",
+        "bbox": [
+            233,
+            281,
+            766,
+            448
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            477,
+            336,
+            492
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) has made great progress in a variety of domains, from playing games such as Pong and Go (Mnih et al., 2015; Silver et al., 2017) to automating robotic locomotion (Schulman et al., 2015; Heess et al., 2017), dexterous manipulation (Florensa et al., 2017b; OpenAI et al., 2018), and perception (Nair et al., 2018; Florensa et al., 2018). Yet, most work in RL is still learning from scratch when faced with a new problem. This is particularly inefficient when tackling multiple related tasks that are hard to solve due to sparse rewards or long horizons. ",
+        "bbox": [
+            174,
+            510,
+            823,
+            593
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A promising technique to overcome this limitation is hierarchical reinforcement learning (HRL) (Sutton et al., 1999). In this paradigm, policies have several modules of abstraction, allowing to reuse subsets of the modules. The most common case consists of temporal hierarchies (Precup, 2000; Dayan & Hinton, 1993), where a higher-level policy (manager) takes actions at a lower frequency, and its actions condition the behavior of some lower level skills or sub-policies. When transferring knowledge to a new task, most prior works fix the skills and train a new manager on top. Despite having a clear benefit in kick-starting the learning in the new task, having fixed skills can considerably cap the final performance on the new task (Florensa et al., 2017a). Little work has been done on adapting pre-trained sub-policies to be optimal for a new task. ",
+        "bbox": [
+            174,
+            599,
+            825,
+            726
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we develop a new framework for simultaneously adapting all levels of temporal hierarchies. First, we derive an efficient approximated hierarchical policy gradient. The key insight is that, despite the decisions of the manager being unobserved latent variables from the point of view of the Markovian environment, from the perspective of the sub-policies they can be considered as part of the observation. We show that this provides a decoupling of the manager and sub-policy gradients, which greatly simplifies the computation in a principled way. It also theoretically justifies a technique used in other prior works (Frans et al., 2018). Second, we introduce a sub-policy specific baseline for our hierarchical policy gradient. We prove that this baseline is unbiased, and our experiments reveal faster convergence, suggesting efficient gradient variance reduction. Then, we introduce a more stable way of using this gradient, Hierarchical Proximal Policy Optimization (HiPPO). This method helps us take more conservative steps in our policy space (Schulman et al., 2017), critical in hierarchies because of the interdependence of each layer. Results show that HiPPO is highly efficient both when learning from scratch, i.e. adapting randomly initialized skills, and when adapting pretrained skills on a new task. Finally, we evaluate the benefit of randomizing the time-commitment of the sub-policies, and show it helps both in terms of final performance and zero-shot adaptation on similar tasks. ",
+        "bbox": [
+            174,
+            732,
+            825,
+            885
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            181,
+            338,
+            196
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We define a discrete-time finitehorizon discounted Markov decision process (MDP) by a tuple $\\begin{array} { r l } { M } & { { } = } \\end{array}$ $( \\boldsymbol { \\bar { s } } , \\boldsymbol { \\mathcal { A } } , \\mathcal { P } , \\boldsymbol { r } , \\rho _ { 0 } , \\gamma , \\boldsymbol { \\dot { H } } )$ , where $s$ is a state set, $\\mathcal { A }$ is an action set, $\\mathcal { P } :$ $S \\times \\mathcal { A } \\times \\mathcal { S } \\to \\mathbb { R } _ { + }$ is the transition probability distribution, $\\gamma ~ \\in ~ [ 0 , 1 ]$ is a discount factor, and $H$ the horizon. Our objective is to find a stochastic policy $\\pi _ { \\theta }$ that maximizes the expected discounted return within the MDP, $\\begin{array} { r } { \\eta ( \\pi _ { \\theta } ) = \\mathbb { E } _ { \\tau } [ \\sum _ { t = 0 } ^ { H } \\gamma ^ { t } r ( s _ { t } , a _ { t } ) ] } \\end{array}$ We use ${ \\boldsymbol \\tau } = ( s _ { 0 } , a _ { 0 } , . . . , )$ to denote the entire state-action trajectory, where $s _ { 0 } ~ \\sim ~ \\rho _ { 0 } ( s _ { 0 } )$ , $a _ { t } \\sim \\dot { \\pi } _ { \\theta } ( a _ { t } | s _ { t } )$ , and $s _ { t + 1 } \\sim \\mathcal { P } ( s _ { t + 1 } | s _ { t } , a _ { t } )$ . ",
+        "bbox": [
+            174,
+            214,
+            421,
+            440
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ba3bb08e3131bf7c0453118b8ce8cd1f7c6946bcc17a481702825cd49f02921.jpg",
+        "image_caption": [
+            "Figure 1: Temporal hierarchy studied in this paper. A latent code $z _ { t }$ is sampled from the manager policy $\\bar { \\pi _ { \\boldsymbol { \\theta } _ { h } } ( \\boldsymbol { z } _ { t } | \\boldsymbol { s } _ { t } ) }$ every $p$ time-steps, using the current observation $s _ { k p }$ . The actions $a _ { t }$ are sampled from the sub-policy $\\pi _ { \\boldsymbol { \\theta } _ { l } } ( a _ { t } | \\bar { s } _ { t } , z _ { k p } )$ conditioned on the same latent code from $t = k p$ to $( k \\bar { + } 1 ) p - 1$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            454,
+            189,
+            810,
+            347
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a method to learn a hierarchical policy and efficiently adapt all the levels in the hierarchy to perform a new task. We study hierarchical policies composed of a higher level, or manager $\\pi _ { \\boldsymbol { \\theta } _ { h } } \\big ( \\boldsymbol { z } _ { t } | \\boldsymbol { s } _ { t } \\big )$ , and a lower level, or sub-policy $\\pi _ { \\theta _ { l } } ( a _ { t ^ { \\prime } } | z _ { t } , s _ { t ^ { \\prime } } )$ . The higher level does not take actions in the environment directly, but rather outputs a command, or latent variable $z _ { t } \\in \\mathcal { Z }$ , that conditions the behavior of the lower level. We focus on the common case where ${ \\mathcal { Z } } = \\mathbb { Z } _ { n }$ making the manager choose among $n$ sub-policies, or skills, to execute. The manager typically operates at a lower frequency than the sub-policies, only observing the environment every $p$ time-steps. When the manager receives a new observation, it decides which low level policy to commit to for $p$ environment steps by the means of a latent code $z$ . Figure 1 depicts this framework where the high level frequency $p$ is a random variable, which is one of the contribution of this paper as described in Section 4.4. Note that the class of hierarchical policies we work with is more restrictive than others like the options framework, where the time-commitment is also decided by the policy. Nevertheless, we show that this loss in policy expressivity acts as a regularizer and does not prevent our algorithm from surpassing other state-of-the art methods. ",
+        "bbox": [
+            173,
+            445,
+            825,
+            640
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            661,
+            344,
+            678
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There has been growing interest in HRL for the past few decades (Sutton et al., 1999; Precup, 2000), but only recently has it been applied to high-dimensional continuous domains as we do in this work (Kulkarni et al., 2016; Daniel et al., 2016). To obtain the lower level policies, or skills, most methods exploit some additional assumptions, like access to demonstrations (Le et al., 2018; Merel et al., 2019; Ranchod et al., 2015; Sharma et al., 2018), policy sketches (Andreas et al., 2017), or task decomposition into sub-tasks (Ghavamzadeh & Mahadevan, 2003; Sohn et al., 2018). Other methods use a different reward for the lower level, often constraining it to be a “goal reacher” policy, where the signal from the higher level is the goal to reach (Nachum et al., 2018; Levy et al., 2019; Vezhnevets et al., 2017). These methods are very promising for state-reaching tasks, but might require access to goal-reaching reward systems not defined in the original MDP, and are more limited when training on tasks beyond state-reaching. Our method does not require any additional supervision, and the obtained skills are not constrained to be goal-reaching. ",
+        "bbox": [
+            173,
+            694,
+            825,
+            861
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "When transferring skills to a new environment, most HRL methods keep them fixed and simply train a new higher-level on top (Hausman et al., 2018; Heess et al., 2016). Other work allows for building on previous skills by constantly supplementing the set of skills with new ones (Shu et al., 2018), but they require a hand-defined curriculum of tasks, and the previous skills are never fine-tuned. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our algorithm allows for seamless adaptation of the skills, showing no trade-off between leveraging the power of the hierarchy and the final performance in a new task. Other methods use invertible functions as skills (Haarnoja et al., 2018), and therefore a fixed skill can be fully overwritten when a new layer of hierarchy is added on top. This kind of “fine-tuning” is promising, although similar to other works (Peng et al., 2019), they do not apply it to temporally extended skills as we do here. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One of the most general frameworks to define temporally extended hierarchies is the options framework (Sutton et al., 1999), and it has recently been applied to continuous state spaces (Bacon et al., 2017). One of the most delicate parts of this formulation is the termination policy, and it requires several regularizers to avoid skill collapse (Harb et al., 2017; Vezhnevets et al., 2016). This modification of the objective may be difficult to tune and affects the final performance. Instead of adding such penalties, we propose to have skills of a random length, not controlled by the agent during training of the skills. The benefit is two-fold: no termination policy to train, and more stable skills that transfer better. Furthermore, these works only used discrete action MDPs. We lift this assumption, and show good performance of our algorithm in complex locomotion tasks. There are other algorithms recently proposed that go in the same direction, but we found them more complex, less principled (their per-action marginalization cannot capture well the temporal correlation within each option), and without available code or evidence of outperforming non-hierarchical methods (Smith et al., 2018). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The closest work to ours in terms of final algorithm structure is the one proposed by Frans et al. (2018). Their method can be included in our framework, and hence benefits from our new theoretical insights. We introduce a modification that is shown to be highly beneficial: the random timecommitment mentioned above, and find that our method can learn in difficult environments without their complicated training scheme. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 EFFICIENT HIERARCHICAL POLICY GRADIENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When using a hierarchical policy, the intermediate decision taken by the higher level is not directly applied in the environment. Therefore, technically it should not be incorporated into the trajectory description as an observed variable, like the actions. This makes the policy gradient considerably harder to compute. In this section we first prove that, under mild assumptions, the hierarchical policy gradient can be accurately approximated without needing to marginalize over this latent variable. Then, we derive an unbiased baseline for the policy gradient that can reduce the variance of its estimate. Finally, with these findings, we present our method, Hierarchical Proximal Policy Optimization (HiPPO), an on-policy algorithm for hierarchical policies, allowing learning at all levels of the policy jointly and preventing sub-policy collapse. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 APPROXIMATE HIERARCHICAL POLICY GRADIENT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Policy gradient algorithms are based on the likelihood ratio trick (Williams, 1992) to estimate the gradient of returns with respect to the policy parameters as ",
+        "bbox": [
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+            641,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19460e68894e2a01fbb52ab7b6bd9fcaeb7d7d5ed53b22bc602da293098f807a.jpg",
+        "text": "$$\n\\begin{array} { c l } { \\displaystyle \\nabla _ { \\theta } \\eta ( \\pi _ { \\theta } ) = \\mathbb E _ { \\tau } \\big [ \\nabla _ { \\theta } \\log P ( \\tau ) R ( \\tau ) \\big ] \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { n } \\nabla _ { \\theta } \\log P ( \\tau _ { i } ) R ( \\tau _ { i } ) } \\\\ { \\displaystyle = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { n } \\frac { 1 } { H } \\sum _ { t = 1 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } ) R ( \\tau _ { i } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In a temporal hierarchy, a hierarchical policy with a manager $\\pi _ { \\boldsymbol { \\theta } _ { h } } \\big ( \\boldsymbol { z } _ { t } | \\boldsymbol { s } _ { t } \\big )$ selects every $p$ time-steps one of $n$ sub-policies to execute. These sub-policies, indexed by $z \\in \\mathbb { Z } _ { n }$ , can be represented as a single conditional probability distribution over actions $\\pi _ { \\boldsymbol { \\theta } _ { l } } ( a _ { t } | \\boldsymbol { z } _ { t } , \\boldsymbol { s } _ { t } )$ . This allows us to not only use a given set of sub-policies, but also leverage skills learned with Stochastic Neural Networks (SNNs) (Florensa et al., 2017a). Under this framework, the probability of a trajectory $\\tau = ( s _ { 0 } , a _ { 0 } , s _ { 1 } , \\dots , s _ { H } )$ can be written as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11ad076417ec9919c11a60634cea9cc17c0260275d1d48b28bf5d29bdfd18a9f.jpg",
+        "text": "$$\nP ( \\tau ) = \\Bigg ( \\prod _ { k = 0 } ^ { H / p } \\Big [ \\sum _ { j = 1 } ^ { n } \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \\Big ] \\Bigg ) \\Bigg [ P ( s _ { 0 } ) \\prod _ { t = 1 } ^ { H } P ( s _ { t + 1 } | s _ { t } , a _ { t } ) \\Bigg ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The mixture action distribution, which presents itself as an additional summation over skills, prevents additive factorization when taking the logarithm, as from Eq. 1 to 2. This can yield numerical instabilities due to the product of the $p$ sub-policy probabilities. For instance, in the case where all the skills are distinguishable all the sub-policies’ probabilities but one will have small values, resulting in an exponentially small value. In the following Lemma, we derive an approximation of the policy gradient, whose error tends to zero as the skills become more diverse, and draw insights on the interplay of the manager actions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. If the skills are sufficiently differentiated, then the latent variable can be treated as part of the observation to compute the gradient of the trajectory probability. Let $\\pi _ { \\boldsymbol { \\theta } _ { h } } ( z | s )$ and $\\pi _ { \\boldsymbol { \\theta } _ { l } } ( a | s , z )$ be Lipschitz functions w.r.t. their parameters, and assume that $0 < \\pi _ { \\theta _ { l } } ( a | s , z _ { j } ) < \\epsilon \\forall j \\neq k p$ , then ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ed339299a39e6e949895b23aa338b11023c6c6915a8ebdf1ecfa3867b0af6638.jpg",
+        "text": "$$\n\\nabla _ { \\theta } \\log P ( \\tau ) = \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { h } } ( z _ { k p } | s _ { k p } ) + \\sum _ { t = 0 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) + \\mathcal { O } ( n H \\epsilon ^ { p - 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. See Appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our assumption can be seen as having diverse skills. Namely, for each action there is just one sub-policy that gives it high probability. In this case, the latent variable can be treated as part of the observation to compute the gradient of the trajectory probability. Many algorithms to extract lowerlevel skills are based on promoting diversity among the skills (Florensa et al., 2017a; Eysenbach et al., 2019), therefore usually satisfying our assumption. We further analyze how well this assumption holds in our experiments section and Table 2. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 UNBIASED SUB-POLICY BASELINE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The policy gradient estimate obtained when applying the log-likelihood ratio trick as derived above is known to have large variance. A very common approach to mitigate this issue without biasing the estimate is to subtract a baseline from the returns (Peters & Schaal, 2008). It is well known that such baselines can be made state-dependent without incurring any bias. However, it is still unclear how to formulate a baseline for all the levels in a hierarchical policy, since an action dependent baseline does introduce bias in the gradient (Tucker et al., 2018). It has been recently proposed to use latent-conditioned baselines (Weber et al., 2019). Here we go further and prove that, under the assumptions of Lemma 1, we can formulate an unbiased latent dependent baseline for the approximate gradient (Eq. 5). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. For any functions $b _ { h } : S  \\mathbb { R }$ and $b _ { l } : \\mathcal { S } \\times \\mathcal { Z } \\to \\mathbb { R }$ we have: ",
+        "bbox": [
+            173,
+            571,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15614f02bd2e16641269ff3847d5e81509a85f9391271d87d9af82980c2e296f.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\tau } [ \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log P ( z _ { k p } | s _ { k p } ) b _ { h } ( s _ { k p } ) ] = 0 \\quad a n d \\quad \\mathbb { E } _ { \\tau } [ \\sum _ { t = 0 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) b _ { l } ( s _ { t } , z _ { k p } ) ] = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            594,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. See Appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now we apply Lemma 1 and Lemma 2 to Eq. 1. By using the corresponding value functions as the function baseline, the return can be replaced by the Advantage function $A ( s _ { k p } , z _ { k p } )$ (see details in Schulman et al. (2016)), and we obtain the following approximate policy gradient expression: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08b5da05b2a61fecfabcab3afd9d5210d474c3c70592f59bc2962c149ec37134.jpg",
+        "text": "$$\n\\hat { g } = \\mathbb { E } _ { \\tau } \\Big [ \\big ( \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { h } } ( z _ { k p } | s _ { k p } ) A ( s _ { k p } , z _ { k p } ) \\big ) + \\big ( \\sum _ { t = 0 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) A ( s _ { t } , a _ { t } , z _ { k p } ) \\big ) \\Big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This hierarchical policy gradient estimate can have lower variance than without baselines, but using it for policy optimization through stochastic gradient descent still yields an unstable algorithm. In the next section, we further improve the stability and sample efficiency of the policy optimization by incorporating techniques from Proximal Policy Optimization (Schulman et al., 2017). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 HIERARCHICAL PROXIMAL POLICY OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using an appropriate step size in policy space is critical for stable policy learning. Modifying the policy parameters in some directions may have a minimal impact on the distribution over actions, whereas small changes in other directions might change its behavior drastically and hurt training ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 HiPPO Rollout ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 HiPPO ",
+        "text_level": 1,
+        "bbox": [
+            575,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: Input: skills $\\pi _ { \\boldsymbol { \\theta } _ { l } } ( a | s , z )$ , manager $\\pi _ { \\boldsymbol { \\theta } _ { h } } ( z | s )$ , time   \ncommitment bounds $P _ { \\mathrm { m i n } }$ and $P _ { \\mathrm { m a x } }$ , horizon $H$   \n2: Reset environment: $s _ { 0 } \\sim \\rho _ { 0 }$ , $t = 0$ .   \n3: while $t < H$ do   \n4: Sample time-commitment $p \\sim \\mathsf { C a t } ( [ P _ { \\operatorname* { m i n } } , P _ { \\operatorname* { m a x } } ] )$   \n5: Sample skill $z _ { t } \\sim \\pi _ { \\theta _ { h } } ( \\cdot | s _ { t } )$   \n6: for $t ^ { \\prime } = t \\ldots ( t + p )$ do   \n7: Sample action $a _ { t ^ { \\prime } } \\sim \\pi _ { \\theta _ { l } } ( \\cdot | s _ { t ^ { \\prime } } , z _ { t } )$   \n8: Observe new state $s _ { t ^ { \\prime } + 1 }$ and reward $\\boldsymbol { r } _ { t ^ { \\prime } }$   \n9: end for   \n10: $t \\gets t + p$   \n11: end while   \n12: Output: $\\left( s _ { 0 } , z _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , \\ldots , s _ { H } , z _ { H } , a _ { H } , s _ { H + 1 } \\right)$ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: Input: skills $\\pi _ { \\boldsymbol { \\theta } _ { l } } ( a | s , z )$ , manager $\\pi _ { \\boldsymbol { \\theta } _ { h } } ( z | s )$ , horizon $H$ , learning rate $\\alpha$   \n2: while not done do   \n3: for actor $= 1$ , 2, ..., N do   \n4: Obtain trajectory with HiPPO Rollout   \n5: Estimate advantages $\\hat { A } ( a _ { t ^ { \\prime } } , s _ { t ^ { \\prime } } , z _ { t } )$ and $\\bar { A } ( z _ { t } , s _ { t } )$   \n6: 7: $\\theta  \\theta + \\alpha \\nabla _ { \\theta } L _ { H i P P O } ^ { C L I P } ( \\theta )$   \n8: end while ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "efficiency (Kakade, 2002). Trust region policy optimization (TRPO) uses a constraint on the KLdivergence between the old policy and the new policy to prevent this issue (Schulman et al., 2015). Unfortunately, hierarchical policies are generally represented by complex distributions without closed form expressions for the KL-divergence. Therefore, to improve the stability of our hierarchical policy gradient we turn towards Proximal Policy Optimization (PPO) (Schulman et al., 2017). PPO is a more flexible and compute-efficient algorithm. In a nutshell, it replaces the KL-divergence constraint with a cost function that achieves the same trust region benefits, but only requires the computation of the likelihood. Letting $\\begin{array} { r } { w _ { t } ( \\theta ) = \\frac { \\pi _ { \\theta } \\left( a _ { t } | s _ { t } \\right) } { \\pi _ { \\theta _ { o l d } } \\left( a _ { t } | s _ { t } \\right) } } \\end{array}$ , the PPO objective is: ",
+        "bbox": [
+            173,
+            338,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a4ea783cffefe4f30f89170f2c960bc5113bd09deb55a6087f21e4398edacf82.jpg",
+        "text": "$$\nL ^ { C L I P } ( \\theta ) = \\mathbb { E } _ { t } \\operatorname* { m i n } \\left\\{ w _ { t } ( \\theta ) A _ { t } , \\mathrm { ~ c ~ l ~ i p } ( w _ { t } ( \\theta ) , 1 - \\epsilon , 1 + \\epsilon ) A _ { t } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can adapt our approximated hierarchical policy gradient with the same approach by letting $\\begin{array} { r } { w _ { h , k p } ( \\theta ) \\ = \\ \\frac { \\pi _ { \\theta _ { h } } ( z _ { k p } | s _ { k p } ) } { \\pi _ { \\theta _ { h , o l d } } ( z _ { k p } | s _ { k p } ) } } \\end{array}$ and $\\begin{array} { r } { w _ { l , t } ( \\theta ) = \\frac { \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) } { \\pi _ { \\theta _ { l , o l d } } ( a _ { t } | s _ { t } , z _ { k p } ) } } \\end{array}$ , and using the super-index clip to denote the clipped objective version, we obtain the new surrogate objective: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d289e3c67898d45d0d1073e28454ecb3f38aafc2eaf3e9232db60a11898a18e9.jpg",
+        "text": "$$\n\\begin{array} { c } { { { \\displaystyle { \\cal L } _ { H i P P O } ^ { C L I P } } ( \\theta ) = \\mathbb { E } _ { \\tau } \\Big [ \\displaystyle { \\sum _ { k = 0 } ^ { H / p } \\operatorname* { m i n } \\left\\{ w _ { h , k p } ( \\theta ) A ( s _ { k p } , z _ { k p } ) , w _ { h , k p } ^ { \\mathrm { c l i p } } ( \\theta ) A ( s _ { k p } , z _ { k p } ) \\right\\} } \\qquad } } \\\\ { { + \\displaystyle { \\sum _ { t = 0 } ^ { H } \\operatorname* { m i n } \\left\\{ w _ { l , t } ( \\theta ) A ( s _ { t } , a _ { t } , z _ { k p } ) , w _ { l , t } ^ { \\mathrm { c l i p } } ( \\theta ) A ( s _ { t } , a _ { t } , z _ { k p } ) \\right\\} } \\Big ] } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We call this algorithm Hierarchical Proximal Policy Optimization (HiPPO). Next, we introduce a critical additions: a switching of the time-commitment between skills. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.4 VARYING TIME-COMMITMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Most hierarchical methods either consider a fixed time-commitment to the lower level skills (Florensa et al., 2017a; Frans et al., 2018), or implement the complex options framework (Precup, 2000; Bacon et al., 2017). In this work we propose an in-between, where the time-commitment to the skills is a random variable sampled from a fixed distribution Categorical $( T _ { \\mathrm { m i n } } , T _ { \\mathrm { m a x } } )$ just before the manager takes a decision. This modification does not hinder final performance, and we show it improves zero-shot adaptation to a new task. This approach to sampling rollouts is detailed in Algorithm 1. The full algorithm is detailed in Algorithm 2. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We designed our experiments to answer the following questions: 1) How does HiPPO compare against a flat policy when learning from scratch? 2) Does it lead to policies more robust to environment changes? 3) How well does it adapt already learned skills? and 4) Does our skill diversity assumption hold in practice? ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/698cac3c55e1a99afc5dbc3e8a69b575139080c5784a5c1282ae08d45458591a.jpg",
+        "image_caption": [
+            "Figure 2: Environments used to evaluate the performance of our method. Every episode has a different configuration: wall heights for (a)-(b), ball positions for (c)-(d) "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 3: Analysis of different time-commitment strategies on learning from scratch. "
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+    {
+        "type": "text",
+        "text": "5.1 TASKS ",
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+        "type": "text",
+        "text": "We evaluate our approach on a variety of robotic locomotion and navigation tasks. The Block environments, depicted in Fig. 2a-2b, have walls of random heights at regular intervals, and the objective is to learn a gait for the Hopper and Half-Cheetah robots to jump over them. The agents observe the height of the wall ahead and their proprioceptive information (joint positions and velocities), receiving a reward of $+ 1$ for each wall cleared. The Gather environments, described by Duan et al. (2016), require agents to collect apples (green balls, $+ 1$ reward) while avoiding bombs (red balls, -1 reward). The only available perception beyond proprioception is through a LIDAR-type sensor indicating at what distance are the objects in different directions, and their type, as depicted in the bottom left corner of Fig. 2c-2d. This is challenging hierarchical task with sparse rewards that requires simultaneously learning perception, locomotion, and higher-level planning capabilities. We use the Snake and Ant robots in Gather. Details for all robotic agents are provided in Appendix B. ",
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+        "type": "text",
+        "text": "5.2 LEARNING FROM SCRATCH AND TIME-COMMITMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we study the benefit of using our HiPPO algorithm instead of standard PPO on a flat policy (Schulman et al., 2017). The results, reported in Figure 3, demonstrate that training from scratch with HiPPO leads to faster learning and better performance than flat PPO. Furthermore, we show that the benefit of HiPPO does not just come from having temporally correlated exploration: PPO with action repeat converges at a lower performance than our method. HiPPO leverages the time-commitment more efficiently, as suggested by the poor performance of the ablation where we set $p = 1$ , when the manager takes an action every environment step as well. Finally, Figure 4 shows the effectiveness of using the presented skill-dependent baseline. ",
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+    {
+        "type": "text",
+        "text": "5.3 COMPARISON TO OTHER METHODS ",
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+    {
+        "type": "text",
+        "text": "We compare HiPPO to current state-of-the-art hierarchical methods. First, we evaluate HIRO (Nachum et al., 2018), an off-policy RL method based on training a goal-reaching lower level policy. Fig. 5 shows that HIRO achieves poor performance on our tasks. As further detailed in Appendix D, this algorithm is sensitive to access to ground-truth information, like the exact $( x , y )$ position of the robot in Gather. In contrast, our method is able to perform well directly from the raw sensory inputs described in Section 5.1. We evaluate Option-Critic (Bacon et al., 2017), a variant of the options framework (Sutton et al., 1999) that can be used for continuous action-spaces. It fails to learn, and we hypothesize that their algorithm provides less time-correlated exploration and learns less diverse skills. We also compare against MLSH (Frans et al., 2018), which repeatedly samples new environment configurations to learn primitive skills. We take these hyperparameters from their Ant Twowalk experiment: resetting the environment configuration every 60 iterations, a warmup period of 20 during which only the manager is trained, and a joint training period of 40 during which both manager and skills are trained. Our results show that such a training scheme does not provide any benefits. Finally, we provide a comparison to a direct application of our Hierarchical Vanilla Policy Gradient (HierVPG) algorithm, and we see that the algorithm is unstable without PPO’s trust-region-like technique. ",
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+        "image_caption": [
+            "Figure 4: Using a skill-conditioned baseline, as defined in Section 4.2, generally improves performance of HiPPO when learning from scratch. "
+        ],
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+        "image_caption": [
+            "Figure 5: Comparison of HiPPO and HierVPG to prior hierarchical methods on learning from scratch. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.4 ROBUSTNESS TO DYNAMICS PERTURBATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We investigate the robustness of HiPPO to changes in the dynamics of the environment. We perform several modifications to the base Snake Gather and Ant Gather environments. One at a time, we change the body mass, dampening of the joints, body inertia, and friction characteristics of both robots. The results, presented in Table 1, show that HiPPO with randomized period Categorical $( [ T _ { \\operatorname* { m i n } } , T _ { \\operatorname* { m a x } } ] )$ is able to better handle these dynamics changes. In terms of the drop in policy performance between the training environment and test environment, it outperforms HiPPO with fixed period on 6 out of 8 related tasks. These results suggest that the randomized period exposes the policy to a wide range of scenarios, which makes it easier to adapt when the environment changes. ",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Zero-shot transfer performance. The final return in the initial environment is shown, as well as the average return over 25 rollouts in each new modified environment. "
+        ],
+        "table_body": "<table><tr><td>Gather</td><td>Algorithm</td><td>Initial</td><td>Mass</td><td>Dampening</td><td>Inertia</td><td>Friction</td></tr><tr><td rowspan=\"3\">Snake</td><td>Flat PPO</td><td>2.72</td><td>3.16 (+16%)</td><td>2.75 (+1%)</td><td>2.11 (-22%)</td><td>2.75 (+1%)</td></tr><tr><td>HiPPO,p = 10</td><td>4.38</td><td>3.28 (-25%)</td><td>3.27 (-25%)</td><td>3.03 (-31%)</td><td>3.27 (-25%)</td></tr><tr><td>HiPPO random p</td><td>5.11</td><td>4.09 (-20%)</td><td>4.03 (-21%)</td><td>3.21 (-37%)</td><td>4.03 (-21%)</td></tr><tr><td rowspan=\"3\">Ant</td><td>Flat PPO</td><td>2.25</td><td>2.53 (+12%)</td><td>2.13 (-5%)</td><td>2.36 (+5%)</td><td>1.96 (-13%)</td></tr><tr><td>HiPPO,p = 10</td><td>3.84</td><td>3.31 (-14%)</td><td>3.37 (-12%)</td><td>2.88 (-25%)</td><td>3.07 (-20%)</td></tr><tr><td>HiPPO random p</td><td>3.22</td><td>3.37 (+5%)</td><td>2.57 (-20%)</td><td>3.36 (+4%)</td><td>2.84 (-12%)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.5 ADAPTATION OF PRE-TRAINED SKILLS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For the Block task, we use DIAYN (Eysenbach et al., 2019) to train 6 differentiated subpolicies in an environment without any walls. Here, we see if these diverse skills can improve performance on a downstream task that’s out of the training distribution. For Gather, we take 6 pretrained subpolicies encoded by a Stochastic Neural Network (Tang & Salakhutdinov, 2013) that was trained in a diversity-promoting environment (Florensa et al., 2017a). We fine-tune them with HiPPO on the Gather environment, but with an extra penalty on the velocity of the Center of Mass. This can be understood as a preference for cautious behavior. This requires adjustment of the sub-policies, which were trained with a proxy reward encouraging them to move as far as possible (and hence quickly). Fig. 6 shows that using HiPPO to simultaneously train a manager and fine-tune the skills achieves higher final performance than fixing the sub-policies and only training a manager with PPO. The two initially learn at the same rate, but HiPPO’s ability to adjust to the new dynamics allows it to reach a higher final performance. Fig. 6 also shows that HiPPO can fine-tune the same given skills better than Option-Critic (Bacon et al., 2017), MLSH (Frans et al., 2018), and HIRO (Nachum et al., 2018). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/70943baaa443c6192a1e00d29a7bad11943570b86e84a71b5f3b01009b4b31d7.jpg",
+        "image_caption": [
+            "Figure 6: Benefit of adapting some given skills when the preferences of the environment are different from those of the environment where the skills were originally trained. Adapting skills with HiPPO has better learning performance than leaving the skills fixed or learning from scratch. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5.6 SKILL DIVERSITY ASSUMPTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In Lemma 1, we derived a more efficient and numerically stable gradient by assuming that the sub-policies are diverse. In this section, we empirically test the validity of our assumption and the quality of our approximation. We run the HiPPO algorithm on Ant Gather and Snake Gather both from scratch and with given pretrained skills, as done in the previous section. In Table 2, we report the average maximum probability under other sub-policies, corresponding to $\\epsilon$ from the assumption. In all settings, this is on the order of magnitude of 0.1. Therefore, under the $p \\approx 1 0$ that we use in our experiments, the term we neglect has a factor $\\epsilon ^ { p - 1 } = 1 0 ^ { - 1 0 }$ . It is not surprising then that the average cosine similarity between the full gradient and our approximation is almost 1, as reported in Table 2. ",
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+    {
+        "type": "table",
+        "img_path": "images/4d67f6dc4e5ab93ac4dd7aa6cd52ca18a1f5d02786d3e925c40e6246048ce4bd.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Gather</td><td>Algorithm</td><td>Cosine Sim.</td><td>maxz&#x27;+zkp T0(（at|st,2&#x27;）</td><td>π0(at|st,2kp)</td></tr><tr><td rowspan=\"2\">Snake</td><td>HiPPO on given skills</td><td>0.98±0.01</td><td>0.09± 0.04</td><td>0.44 ± 0.03</td></tr><tr><td>HiPPO on random skills</td><td>0.97 ± 0.03</td><td>0.12 ± 0.03</td><td>0.32 ± 0.04</td></tr><tr><td rowspan=\"2\">Ant</td><td>HiPPO on given skills</td><td>0.96±0.04</td><td>0.11 ± 0.05</td><td>0.40±0.08</td></tr><tr><td>HiPPO on random skills</td><td>0.94 ± 0.03</td><td>0.13 ± 0.05</td><td>0.31 ± 0.09</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Empirical evaluation of Lemma 1. In the middle and right columns, we evaluate the quality of our assumption by computing the largest probability of a certain action under other skills (\u000f), and the action probability under the actual latent. We also report the cosine similarity between our approximate gradient and the exact gradient from Eq. 3. The mean and standard deviation of these values are computed over the full batch collected at iteration 10. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we examined how to effectively adapt temporal hierarchies. We began by deriving a hierarchical policy gradient and its approximation. We then proposed a new method, HiPPO, that can stably train multiple layers of a hierarchy jointly. The adaptation experiments suggest that we can optimize pretrained skills for downstream environments, and learn emergent skills without any unsupervised pre-training. We also demonstrate that HiPPO with randomized period can learn from scratch on sparse-reward and long time horizon tasks, while outperforming non-hierarchical methods on zero-shot transfer. ",
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+    {
+        "type": "text",
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+        "text": "Ronald J Williams. Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning. Machine Learning, 8(3-4):229–256, 1992. ",
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+    {
+        "type": "text",
+        "text": "A HYPERPARAMETERS AND ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The Block environments used a horizon of 1000 and a batch size of 50,000, while Gather used a batch size of 100,000. Ant Gather has a horizon of 5000, while Snake Gather has a horizon of 8000 due to its larger size. For all experiments, both PPO and HiPPO used learning rate $3 \\times 1 0 ^ { - 3 }$ , clipping parameter $\\epsilon = 0 . 1$ , 10 gradient updates per iteration, and discount $\\gamma = 0 . 9 9 9$ . The learning rate, clipping parameter, and number of gradient updates come from the OpenAI Baselines implementation. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "HiPPO used $n = 6$ sub-policies. HiPPO uses a manager network with 2 hidden layers of 32 units, and a skill network with 2 hidden layers of 64 units. In order to have roughly the same number of parameters for each algorithm, flat PPO uses a network with 2 hidden layers with 256 and 64 units respectively. For HiPPO with randomized period, we resample $p \\sim \\mathrm { U n i f o r m } \\{ 5 , 1 5 \\}$ every time the manager network outputs a latent, and provide the number of timesteps until the next latent selection as an input into both the manager and skill networks. The single baselines and skill-dependent baselines used a MLP with 2 hidden layers of 32 units to fit the value function. The skill-dependent baseline receives, in addition to the full observation, the active latent code and the time remaining until the next skill sampling. All runs used five random seeds. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ROBOT AGENT DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            368,
+            450,
+            385
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hopper is a 3-link robot with a 14-dimensional observation space and a 3-dimensional action space. Half-Cheetah has a 20-dimensional observation space and a 6-dimensional action space. We evaluate both of these agents on a sparse block hopping task. In addition to observing their own joint angles and positions, they observe the height and length of the next wall, the $\\mathbf { X }$ -position of the next wall, and the distance to the wall from the agent. We also provide the same wall observations for the previous wall, which the agent can still interact with. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Snake is a 5-link robot with a 17-dimensional observation space and a 4-dimensional action space. Ant is a quadrupedal robot with a 27-dimensional observation space and a 8-dimensional action space. Both Ant and Snake can move and rotate in all directions, and Ant faces the added challenge of avoiding falling over irrecoverably. In the Gather environment, agents also receive 2 sets of 10-dimensional lidar observations, whcih correspond to separate apple and bomb observations. The observation displays the distance to the nearest apple or bomb in each $3 6 ^ { \\circ }$ bin, respectively. All environments are simulated with the physics engine MuJoCo (Todorov et al., 2012). ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C PROOFS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. If the skills are sufficiently differentiated, then the latent variable can be treated as part of the observation to compute the gradient of the trajectory probability. Concretely, if $\\pi _ { \\boldsymbol { \\theta } _ { h } } ( z | s )$ and $\\pi _ { \\boldsymbol { \\theta } _ { l } } ( a | s , z )$ are Lipschitz in their parameters, and $0 \\overset { \\cdot } { < } \\pi _ { \\theta _ { l } } ( \\bar { a } _ { t } \\vert s _ { t } , z _ { j } ) < \\epsilon \\forall j \\neq k p$ , then ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8236ffd329f35ab1a4e2bbf2be468b118543ccd8771a64769daab5478937d004.jpg",
+        "text": "$$\n\\nabla _ { \\theta } \\log P ( \\tau ) = \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { h } } ( z _ { k p } | s _ { k p } ) + \\sum _ { t = 1 } ^ { p } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) + \\mathcal { O } ( n H \\epsilon ^ { p - 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. From the point of view of the MDP, a trajectory is a sequence $\\begin{array} { r l } { \\tau } & { { } = } \\end{array}$ $( s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , \\ldots , a _ { H - 1 } , s _ { H } )$ . Let’s assume we use the hierarchical policy introduced above, with a higher-level policy modeled as a parameterized discrete distribution with $n$ possible outcomes $\\pi _ { \\boldsymbol { \\theta } _ { h } } ( \\bar { z } | s ) = C \\bar { a } t e g o r i c a l _ { \\boldsymbol { \\theta } _ { h } } ( n )$ . We can expand $P ( \\tau )$ into the product of policy and environment dynamics terms, with $z _ { j }$ denoting the $j$ th possible value out of the $n$ choices, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4839a99401af73d6ff79c4b0652b05affe740477308089090578cf66fc05d14c.jpg",
+        "text": "$$\nP ( \\tau ) = \\Bigg ( \\prod _ { k = 0 } ^ { H / p } \\Big [ \\sum _ { j = 1 } ^ { n } \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \\Big ] \\Bigg ) \\Bigg [ P ( s _ { 0 } ) \\prod _ { t = 1 } ^ { H } P ( s _ { t + 1 } | s _ { t } , a _ { t } ) \\Bigg ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Taking the gradient of $\\log { P ( \\tau ) }$ with respect to the policy parameters $\\theta = [ \\theta _ { h } , \\theta _ { l } ]$ , the dynamics terms disappear, leaving: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f276b6f27c4dc828b0831b7c849b0d43aa95a8b015db1c03dbfa182471890c1.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle 7 _ { \\theta } \\log P ( \\tau ) = \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log \\Big ( \\sum _ { j = 1 } ^ { n } \\pi _ { \\theta _ { l } } ( z _ { j } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { s , \\theta } ( a _ { t } | s _ { t } , z _ { j } ) \\Big ) } } \\\\ { { \\displaystyle \\qquad = \\sum _ { k = 0 } ^ { H / p } \\frac { 1 } { \\sum _ { j = 1 } ^ { n } \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) } \\sum _ { j = 1 } ^ { n } \\nabla _ { \\theta } \\Big ( \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \\Big ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The sum over possible values of $z$ prevents the logarithm from splitting the product over the $p$ -step sub-trajectories. This term is problematic, as this product quickly approaches 0 as $p$ increases, and suffers from considerable numerical instabilities. Instead, we want to approximate this sum of products by a single one of the terms, which can then be decomposed into a sum of logs. For this we study each of the terms in the sum: the gradient of a sub-trajectory probability under a specific latent $\\begin{array} { r } { \\nabla _ { \\theta } \\Big ( \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , \\bar { z } _ { j } ) \\Big ) } \\end{array}$ . Now we can use the assumption that the skills are easy to distinguish, $0 < \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) < \\epsilon \\forall j \\neq k p$ . Therefore, the probability of the sub-trajectory under a latent different than the one that was originally sampled $z _ { j } \\neq z _ { k p }$ , is upper bounded by $\\epsilon ^ { p }$ . Taking the gradient, applying the product rule, and the Lipschitz continuity of the policies, we obtain that for all $z _ { j } \\neq z _ { k p }$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b9a34f2fdf068bcd97ef37491aa402bf723c3fe42170b8de2f627b0edde54f95.jpg",
+        "text": "$$\n\\begin{array} { l l } { { \\displaystyle 7 _ { \\theta } \\Big ( \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) } } & { { \\displaystyle \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \\Big ) = \\nabla _ { \\theta } \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) } } & { { \\displaystyle \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) } + } \\\\ { { } } & { { \\displaystyle \\sum _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { h } } ( z _ { j } | s _ { k p } ) \\big ( \\nabla _ { \\theta } \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { j } ) \\big ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { l } } ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } ) \\Big \\} } } \\\\ { { } } & { { \\displaystyle = \\mathcal { O } ( p \\epsilon ^ { p - 1 } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Thus, we can across the board replace the summation over latents by the single term corresponding to the latent that was sampled at that time. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef9c9b65e495366dbe304e726544603ed27c5738be621feb6ee872f115d9e45d.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\zeta _ { \\theta } \\log P ( \\tau ) = \\sum _ { k = 0 } ^ { H / p } \\frac { 1 } { \\pi _ { \\theta _ { h } } ( z _ { k p } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) } \\nabla _ { \\theta } \\Big ( P ( z _ { k p } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) \\Big ) } } \\\\ { { \\displaystyle \\qquad = \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log \\Big ( \\pi _ { \\theta _ { h } } ( z _ { k p } | s _ { k p } ) \\prod _ { t = k p } ^ { ( k + 1 ) p - 1 } \\pi _ { \\theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) \\Big ) + \\mathcal { O } ( n H \\epsilon ^ { p - 1 } ) } } \\\\ { { \\displaystyle \\qquad = \\mathbb { E } _ { \\tau } \\bigg [ \\Big ( \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { h } } \\big ( z _ { k p } | s _ { k p } ) + \\sum _ { t = 1 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { t } } ( a _ { t } | s _ { t } , z _ { k p } ) \\Big ) \\bigg ] + \\mathcal { O } ( n H \\epsilon ^ { p - 1 } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, this is exactly $\\nabla _ { \\theta } P ( s _ { 0 } , z _ { 0 } , a _ { 0 } , s _ { 1 } , \\dots )$ . In other words, it’s the gradient of the probability of that trajectory, where the trajectory now includes the variables $z$ as if they were observed. ",
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. For any functions $b _ { h } : S  \\mathbb { R }$ and $b _ { l } : \\mathcal { S } \\times \\mathcal { Z } \\to \\mathbb { R }$ we have: ",
+        "bbox": [
+            173,
+            810,
+            651,
+            827
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d658fe55923c196dddf6427886448d1366293efde6af85508c2f627767c0722a.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\tau } [ \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            833,
+            625,
+            878
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/93686aac62a5346e912db3a7431a869e39e9fca59f62cf53945e81803eb84c26.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\tau } [ \\sum _ { t = 0 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { \\theta _ { l } } ( a _ { t } | s _ { t } , z _ { k p } ) b ( s _ { t } , z _ { k p } ) ] = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            885,
+            647,
+            928
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. We can use the tower property as well as the fact that the interior expression only depends on $s _ { k p }$ and $z _ { k p }$ : ",
+        "bbox": [
+            169,
+            103,
+            825,
+            133
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/612e4018f601b97f6ed88be6fd0b1187923e9c922a75bbba9c27054c209db0a2.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathbb { E } _ { \\tau } [ \\displaystyle \\sum _ { k = 0 } ^ { H / p } \\nabla _ { \\theta } \\log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] = \\displaystyle \\sum _ { k = 0 } ^ { H / p } \\mathbb { E } _ { s _ { k p } , z _ { k p } } [ \\mathbb { E } _ { \\tau \\setminus s _ { k p } , z _ { k p } } [ \\nabla _ { \\theta } \\log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] } & { } \\\\ { \\displaystyle } & { ~ = \\displaystyle \\sum _ { k = 0 } ^ { H / p } \\mathbb { E } _ { s _ { k p } , z _ { k p } } [ \\nabla _ { \\theta } \\log P ( z _ { k p } | s _ { k p } ) b ( s _ { k p } ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            140,
+            787,
+            233
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then, we can write out the definition of the expectation and undo the gradient-log trick to prove that the baseline is unbiased. ",
+        "bbox": [
+            173,
+            256,
+            825,
+            285
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1947b0d6b536b0fc78749386a818df6ba5d7a6e7fc59e264d57fb28517339a61.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\nabla _ { \\theta } \\log \\mathfrak { A } _ { \\theta _ { 1 } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } | \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } ) \\hat { \\mathfrak { H } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } ) \\rVert ( \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } ) } & { = \\displaystyle \\operatorname* { l i m } _ { \\le t \\le j } \\int _ { ( s _ { \\phi _ { 1 } , \\phi _ { 2 } } , \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } ) \\in \\mathcal { L } _ { \\mathcal { F } _ { 1 } } } \\nabla _ { ( s _ { \\phi _ { 2 } } , \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } ) \\in \\mathcal { L } _ { \\mathcal { F } _ { 1 } } } \\left. \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } ( \\mathcal { L } _ { \\mathcal { F } _ { 2 } } | \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } ) \\hat { \\mathfrak { H } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } ) \\hat { \\mathfrak { H } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } ) \\hat { \\mathfrak { H } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } ) \\hat { \\mathfrak { H } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } ) \\right. } \\\\ &  = \\displaystyle \\sum _ { s = 0 } ^ { \\mathcal { N } _ { 0 } } \\int _ { s _ { \\phi _ { 2 } } \\in \\mathcal { L } _ { \\mathcal { F } _ { 2 } } } P ( s _ { \\phi _ { 3 } , \\phi _ { 3 } } ) \\hat { \\mathfrak { H } } ( s _ { \\phi _ { 3 } } ) \\int _ { s _ { \\phi _ { 2 } } \\le s _ { \\phi _ { 1 } } } \\left( \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } | \\mathfrak { L } _ { \\mathcal { F } _ { 3 } } \\right) \\nabla _ { \\phi } \\log \\mathfrak { L } _ { \\mathcal { F } _ { 3 } } ( \\mathcal { L } _ { \\mathcal { F } _ { 1 } } | \\mathfrak { L } _ { \\mathcal { F } _ { 3 } } ) \\hat { \\mathfrak { H } } ( \\mathfrak { L } _ { \\mathcal { F } _ { 2 } } | \\hat { \\mathfrak { L } } _ { \\mathcal { F } _ { 3 } } ) \\hat { \\mathfrak { H } } ( \\mathcal { L } _ { \\mathcal { F } _ { 1 } } | \\mathfrak { L } _ { \\mathcal { F } _ { 1 } } ) \\prod _ { s = 0 }  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            290,
+            883,
+            537
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Subtracting a state- and subpolicy- dependent baseline from the second term is also unbiased, i.e. ",
+        "bbox": [
+            171,
+            574,
+            807,
+            590
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0319122e3f168ce1f984ac3a17226570906a297f34d4709ad5b0fee7782177d5.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\tau } [ \\sum _ { t = 0 } ^ { H } \\nabla _ { \\theta } \\log \\pi _ { s , \\theta } ( a _ { t } | s _ { t } , z _ { k p } ) b ( s _ { t } , z _ { k p } ) ] = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            595,
+            648,
+            640
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We’ll follow the same strategy to prove the second equality: apply the tower property, express the expectation as an integral, and undo the gradient-log trick. ",
+        "bbox": [
+            171,
+            652,
+            825,
+            683
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0bcd0c01c8a3bd5291666c428ad09de4e7cc78a3e475cd0f57bc559647919e8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { x } | \\displaystyle \\sum _ { t = 1 } ^ { M } \\nabla \\cdot \\nabla ^ { \\theta } \\log \\pi _ { \\theta } \\{ \\alpha _ { t } | s _ { t } , z _ { t , p } \\} b ( s _ { t } , z _ { t , p } ) \\Big | } \\\\ &  \\begin{array} { r l } &  \\displaystyle = \\sum _ { t = 0 } ^ { H } \\sum _ { \\alpha _ { t } , \\alpha _ { t } , z _ { t , p } \\} \\mathbb { E } _ { \\Gamma ( s ) \\cup \\mathcal { L } _ { q } \\cup \\mathcal { L } _ { q } } [ \\nabla _ { s } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) ] \\Big | } \\\\ &  \\displaystyle = \\sum _ { t = 0 } ^ { H } \\sum _ { \\alpha _ { t } , \\alpha _ { t } , z _ { t , p } \\} \\big [ \\nabla _ { s } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \\big ] \\big ( \\nabla _ { s } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \\big ) \\big | } \\\\ &  \\displaystyle = \\sum _ { t = 0 } ^ { H } \\mathbb { E } _ { \\kappa _ { t } , \\alpha _ { t } , z _ { t , p } \\} \\big [ \\nabla _ { s } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \\big ] \\big ( \\nabla _ { s } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) \\big ) } \\\\ & { \\displaystyle = \\sum _ { t = 0 } ^ { H } \\int _ { ( a _ { t } , z _ { t , p } ) } P ( s _ { t } , z _ { t , p } ) b ( s _ { t } , z _ { t , p } ) \\int _ { a _ { t } } \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( a _ { t } | s _ { t } , z _ { t , p } ) b ( a _ { t } | s _ { t } , z _ { t , p } ) d a _ { t } d a _ { t } d a _ { t } d a _ { t } } \\\\ &  \\displaystyle = \\sum _ { t = 0 } ^ { H } \\int _ { ( a _ { t } , z _ { t , p } ) } P ( s _ { t } , z _ { t , p } \\end{array} \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            688,
+            813,
+            928
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ee0ef8b7d618f7c26e204fd303b0f1de7942fc7e9d8b5b2ff23a13be001b23f1.jpg",
+        "image_caption": [
+            "Figure 7: HIRO performance on Ant Gather with and without access to the ground truth $( x , y )$ , which it needs to communicate useful goals. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            385,
+            99,
+            611,
+            229
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D HIRO SENSITIVITY TO OBSERVATION-SPACE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            295,
+            578,
+            310
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In this section we provide a more detailed explanation of why HIRO (Nachum et al., 2018) performs poorly under our environments. As explained in our related work section, HIRO belongs to the general category of algorithms that train goal-reaching policies as lower levels of the hierarchy (Vezhnevets et al., 2017; Levy et al., 2017). These methods rely on having a goal-space that is meaningful for the task at hand. For example, in navigation tasks they require having access to the $( x , y )$ position of the agent such that deltas in that space can be given as meaningful goals to move in the environment. Unfortunately, in many cases the only readily available information (if there’s no GPS signal or other positioning system installed) are raw sensory inputs, like cameras or the LIDAR sensors we mimic in our environments. In such cases, our method still performs well because it doesn’t rely on the goal-reaching extra supervision that is leveraged (and detrimental in this case) in HIRO and similar methods. In Figure 7, we show that knowing the ground truth location is critical for its success. We have reproduced the HIRO results in Fig. 7 using the published codebase, so we are convinced that our results showcase a failure mode of HIRO. ",
+        "bbox": [
+            173,
+            325,
+            826,
+            506
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E HYPERPARAMETER SENSITIVITY PLOTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            527,
+            542,
+            542
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/bf71f0947b79c1dacc6adb6a4abac4033bdc6c446907c6564290d12f3e9db82b.jpg",
+        "image_caption": [
+            "Figure 8: Sensitivity of HiPPO to variation in the time-commitment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            266,
+            559,
+            723,
+            693
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/6e231db06c188eafb086f33848ba035a6839f3368bff333cca0e31a26f509a6d.jpg",
+        "image_caption": [
+            "Figure 9: Sensitivity of HiPPO to variation in the number of skills. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            737,
+            720,
+            869
+        ],
+        "page_idx": 14
+    }
+]

parse/train/J_pvI6ap5Mn/J_pvI6ap5Mn.md

@@ -0,0 +1,488 @@
+# TRANSFER LEARNING OF GRAPH NEURAL NETWORKS WITH EGO-GRAPH INFORMATION MAXIMIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Graph neural networks (GNNs) have been shown with superior performance in various applications, but training dedicated GNNs can be costly for large-scale graphs. Some recent work started to study the pre-training of GNNs. However, none of them provide theoretical insights into the design of their frameworks, or clear requirements and guarantees towards the transferability of GNNs. In this work, we establish a theoretically grounded and practically useful framework for the transfer learning of GNNs. Firstly, we propose a novel view towards the essential graph information and advocate the capturing of it as the goal of transferable GNN training, which motivates the design of EGI (ego-graph information maximization) to analytically achieve this goal. Secondly, we specify the requirement of structurerespecting node features as the GNN input, and conduct a rigorous analysis of GNN transferability based on the difference between the local graph Laplacians of the source and target graphs. Finally, we conduct controlled synthetic experiments to directly justify our theoretical conclusions. Extensive experiments on realworld networks towards role identification show consistent results in the rigorously analyzed setting of direct-transfering (freezing parameters), while those towards large-scale relation prediction show promising results in the more generalized and practical setting of transfering with fine-tuning.
+
+# 1 INTRODUCTION
+
+Graph neural networks (GNNs) have been intensively studied recently (Kipf & Welling, 2017; Keriven & Peyré, 2019; Chen et al., 2019; Oono & Suzuki, 2020; Huang et al., 2018), due to their established performance towards various real-world tasks (Hamilton et al., 2017; Ying et al., 2018b; Velickovic et al., 2018), as well as close connections to spectral graph theory (Defferrard et al., 2016; Bruna et al., 2014; Hammond et al., 2011). While most GNN architectures are not very complicated, the training of GNNs can still be costly regarding both memory and computation resources on real-world large-scale graphs (Chen et al., 2018; Ying et al., 2018a). Moreover, it is intriguing to transfer learned structural information across different graphs and even domains in settings like few-shot learning (Vinyals et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017). Therefore, several very recent studies have been conducted on the transferability of GNNs, which focus on the setting of pre-training plus fine-tuning (Hu et al., 2019a,b, 2020; Wu et al., 2020). However, it is unclear in what situations the models will excel or fail especially when the pre-training and fine-tuning tasks are different. To provide rigorous analysis and guarantee on the transferability of GNNs, we focus on the setting of direct-transfering between the source and target graphs, under an analogous setting of “domain adaptation” (Ben-David et al., 2007).
+
+In this work, we establish a theoretically grounded framework for the transfer learning of GNNs, and leverage it to design a practically transferable GNN model. Figure 1 gives an overview of our framework. It is based on a novel view of a graph as samples from the joint distribution of its $\mathbf { k }$ -hop ego-graph structures and node features, which allows us to define graph information and similarity, so as to analyze GNN transferability (§2). This view motivates us to design EGI, a novel GNN model based on ego-graph information maximization, which is effective in capturing the graph information as we define $( \ S 2 . 1 )$ . Then we further specify the requirement on transferable node features and analyze the transferability of EGI that is dependent on the local graph Laplacians of source and target graphs (§2.2).
+
+![](images/945736827b0c6721f1d4f23fa94eac30328beb3559853d5752b261a4480d6aba.jpg)  
+Figure 1: Overview of our GNN transfer learning framework: (1) we represent graph as a combination of its 1-hop ego-graph and node feature distributions; (2) we design a transferable GNN regarding the capturing of such essential graph information; (3) we establish a rigorous guarantee of GNN transferability based on the requirement on nodes features and difference between graph structures.
+
+All of our theoretical conclusions have been directly validated through controlled synthetic experiments (Table 1), where we use structural-equivalent role identification in a direct-transfering setting to analyze the impacts of different model designs, node features and source-target structure similarities on GNN transferability. In $\ S 3$ , we conduct real-world experiments on multiple publicly available network datasets. On the Airport and Gene graphs (§3.1), we closely follow the settings of our synthetic experiments and observe consistent but more detailed results supporting the design of EGI and the utility of our theoretical analysis. On the YAGO graphs (§3.2), we further evaluate EGI on the more generalized and practical setting of transfer learning with task-specific fine-tuning. We find our theoretical insights still indicative in such scenarios, where EGI consistently outperforms state-of-the-art GNN models and transfer learning frameworks with significant margins.
+
+# 2 TRANSFERABLE GRAPH NEURAL NETWORKS
+
+Based on the connection between GNN and spectral graph theory (Kipf & Welling, 2017), we describe the output of a GNN as a combination of its input node features, fixed graph Laplacian and learnable graph filters. The goal of training a GNN is then to improve its utility by learning the graph filters that are compatible with the other two components towards specific tasks.
+
+In the graph transfer learning setting where downstream tasks are often unknown during pre-training, we argue that the general utility of a GNN should be optimized and quantified w.r.t. its ability of capturing the essential graph information in terms of the joint distribution of its link structures and node features, which motivates us to design a novel ego-graph information maximization model (EGI) (§2.1). The general transferability of a GNN is then quantified by the gap between its abilities to model the source and target graphs. Under reasonable requirements such as using structure-respecting node features as the GNN input, we analyze this gap for EGI based on the structural difference between two graphs $w . r . t .$ . their local graph Laplacians $( \ S 2 . 2 )$ .
+
+# 2.1 TRANSFERABLE GNN VIA EGO-GRAPH INFORMATION MAXIMIZATION
+
+In this work, we focus on the direct-transfering setting where a GNN is pre-trained on a source graph $G _ { a }$ in an unsupervised fashion and applied on a target graph $G _ { b }$ without fine-tuning.1 Consider a graph $G = \{ \bar { V , } E \}$ , where the set of nodes $V$ are associated with certain features and the set of links $E$ form certain structures. Intuitively, the transfer learning will be successful only if both the features and structures of $G _ { a }$ and $G _ { b }$ are similar in some ways, so that the graph filters of a GNN learned on $G _ { a }$ are compatible with the features and structures of $G _ { b }$ .
+
+Motivated by the concept of $\mathbf { k }$ -layer expansion sub-graph in (Bai & Hancock, 2016), we introduce a novel view of a graph as samples from the joint distribution of its $k$ -hop ego-graph structures and node features. This view allows us to give concrete definitions towards structural information of graphs in the transfer learning setting, which facilitates the measuring of similarity (difference) among graphs.
+
+Definition 2.1 (K-hop ego-graph). We call $a$ graph $g _ { i } = \{ V ( g _ { i } ) , E ( g _ { i } ) \}$ a $k$ -hop ego-graph centered at $v _ { i }$ if it has a $k$ -layer centroid expansion (Bai & Hancock, 2016) such that the greatest shortest path rooted from $v _ { i }$ has length $k ,$ , i.e., $k = \mathrm { m a x } _ { v _ { j } \in V } | S ( v _ { i } , v _ { j } ) |$ , where $S ( v _ { i } , v _ { j } )$ is the shortest path between $v _ { i }$ and $v _ { j }$ .
+
+For an ordered $\mathrm { k }$ -hop ego-graph, we denote $v _ { p , q }$ as the $q$ -th node in the $p$ -th layer of the ego-graph (i.e., $| S _ { i } ( v _ { i } , v _ { p , q } ) | = \bar { p } )$ , where $p = 0 , \ldots , k$ , and $e _ { v v ^ { \prime } }$ as the edge between $v _ { p , q }$ and $v _ { p + 1 , q ^ { \prime } }$ .
+
+Definition 2.2 (Structural information). Let $\mathcal { G }$ be a topological space of sub-graphs (Verma & Zhang, 2019). We view a graph $G$ as samples of $k$ -hop ego-graphs $G = \{ g _ { i } \} _ { i = 1 } ^ { n }$ drawn i.i.d. from $\mathcal { G }$ with probability $\mu ,$ , i.e., $g _ { i } \stackrel { \mathrm { i . i . d . } } { \sim } \mu \forall i = 1 , \cdots , n .$ . The structural information of $G$ is then defined to be the combination of the distribution $\mu$ and the set of spectrum of $\{ g _ { i } \} _ { i = 1 } ^ { n }$ .
+
+The structural information of a graph $G$ can be characterized by $\{ g _ { i } \} _ { v _ { i } \in V }$ and its empirical distribution, where each $g _ { i }$ is a $\mathbf { k }$ -hop ego-graph of $G$ centered at node $v _ { i }$ with $V ( g _ { i } ) = \{ u \in V ( G ) :$ $S ( u , v _ { i } ) \leq k \}$ , and edges $E ( g _ { i } ) \stackrel { - } { = } \{ \bar { e } _ { u v } \stackrel { - } { \in } E ( G ) : u , v \in V ( g _ { i } ) \}$ . As shown in Figure 1, three graphs $G _ { 0 }$ , $G _ { 1 }$ and $G _ { 2 }$ are characterized by a set of 1-hop ego-graphs and their empirical distributions, which allows us to quantify the structural similarity among graphs as shown in $\ S 2 . 2$ (i.e., $G _ { 0 }$ is more similar to $G _ { 1 }$ than $G _ { 2 }$ under such characterization).
+
+In practice, the nodes in a graph $G$ are characterized not only by their $\mathbf { k }$ -hop ego-graph structures but also their associated node features. Therefore, $G$ should be regarded as samples $\{ ( g _ { i } , x _ { i } ) \} ^ { n } \in \mathcal { G } \times \mathcal { X }$ , drawn with the joint distribution $p$ on the product space of $\mathcal { G }$ and a node feature space $\mathcal { X }$ . To capture such joint distributions of structural information and node features, we design ego-graph information maximization (EGI), which recursively reconstructs the $\mathbf { k }$ -hop ego-graph of each node based on their features in an unsupervised fashion.
+
+Ego-Graph Information Maximization. Assume we are given a set of ego-graphs $\{ ( g _ { i } , x _ { i } ) \} _ { i }$ with empirical joint distribution $\mathbb { P }$ . Similarly with the “local” version of DIM (Hjelm et al., 2019), we define $\mathbb { U } _ { \Psi ( g _ { i } , x _ { i } ) }$ as the empirical distribution of the embedding produced by the GNN encoder $\Psi$ for the the center node $v _ { i }$ of ego-graph $g _ { i }$ . Unlike DGI (Velickovic et al., 2019) that models the local-global mutual information (MI), EGI optimizes $\Psi$ to maximize the $\mathbf { M I }$ of ${ \mathcal { T } } ( g _ { i } , \Psi ( g _ { i } , x _ { i } ) )$ , which is directly between the structural input and output of GNN, with a focus on the structural information $g _ { i }$ . Specifically, we use the Jensen-Shannon MI estimator in (Hjelm et al., 2019),
+
+$\mathcal { L } _ { \mathrm { E G I } } = - \mathcal { T } ^ { \mathrm { ( J S D ) } } \left( \boldsymbol { \mathcal { G } } , \boldsymbol { \Psi } \right) = \mathbb { E } _ { \mathbb { P } \times \tilde { \mathbb { U } } } \left[ \mathrm { s p } \left( T _ { \mathcal { D } , \Psi } \big ( g _ { i } , \Psi \big ( g _ { i } ^ { \prime } , x _ { i } ^ { \prime } \big ) \big ) \right) \right] - \mathbb { E } _ { \mathbb { P } } \left[ - \mathrm { s p } \left( - T _ { \mathcal { D } , \Psi } \big ( g _ { i } , \Psi \big ( g _ { i } , x _ { i } \big ) \big ) \right) \right] ,$ (1) $T _ { \mathcal { D } , \Psi } = \mathcal { D } \circ ( g _ { i } , \Psi ( g _ { i } , x _ { i } ) )$ , where $\mathcal { D }$ is a discriminator ${ \mathcal { D } } : g _ { i } \times \Psi ( g _ { i } , x _ { i } ) \to \mathbb { R } ^ { + }$ . In Eq. 1, during the training of $\mathcal { D }$ , the input space of $\mathcal { D }$ is at least as large as the number of graph permutations $| V ( g _ { i } ) | !$ . Instead of enumerating all possible graphs $g _ { i } ^ { \prime }$ , we fix $g _ { i }$ and sample GNN’s output $\Psi ( g _ { i } ^ { \prime } , x _ { i } ^ { \prime } )$ from the marginal distribution $\tilde { \mathbb { U } }$ by uniformly sampling $( g _ { i } ^ { \prime } , x _ { i } ^ { \prime } ) \sim \tilde { \mathbb { P } } , \tilde { \mathbb { P } } = \mathbb { P }$ . The correspondence between sampling $( g _ { i } ^ { \prime } , x _ { i } ^ { \prime } ) \sim \tilde { \mathbb { P } }$ and $g _ { i } ^ { \prime } \sim \mathcal { G }$ is discussed in Remark 2 when node features are strcuture-respecting (Def. 2.3).
+
+Formally, we characterize the decision process of $\mathcal { D }$ with a fixed graph ordering, i.e., BFS-ordering $\pi$ over edges $E ( g _ { i } )$ . $\mathcal { D }$ is a GNN scoring function over an edge sequence $E ^ { \pi } : \left\{ e _ { 1 } , e _ { 2 } , . . . , e _ { n } \right\}$ , which makes predictions on BFS-ordered edges. Let $z _ { i } = \Psi ( g _ { i } , x _ { i } )$ , then we have,
+
+$$
+\mathcal { D } ( g _ { i } , z _ { i } ) = \sum _ { p = 0 } ^ { k } \sum _ { q = 1 } ^ { | V _ { p } ( g _ { i } ) | } \log \mathcal { D } ( e _ { \tilde { v } v } | h _ { p , q } ^ { \tilde { q } } , x _ { p , q } ^ { i } , z _ { i } ) ,
+$$
+
+where $h$ is the hidden representation output by $\mathcal { D }$ , $e _ { \tilde { v } v } \in E ( g _ { i } )$ is an edge between node $\tilde { v }$ in layer $p$ and $v$ in layer $p + 1$ , following the notation defined below Def 2.1. More specifically, we have
+
+$$
+\begin{array} { r } { \mathcal { D } ( e _ { \tilde { v } v } | h _ { p , q } ^ { \tilde { q } } , x _ { p , q } ^ { i } , z _ { i } ) = \sigma \left( U ^ { T } \cdot \tau \left( W ^ { T } [ h _ { p , q } ^ { \tilde { q } } | | x _ { p , q } ^ { i } | | z _ { i } ] \right) \right) , } \end{array}
+$$
+
+where $\sigma$ and $\tau$ are Sigmoid and ReLU activation functions, respectively. Thus, the discriminator is asked to distinguish positive $( e _ { \tilde { v } v } , \Psi ( g _ { i } , x _ { i } ) )$ and negative pair $( e _ { \tilde { v } v } , \Psi ( g _ { i } ^ { \prime } , x _ { i } ^ { \prime } ) )$ that consists of an observed edge and positive/negative center node embeddings $\Psi ( \cdot )$ .
+
+Due to the fact that the output of a $\mathbf { k }$ -layer GNN only depends on a $\mathbf { k }$ -hop ego-graphs, EGI can be trained in parallel by sampling batches of $g _ { i }$ ’s. Besides, the training objective of EGI is transferable as long as $( g _ { i } , x _ { i } )$ across source graph $G _ { a }$ and $G _ { b }$ satisfies the conditions given in $\ S 2 . 2$ . More details about the model are in Appendix $\ S _ { \mathbf { B } }$ and source code in the Supplementary Materials.
+
+Connection with existing work. To provide more insights into the EGI objective, we also present it as a dual problem of ego-graph reconstruction. Recall our definition of ego-graph mutual information ${ \mathcal { T } } ( g _ { i } , \Psi ( g _ { i } , x _ { i } ) )$ . It can be related to an ego-graph reconstruction loss $R ( g _ { i } | \Psi ( g _ { i } , x _ { i } ) )$ as
+
+When EGI is maximizing the mutual information, it simultaneously minimizes the upper error bound of reconstructing an ego-graph $g _ { i }$ . In this view, the key difference between EGI and GVAE (Kipf & Welling, 2016) is they assume each edge in a graph to be observed independently during the reconstruction, while we assume the edges in an ego-graph to be observed jointly. Moreover, existing mutual information based GNNs such as DGI (Velickovic et al., 2019) and GMI (Peng et al., 2020) explicitly measure the mutual information between node features $x$ and GNN output $\Psi$ . In this way, they tend to capture node features instead of graph structures, which we deem more essential in graph transfer learning as discussed in $\ S 2 . 2$ .
+
+Supportive observations. In the first three columns of Table 1, in both cases of transfering GNNs between similar graphs (F-F) and dissimilar graphs (B-F), EGI significantly outperforms all competitors when using node degree one-hot encoding as transferable node features. In particular, the performance gains over the untrained GIN and GCN show the effectiveness of training and transfering, and our gains are always larger than the two state-of-the-art unsupervised GNNs. Such results clearly indicate advantageous structure preserving capability and transferability of EGI.
+
+# 2.2 TRANSFERABILITY ANALYSI BASED ON LOCAL GRAPH LAPLACIANS
+
+We now study the transferability of a GNN (in particular, EGI) between the source graph $G _ { a }$ and target graph $G _ { b }$ based on the graph similarity between $G _ { a }$ and $G _ { b }$ . We firstly establish the requirement towards node features, under which we then focus on analyzing the transferability of EGI w.r.t. the structural information of $G _ { a }$ and $G _ { b }$ .
+
+Recall our view of the GNN output as a combination of its input node features, fixed graph Laplacian and learnable graph filters. The utility of a GNN is determined by the compatibility among the three. In order to fulfill such compatibility, we require the node features to be structure-respecting:
+
+node Then with a set of node featuree say the node features on , where respect $g _ { i }$ -th hop of or any no $v _ { i }$ $\bar { \{ x _ { p , q } ^ { i } \} } _ { p = 0 , q = 1 } ^ { k , | V _ { p } ( g _ { i } ) | }$ $V _ { p } ( g _ { i } )$ $p$ $g _ { i }$ $g _ { i }$ $\dot { x } _ { p , q } ^ { i } = [ f ( g _ { i } ) ] _ { p , q } \in \mathbb { R } ^ { d }$ $v _ { q } \in V _ { p } ( g _ { i } )$ , where $f : \mathcal { G }  \mathbb { R } ^ { d \times \lvert V ( g _ { i } ) \rvert }$ is a function. In the strict case, $f$ should be injective.
+
+In its essence, Def 2.3 requires the node features to be a function of the graph structures, which is sensitive to changes in the graph structures, and in an ideal case, injective to the graph structures. In this way, when the learned graph filters of a transfered GNN is compatible to the structure of $G$ , they are also compatible to the node features of $G$ . As we will explain in Remark 2 of Theorem 2.1, this requirement is also essential for the analysis of our GNN transferability which eventually only depends on the structural difference between two graphs.
+
+In practice, commonly used node features like node degrees, PageRank scores (Page et al., 1999), spectral embeddings (Chung & Graham, 1997), and many pre-computed unsupervised network embeddings (Perozzi et al., 2014; Tang et al., 2015; Grover & Leskovec, 2016) are all structurerespecting in nature. However, other commonly used node features like random vectors (Yang et al., 2019) or uniform vectors (Xu et al., 2019) are not and thus non-transferable. When organic node attributes are available, they are transferable as long as the concept of homophily (McPherson et al., 2001) applies, which also implies Def 2.3, but we do not have a rigorous analysis on it yet.
+
+Supportive observations. In the fifth and sixth columns in Table 1, where we use uniform embedding as non-transferable node features to contrast with the first three columns, there is almost no or even negative transferability for all compared methods when non-transferable features are used, as the performance of trained GNNs are similar to or worse than their untrained baselines.
+
+With our view of graphs and requirement on node features both established, now we derive the following theorem by characterizing the performance difference of EGI on two graphs based on Eq. 1.
+
+Theorem 2.1 (GNN transferability). Let ${ \cal G } _ { a } = \{ ( g _ { i } , x _ { i } ) \} _ { i = 1 } ^ { n }$ and $G _ { b } = \{ ( g _ { i ^ { \prime } } , x _ { i ^ { \prime } } ) \} _ { i ^ { \prime } = 1 } ^ { m }$ be two graphs. Then denote $L _ { g _ { i } }$ as the (normalised) graph Laplacian of $g _ { i } \forall i = 1 , \cdot \cdot \cdot , n$ , and let the node features of $g _ { i }$ be structure-respecting and normalized (similarly for $g _ { i ^ { \prime } }$ ). Consider GNN $\Psi _ { \theta }$ with $k$ layers and a $I$ -hop polynomial filter $\phi _ { \theta }$ . With reasonable assumptions on the local spectrum of $G _ { a }$ and $G _ { b }$ , the empirical performance difference of $\Psi _ { \theta }$ with $\phi _ { \theta }$ evaluated on $\mathcal { L } _ { \mathrm { E G I } }$ satisfies
+
+$$
+| \mathcal { L } _ { \mathrm { E G I } } ( G _ { a } ) - \mathcal { L } _ { \mathrm { E G I } } ( G _ { b } ) | \leq \mathcal { O } \left( M + \frac { 1 } { n m } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { m } \| \lambda ( L _ { g _ { i } } ) - \lambda ( L _ { g _ { i ^ { \prime } } } ) \| _ { 2 } \right) ,
+$$
+
+where $M$ is a constant dependant on $k$ $, \phi _ { \theta } , \{ L _ { g _ { i } } \} , \{ L _ { g _ { i ^ { \prime } } } \} , \{ x _ { i } \} , \{ x _ { i ^ { \prime } } \} ,$ , and finally $\lambda ( L _ { g _ { i } } )$ denotes the ordered eigenvalues of the graph Laplacian of $g _ { i } \in G _ { a }$ (similarly for $g _ { i ^ { \prime } }$ ).
+
+Proof. The full proof is detailed in Appendix $\ S \mathrm { A }$ .
+
+Remark 1. Our view of a graph $G$ as samples of $k$ -hop ego-graphs is important, as it allows us to make node-wise characterization of GNN similarly as in (Verma & Zhang, 2019). It also allows us to set the depth of ego-graphs in the analysis to be the same as the number of GNN layers $( k )$ , since the GNN embedding of each node mostly depends on its $k$ -hop ego-graph instead of the whole graph.
+
+Remark 2. For Eq. 1, Def 2.3 ensures the sampling of GNN embedding at a node always corresponds to sampling an ego-graph from $\mathcal { G }$ , which reduces to uniformly sampling from $G = \{ g _ { i } \} _ { i = 1 } ^ { n }$ under the setting of Theorem 2.1. Therefore, the requirement of Def 2.3 in the context of Theorem 2.1 guarantees the analysis to be only depending on the structural information of the graph.
+
+The analysis in Theorem 2.1 naturally instantiates our insight about the correspondence between structural similarity and GNN transferability. It tells us how well a GNN trained on $G _ { a }$ can work on $G _ { b }$ by only checking the local graph Laplacians of $G _ { a }$ and $G _ { b }$ without actually training the model.
+
+In practice, the computation of eigenvalues on the small ego-graphs can be rather efficient (Arora et al., 2005), and we do not need to enumerate all pairs of ego-graphs. Suppose we need to sample $M$ pairs of $\mathrm { k }$ -hop ego-graphs to compare two large graphs, and the average size of ego-graphs are $L$ , then the overall complexity of computing Eq. 5 is $\bar { \mathcal { O } } ( \bar { M } L ^ { 2 } )$ , where $M$ is often less than 1K and $L$ less than 50.
+
+Supportive observations. In Table 1, in the $\bar { d }$ columns, we compute the average structural difference between two Forest-fire graphs $( \bar { d } ( F , F ) )$ and between Barabasi and Forest-fire graphs $( \bar { d } ( B , F ) )$ , based on the RHS of Eq. 5. The results validate our usage of the two graph models to generate structurally different graphs, while also verify our novel view of graphs and the way we propose based on it to characterize structural information of graphs. We further highlight in the $\Delta$ columns the performance difference between the GNNs transfered from Forest-fire graphs and Barabasi graphs to Forest-fire graphs. Since Forest-fire graphs are more similar to Forest-fire graphs than Barabasi graphs (as verified in the $\bar { d }$ columns), we expect $\Delta$ to be positive and large, indicating more positive transfer between the more similar graphs. Indeed, the behaviors of EGI align well with the expectation, which indicates its well-understood transferability and the utility of our theoretical analysis.
+
+Table 1: Synthetic experiments of identifying structural equivalent nodes. We randomly generate 40 graphs with the Forest-fire model (F) (Leskovec et al., 2005) and 40 graphs with the Barabasi model (B) (Albert & Barabási, 2002), The GNN models we use include the untrained encoders of GCN (Kipf & Welling, 2017) and GIN (Xu et al., 2019) with random parameters (baselines with only the neighborhood aggregation function), GVAE with GCN encoder (Kipf & Welling, 2016), DGI with GIN encoder (Velickovic et al., 2019), and EGI with GIN encoder. We train GVAE, DGI and EGI on one graph from either set (F and B), and test them on the rest of Forest-fire graphs (F). More details about the results and dataset can be found in Appendix $\ S { \bf C } . 1$ .   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">transferable features</td><td colspan="3">non-transferable feature</td><td rowspan="2">structural difference d(F,F)</td><td rowspan="2">d(B.F)</td></tr><tr><td>F-F</td><td>B-F</td><td>△</td><td>F-F</td><td>B-F</td><td>△</td></tr><tr><td>GCN (untrained)</td><td>0.478</td><td>0.478</td><td>/</td><td>0.229</td><td>0.229</td><td>1</td><td rowspan="4"></td><td rowspan="4"></td></tr><tr><td>GIN (untrained)</td><td>0.572</td><td>0.572</td><td>/</td><td>0.358</td><td>0.358</td><td>/</td></tr><tr><td>GVAE (GCN)</td><td>0.498</td><td>0.432</td><td>+0.066</td><td>0.240</td><td>0.239</td><td>0.001</td></tr><tr><td>DGI (GIN)</td><td>0.578</td><td>0.591</td><td>-0.013</td><td>0.394</td><td>0.213</td><td>+0.181</td></tr><tr><td>EGI (GIN)</td><td>0.710</td><td>0.616</td><td>+0.094</td><td>0.376</td><td>0.346</td><td>+0.03</td><td rowspan="2"></td><td rowspan="2"></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 3 REAL DATA EXPERIMENTS
+
+Baselines. We compare the proposed model with existing unsupervised GNNs and pre-training GNN frameworks. The unsupervised GNNs are the same as used in our synthetic experiments, i.e., GVAE with GCN encoder (Kipf & Welling, 2016) and DGI with GIN encoder (Velickovic et al., 2019). The pre-training GNN frameworks include Mask-GIN and ContextPred-GIN, two node-level pre-training models proposed in (Hu et al., 2019a)2. Besides, Structural Pre-train (Hu et al., 2019b) also conducts unsupervised node-level pre-training with structural features like node degrees and clustering coefficients.
+
+Protocols. By default, we use node degree one-hot encoding as the transferable feature across all different graphs. As stated before, other transferable features like spectral and other pre-computed node embeddings are also applicable. We focus on the setting where the downstream tasks on target graphs are unspecified but assumed to be structure-relevant, and thus pre-train the GNNs on source graphs in an unsupervised fashion.3 In terms of evaluation, we design two realistic experimental settings: (1) Direct-transfering on the more structure-relevant task of role identification without given node features to directly evaluate the utility and transferability of EGI. (2) Few-shot learning on relation prediction with task-specific node features to evaluate the generalization ability of EGI.
+
+# 3.1 DIRECT-TRANSFERING ON ROLE IDENTIFICATION
+
+First, we use the role identification without node features in a direct-transfering setting as a reliable proxy to evaluate transfer learning performance regarding different pre-training objectives. Role in a network is defined as nodes with similar structural behaviors, such as clique members, hub and bridge (Henderson et al., 2012). Across graphs in the same domain, we assume the definition of role to be consistent, and the task of role identification is highly structure-relevant, which can directly reflect the transferability of different methods and allows us to conduct the analysis according to Theorem 2.1. Upon convergence of pre-training each model on the source graphs, we directly apply them on the target graphs and further train a multi-layer perceptron (MLP) upon their outputs. The GNN parameters are freezing during the MLP training. We refer to this strategy as direct-transfering since there is no fine-tuning of the models after transfering to the target graphs.
+
+We use two real-world network datasets with role-based node labels: (1) Airport (Ribeiro et al., 2017) contains three networks from different regions– Brazil, USA and Europe. Each node is an airport and each link is the flight between airports. The airports are assigned with external labels based on their level of popularity. (2) Gene (Yang et al., 2019) contains the gene interactions regarding 50 different cancers. Each gene has a binary label indicating whether it is a transcription factor.
+
+The experimental setup on the Airport dataset closely resembles that of our synthetic experiments in Table 1, but with real data and more detailed comparisons. We train all models (except for the untrained ones) on the Europe network, and test them on all three networks. The results are presented in Table 2. We notice that the node degree features themselves (with MLP) show reasonable performance in all three networks, which is not surprising since the popularity-based airport role labels are highly relevant to node degrees. The untrained GIN encoder yields a significant margin over both node degrees and the untrained vanilla GCN encoder, indicating the importance of proper aggregation mechanisms. While training of the GCN (through GVAE) and GIN (through DGI) can further improve the performance on the source graph, EGI shows the best performance there with the structure-respecting node degree features (59.15), corroborating the claimed effectiveness of EGI in capturing the essential graph information as we stress in $\ S 2$ .
+
+When transfering the models to USA and Brazil networks, EGI further achieves the best performance compared with all baselines when node degree features are used (64.55 and 73.15), which reflects the most significant positive transfer. Interestingly, direct application of GVAE and DGI without the consideration of essential graph information as we stress leads to rather limited and even negative transferrability (through comparison against the untrained GCN and GIN encoders). The recently proposed transfer learning frameworks for GNN like Mask-GIN and Structural Pre-train are able to mitigate negative transfer to some extent, but their performances are still inferior to EGI. We believe this is because their models do not aim to capture the underlying ego-graph distributions as we deem important, so they are prune to learn the graph-specific information that is less transferable across different graphs. Similarly as in Table 1, we also compute the structural difference among three networks $w . r . t .$ . to RHS of Eq. 5. The structural difference is 12.03 between the Europe and USA networks, and 12.14 between the Europe and Brazil datasets, which are pretty close. Consequently, the transferability of EGI regarding its performance gain over the untrained GIN baseline is $4 . 8 \%$ on the USA network and $4 . 4 \%$ on the Brazil network, which are also pretty close. Such observations once again align well with our conclusion in Theorem 2.1 that the transferability of EGI is closely related to the structural different between source and target graphs.
+
+![](images/1190e57a9ca7bf124e98a12cc3fb78d2e3b3878e0b9c2f02170f2472a94b6e90.jpg)  
+Figure 2: Role identification on the Gene dataset. Due to severe label imbalance that vanishes the performance gaps, we only use the 7 brain cancer networks that have a more consistent balance of labels. We visualize the source graph $G _ { 0 }$ and two example target graphs that are relatively more similar $\left( G _ { 5 } \right)$ and different $\left( G _ { 6 } \right)$ with $G _ { 0 }$
+
+Table 2: Results of role identification with direct-transfering on the Airport dataset. The performance reported $( \% )$ are the average over $1 0 0 \mathrm { r u n s }$ . The scores marked with ∗∗ passed t-test with $\mathsf { p } < 0 . 0 1$ over the second best results. More details about the results and dataset can be found in Appendix $\ S { \bf C } . 2$ .   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Europe (source)</td><td colspan="2">USA (target)</td><td colspan="2">Brazil (target)</td></tr><tr><td>node degree</td><td>uniform</td><td>node degree</td><td>uniform</td><td>node degree</td><td>uniform</td></tr><tr><td>MLP</td><td>52.81</td><td>20.59</td><td>55.67</td><td>20.22</td><td>67.11</td><td>19.63</td></tr><tr><td>GCN (untrained)</td><td>52.96</td><td>20.11</td><td>55.30</td><td>22.07</td><td>68.30</td><td>17.63</td></tr><tr><td>GIN (untrained)</td><td>55.75</td><td>53.88</td><td>61.56</td><td>58.32</td><td>70.04</td><td>70.37</td></tr><tr><td>GVAE(GCN) (Kipf &amp; Welling,2016)</td><td>53.90</td><td>21.12</td><td>55.51</td><td>22.39</td><td>66.33</td><td>17.70</td></tr><tr><td>DGI (GIN) (Velickovic et al.,2019)</td><td>57.75</td><td>22.13</td><td>54.90</td><td>21.76</td><td>67.93</td><td>18.78</td></tr><tr><td>Mask-GIN(Hu et al.,2019a)</td><td>56.37</td><td>55.53</td><td>60.82</td><td>54.64</td><td>66.71</td><td>74.54</td></tr><tr><td>ContextPred-GIN (Hu et al., 2019a)</td><td>52.69</td><td>49.95</td><td>50.38</td><td>54.75</td><td>62.11</td><td>70.66</td></tr><tr><td>Structural Pre-train (Hu et al.,2019b)</td><td>56.00</td><td>53.83</td><td>62.17</td><td>57.49</td><td>68.78</td><td>72.41</td></tr><tr><td>EGI (GIN)</td><td>59.15**</td><td>54.98</td><td>64.55**</td><td>57.40</td><td>73.15**</td><td>70.00</td></tr></table>
+
+On the Gene dataset, with more graphs available, we focus on EGI to further analyze the utility of Eq. 5 in Theorem 2.1, regarding the connection between the structural difference of two graphs and the performance gap of EGI on them. As shown in Figure 2, we train EGI on one graph and test it on six different graphs. The $x$ -axis shows the structural difference measured w.r.t. the RHS of Eq. 5, and $y$ -axis shows the performance loss compared with an untrained GIN. The positive correlation between two quantities is obvious. Specifically, when the structural difference is small, positive transfer is observed as the performance of transfered EGI is better than untrained GIN, and when the structural difference becomes large, negative transfer is observed. Note that, at its current stage, Eq. 5 in Theorem 5 mainly gives a relative indication on the transferability of EGI, because the absolute values of structural difference may vary a lot across different datasets.
+
+# 3.2 FEW-SHOT LEARNING ON RELATION PREDICTION
+
+Here we evaluate EGI in the more generalized and practical setting of few-shot learning on the less structure-relevant task of relation prediction, with task-specific node features and fine-tuning. The source graph contains a cleaned full dump of 579K entities from YAGO (Suchanek et al., 2007), and we investigate 20-shot relation prediction on a target graph with 24 relation types, which is a sub-graph of 115K entities sampled from the same dump. In post-fine-tuning, the models are pre-trained with an unsupervised loss on the source graph and fine-tuned with the task-specific loss on the target graph. In joint-fine-tuning, the same pre-trained models are jointly optimized w.r.t. the unsupervised pre-training loss and task-specific fine-tuning loss on the target graph. In Table 3, we observe most of the existing models fail to transfer across pre-training and fine-tuning tasks, especially in the joint-fine-tuning setting. In particular, both Mask-GIN and ContextPred-GIN rely a lot on task-specific fine-tuning, while EGI focuses on the capturing of similar ego-graph structures that are transferable across graphs. As a consequence, EGI significantly outperforms all compared methods in both settings.
+
+Table 3: Performance of few-shot relation prediction on YAGO. Structural Pre-train (Hu et al., 2019b) can not scale to the YAGO graphs with $1 0 0 \mathrm { K } +$ nodes. More details can be found in Appendix $\ S { \bf C } . 3$ .   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">post-fine-tuning</td><td colspan="2"> joint-fine-tuning</td></tr><tr><td>AUROC</td><td>MRR</td><td>AUROC</td><td>MRR</td></tr><tr><td>No pre-train</td><td>0.6866</td><td>0.5962</td><td>N.A.</td><td>N.A</td></tr><tr><td>GVAE (Kipf &amp; Welling,2016)</td><td>0.7009</td><td>0.6009</td><td>0.6786</td><td>0.5676</td></tr><tr><td>DGI (Velickovic et al., 2019)</td><td>0.6885</td><td>0.5861</td><td>0.6880</td><td>0.5366</td></tr><tr><td>Mask-GIN (Hu et al.,2019a)</td><td>0.7041</td><td>0.6242</td><td>0.6720</td><td>0.5603</td></tr><tr><td>ContextPred-GIN (Hu et al., 2019a)</td><td>0.6882</td><td>0.6589</td><td>0.5293</td><td>0.3367</td></tr><tr><td>EGI</td><td>0.7389**</td><td>0.6695</td><td>0.7870**</td><td>0.7289**</td></tr></table>
+
+# 4 RELATED WORK
+
+Representation learning on graphs has been studied for decades, with earlier spectral-based methods (Belkin & Niyogi, 2002; Roweis & Saul, 2000; Tenenbaum et al., 2000) theoretically grounded but hardly scaling up to graphs with over a thousand of nodes. With the emergence of neural networks, unsupervised network embedding methods based on the Skip-gram objective (Mikolov et al., 2013) have replenished the field (Tang et al., 2015; Grover & Leskovec, 2016; Perozzi et al., 2014; Ribeiro et al., 2017). Equipped with efficient structural sampling (random walk, neighborhood, etc.) and negative sampling schemes, these methods are easily parallelizable and scalable to graphs with thousands to millions of nodes. However, these models are essentially transductive as they compute fully parameterized embeddings only for nodes seen during training, which are impossible to be transfered to unseen graphs.
+
+More recently, researchers introduce the family of graph neural networks (GNNs) that are capable of inductive learning and generalizing to unseen nodes given meaningful node features (Kipf & Welling, 2017; Defferrard et al., 2016; Hamilton et al., 2017). Yet, most existing GNNs require task-specific labels for training in a semi-supervised fashion to achieve satisfactory performance (Kipf & Welling, 2017; Hamilton et al., 2017; Velickovic et al., 2018; Chen et al., 2018), and their usage is limited to single graphs where the downstream task is fixed. To this end, several unsupervised GNNs are presented, such as the auto-encoder-based ones like GVAE (Kipf & Welling, 2016) and GNFs (Liu et al., 2019), as well as the deep-infomax-based ones like DGI (Velickovic et al., 2019) and InfoGraph (Sun et al., 2019). Their potential in the transfer learning of GNN remains unclear when the node features and link structures vary across different graphs.
+
+Although the architectures of GNNs are not very complicated, training a dedicated model for each graph can still be cumbersome (Chen et al., 2018; Ying et al., 2018a). Moreover, as pre-training neural networks are proven to be successful in other domains (Devlin et al., 2019; He et al., 2016), the idea is intriguing to transfer well-trained GNNs from relevant source graphs to improve the modeling of target graphs or enable few-shot learning (Vinyals et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017) when labeled data are scarce. In the light of this, pioneering works have studied both generative (Hu et al., 2020) and discriminative (Hu et al., 2019a,b) GNN pre-training schemes. Among these work, though Graph Contrastive Coding (Qiu et al., 2020) shares similar structural view as ours, it utilizes contrastive learning in the embedding space instead of structural space as EGI. Unsupervised domain adaptive GCNs (Wu et al., 2020) study the domain adaption problem while source and target tasks are homogenous. Previous pre-training and self-supervised GNNs lack a rigorous analysis towards their transferability and thus have unpredictable effectiveness.
+
+# 5 CONCLUSION
+
+To the best of our knowledge, this is the first research effort towards establishing a theoretically grounded framework to analyze GNN transferability, which we also demonstrate to be practically useful for guiding the design and conduct of transfer learning with GNNs. For future work, it is intriguing to further strengthen the bound with relaxed assumptions, rigorously extend it to the more complicated and less restricted settings regarding node features and downstream tasks, as well as analyze and improve the proposed framework over more transfer learning scenarios and datasets.
+
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+
+# A THEORY DETAILS
+
+From the $\mathcal { L } _ { \mathrm { E G I } }$ objective, we have assumed $g _ { i } \stackrel { i . i . d . } { \sim } \mu , x _ { i } \stackrel { i . i . d . } { \sim } \nu$ , and $( g _ { i } , x _ { i } ) \stackrel { i . i . d . } { \sim } p$ . Then with graph $G$ , we have access to the empirical distributions of the three. So the sampling reduces to bootstrapping in the procedure of evaluating the objective.
+
+Note that, in Eq. 2 of the main paper, we used a $d$ dimensional hidden state $h _ { p , q } ^ { \tilde { q } }$ , specified in Eq. 13 to denote an edge encoding derived from the structure of the ego-graph and the associated source node feature from $( p - 1 )$ -th layer. For simplicity, we consider the concatenated vector $f ( x ^ { i } ) \| z _ { i }$ , where $f ( x ^ { i } ) = h _ { p , q } ^ { \tilde { q } } \lVert x _ { p , q } ^ { i }$ and $\dot { h _ { p , q } ^ { q } } , x _ { p , q } ^ { i }$ are as defined in the EGI model and in 13. Additionally, since both of $h _ { p , q } ^ { \tilde { q } }$ and $x _ { p , q } ^ { i }$ are normalised, $f$ is bounded.
+
+Finally, as we are considering GNN with $k$ layers, its computation only depends on the $\mathbf { k }$ -hop egographs of $G$ , which is an important consideration when unfolding the embedding of GNN at a centre node.
+
+# A.1 PROOF FOR THEOREM 3.1
+
+Lemma A.1. For any $A \in \mathbb { R } ^ { m \times n }$ , where $m \geq n$ , and $A$ is a submatrix of $B \in \mathbb { R } ^ { m ^ { \prime } \times n }$ , where $m < m ^ { \prime }$ , we have
+
+$$
+\| A \| _ { 2 } \leq \| B \| _ { 2 } .
+$$
+
+Proof. Note that, $A A ^ { T }$ is a principle matrix of $B B ^ { T }$ , i.e., $A A ^ { T }$ is obtained by removing the same set of rows and columns from $B B ^ { \hat { T } }$ . Then, by Eigenvalue Interlacing Theorem (Hwang (2004)) and the fact that $A ^ { T } A$ and $A A ^ { T }$ have the same set of non-zero singular values, the matrix operator norm satisfies $\| A \| _ { 2 } = \sqrt { \lambda _ { \operatorname* { m a x } } ( A ^ { T } A ) } = \sqrt { \lambda _ { \operatorname* { m a x } } ( A A ^ { T } ) } \leq \sqrt { \lambda _ { \operatorname* { m a x } } ( B B ^ { T } ) } = \| B \| _ { 2 } .$ □
+
+We restate Theorem 3.1 from the main paper as below.
+
+Theorem A.2 (GNN transferability). Let ${ \cal G } _ { a } = \{ ( g _ { i } , x _ { i } ) \} _ { i = 1 } ^ { n }$ and $G _ { b } = \{ ( g _ { i ^ { \prime } } , x _ { i ^ { \prime } } ) \} _ { i ^ { \prime } = 1 } ^ { m }$ be two graphs. Then denote $L _ { g _ { i } }$ as the (normalised) graph Laplacian of $g _ { i } \ \forall i = 1 , \cdot \cdot \cdot , n$ , and let the node features of $g _ { i }$ be structure-respecting and normalized (similarly for $g _ { i ^ { \prime } }$ ). Consider GNN $\Psi _ { \theta }$ with $k$ layers and a $^ { l }$ -hop polynomial filter $\phi _ { \theta }$ , the empirical performance difference of $\Psi _ { \theta }$ with $\phi _ { \theta }$ evaluated on $\mathcal { L } _ { \mathrm { E G I } }$ satisfies
+
+$$
+| \mathcal { L } _ { \mathrm { E G I } } ( G _ { a } ) - \mathcal { L } _ { \mathrm { E G I } } ( G _ { b } ) | \leq \mathcal { O } \left( M + \frac { 1 } { n m } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { m } \lambda _ { \operatorname* { m a x } } ( L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } ) ^ { 1 / 2 } \right) ,
+$$
+
+where $M$ is a constant dependant on $k$ $~ ; ~ \phi _ { \theta } , ~ \{ L _ { g _ { i } } \} , ~ \{ L _ { g _ { i ^ { \prime } } } \} , ~ \{ x _ { i } \} , ~ \{ x _ { i ^ { \prime } } \}$ . In addition, $i f \exists U \in$ $O ( n \vee m ) ^ { 4 } s . t .$ .,
+
+$$
+U L _ { g _ { i } } U ^ { T } = D i a g ( \lambda ( L _ { g _ { i } } ) ) , U L _ { g _ { i ^ { \prime } } } U ^ { T } = D i a g ( \lambda ( L _ { g _ { i ^ { \prime } } } ) )
+$$
+
+we have $\begin{array} { r } { \mathcal { O } \left( M + \frac { 1 } { n m } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { m } \| \lambda ( L _ { g _ { i } } ) - \lambda ( L _ { g _ { i ^ { \prime } } } ) \| _ { 2 } \right) } \end{array}$ , where $\lambda ( L _ { g _ { i } } )$ denotes the ordered eigenvalues of the graph Laplacian of $g _ { i } \in G _ { a }$ (similarly for $g _ { i ^ { \prime } }$ ).
+
+Proof. We denote $\sigma _ { s } ( t ) = \log ( 1 + e ^ { t } )$ , the softplus activation function, which is 1-Lipschitz continuous. Now,
+
+$$
+\begin{array} { l } { | \mathcal { L } _ { \mathrm { E G I } } ( G ) - \mathcal { L } _ { \mathrm { E G I } } ( G ^ { \prime } ) | } \\ { = \displaystyle \left| \frac { 1 } { n ^ { 2 } } \sum _ { i , j = 1 } ^ { n } ( \mathcal { D } ( g _ { i } , z _ { j } ) ) - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( - ( - \mathcal { D } ( g _ { i } , z _ { i } ) ) - ( \frac { 1 } { m ^ { 2 } } \sum _ { i ^ { \prime } , j ^ { \prime } = 1 } ^ { m } ( \mathcal { D } ( g _ { i ^ { \prime } } , z _ { j ^ { \prime } } ) ) - \frac { 1 } { m } \sum _ { i ^ { \prime } = 1 } ^ { m } ( - \mathcal { D } ( g _ { i ^ { \prime } } , z _ { i ^ { \prime } } ) ) ) \right| } \\ { \displaystyle \lesssim \frac { 1 } { n ^ { 2 } m ^ { 2 } } \sum _ { i , j = 1 } ^ { n } \sum _ { i ^ { \prime } , j ^ { \prime } = 1 } ^ { m } | \mathcal { D } ( g _ { i } , z _ { j } ) - \mathcal { D } ( g _ { i ^ { \prime } } , z _ { j ^ { \prime } } ) | + \frac { 1 } { n m } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { m } | \mathcal { D } ( g _ { i } , z _ { i } ) - \mathcal { D } ( g _ { i ^ { \prime } } , z _ { i ^ { \prime } } ) | } \\ { = \displaystyle \frac { 1 } { n ^ { 2 } m ^ { 2 } } \sum _ { i , j = 1 } ^ { n } \sum _ { i ^ { \prime } , j ^ { \prime } = 1 } ^ { m } A + \frac { 1 } { n m } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { m } B . } \end{array}
+$$
+
+First we consider $B$ . Recall that, $V _ { p } ( g _ { i } )$ is the set of nodes in layer $p$ of $g _ { i }$ ,
+
+$$
+\mathcal { D } ( g _ { i } , z _ { i } ) = \sum _ { p = 1 } ^ { k } \sum _ { q = 1 } ^ { | V _ { p } ( g _ { i } ) | } \log ( \sigma _ { s i g } \left( U ^ { T } \tau \left( W ^ { T } [ f ( x ^ { i } ) \| z _ { i } ] \right) \right) ) ,
+$$
+
+where $\begin{array} { r } { \sigma _ { s i g } ( t ) = \frac { 1 } { 1 + e ^ { - t } } } \end{array}$ is the sigmoid function, $\tau$ is some $\gamma _ { \tau }$ -Lipschitz activation function and [·k·] denotes the concatenation of two vectors. Then we have
+
+$$
+U ^ { T } \tau \left( W ^ { T } [ f ( x ^ { i } ) \| z _ { i } ] \right) = U ^ { T } \tau \left( W _ { 1 } ^ { T } f ( x ^ { i } ) + W _ { 2 } ^ { T } z _ { i } \right) .
+$$
+
+WLOG, assume $d _ { p } = | V _ { p } ( g _ { i } ) | = | V _ { p } ( g _ { i ^ { \prime } } ) | \ \forall u = 1 , \cdots , k$ . In addition, since $\log ( \sigma _ { s i g } ( t ) ) =$ $- \log ( 1 + e ^ { - t } ) = { \dot { - } } \sigma _ { s } ( { \dot { - t } } )$ , which is 1-Lipschitz, it gives
+
+$$
+\begin{array} { r l } & { B \leq \displaystyle \sum _ { p = 1 } ^ { k } \displaystyle \sum _ { q = 1 } ^ { d _ { p } } | U ^ { T } \tau \left( W _ { 1 } ^ { T } f ( x ^ { i } ) + W _ { 2 } ^ { T } z _ { i } \right) - U ^ { T } \tau \left( W _ { 1 } ^ { T } f ( x ^ { i ^ { \prime } } ) + W _ { 2 } ^ { T } z _ { i ^ { \prime } } \right) | } \\ & { \quad \leq \gamma _ { \tau } s _ { U } \displaystyle \sum _ { p = 1 } ^ { k } \sum _ { q = 1 } ^ { d _ { p } } ( \| W _ { 1 } ^ { T } f ( x ^ { i } ) - W _ { 1 } ^ { T } f ( x ^ { i ^ { \prime } } ) \| _ { 2 } + \| W _ { 2 } ^ { T } z _ { i } - W _ { 2 } ^ { T } z _ { i ^ { \prime } } \| _ { 2 } ) } \\ & { \qquad \leq \gamma _ { \tau } s _ { U } s _ { W } \displaystyle \sum _ { p = 1 } ^ { k } \sum _ { q = 1 } ^ { d _ { p } } ( \| f ( x ^ { i } ) - f ( x ^ { i ^ { \prime } } ) \| _ { 2 } + \| z _ { i } - z _ { i ^ { \prime } } \| _ { 2 } ) , } \end{array}
+$$
+
+where $s _ { U }$ is the largest singular value of $U$ , and similarly $s _ { W } = s _ { W _ { 1 } } \vee s _ { W _ { 2 } }$ . Since we assumed the node features are normalised, then $\| f ( x ^ { i } ) - f ( x ^ { i ^ { \prime } } ) \| _ { 2 } \leq c _ { D }$ .
+
+From Eq. 7, we only care about $x _ { i }$ ’s embedding obtained from a $k$ -layer GNN with 1-hop polynomial (linear in $L$ ) filter. Inspired by the characterization of GNN from a node-wise view in Verma & Zhang (2019), we similarly denote the embedding of node $x _ { i } \forall i = 1 , \cdots , n$ in the final layer of the GNN as
+
+$$
+z _ { i } ^ { k } = z _ { i } = \Psi _ { \theta } ( x _ { i } ) = \sigma ( \sum _ { j \in \mathcal { N } ( x _ { i } ) } e . _ { j } z _ { j } ^ { k - 1 } ) \in \mathbb { R } ^ { d } ,
+$$
+
+where $e _ { \cdot j } = [ \phi _ { \theta } ( L ) ] _ { \cdot j } \in \mathbb { R }$ . We may denote $z _ { i } ^ { \ell } \in \mathbb { R } ^ { d }$ similarly for $\ell = 1 , \cdots , k - 1$ , and $z _ { i } ^ { 0 } = x _ { i } \in$ $\mathbb { S } ^ { d - 1 }$ the node feature of node $x _ { i }$ . With the assumption of GNN stated in the statement, it is clear that only the $\mathrm { k }$ -hop ego-graph $g _ { i }$ centered at $x _ { i }$ is needed to compute $z _ { i } ^ { k }$ for any $i = 1 , \cdots , n$ instead of the whole of $G$ . With such observation in mind, let us denote the matrix of node embeddings of $g _ { i }$ at the $\ell$ th layer as $( z _ { p , q } ^ { i ( \ell ) } ) \in \mathbb { R } ^ { | V ( g _ { i } ) | \times d }$ , for $\ell = 1 , \cdots , k$ ; and let $( z _ { p , q } ^ { i ( 0 ) } ) \equiv ( x _ { p , q } ^ { i } ) \in ( \mathbb { S } ^ { d - 1 } ) ^ { | \bar { V } ( g _ { i } ) | }$ denote the matrix of node features in the $k$ -hop ego-graph $g _ { i }$ . In addition, we denote $( z _ { p , q } ^ { i ( \ell ) } ) _ { p \leq t }$ to be the submatrix that is obtained by selecting rows that corresponds to $v \in V _ { p } ( g _ { i } )$ for $p = 0 , \cdots , t \leq k$ Similarly for $g _ { i ^ { \prime } }$ .
+
+Moreover, let us denote $\phi _ { \theta } ( L _ { g _ { i } } ) \equiv [ \phi _ { \theta } ( L ) ] _ { g _ { i } }$ , i.e., the filtered full graph Laplacian of $G$ subsetted by the $\mathbf { k }$ -hop ego-graph $g _ { i }$ . Then, let $\phi _ { \theta } ( L _ { g _ { i } } ) _ { p \leq t }$ denotes the submatrix that is obtained by selecting rows and columns that corresponds to $v \in V _ { p } ( g _ { i } )$ for $p = 0 , \cdots , t \leq k$ . Similarly for $g _ { i ^ { \prime } }$ .
+
+Therefore, by Lemma A.1, for any $\ell = 1 , \cdots , k$ , the following holds
+
+$$
+\| ( z _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) _ { p \leq t } - ( z _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) _ { p \leq t } \| _ { 2 } \leq \| ( z _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) _ { p \leq t + 1 } - ( z _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) _ { p \leq t + 1 } \| _ { 2 } .
+$$
+
+Assume $\| ( z _ { p , q } ^ { i ^ { \prime } ( \ell - 1 ) } ) \| _ { 2 } \leq c _ { z } < \infty \forall \ell$ . Now, at the final layer,
+
+$$
+\begin{array} { r l } & { \| z _ { i } - z _ { i ^ { \prime } } \| _ { 2 } = \| ( z _ { p , q } ^ { i ^ { \prime } ( k ) } ) _ { p = 0 } - ( z _ { p , q } ^ { i ^ { \prime } ( k ) } ) _ { p = 0 } \| _ { 2 } } \\ & { \le \| [ \sigma ( \phi _ { \theta } ( L _ { g _ { i } } ) _ { p \le 1 } ( z _ { p , q } ^ { i ( k - 1 ) } ) _ { p \le 1 } ) - \sigma ( \phi _ { \theta } ( L _ { g _ { i ^ { \prime } } } ) _ { p \le 1 } ( z _ { p , q } ^ { i ^ { \prime } ( k - 1 ) } ) _ { p \le 1 } ) ] _ { p = 0 } \| _ { 2 } } \\ & { \le \gamma _ { \sigma } \| \phi _ { \theta } ( L _ { g _ { i } } ) _ { p \le 1 } ( z _ { p , q } ^ { i ( k - 1 ) } ) _ { p \le 1 } - \phi _ { \theta } ( L _ { g _ { i ^ { \prime } } } ) _ { p \le 1 } ( z _ { p , q } ^ { i ^ { \prime } ( k - 1 ) } ) _ { p \le 1 } \| _ { 2 } } \\ & { \le \gamma _ { \sigma } \| \phi _ { \theta } ( L _ { g _ { i } } ) _ { p \le 1 } \| _ { 2 } \| ( z _ { p , q } ^ { i ( k - 1 ) } ) _ { p \le 1 } - ( z _ { p , q } ^ { i ^ { \prime } ( k - 1 ) } ) _ { p \le 1 } \| _ { 2 } } \\ & { + \gamma _ { \sigma } \| ( z _ { p , q } ^ { i ^ { \prime } ( k - 1 ) } ) _ { p \le 1 } \| _ { 2 } \| \phi _ { \theta } ( L _ { g _ { i } } ) _ { p \le 1 } - \phi _ { \theta } ( L _ { g _ { i ^ { \prime } } } ) _ { p \le 1 } \| _ { 2 } } \\ & { \le \gamma _ { \sigma } \| \phi _ { \theta } ( L _ { g _ { i } } ) \| _ { 2 } \| ( z _ { p , q } ^ { i ( k - 1 ) } ) _ { p \le 1 } - ( z _ { p , q } ^ { i ^ { \prime } ( k - 1 ) } ) _ { p \le 1 } \| _ { 2 } . } \end{array}
+$$
+
+In general, for $\ell = 1 , \cdots , k - 1$ , the following holds with $t = k - \ell$ ,
+
+$$
+\begin{array} { r l } & { \quad \| ( z _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) _ { p \leq t } - ( z _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) _ { p \leq t } \| _ { 2 } } \\ & { \leq \gamma _ { \sigma } \| \phi _ { \theta } ( L _ { g _ { i } } ) _ { p \leq t + 1 } ( z _ { p , q } ^ { i ( \ell - 1 ) } ) _ { p \leq t + 1 } - \phi _ { \theta } ( L _ { g _ { i ^ { \prime } } } ) _ { p \leq t + 1 } ( z _ { p , q } ^ { i ^ { \prime } ( \ell - 1 ) } ) _ { p \leq t + 1 } \| _ { 2 } } \\ & { \leq \gamma _ { \sigma } \| \phi _ { \theta } ( L _ { g _ { i } } ) \| _ { 2 } \| ( z _ { p , q } ^ { i ( \ell - 1 ) } ) _ { p \leq t + 1 } - ( z _ { p , q } ^ { i ^ { \prime } ( \ell - 1 ) } ) _ { p \leq t + 1 } \| _ { 2 } + \gamma _ { \sigma } c _ { z } \| \phi _ { \theta } ( L _ { g _ { i } } ) - \phi _ { \theta } ( L _ { g _ { i ^ { \prime } } } ) \| _ { 2 } . } \end{array}
+$$
+
+Then we equivalently write Eq. 9 as $E _ { \ell } \leq b E _ { \ell - 1 } + a$ , which gives
+
+$$
+E _ { \ell } \leq b ^ { \ell } E _ { 1 } + \frac { b ^ { \ell } + 1 } { b - 1 } a .
+$$
+
+Then, with $( x _ { p , q } ^ { i } ) = ( z _ { p , q } ^ { i ( 0 ) } )$ , we see the following is only dependant on the structure of $g _ { i }$ and $g _ { i ^ { \prime } }$
+
+$$
+\begin{array} { r l } & { \| ( \boldsymbol { z } _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) - ( \boldsymbol { z } _ { p , q } ^ { i ^ { \prime } ( \ell ) } ) \| _ { 2 } \leq \gamma _ { \sigma } ^ { \ell } \| \phi _ { \theta } ( L _ { g _ { i } } ) \| _ { 2 } ^ { \ell } \| ( \boldsymbol { x } _ { p , q } ^ { i } ) - ( \boldsymbol { x } _ { p , q } ^ { i ^ { \prime } } ) \| _ { 2 } } \\ & { \phantom { = } + \frac { \gamma _ { \sigma } ^ { \ell } \| \phi _ { \theta } ( L _ { g _ { i } } ) \| _ { 2 } ^ { \ell } + 1 } { \gamma _ { \sigma } \| \phi _ { \theta } ( L _ { g _ { i } } ) \| _ { 2 } - 1 } \gamma _ { \sigma } c _ { z } \| \phi _ { \theta } ( L _ { g _ { i } } ) - \phi _ { \theta } ( L _ { g _ { i ^ { \prime } } } ) \| _ { 2 } . } \end{array}
+$$
+
+Since the features are normalised, and so are the graph Laplacians, we have $\| \phi _ { \theta } ( L _ { g _ { i } } ) \| _ { 2 } \leq c _ { L }$ and $\| ( x _ { p , q } ^ { i } ) - ( x _ { p , q } ^ { i ^ { \prime } } ) ) \| _ { 2 } \leq c _ { x }$ . Then with Eq. 8, we have
+
+$$
+\begin{array} { r l } & { \| z _ { i } - z _ { i ^ { \prime } } \| _ { 2 } \leq \gamma _ { \sigma } ^ { k } c _ { L } ^ { k } c _ { x } + \frac { \gamma _ { \sigma } ^ { k } c _ { L } ^ { k } + 1 } { \gamma _ { \sigma } c _ { L } - 1 } \gamma _ { \sigma } \gamma _ { \theta } c _ { z } \| L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } \| _ { 2 } } \\ & { \qquad \leq c _ { \gamma , \Psi } ( M + \| L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } \| _ { 2 } ) } \\ & { \qquad = c _ { \gamma , \Psi } ( M + \lambda _ { \operatorname* { m a x } } ( L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } ) ^ { 1 / 2 } ) . } \end{array}
+$$
+
+Now, by Eq. 7, we have
+
+$$
+B \leq k d _ { \operatorname* { m a x } } \gamma _ { \tau } s ( c _ { D } + c _ { \gamma , \Psi } ( M + \lambda _ { \operatorname* { m a x } } ( L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } ) ^ { 1 / 2 } ) ) ,
+$$
+
+where $d _ { \operatorname* { m a x } } = \operatorname* { m a x } _ { p } d _ { p }$ . Similarly, the above holds for $A$ , since from Eq. 8, the node features and embedded features are bounded by separate terms. We therefore arrive at
+
+$$
+\begin{array} { r l } { | \mathcal { L } _ { \mathrm { E G I } } ( G ) - \mathcal { L } _ { \mathrm { E G I } } ( G ^ { \prime } ) | \leq 2 k d _ { \operatorname* { m a x } } \gamma _ { \tau } c _ { \gamma , \Psi } s ( M ^ { \prime } + \cfrac { 1 } { n m } \displaystyle \sum _ { i = 1 } ^ { n } \displaystyle \sum _ { i ^ { \prime } = 1 } ^ { m } \lambda _ { \operatorname* { m a x } } ( L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } ) ^ { 1 / 2 } ) ) } & { } \\ { \leq 2 k d _ { \operatorname* { m a x } } \gamma _ { \tau } c _ { \gamma , \Psi } s ( M ^ { \prime } + \cfrac { 1 } { n m } \displaystyle \sum _ { i = 1 } ^ { n } \displaystyle \sum _ { i ^ { \prime } = 1 } ^ { m } \| L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } \| _ { F } ) ) . } & { } \end{array}
+$$
+
+Moreover, by Von Neumann’s Trace Inequality Grigorieff (1991), if $\exists U \in O ( \beta ) ^ { 5 }$ , where $\beta =$ $\textstyle \sum _ { p = 0 } ^ { k } d _ { p }$ , s.t.
+
+$$
+U L _ { g _ { i } } U ^ { T } = \mathrm { D i a g } ( \lambda ( L _ { g _ { i } } ) ) , \quad U L _ { g _ { i ^ { \prime } } } U ^ { T } = \mathrm { D i a g } ( \lambda ( L _ { g _ { i ^ { \prime } } } ) ) ,
+$$
+
+we have $\Vert L _ { g _ { i } } - L _ { g _ { i ^ { \prime } } } \Vert _ { F } = \Vert \lambda ( L _ { g _ { i } } ) - \lambda ( L _ { g _ { i ^ { \prime } } } ) \Vert _ { 2 }$ , then E $q . \ 1 1 \leq c _ { \gamma , \Psi } ( M + \| \lambda ( L _ { g _ { i } } ) - \lambda ( L _ { g _ { i ^ { \prime } } } ) \| _ { 2 } ) .$
+
+Therefore Eq. 11 becomes
+
+$$
+| \mathcal { L } _ { \mathrm { E G I } } ( G ) - \mathcal { L } _ { \mathrm { E G I } } ( G ^ { \prime } ) | \leq 2 k d _ { \operatorname* { m a x } } \gamma _ { \tau } c _ { \gamma , \Psi } s ( M ^ { \prime } + \frac { 1 } { n m } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { m } \| \lambda ( L _ { g _ { i } } ) - \lambda ( L _ { g _ { i ^ { \prime } } } ) \| _ { 2 } ) .
+$$
+
+Note that, our view of structural information is closely related to graph kernels (Bai & Hancock, 2016) and graph perturbation (Verma & Zhang, 2019). Specifically, our Def 2.1 is motivated by the concept of k-layer expansion sub-graph in (Bai & Hancock, 2016). However, (Bai & Hancock, 2016) used the Jensen-Shannon divergence between pairwise representations of sub-graphs to define a depth-based sub-graph kernel, while we depict $G$ as samples of its ego-graphs. In this sense, our view is related to the setup in (Verma & Zhang, 2019), which derived a uniform algorithmic stability bound of a 1-layer GNN under 1-hop structure perturbation of $G$ .
+
+In the setting of domain adaptation, (Ben-David et al., 2007) draws a connection between the difference in the distributions of source and target domains and the model transferability, and learns a transferable model by minimizing such distribution differences. This coincides with our approach of connecting the structure difference of two graphs in terms of k-hop subgraph distributions and the transferability of GNNs in the above theory.
+
+# B MODEL DETAILS
+
+Following the same notations used in the paper, EGI consists of a GNN encoder $\Psi$ and a GNN discriminator $\mathcal { D }$ . In general, the GNN encoder $\Psi$ and decoder $\mathcal { D }$ can be any existing GNN models. For each ego-graph and its node features $\{ g _ { i } , x _ { i } \}$ , the GNN encoder returns node embedding $z _ { i }$
+
+for the center node $v _ { i }$ . As mentioned in Eq. 2 in the main paper, the GNN discriminator $\mathcal { D }$ makes edge-level predictions as follows,
+
+$$
+\begin{array} { r } { \mathcal { D } ( e _ { \tilde { v } v } | h _ { p , q } ^ { \tilde { q } } , x _ { p , q } ^ { i } , z _ { i } ) = \sigma \left( U ^ { T } \cdot \tau \left( W ^ { T } [ h _ { p , q } ^ { \tilde { q } } | | x _ { p , q } ^ { i } | | z _ { i } ] \right) \right) , } \end{array}
+$$
+
+where $e _ { \tilde { v } v } \in E ( g _ { i } )$ and $h _ { p , q } ^ { \tilde { q } } \in \mathbb { R } ^ { d }$ is the representation for edge $e _ { \tilde { v } v }$ between node $v _ { p - 1 , \tilde { q } }$ in hop $p - 1$ and $v _ { p , q }$ in hop $p$ . Specifically, we denote the source node at $p - 1$ hop as $\tilde { q } \in \tilde { Q } _ { p , q } , \tilde { Q } _ { p , q } = \{ \tilde { q } :$ $v _ { p - 1 , \tilde { q } } \in \mathring { V _ { p - 1 } } ( g _ { i } ) , e _ { ( p - 1 , \tilde { q } ) ( p , q ) } \in E ( g _ { i } ) \} .$ . Hence, the edge prediction relies on the combination of center node embedding $z _ { i }$ , destination node feature $x _ { p , q } ^ { i }$ and edge message $h _ { p , q } ^ { \tilde { q } }$ .
+
+![](images/78244400e465c2f93dcfab6aaa6d1990ba9d87a58f167864090264378f9cd696.jpg)  
+Figure 3: The overall EGI training framework.
+
+In Figure 3, $\{ g _ { i } , x _ { i } \}$ and $\{ g _ { i } ^ { \prime } , x _ { i } ^ { \prime } \}$ are the positive and negative training samples w.r.t ego-graph topology $g _ { i }$ . The discriminator $\mathcal { D }$ operates on a reversed ego-graph $\tilde { g } _ { i }$ comparing encoder’s forward propagation on $g _ { i }$ . It starts from the center node $v _ { i }$ and compute the hidden representation $m _ { p - 1 , \tilde { q } }$ for node $v _ { p - 1 , q }$ at each hop. The edge message $h _ { p , q } ^ { \tilde { q } }$ is calculated between source node’s hidden representation $m _ { p - 1 , \tilde { q } }$ and destination node features $x _ { p , q }$ .
+
+$$
+h _ { p , q } ^ { \tilde { q } } = \mathrm { R e L U } \left( W _ { p } ^ { T } \left( m _ { p - 1 , \tilde { q } } + x _ { p , q } ^ { i } \right) \right) , m _ { p - 1 , \tilde { q } } = \frac { 1 } { | \tilde { Q } _ { p - 1 , \tilde { q } } | } \sum _ { q ^ { \prime } \in \tilde { Q } _ { p - 1 \tilde { q } } } h _ { p - 1 , \tilde { q } } ^ { q ^ { \prime } }
+$$
+
+When $p = 1$ , every edge origins from the center node $v _ { i }$ and $m _ { 0 , q ^ { \prime } }$ is the center node feature $\boldsymbol { x } _ { v _ { i } }$
+
+In every batch, we sample a set of ego-graphs and their node features $\{ g _ { i } , x _ { i } \}$ . During the forward pass of encoder $\Psi$ , it aggregates from neighbor nodes to the center node $v _ { i }$ . Then, the discriminator calculates the edge embedding in Eq. 12 from center node $v _ { i }$ to its neighbors and make edge-level predictions– fake or true. The training framework of EGI is depicted in Figure 3 and Algorithm 1.
+
+We implement our method and all of the baselines using the same encoders $\Psi \colon 2 \cdot$ -layer GIN ( $\mathrm { X u }$ et al., 2019) for synthetic and role identification experiments, 2-layer GraphSAGE (Hamilton et al., 2017) for the relation prediction experiments. We set hidden dimension as 32 for both synthetic and role identification experiments, For relation prediction fine-tuning task, we set hidden dimension as 256. We train EGI in a mini-batch fashion since all the information for encoder and discriminators are within the $\mathbf { k }$ -hop ego-graph $g _ { i }$ and its features $x _ { i }$ . Further, we conduct neighborhood sampling and set maximum neighbors as 10 to speed up the parrallel training. The space and time complexity of EGI is $O ( B N ^ { K } )$ , where $B$ is the batch size, $N$ is the number of the neighbors and $\mathrm { k }$ is the number of hops of ego-graphs. Notice that both the encoder $\Psi$ and discriminator $\mathcal { D }$ propagate message on the $\mathbf { k }$ -hop ego-graphs, so the extra computation cost of $\mathcal { D }$ compared with a common GNN module is a constant multiplier over the original one. The scalability of EGI on million scale YAGO network is reported in section C.3.
+
+# B.1 TRANSFER LEARNING SETTINGS
+
+The goal of transfer learning is to train a model on a dataset or task, and use it on another. In our graph learning setting, we focus on training the model on one graph and using it on another. In particular, we focus our study on the setting of direct-transfering, where the model learned on the source graph is directly applied on the target graph without fine-tuning. We study this setting because
+
+# Algorithm 1: Pseudo code for training EGI
+
+1 The GNN encoder $\Psi$ and the GNN discriminator $\mathcal { D }$ , k-hop ego graph and features $\{ g _ { i } , x _ { i } \}$ ;   
+2 $/ *$ EGI-training starts $^ { * }$   
+3 while $\mathcal { L } _ { \mathrm { E G I } }$ not converges do   
+4 Sample M ego-graphs $\{ ( g _ { 1 } , x _ { 1 } ) , . . . , ( g _ { M } , x _ { M } ) \}$ from empirical distribution $\mathbb { P }$ without   
+replacement, and obtained their positive and negative node embeddings $z _ { i } , z _ { i } ^ { \prime }$ through $\Psi$   
+$z _ { i } = \Psi ( g _ { i } , x _ { i } ) , z _ { i } ^ { \prime } = \Psi ( g _ { i } ^ { \prime } , x _ { i } ^ { \prime } )$ ,   
+$/ *$ Initialize positive and negative expectation in Eq. 1 in the main paper\*/   
+5 $E _ { p o s } = 0 , E _ { n e g } = 0$   
+6 for $p = { \cal I }$ to $k$ do   
+7 $/ { } ^ { * }$ Compute JSD on edges at each hop\*/   
+8 for $e _ { ( p - 1 , \tilde { q } ) ( p , q ) } \in E ( g _ { i } )$ do   
+9 generate edge embedding $h _ { p , q } ^ { \tilde { q } }$ in Eq. (13) ;   
+10 $E _ { \mathrm { p o s } } = E _ { \mathrm { p o s } } + \sigma \left( U ^ { T } \cdot \tau \left( \dot { W } ^ { \bar { T } } [ h _ { p , q } ^ { \tilde { q } } | | x _ { p , q } ^ { i } | | z _ { i } ] \right) \right)$   
+11 $E _ { \mathrm { n e g } } = E _ { \mathrm { n e g } } + \sigma \left( U ^ { T } \cdot \tau \left( W ^ { T } [ h _ { p , q } ^ { \tilde { q } } | | x _ { p , q } ^ { i } | | z _ { i } ^ { \prime } ] \right) \right)$   
+12 end   
+13 end   
+14 $/ { * }$ Compute batch loss\*/   
+15 $\mathcal { L } _ { \mathrm { E G I } } = E _ { \mathrm { n e g } } - E _ { \mathrm { p o s } }$   
+16 $/ { * }$ Update Ψ, D \*/   
+17 $\theta _ { \Psi } \stackrel { + } {  } - \nabla _ { \Psi } \mathcal { L } _ { \mathrm { E G I } } , \theta _ { \mathcal { D } } \stackrel { + } {  } - \nabla _ { \mathcal { D } } \mathcal { L } _ { \mathrm { E G I } }$   
+18 end
+
+it allows us to directly measure the transferability of GNNs, which is not affected by the fine-tuning process on the target graph. In other words, the fine-tuning process introduces significant uncertainty to the analysis, because there is no guarantee on how much the fine-tuned GNN is different from the pre-trained one. Depending on specific tasks and labels distributions on the two graphs, the fine-tuned GNN might be quite similar to the pre-trained one, or it can be significantly different. It is then very hard to analyze how much the pre-trained GNN itself is able to help. Another reason is about efficiency. The fine-tuning of GNNs requires the same environment set-up and computation resource as training GNNs from scratch, although it may take less training time eventually if pre-training is effective. It is intriguing if this whole process can be eliminated when we guarantee the performance with direct-transfering.
+
+In our experiments, we also study the setting of transfer learning with fine-tuning, particularly on the real-world large-scale YAGO graphs. Since we aim to study the general transferability of GNNs not bounded to specific tasks, we always pre-train GNNs with the unsupervised pre-training objective on source graphs. Then we enable two types of fine-tuning. The first one is post-fine-tuning $\mathcal { L } = \mathcal { L } _ { s } $ ), where the pre-trained GNNs are fine-tuned with the supervised task specific objective $\mathcal { L } _ { s }$ on the target graphs. The second on is joint-fine-tuning $\mathcal { L } = \mathcal { L } _ { s } + \mathcal { L } _ { u } )$ , where pre-training is the same, but fine-tuning is done w.r.t. both the pre-training objective $\mathcal { L } _ { u }$ and task specific objective $\mathcal { L } _ { s }$ on target graphs in a semi-supervised learning fashion. The unsupervised pre-training objective $\mathcal { L } _ { u }$ of EGI is Algorithm 1, while those of the compared algorithms are as defined in their papers. The supervised fine-tuning objective $\mathcal { L } _ { s }$ is the same as in the DistMult paper (Yang et al., 2014) for all algorithms.
+
+# C EXPERIMENT DETAILS
+
+# C.1 SYNTHETIC EXPERIMENTS
+
+Data. As mentioned in the main paper, we use two traditional graph generation models for synthetic data generation: (1) barabasi-albert graph (Barabási & Albert, 1999) and (2) forest-fire graph (Leskovec et al., 2005). We generate 40 graphs each with 100 nodes with each model. We control the parameters of two models to generate two graphs with different ego-graph distributions. Specifically, we set the number of attached edges as 2 for barabasi-albert model and set $p _ { \mathrm { f o r w a r d } } = 0 . 4$ $p _ { \mathrm { b a c k w a r d } } = 0 . 3$ for forest-fire model. In Figure 4a and 4b, we show example graphs from two families in our datasets. They have the same size but different appearance which leads to our study on the transferability gap in Table 1 in the main paper. The accuracy of this task defined as the percentage of nearest neighbors for target node in the embedding space that are structure-equivalent, i.e. #correct $\mathbf { k }$ -nn neighbors / #ground truth equivalent nodes.
+
+![](images/d3a15bcb740875167a635a7c53285ffefcd70103039cef540b267f21d00debab.jpg)  
+Figure 4: Visualizations of the graphs and labels we use in the synthetic experiments.
+
+Results. The structural equivalence label is obtained by a 2-hop WL-test (Weisfeiler & Lehman, 1968) on the ego-graphs. If two nodes have the same 2-hop ego-graphs, they will be assigned the same label. In the example of Figure 4c, the nodes labeled with same number (e.g. 2, 4) have the isomorphic 2-hop ego-graphs. Note that this task is exactly solvable when node features and GNN architectures are powerful enough like GIN (Xu et al., 2019). In order to show the performance difference among different methods, we set the length of one-hot node degree encoding to 3 (all nodes with degrees higher than 3 have the same encoding). Here, we present the performance comparison with different length of degree encodings (d) in Table 4. When the capacity of initial node features is high $\mathrm { ( d = 1 0 }$ ), the transfer learning gap diminishes between different methods and different graphs because the structural equivalence problem can be exactly solved by neighborhood aggregations. However, when the information in initial node features is limited, the advantage of EGI in learning and transfering the graph structural information is obvious. In Table 5, we also show the performance of different transferable and non-transferable features, i.e. node embedding (Perozzi et al., 2014) and random feature vectors. The observation is similar with Table 1 in the main paper: the transferable feature can reflect the performance gap between similar and dissimilar graphs while non-transferable features can not.
+
+In both Table 4 and 7 here as well as Table 1 in the main paper, we report the structural difference among graphs in the two sets ( ¯d) calculated w.r.t. the term 1nm Pni=1 Pmi0=1 kλ(Lgi ) − λ(Lgi0 )k2 on similar to the other Forest fire graphs, while less similar to the Barabasi graphs, as can be verified from Figure 4a and 4b. Our bound in Theorem 3.1 then tells us that the GNNs (in particular, EGI) should be more transferable in the F-F case than B-F. This is verified in Table 4 and 5 when using the transferable node features of degree encoding with limited dimension $\left( \mathrm { d } \mathrm { = } 3 \right)$ as well as DeepWalk embedding, as EGI trained on Forest fire graphs performs significantly better on Forest fire graphs than on Barabasi graphs (with $+ 0 . 0 9 4$ and $+ 0 . 0 5 7$ differences, respectively).
+
+Table 4: Synthetic experiments of identifying structural-equivalent nodes with different degree encoding dimensions.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">#dim degree encoding d= 3</td><td colspan="3"># dim degree encoding d = 10</td><td colspan="2">structural difference</td></tr><tr><td>F-F</td><td>B-F</td><td>△</td><td>F-F</td><td>B-F</td><td>△</td><td>d(F,F)</td><td>d(B,F)</td></tr><tr><td>GCN (untrained)</td><td>0.478</td><td>0.478</td><td>/</td><td>0.940</td><td>0.940</td><td>/</td><td></td><td></td></tr><tr><td>GIN (untrained)</td><td>0.572</td><td>0.572</td><td>/</td><td>0.940</td><td>0.940</td><td>/</td><td></td><td></td></tr><tr><td>GVAE (GCN)</td><td>0.498</td><td>0.432</td><td>+0.066</td><td>0.939</td><td>0.937</td><td>0.002</td><td>1.78</td><td>2.17</td></tr><tr><td>DGI (GIN)</td><td>0.578</td><td>0.591</td><td>-0.013</td><td>0.939</td><td>0.941</td><td>-0.002</td><td></td><td></td></tr><tr><td>EGI (GIN)</td><td>0.710</td><td>0.616</td><td>+0.094</td><td>0.942</td><td>0.942</td><td>0</td><td></td><td></td></tr></table>
+
+# C.2 REAL-WORLD ROLE IDENTIFICATION EXPERIMENTS
+
+Data. We report the number of nodes, edges and classes for both airport and gene dataset. The numbers for the Gene dataset are the aggregations of the total 52 gene networks in the dataset. For the three airport networks, Figure 5 shows the power-law degree distribution on log-log scale. The class labels are between 0 to 3 reflecting the level of the airport activities (Ribeiro et al., 2017). For the Gene dataset, we matched the gene names in the TCGA dataset (Yang et al., 2019) to the list of transcription factors on wikipedia6. $7 5 \%$ of the genes are marked as 1 (transcription factors) and some gene graphs have extremely imbalanced class distributions. So we conduct experiments on the relatively balanced gene graphs of brain cancers (Figure 2 in the main paper). Both datasets do not have organic node attributes. The role-based node labels are highly relevant to their local graph structures, but are not trivially computable such as from node degrees.
+
+Table 5: Synthetic experiments of identifying structural-equivalent nodes with different transferable and nontransferable features.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Deep Walk embedding</td><td colspan="3">random vectors</td><td rowspan="2">structural difference d(F,F)</td><td rowspan="2">d(B.F)</td></tr><tr><td>F-F</td><td>B-F</td><td>△</td><td>F-F</td><td>B-F</td><td>△</td></tr><tr><td>GCN (untrained)</td><td>0.658</td><td>0.658</td><td>/</td><td>0.246</td><td>0.246</td><td>1</td><td rowspan="4"></td><td rowspan="4">2.17</td></tr><tr><td>GIN (untrained)</td><td>0.663</td><td>0.663</td><td>/</td><td>0.520</td><td>0.520</td><td>/</td></tr><tr><td>GVAE (GCN)</td><td>0.713</td><td>0.659</td><td>+0.054</td><td>0.266</td><td>0.264</td><td>0.002</td></tr><tr><td>DGI (GIN)</td><td>0.640</td><td>0.613</td><td>+0.027</td><td>0.512</td><td>0.576</td><td>-0.064</td></tr><tr><td>EGI (GIN)</td><td>0.772</td><td>0.715</td><td>+0.057</td><td>0.507</td><td>0.485</td><td>+0.022</td><td rowspan="2"></td><td rowspan="2"></td></tr></table>
+
+Table 6: Overall Dataset Statistics   
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Edges</td><td># Classes</td></tr><tr><td>Europe</td><td>399</td><td>5,995</td><td>4</td></tr><tr><td>USA</td><td>1,190</td><td>13,599</td><td>4</td></tr><tr><td>Brazil</td><td>131</td><td>1,074</td><td>4</td></tr><tr><td>Gene</td><td>9,228</td><td>57,029</td><td>2</td></tr></table>
+
+![](images/6fccec6456a027a0fdad1ea2e71561d242a41cb1b551cbda7aec9a273a7a7c0f.jpg)  
+Figure 5: Visualizations of power-law degree distribution on three airport dataset.
+
+Results. As we can observe from Figure 5, the three airport graphs have quite different sizes and structures (e.g., regarding edge density and connectivity pattern). Thus, the absolute classification accuracy in both Table 2 in the main paper and Table 7 here varies across different graphs. However, as we mention in the main paper, the structural difference we compute based on Eq. 5 in Theorem 3.1 is close among the Europe-USA and Europe-Brazil graph pairs (12.03 and 12.14), which leads to close transferability of EGI from Europe to USA and Brazil. This indicates the effectiveness of our view over essential structural information.
+
+Note that, the results present in Table 7 are the accuracy of GNNs directly trained and evaluated on each network without transfering. Therefore, only the Europe column has the same results as in Table 2 in the main paper, while the USA and Brazil columns can be regarded as providing an upper-bound performance of GNN transfered from other graphs. As we can see, EGI gives the closest results from Table 2 in the main paper to Table 7 here, demonstrating the its plausible transferability. The scores are so close, showing a possibility to skip fine-tuning when the source and target graphs are similar enough. Also note that, although the variances are pretty large (which is also observed in other works like (Ribeiro et al., 2017) since the networks are small), our t-tests have shown the improvements of EGI to be significant.
+
+Table 7: Role identification that identifies structurally similar nodes on real-world networks. The performance reported are the average and standard deviation for 10 runs. Our classification accuracy on three datasets all passed the t-test $_ { ( \mathrm { p < 0 . 0 1 } ) }$ with the second best result in the table.   
+C.3 REAL-WORLD LARGE-SCALE RELATION PREDICTION EXPERIMENTS   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Airport (Ribeiro et al., 2017)</td></tr><tr><td>Europe</td><td>USA</td><td>Brazil</td></tr><tr><td>node degree</td><td>52.81% ± 5.81%</td><td>55.67% ± 3.63%</td><td>67.11% ± 7.58%</td></tr><tr><td>GCN (random-init)</td><td>52.96% ± 4.51%</td><td>56.18% ± 3.82%</td><td>55.93% ±1.38%</td></tr><tr><td>GIN (random-init)</td><td>55.75% ± 5.84%</td><td>62.77% ± 2.35%</td><td>69.26% ± 9.08%</td></tr><tr><td>GVAE(GIN) (Kipf &amp;Welling,2016)</td><td>53.90% ±4.65%</td><td>58.99% ±2.44%</td><td>55.56% ± 6.83%</td></tr><tr><td>DGI (GIN) (Velickovic et al.,2019)</td><td>57.75% ± 4.47%</td><td>62.44% ± 4.46%</td><td>68.15% ± 6.24%</td></tr><tr><td>Mask-GIN (Hu et al.,2019a)</td><td>56.37% ± 5.07%</td><td>63.78% ± 2.79%</td><td>61.85% ±10.74%</td></tr><tr><td>ContextPred-GIN (Hu et al.,2019a)</td><td>52.69% ±6.12%</td><td>56.22% ± 4.05%</td><td>58.52% ±10.18%</td></tr><tr><td>Structural Pre-train (Hu et al.,2019b)</td><td>56.00% ± 4.58%</td><td>62.29% ± 3.51%</td><td>71.48% ± 9.38 %</td></tr><tr><td>EGI (GIN)</td><td>59.15% ± 4.44%</td><td>65.88% ± 3.65%</td><td>74.07% ± 5.49%</td></tr></table>
+
+Data. As shown in Table 8, the source graph we use to pre-train GNNs is the full graph cleaned from the YAGO dump (Suchanek et al., 2007), where we assume the relations among entities are unknown. The target graph we use is a subgraph uniformed sampled from the same YAGO dump (we sample the nodes and then include all edges among the sampled nodes). The similar ratio between number of nodes and edges can be observed in Table 8. On the target graph, we also have the access to 24 different relations (Shi et al., 2018) such as isAdvisedBy, isMarriedTo and so on. Such relation labels are still relevant to the graph structures, but the relevance is lower compared with the structural role labels. We use the 256-dim degree encoding as node features for pre-training on the source graph, then we use the 128-dim positional embedding generated by LINE (Tang et al., 2015) for fine-tuning on the target graph, to explicitly make the features differ across source and target graphs.
+
+Results. In Section B.1, we introduced two different types of fine-tuning, i.e., post-fine-tuning and joint-fine-tuning. For both types of fine-tuning, we add one feature encoder $\mathcal { E }$ before feeding it into the GNNs for two purposes. First, the target graph fine-tuning feature usually has different dimensions with the pre-training features, such as the node degree encoding we use. Second, the semantics and distributions of fine-tuning features can be different from pre-training features. The feature encoder aims to bridge the gap between feature difference in practice. The supervised loss used in this experiment is the same as in DistMult (Yang et al., 2014). In particular, the bilinear score function is calculated as $s ( h , r , t ) = z _ { h } ^ { T } M _ { r } z _ { t }$ , where $M _ { r }$ is a diagonal matrix for each relation $r$ , $z _ { h }$ and $z _ { t }$ the the embedding of GNN encoder $\Psi$ for head and tail entities. The experiments were run on GTX1080 with 12G memories. We report the average training time per epoch of our algorithm in pre-training and fine-tuning stage in Table 8 as well. The pre-training and fine-tuning takes about 40 epochs and 10 epochs to converge, respectively. In Table 8, we also present the per-epoch training time of EGI. EGI takes about 338 seconds per epoch for optimizing the ego-graph information maximization objective on YAGO-source. As we can see, fine-tuning also takes significant time compared to pre-training, which strengthens our arguments about avoiding or reducing fine-tuning through structural analysis. We implement all baselines within the same pipeline, and the runtimes are all at the same scale.
+
+Table 8: dataset statistics and running time of EGI   
+
+<table><tr><td>Dataset</td><td># Nodes</td><td>#Edges</td><td>#Relations</td><td># Train/Test</td><td>Training time per epoch</td></tr><tr><td>YAGO-Source</td><td>579,721</td><td>2,191,464</td><td>/</td><td>/</td><td>338 seconds</td></tr><tr><td>YAGO-Target</td><td>115,186</td><td>409,952</td><td>24</td><td>480/409,472</td><td>134 seconds</td></tr></table>

parse/train/SJxu5iR9KQ/SJxu5iR9KQ.md

@@ -0,0 +1,398 @@
+# LEARNING TO SCHEDULE COMMUNICATION IN MULTI-AGENT REINFORCEMENT LEARNING
+
+Daewoo Kim, Sangwoo Moon, David Hostallero, Wan Ju Kang, Taeyoung Lee,   
+Kyunghwan Son & Yung Yi   
+School of Electrical Engineering, KAIST   
+Daejeon, South Korea   
+{dwkim, swmoon, ddhostallero, wjkang, taeyoung.lee, khson}@lanada.kaist. ac.kr, yiyung@kaist.edu
+
+# ABSTRACT
+
+Many real-world reinforcement learning tasks require multiple agents to make sequential decisions under the agents’ interaction, where well-coordinated actions among the agents are crucial to achieve the target goal better at these tasks. One way to accelerate the coordination effect is to enable multiple agents to communicate with each other in a distributed manner and behave as a group. In this paper, we study a practical scenario when $( i )$ the communication bandwidth is limited and (ii) the agents share the communication medium so that only a restricted number of agents are able to simultaneously use the medium, as in the state-of-the-art wireless networking standards. This calls for a certain form of communication scheduling. In that regard, we propose a multi-agent deep reinforcement learning framework, called SchedNet, in which agents learn how to schedule themselves, how to encode the messages, and how to select actions based on received messages. SchedNet is capable of deciding which agents should be entitled to broadcasting their (encoded) messages, by learning the importance of each agent’s partially observed information. We evaluate SchedNet against multiple baselines under two different applications, namely, cooperative communication and navigation, and predator-prey. Our experiments show a non-negligible performance gap between SchedNet and other mechanisms such as the ones without communication and with vanilla scheduling methods, e.g., round robin, ranging from $32 \%$ to $43 \%$ .
+
+# 1 INTRODUCTION
+
+Reinforcement Learning (RL) has garnered renewed interest in recent years. Playing the game of Go (Mnih et al., 2015), robotics control (Gu et al., 2017; Lillicrap et al., 2015), and adaptive video streaming (Mao et al., 2017) constitute just a few of the vast range of RL applications. Combined with developments in deep learning, deep reinforcement learning (Deep RL) has emerged as an accelerator in related fields. From the well-known success in single-agent deep reinforcement learning, such as Mnih et al. (2015), we now witness growing interest in its multi-agent extension, the multi-agent reinforcement learning (MARL), exemplified in Gupta et al. (2017); Lowe et al. (2017); Foerster et al. (2017a); Omidshafiei et al. (2017); Foerster et al. (2016); Sukhbaatar et al. (2016); Mordatch & Abbeel (2017); Havrylov & Titov (2017); Palmer et al. (2017); Peng et al. (2017); Foerster et al. (2017c); Tampuu et al. (2017); Leibo et al. (2017); Foerster et al. (2017b). In the MARL problem commonly addressed in these works, multiple agents interact in a single environment repeatedly and improve their policy iteratively by learning from observations to achieve a common goal. Of particular interest is the distinction between two lines of research: one fostering the direct communication among agents themselves, as in Foerster et al. (2016); Sukhbaatar et al. (2016) and the other coordinating their cooperative behavior without direct communication, as in Foerster et al. (2017b); Palmer et al. (2017); Leibo et al. (2017).
+
+In this work, we concern ourselves with the former. We consider MARL scenarios wherein the task at hand is of a cooperative nature and agents are situated in a partially observable environment, but each endowed with different observation power. We formulate this scenario into a multi-agent sequential decision-making problem, such that all agents share the goal of maximizing the same discounted sum of rewards. For the agents to directly communicate with each other and behave as a coordinated group rather than merely coexisting individuals, they must carefully determine the information they exchange under a practical bandwidth-limited environment and/or in the case of high-communication cost. To coordinate this exchange of messages, we adopt the centralized training and distributed execution paradigm popularized in recent works, e.g., Foerster et al. (2017a); Lowe et al. (2017); Sunehag et al. (2018); Rashid et al. (2018); Gupta et al. (2017).
+
+In addition to bandwidth-related constraints, we take the issues of sharing the communication medium into consideration, especially when agents communicate over wireless channels. The stateof-the-art standards on wireless communication such as Wi-Fi and LTE specify the way of scheduling users as one of the basic functions. However, as elaborated in Related work, MARL problems involving scheduling of only a restricted set of agents have not yet been extensively studied. The key challenges in this problem are: (i) that limited bandwidth implies that agents must exchange succinct information: something concise and yet meaningful and (ii) that the shared medium means that potential contenders must be appropriately arbitrated for proper collision avoidance, necessitating a certain form of communication scheduling, popularly referred to as MAC (Medium Access Control) in the area of wireless communication. While stressing the coupled nature of the encoding/decoding and the scheduling issue, we zero in on the said communication channel-based concerns and construct our neural network accordingly.
+
+Contributions In this paper, we propose a new deep multi-agent reinforcement learning architecture, called SchedNet, with the rationale of centralized training and distributed execution in order to achieve a common goal better via decentralized cooperation. During distributed execution, agents are allowed to communicate over wireless channels where messages are broadcast to all agents in each agent’s communication range. This broadcasting feature of wireless communication necessitates a Medium Access Control (MAC) protocol to arbitrate contending communicators in a shared medium. CSMA (Collision Sense Multiple Access) in Wi-Fi is one such MAC protocol. While prior work on MARL to date considers only the limited bandwidth constraint, we additionally address the shared medium contention issue in what we believe is the first work of its kind: which nodes are granted access to the shared medium. Intuitively, nodes with more important observations should be chosen, for which we adopt a simple yet powerful mechanism called weight-based scheduler (WSA), designed to reconcile simplicity in training with integrity of reflecting real-world MAC protocols in use (e.g., 802.11 Wi-Fi). We evaluate SchedNet for two applications: cooperative communication and navigation and predator/prey and demonstrate that SchedNet outperforms other baseline mechanisms such as the one without any communication or with a simple scheduling mechanism such as round robin. We comment that SchedNet is not intended for competing with other algorithms for cooperative multi-agent tasks without considering scheduling, but a complementary one. We believe that adding our idea of agent scheduling makes those algorithms much more practical and valuable.
+
+Related work We now discuss the body of relevant literature. Busoniu et al. (2008) and Tan (1993) have studied MARL with decentralized execution extensively. However, these are based on tabular methods so that they are restricted to simple environments. Combined with developments in deep learning, deep MARL algorithms have emerged (Tampuu et al., 2017; Foerster et al., 2017a; Lowe et al., 2017). Tampuu et al. (2017) uses a combination of DQN with independent Q-learning. This independent learning does not perform well because each agent considers the others as a part of environment and ignores them. Foerster et al. (2017a); Lowe et al. (2017); Gupta et al. (2017); Sunehag et al. (2018), and Foerster et al. (2017b) adopt the framework of centralized training with decentralized execution, empowering the agent to learn cooperative behavior considering other agents’ policies without any communication in distributed execution.
+
+It is widely accepted that communication can further enhance the collective intelligence of learning agents in their attempt to complete cooperative tasks. To this end, a number of papers have previously studied the learning of communication protocols and languages to use among multiple agents in reinforcement learning. We explore those bearing the closest resemblance to our research. Foerster et al. (2016); Sukhbaatar et al. (2016); Peng et al. (2017); Guestrin et al. (2002), and Zhang & Lesser (2013) train multiple agents to learn a communication protocol, and have shown that communicating agents achieve better rewards at various tasks. Mordatch & Abbeel (2017) and Havrylov & Titov (2017) investigate the possibility of the artificial emergence of language. Coordinated RL by Guestrin et al. (2002) is an earlier work demonstrating the feasibility of structured communication and the agents’ selection of jointly optimal action.
+
+Only DIAL (Foerster et al., 2016) and Zhang & Lesser (2013) explicitly address bandwidth-related concerns. In DIAL, the communication channel of the training environment has a limited bandwidth, such that the agents being trained are urged to establish more resource-efficient communication protocols. The environment in Zhang & Lesser (2013) also has a limited-bandwidth channel in effect, due to the large amount of exchanged information in running a distributed constraint optimization algorithm. Recently, Jiang & Lu (2018) proposes an attentional communication model that allows some agents who request additional information from others to gather observation from neighboring agents. However, they do not explicitly consider the constraints imposed by limited communication bandwidth and/or scheduling due to communication over a shared medium.
+
+To the best of our knowledge, there is no prior work that incorporates an intelligent scheduling entity in order to facilitate inter-agent communication in both a limited-bandwidth and shared medium access scenarios. As outlined in the introduction, intelligent scheduling among learning agents is pivotal in the orchestration of their communication to better utilize the limited available bandwidth as well as in the arbitration of agents contending for shared medium access.
+
+# 2 BACKGROUND
+
+Reinforcement Learning We consider a standard RL formulation based on Markov Decision Process (MDP). An MDP is a tuple $< S , { \mathcal { A } } , r , P , \gamma >$ where $s$ and $\mathcal { A }$ are the sets of states and actions, respectively, and $\gamma \in [ 0 , 1 ]$ is the discount factor. A transition probability function $P : \mathcal { S } \times \mathcal { A }  \mathcal { S }$ maps states and actions to a probability distribution over next states, and $r : S \times \mathcal { A }  \mathbb { R }$ denotes the reward. The goal of RL is toing the expected discounted return $\pi : { \mathcal { S } }  A$ that solves the MDP by maximiz-The policy induces a value function $\begin{array} { r } { R _ { t } = \mathbb { E } [ \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } r _ { t + k } | \pi ] } \end{array}$ $V ^ { \bar { \pi } } ( s ) = \mathbb { E } _ { \pi } [ R _ { t } | s _ { t } = s ]$ , and an action value function $Q ^ { \pi } ( \bar { s , a } ) = \mathbb { E } _ { \pi } [ R _ { t } | s _ { t } = s , a _ { t } = a ]$ .
+
+Actor-critic Method The main idea of the policy gradient method is to optimize the policy, parametrized by $\theta ^ { \pi }$ , in order to maximize the objective $J ( \theta ) = \mathbb { E } _ { s \sim p ^ { \pi } , a \sim \pi _ { \theta } } [ R ]$ by directly adjusting the parameters in the direction of the gradient. By the policy gradient theorem Sutton et al. (2000), the gradient of the objective is:
+
+$$
+\nabla _ { \theta } J ( \pi _ { \theta } ) = \mathbb { E } _ { s \sim \rho ^ { \pi } , a \sim \pi _ { \theta } } [ \nabla _ { \theta } \log \pi _ { \theta } ( a | s ) Q ^ { \pi } ( s , a ) ] ,
+$$
+
+where $\rho ^ { \pi }$ is the state distribution. Our baseline algorithmic framework is the actor-critic approach Konda & Tsitsiklis (2003). In this approach, an actor adjusts the parameters $\theta$ of the policy $\pi _ { \boldsymbol { \theta } } ( s )$ by gradient ascent. Instead of the unknown true action-value function $Q ^ { \pi } ( s , a )$ , its approximated version $Q ^ { w } ( s , a )$ is used with parameter $w$ . A critic estimates the action-value function $Q ^ { w } ( s , a )$ using an appropriate policy evaluation algorithm such as temporal-difference learning Tesauro (1995). To reduce the variance of the gradient updates, some baseline function $b ( s )$ is often subtracted from the action value, thereby resulting in $Q ^ { \pi } ( s , a ) - b ( s )$ Sutton $\&$ Barto (1998). A popular choice for this baseline function is the state value $V ( s )$ , which indicates the inherent “goodness” of the state. This difference between the action value and the state value is often dubbed as the advantage $A ( s , a )$ whose TD-error-based substitute $\delta _ { t } = r _ { t } + \gamma V \left( s _ { t + 1 } \right) - V \left( s _ { t } \right)$ is an unbiased estimate of the advantage as in Mnih et al. (2016). The actor-critic algorithm can also be applied to a deterministic policy $\mu _ { \theta } : { \mathcal { S } }  A$ . By the deterministic policy gradient theorem Silver et al. (2014), we update the parameters as follows:
+
+$$
+\nabla _ { \theta } J ( \mu _ { \theta } ) = \mathbb { E } _ { s \sim \rho ^ { \mu } } [ \nabla _ { \theta } \mu _ { \theta } ( s ) \nabla _ { a } Q ^ { \mu } ( s , a ) | _ { a = \mu _ { \theta } ( s ) } ] .
+$$
+
+MARL: Centralized Critic and Distributed Actor (CCDA) We formalize MARL using DECPOMDP (Oliehoek et al., 2016), which is a generalization of MDP to allow a distributed control by multiple agents who may be incapable of observing the global state. A DEC-POMDP is described by a tuple $< \ S , { \mathcal { A } } , r , P , \Omega , { \mathcal { O } } , \gamma \ >$ . We use bold-face fonts in some notations to highlight the context of multi-agents. Each agent $i \in \mathcal N$ chooses an action $a _ { i } ~ \in ~ { \mathcal { A } }$ , forming a joint action vector $\mathbf { \sigma } _ { a } = [ a _ { i } ] \in \mathbf { \bar { \mathcal { A } } } ^ { n }$ and has partial observations $o _ { i } \in \Omega$ according to some observation function $\mathcal { O } ( s , i ) : \mathcal { S } \times \mathcal { N } \mapsto \Omega$ . $P ( s ^ { \prime } | s , a ) : \mathcal { S } \times \mathcal { A } ^ { n } \mapsto [ 0 , 1 ]$ is the transition probability function. All agents share the same reward $r ( s , u ) : \mathcal { S } \times \mathcal { A } ^ { n } \mapsto \mathbb { R }$ . Each agent $i$ takes action $a _ { i }$ based on its own policy $\pi ^ { i } ( a _ { i } | o _ { i } )$ . As mentioned in Section 1, our particular focus is on the centralized training and distributed execution paradigm, where the actor-critic approach is a good fit to such a paradigm. Since the agents should execute in a distributed setting, each agent, say $i$ , maintains its own actor that selects $i$ ’s action based only on what is partially observed by $i$ . The critic is naturally responsible for centralized training, and thus works in a centralized manner. Thus, the critic is allowed to have the global state $\pmb { s }$ as its input, which includes all agents’ observations and extra information from the environment. The role of the critic is to “criticize” individual agent’s actions. This centralized nature of the critic helps in providing more accurate feedback to the individual actors with limited observation horizon. In this case, each agent’s policy, $\pi ^ { i }$ , is updated by a variant of (1) as:
+
+$$
+\nabla _ { \theta } J ( \pi _ { \theta } ^ { i } ) = \mathbb { E } _ { s \sim \rho ^ { \pi } , a \sim \pi _ { \theta } } [ \nabla _ { \theta } \log \pi _ { \theta } ^ { i } ( a _ { i } | o _ { i } ) ( r + \gamma V ( s _ { t + 1 } ) - V ( s _ { t } ) ) ] .
+$$
+
+# 3 METHOD
+
+# 3.1 COMMUNICATION ENVIRONMENT AND PROBLEM
+
+In practical scenarios where agents are typically separated but are able to communicate over a shared medium, e.g., a frequency channel in wireless communications, two important constraints are imposed: bandwidth and contention for medium access (Rappaport, 2001). The bandwidth constraint entails a limited amount of bits per unit time, and the contention constraint involves having to avoid collision among multiple transmissions due to the natural aspect of signal broadcasting in wireless communication. Thus, only a restricted number of agents are allowed to transmit their messages each time step for a reliable message transfer. In this paper, we use a simple model to incorporate that the aggregate information size per time step is limited by $L _ { \mathrm { b a n d } }$ bits and that only $K _ { \mathrm { s c h e d } }$ out of $n$ agents may broadcast their messages.
+
+Weight-based Scheduling Noting that distributed execution of agents is of significant importance, there may exist a variety of scheduling mechanisms to schedule $K _ { \mathrm { s c h e d } }$ agents in a distributed manner. In this paper, we adopt a simple algorithm that is weight-based, which we call WSA (Weightbased Scheduling Algorithm). Once each agent decides its own weight, the agents are scheduled based on their weights following a class of the pre-defined rules. We consider the following two specific ones among many different proposals due to simplicity, but more importantly, good approximation of wireless scheduling protocols in practice.
+
+$\circ \ T o p ( k )$ . Selecting top $k$ agents in terms of their weight values. ◦ Softmax $( k )$ . Computing softmax values $\begin{array} { r } { \sigma _ { i } ( \pmb { w } ) = \frac { e ^ { w _ { i } } } { \sum _ { j = 1 } ^ { n } e ^ { w _ { j } } } } \end{array}$ for each agent $i$ , and then randomly selecting $k$ agents acoording to the probability distribution $[ \sigma _ { i } ( { \pmb w } ) ] _ { i = 1 } ^ { n }$ .
+
+Since distributed execution is one of our major operational constraints in SchedNet or other CTDEbased MARL algorithms, $\mathrm { T o p } ( k )$ and Softmax $( k )$ should be realizable via a weight-based mechanism in a distributed manner. In fact, this has been an active research topic to date in wireless networking, where many algorithms exist (Tassiulas & Ephremides, 1992; Yi et al., 2008; Jiang & Walrand, 2010). Due to space limitation, we present how to obtain distributed versions of those two rules based on weights in our supplementary material. To summarize, using so-called CSMA (Carrier Sense Multiple Access) (Kurose, 2005), which is a fully distributed MAC scheduler and forms a basis of Wi-Fi, given agents’ weight values, it is possible to implement $\mathrm { T o p } ( k )$ and Softmax $( k )$ .
+
+Our goal is to train agents so that every time each agent takes an action, only $K _ { \mathrm { s c h e d } }$ agents can broadcast their messages with limited size $L _ { \mathrm { b a n d } }$ with the goal of receiving the highest cumulative reward via cooperation. Each agent should determine a policy described by its scheduling weights, encoded communication messages, and actions.
+
+# 3.2 ARCHITECTURE
+
+To this end, we propose a new deep MARL framework with scheduled communications, called SchedNet, whose overall architecture is depicted in Figure 1. SchedNet consists of the following three components: $( i )$ actor network, (ii) scheduler, and (iii) critic network. This section is devoted to presenting the architecture only, whose details are presented in the subsequent sections.
+
+Neural networks The actor network is the collection of $n$ per-agent individual actor networks, where each agent $i$ ’s individual actor network consists of a triple of the following networks: a
+
+message encoder, an action selector, and a weight generator, as specified by:
+
+message encoder $f _ { \mathrm { e n c } } ^ { i } : o _ { i } \mapsto m _ { i }$ , action selector $f _ { \mathrm { a s } } ^ { i } : \left( o _ { i } , m \otimes c \right) \mapsto u _ { i }$ weight generator $f _ { \mathrm { w g } } ^ { i } : o _ { i } \mapsto { w } _ { i }$ .
+
+Here, $m = [ m _ { i } ] _ { n } ^ { 1 }$ is the vector of each $i$ ’s encoded message $m _ { i }$ . An agent schedule vector $c = [ c _ { i } ] _ { n }$ , $c _ { i } \in \{ 0 , \bar { 1 } \}$ represents whether each agent is scheduled. Note that agent $i$ ’s encoded message $m _ { i }$ is generated by a neural network $f _ { \mathrm { e n c } } ^ { i } : o _ { i } \mapsto m _ { i }$ . The operator “ $\mathbf { \vec { \otimes } } ^ { , }$ ” concatenates all the sched
+
+![](images/0b09eba9fc5256f5eb0c04d6fb816892e9f95d8e9ee835b551f1b08022a7efdf.jpg)  
+Figure 1: Architecture of SchedNet with three agents. Agents 1 and 3 have been scheduled for this time step.
+
+uled agents’ messages. For example, for $m = [ 0 1 0 , 1 1 1 , 1 0 1 ]$ and $c = [ 1 1 0 ]$ , $\pmb { m } \otimes \pmb { c } = 0 1 0 1 1 1$ . This concatenation with the schedule profile $^ c$ means that only those agents scheduled in $^ c$ may broadcast their messages to all other agents. We denote by $\dot { \theta _ { \mathrm { a s } } ^ { i } } , \ : \theta _ { \mathrm { w g } } ^ { i } .$ , and $\theta _ { \mathrm { e n c } } ^ { i }$ the parameters of the action selector, the weight generator, and the encoder of agent $i$ , respectively, where we let $\theta _ { \mathrm { a s } } = [ \theta _ { \mathrm { a s } } ^ { i } ] _ { n }$ , and similarly define $\theta _ { \mathrm { w g } }$ and $\theta _ { \mathrm { e n c } }$ .
+
+Coupling: Actor and Scheduler Encoder, weight generator and the scheduler are the modules for handling the constraints of limited bandwidth and shared medium access. Their common goal is to learn the state-dependent “importance” of individual agent’s observation, encoders for generating compressed messages and the scheduler for being used as a basis of an external scheduling mechanism based on the weights generated by per-agent weight generators. These three modules work together to smartly respond to time-varying states. The action selector is trained to decode the incoming message, and consequently, to take a good action for maximizing the reward. At every time step, the schedule profile $^ c$ varies depending on the observation of each agent, so the incoming message $_ { \mathbf { \nabla } } \mathbf { m }$ comes from a different combination of agents. Since the agents can be heterogeneous and they have their own encoder, the action selector must be able to make sense of incoming messages from different senders. However, the weight generator’s policy changes, the distribution of incoming messages also changes, which is in turn affected by the pre-defined WSA. Thus, the action selector should adjust to this changed scheduling. This also affects the encoder in turn. The updates of the encoder and the action selector trigger the update of the scheduler again. Hence, weight generators, message encoders, and action selectors are strongly coupled with dependence on a specific WSA, and we train those three networks at the same time with a common critic.
+
+Scheduling logic The schedule profile $^ c$ is determined by the WSA module, which is mathematically a mapping from all agents’ weights $\textbf { \em w }$ (generated by $f _ { \mathrm { w g } } ^ { i } )$ to $^ c$ . Typical examples of these mappings are $T o p ( k )$ and Softmax $( k )$ , as mentioned above. The scheduler of each agent is trained appropriately depending on the employed WSA algorithm.
+
+# 3.3 TRAINING AND EXECUTION
+
+In the centralized training with distributed execution, for a given WSA, we include all components and modules in Figure 1 to search for $\pmb { \theta } _ { \mathrm { a s } }$ , $\theta _ { \mathrm { w g } }$ , and $\pmb { \theta } _ { \mathrm { e n c } }$ , whereas in execution, each agent $i$ runs a certain shared medium access mechanism, well-modeled by a weightbased scheduler, and just needs three agent-specific parameters $\mathsf { \bar { \theta } } _ { \mathrm { a s } } ^ { i }$ $\theta _ { \mathrm { w g } } ^ { i }$ , and $\theta _ { \mathrm { e n c } } ^ { i }$ .
+
+# 3.3.1 CENTRALIZED TRAINING
+
+Centralized critic The actor is trained by dividing it into two parts: $( i )$ message encoders and action selectors, and $( i i )$ weight generators. This partitioning is motivated by the fact that it is hard to update both parts with one backpropagation since WSA is not differentiable. To update the actor, we use a centralized critic parametrized by $\theta _ { \mathrm { c } }$ to estimate the state value function $V _ { \theta _ { \mathrm { c } } } ( s )$ for the action selectors and message encoders, and the action-value function $Q _ { \theta _ { \mathrm { c } } } ^ { \pi } ( s , w )$ for the weight generators. The critic is used only when training, and it can use the global state $\pmb { s }$ , which includes the observation of all agents. All networks in the actor are trained with gradient-based on temporal difference backups. To share common features between $V _ { \theta _ { \mathrm { c } } } ( s )$ and $Q _ { \theta _ { \mathrm { c } } } ^ { \pi } ( s , w )$ and perform efficient training, we use shared parameters in the lower layers of the neural network between the two functions, as shown in Figure 2.
+
+![](images/798880e218eb956e11efe3aab637ba7f35e47f1fc2322d9f353d68b513abc12a.jpg)  
+Figure 2: Architecture of the critic. FC stands for fully connected neural network.
+
+Weight generators We consider the collection of all agents’ WGs as a single neural network $\mu _ { \theta _ { \mathrm { w g } } } ( o )$ mapping from $^ o$ to $\pmb { w }$ , parametrized by $\theta _ { \mathrm { w g } }$ . Noting that $w _ { i }$ is a continuous value, we apply the DDPG algorithm (Lillicrap et al., 2015), where the entire policy gradient of the collection of WGs is given by:
+
+$$
+\nabla _ { \theta _ { \mathrm { w g } } } J ( \theta _ { \mathrm { w g } } , \cdot ) = \mathbb { E } _ { w \sim \mu _ { \theta _ { \mathrm { w g } } } } [ \nabla _ { \theta _ { \mathrm { w g } } } \mu _ { \theta _ { \mathrm { w g } } } ( o ) \nabla _ { w } Q _ { \theta _ { \mathrm { c } } } ( s , w ) | _ { w = \mu _ { \theta _ { \mathrm { w g } } } ( o ) } ] .
+$$
+
+We sample the policy gradient for sufficient amount of experience in the set of all scheduling profiles, i.e., $\begin{array} { r } { \mathcal { C } = \{ c \vert \bar { \sum _ { c _ { i } } } \dot { \leq k } \} } \end{array}$ . The values of $Q _ { \theta _ { \mathrm { c } } } ( \pmb { s } , \pmb { w } )$ are estimated by the centralized critic, where $\pmb { s }$ is the global state corresponding to $^ o$ in a sample.
+
+Message encoders and action selectors The observation of each agent travels through the encoder and the action selector. We thus serialize $f _ { \mathrm { e n c } } ^ { i }$ and $f _ { \mathrm { a s } } ^ { i }$ together and merge the encoders and actions selectors of all agents into one aggregate network $\mathbf { \bar { \pi } } _ { \theta _ { \mathrm { u } } } ( \pmb { u } | \mathbf { o } , \mathbf { c } )$ , which is parametrized by $\theta _ { \mathrm { { u } } } ~ = $ $\{ \theta _ { \mathrm { e n c } } , \theta _ { \mathrm { a s } } \}$ . This aggregate network $\pi _ { \pmb { \theta } _ { \mathrm { u } } }$ learns via backpropagation of actor-critic policy gradients, described below. The gradient of this objective function, which is a variant of (3), is given by
+
+$$
+\begin{array} { r } { \nabla _ { \theta _ { \mathfrak { u } } } J ( \cdot , \theta _ { \mathfrak { u } } ) = \mathbb { E } _ { s \sim \rho ^ { \pi } , { \boldsymbol u } \sim \pi _ { \theta _ { \mathfrak { u } } } } [ \nabla _ { \theta _ { \mathfrak { u } } } \log \pi _ { \theta _ { \mathfrak { u } } } ( { \boldsymbol u } | o , c ) [ r + \gamma V _ { \theta _ { \mathrm { c } } } ( { \boldsymbol s } ^ { \prime } ) - V _ { \theta _ { \mathrm { c } } } ( { \boldsymbol s } ) ] ] , } \end{array}
+$$
+
+where $\pmb { s }$ and $s ^ { \prime }$ are the global states corresponding to the observations at current and next time step. We can get the value of state $V _ { \theta _ { \mathrm { c } } } ( s )$ from the centralized critic and then adjust the parameters $\theta _ { \mathrm { u } }$ via gradient ascent accordingly.
+
+# 3.3.2 DISTRIBUTED EXECUTION
+
+In execution, each agent $i$ should be able to determine the scheduling weight $w _ { i }$ , encoded message $m _ { i }$ , and action selection $u _ { i }$ in a distributed manner. This process must be based on its own observation, and the weights generated by its own action selector, message encoder, and weight generator with the parameters ${ \bar { \theta } _ { \mathrm { a s } } ^ { i } }$ , $\theta _ { \mathrm { e n c } } ^ { i }$ , and $\theta _ { \mathrm { w g } } ^ { i }$ , respectively. After each agent determines its scheduling weight, $K _ { \mathrm { s c h e d } }$ agents are scheduled by WSA, which leads the encoded messages of scheduled agents to be broadcast to all agents. Finally, each agent finally selects an action by using received messages. This process is sequentially repeated under different observations over time.
+
+# 4 EXPERIMENT
+
+Environments To evaluate SchedNet2, we consider two different environments for demonstrative purposes: Predator and Prey (PP) which is used in Stone & Veloso (2000), and Cooperative Communication and Navigation (CCN) which is the simplified version of the one in Lowe et al. (2017). The detailed experimental environments are elaborated in the following subsections as well as in supplementary material. We take the communication environment into our consideration as follows. $k$ out of all agents can have the chance to broadcast the message whose bandwidth3 is limited by $l$ .
+
+Tested algorithms and setup We perform experiments in aforementioned environments. We compare SchedNet with a variant of DIAL,4 (Foerster et al., 2016) which allows communication with limited bandwidth. During the execution of DIAL, the limited number $( k )$ of agents are scheduled following a simple round robin scheduling algorithm, and the agent reuses the outdated messages of non-scheduled agents to make a decision on the action to take, which is called $\mathrm { D I A L } ( k )$ . The other baselines are independent DQN (IDQN) (Tampuu et al., 2017) and COMA (Foerster et al., 2017a) in which no agent is allowed to communicate. To see the impact of scheduling in SchedNet, we compare SchedNet with $( i )$ RR (round robin), which is a canonical scheduling method in communication systems where all agents are sequentially scheduled, and (ii) FC (full communication), which is the ideal configuration, wherein all the agents can send their messages without any scheduling or bandwidth constraints. We also diversify the WSA in SchedNet into: $( i )$ Sched-Softmax(1) and (ii) Sched-Top(1) whose details are in Section 3.1. We train our models until convergence, and then evaluate them by averaging metrics for 1,000 iterations. The shaded area in each plot denotes $9 5 \%$ confidence intervals based on 6-10 runs with different seeds.
+
+![](images/e6b822eda9e2ee339a2ab4a81e3255c547131604a1e9e4a2e2b68aa389b293ea.jpg)  
+Figure 3: Learning curves during the learning of the PP and CCN tasks. The plots show the average time taken to complete the task, where shorter time is better for the agents.
+
+# 4.1 PREDATOR AND PREY
+
+In this task, there are multiple agents who must capture a randomly moving prey. Agents’ observations include position of themselves and the relative positions of prey, if observed. We employ four agents, and they have different observation horizons, where only agent 1 has a $5 \times 5$ view while agents 2, 3, and 4 have a smaller, $3 \times 3$ view. The predators are rewarded when they capture the prey, and thus the performance metric is the number of time steps taken to capture the prey.
+
+Result in PP Figure 3a illustrates the learning curve of 750,000 steps in PP. In FC, since the agents can use full state information even during execution, they achieve the best performance. SchedNet outperforms IDQN and COMA in which communication is not allowed. It is observed that agents first find the prey, and then follow it until all other agents also eventually observe the prey. An agent successfully learns to follow the prey after it observes the prey but that it takes a long time to meet the prey for the first time. If the agent broadcasts a message that includes the location information of the prey, then other agents can find the prey more quickly. Thus, it is natural that SchedNet and DIAL perform better than IDQN or COMA, because they are trained to work with communication. However, DIAL is not trained for working under medium contention constraints. Although DIAL works well when there is no contention constraints, under the condition where only one agent is scheduled to broadcast the message by a simple scheduling algorithm (i.e., RR), the average number of steps to capture the prey in DIAL(1) is larger than that of SchedNet-Top(1), because the outdated messages of non-scheduled agents is noisy for the agents to decide on actions. Thus, we should consider the scheduling from when we train the agents to make them work in a demanding environment.
+
+Impact of intelligent scheduling In Figure 3b, we observe that IDQN, RR, and SchedNetSoftmax(1) lie more or less on a comparable performance tier, with SchedNet-Softmax(1) as the best in the tier. SchedNet-Top(1) demonstrates a non-negligible gap better than the said tier, implying that a deterministic selection improves the agents’ collective rewards the best. In particular, SchedNet-Top(1) improves the performance by $43 \%$ compared to RR. Figure 3b lets us infer that, while all the agents are trained under the same conditions except for the scheduler, the difference in the scheduler is the sole determining factor for the variation in the performance levels. Thus, ablating away the benefit from smart encoding, the intelligent scheduling element in SchedNet can be accredited with the better performance.
+
+Weight-based Scheduling We attempt to explain the internal behavior of SchedNet by investigating instances of temporal scheduling profiles obtained during the execution. We observe that SchedNet has learned to schedule those agents with a farther observation horizon, realizing the rationale of importance-based assignment of scheduling priority also for the PP scenario. Recall that Agent 1 has a wider view and thus tends to obtain valuable observation more frequently. In Figure 4, we see that scheduling chances are distributed over (14, 3, 4, 4) where corresponding average weights are (0.74, 0.27, 0.26, 0.26), implying that those with greater observation power tend to be scheduled more often.
+
+Message encoding We now attempt to understand what the predator agents communicate when performing the task. Figure 5 shows the projections of the messages onto a 2D plane, which is generated by the scheduled agent under SchedNetTop(1) with $l = 2$ . When the agent does not observe the prey (blue circle in Figure), most of the messages reside in the bottom or the left partition of the plot. On the other hand, the messages have large variance when it observes the prey (red $\mathbf { \acute { x } } )$ . This is because the agent should transfer more informative messages that implicitly include the location of the prey, when it observes the prey. Further analysis of the messages is presented in our supplementary material.
+
+![](images/850745e7fceadc7d3b641265f04b0380cb7072c69746ce82cd0412f5bc7c7e99.jpg)  
+Figure 4: Instances of scheduling results over 25 time steps in PP
+
+# 4.2 COOPERATIVE COMMUNICATION AND NAVIGATION
+
+In this task, each agent’s goal is to arrive at a pre-specified destination on its one-dimensional world, and they collect a joint reward when both agents reach their respective destination. Each agent has a zero observation horizon around itself, but it can observe the situation of the other agent. We introduce heterogeneity into the scenario, where the agent-destination distance at the beginning of the task differs across agents. The metric used to gauge the performance is the number of time steps taken to complete the CCN task.
+
+![](images/1654eeaaf861f0438f69594dff4ba9b7d1068969a5a21f02af7785af9e21e2d8.jpg)  
+Figure 5: Encoded messages projected onto 2D plane in PP task
+
+Result in CCN We examine the CCN environment whose results are shown in Figure 3c. SchedNet and other baselines were trained for 200,000 steps. As expected, IDQN takes the longest time, and FC takes the shortest time. RR exhibits mediocre performance, better than IDQN, because agents at least take turns in obtaining the communication opportunity. Of particular interest is SchedNet, outperforming both IDQN and RR with a non-negligible gap. We remark that the deterministic selection with SchedNet-Top(1) slightly beats the probabilistic counterpart, SchedNet-Softmax(1). The $32 \%$ improved gap between RR and SchedNet clearly portrays the effects of intelligent scheduling, as the carefully learned scheduling method of SchedNet was shown to complete the CCN task faster than the simplistic RR.
+
+Scheduling in CCN As Agent 2 is farther from its destination than Agent 1, we observe that Agent 1 is scheduled more frequently to drive Agent 2 to its destination (7 vs. 18), as shown in Figure 6. This evidences that SchedNet flexibly adapts to heterogeneity of agents via scheduling. Towards more efficient completion of the task, a rationale of more scheduling for more important agents should be implemented. This is in accordance with the results obtained from PP environments: more important agents are scheduled more.
+
+![](images/d44ae670a74130e5abba6f6532fb53583eccf90d6394771bbf283d81e80af5a6.jpg)  
+Figure 6: Instances of scheduling results over 25 time steps in CCN
+
+# 5 CONCLUSION
+
+We have proposed SchedNet for learning to schedule inter-agent communications in fullycooperative multi-agent tasks. In SchedNet, we have the centralized critic giving feedback to the actor, which consists of message encoders, action selectors, and weight generators of each individual agent. The message encoders and action selectors are criticized towards compressing observations more efficiently and selecting actions that are more rewarding in view of the cooperative task at hand. Meanwhile, the weight generators are criticized such that $k$ agents with apparently more valuable observation are allowed to access the shared medium and broadcast their messages to all other agents. Empirical results and an accompanying ablation study indicate that the learnt encoding and scheduling behavior each significantly improve the agents’ performance. We have observed that an intelligent, distributed communication scheduling can aid in a more efficient, coordinated, and rewarding behavior of learning agents in the MARL setting.
+
+# ACKNOWLEDGE
+
+This work was supported by Institute for Information communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No.2018-0-00170, Virtual Presence in Moving Objects through 5G)
+
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+SUPPLEMENTARY MATERIAL
+
+# A SCHEDNET TRAINING ALGORITHM
+
+The training algorithm for SchedNet is provided in Algorithm 1. The parameters of the message encoder are assumed to be included in the actor network. Thus, we use the notation $f _ { \mathrm { a s } } ^ { i } ( o ^ { i } , c ) \stackrel { \smile } { = }$ $f _ { \mathrm { a s } } ^ { i } \left( o ^ { i } , f _ { \mathrm { e n c } } ^ { i } \left( o ^ { i } \right) \otimes c \right)$ to simplify the presentation.
+
+# Algorithm 1 SchedNet
+
+1: Initialize actor parameters $\theta _ { u }$ , scheduler parameters $\theta _ { \mathrm { w g } }$ , and critic parameters $\theta _ { c }$   
+2: Initialize target scheduler parameters $\theta _ { \mathrm { w g } } ^ { \prime }$ , and target critic parameters $\theta _ { c } ^ { \prime }$   
+3: for episode $= 1$ to $M$ do   
+4: Observe initial state $\pmb { s }$   
+5: for $t = 1$ to $T$ do   
+6: $\mathbf { } w _ { t } \gets$ the priority $w ^ { i } = f _ { \mathrm { w g } } ^ { i } ( o ^ { i } )$ of each agent $i$   
+7: Get schedule $c _ { t }$ from ${ \pmb w } _ { t }$   
+8: $\mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf  \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf$ the action $u ^ { i } = f _ { \mathrm { a s } } ^ { i } ( o ^ { i } , \pmb { c } _ { t } )$ of each agent $i$   
+9: Execute the actions $\mathbf { \Delta } \mathbf { u } _ { t }$ and observe the reward $r _ { t }$ and next state $\mathbf { } s _ { t + 1 }$   
+10: Store $( s _ { t } , u _ { t } , r _ { t } , s _ { t + 1 } , c _ { t } , w _ { t } )$ in the replay buffer $B$   
+11: Sample a minibatch of $S$ samples $( s _ { k } , u _ { k } , r _ { k } , s _ { k + 1 } , c _ { k } , { \pmb w } _ { k } )$ from $B$   
+12: Set $\begin{array} { r } { \dot { y } _ { k } = r _ { k } + \gamma \bar { V } ( s _ { k + 1 } ) } \end{array}$   
+13: Set $\begin{array} { r } { \hat { y } _ { k } = r _ { k } + \gamma \bar { Q } \big ( \pmb { s } _ { k + 1 } , \bar { f } _ { \mathrm { w g } } ^ { i } \big ( \pmb { o } _ { k + 1 } , \pmb { c } _ { k + 1 } \big ) \big ) } \end{array}$   
+14: Update the critic by minimizing the loss:
+
+$$
+L = \frac { 1 } { S } \sum _ { k } ( ( y _ { k } - V ( s _ { k } ) ) ^ { 2 } + ( \hat { y } _ { k } - Q ( \pmb { s } , \pmb { w } _ { k } ) ) ^ { 2 } )
+$$
+
+Update the actor along with the encoder using sampled policy gradient:
+
+$$
+\nabla _ { \theta _ { \mathrm { u } } } J ( \cdot , \theta _ { \mathrm { u } } ) = \mathbb { E } _ { s \sim \rho ^ { \pi } , u \sim \pi } [ \nabla _ { \theta _ { \mathrm { u } } } \log \pi ( u | o , c ) [ r + \gamma V _ { \theta _ { \mathrm { c } } } ( s ^ { \prime } ) - V _ { \theta _ { \mathrm { c } } } ( s ) ] ]
+$$
+
+Update scheduler using sampled policy gradient:
+
+$$
+\nabla _ { \theta _ { \mathrm { w g } } } J ( \theta _ { \mathrm { w g } } , \cdot ) = \mathbb { E } _ { w \sim \mu } \big [ \nabla _ { \theta _ { \mathrm { w g } } } \mu ( o ) \nabla _ { w } Q _ { \theta _ { \mathrm { c } } } ( s , w ) | _ { w = \mu ( o ) } \big ]
+$$
+
+17: Update target network parameters:
+
+$$
+\begin{array} { r l r } {  { \theta _ { \mathrm { w g } } ^ { \prime }  \tau \theta _ { \mathrm { w g } } + ( 1 - \tau ) \theta _ { \mathrm { w g } } ^ { \prime } } } \\ & { } & { \theta _ { c } ^ { \prime }  \tau \theta _ { c } + ( 1 - \tau ) \theta _ { c } ^ { \prime } } \end{array}
+$$
+
+18: end for   
+19: end for
+
+# B DETAILS OF ENVIRONMENTS AND IMPLEMENTATION
+
+![](images/a8c51b379422ff426a458ab6afa9b36fb546b892af79b1d7da7be41fa7af6e77.jpg)  
+Figure 7: Illustrations of the experimental environment
+
+# B.1 ENVIRONMENTS: PP AND CCN
+
+Predator and prey We assess SchedNet in this predator-prey setting as in Stone & Veloso (2000), illustrated in Figure 7a. This setting involves a discretized grid world and multiple cooperating predators who must capture a randomly moving prey. Agents’ observations include position of themselves and the relative positions of the prey, if observed. The observation horizon of each predator is limited, thereby emphasizing the need for communication. The termination criterion for the task is that all agents observe the prey, as in the right of Figure 7a. The predators are rewarded when the task is terminated. We note that agents may be endowed with different observation horizons, making them heterogeneous. We employ four agents in our experiment, where only agent 1 has a $5 \times 5$ view while agents 2, 3, and 4 have a smaller, $3 \times 3$ view. The performance metric is the number of time steps taken to capture the prey.
+
+Cooperative communication and navigation We adopt and modify the cooperative communication and navigation task in Lowe et al. (2017), where we test SchedNet in a simple one-dimensional grid as in Figure 7b. In CCN, each of the two agents resides in its one-dimensional grid world. Each agent’s goal is to arrive at a pre-specified destination (denoted by the square with a star or a heart for Agents 1 and 2, respectively), and they collect a joint reward when both agents reach their target destination. Each agent has a zero observation horizon around itself, but it can observe the situation of the other agent. We introduce heterogeneity into the scenario, where the agent-destination distance at the beginning of the task differs across agents. In our example, Agent 2 is initially located at a farther place from its destination, as illustrated in Figure 7b. The metric used to gauge the performance of SchedNet is the number of time steps taken to complete the CCN task.
+
+# B.2 EXPERIMENT DETAILS
+
+Table 1 shows the values of the hyperparameters for the CCN and the PP task. We use Adam optimizer to update network parameters and soft target update to update target network. The structure of the networks is the same across tasks. For the critic, we used three hidden layers, and the critic between the scheduler and the action selector shares the first two layers. For the actor, we use one hidden layer; for the encoder and the weight generator, three hidden layers each. Networks use rectified linear units for all hidden layers. Because the complexity of the two tasks differ, we sized the hidden layers differently. The actor network and the critic network for the CCN have hidden layers with 8 units and 16 units, respectively. The actor network and the critic network for the PP have hidden layers with 32 units and 64 units, respectively.
+
+Table 1: List of hyperparameters   
+
+<table><tr><td>Hyperparameter</td><td>Value</td><td>Description</td></tr><tr><td>training step</td><td>750000</td><td>Maximum time steps until the end of training</td></tr><tr><td>episode length</td><td>1000</td><td>Maximum time steps per episode</td></tr><tr><td>discount factor</td><td>0.9</td><td>Importance of future rewards</td></tr><tr><td>learning rate for actor</td><td>0.00001</td><td>Actor network learning rate used by Adam optimizer</td></tr><tr><td>learning rate for critic</td><td>0.0001</td><td>Critic network learning rate used by Adam optimizer</td></tr><tr><td>target update rate</td><td>0.05</td><td>Target network update rate to track learned network</td></tr><tr><td>entropy regularization weight</td><td>0.01</td><td>Weight of regularization to encourage exploration</td></tr></table>
+
+![](images/b234f28f3af394f32d707c8f9b07324b849dcf5700f9b727da648d8fe610602e.jpg)  
+Figure 8: Performance evaluation of SchedNet. The graphs show the average time taken to complete the task, where shorter time is better for the agents.
+
+# C ADDITIONAL EXPERIMENT RESULTS
+
+# C.1 PREDATOR AND PREY
+
+Impact of bandwidth $( L )$ and number of schedulable agents $( K )$ Due to communication constraints, only $k$ agents can communicate and scheduled agents can broadcast their message, each of which has a limited size $l$ due to bandwidth constraints. We see the impact of $l$ and $k$ on the performance in Figure 8a. As $L$ increases, more information can be encoded into the message, which can be used by other agents to take action. Since the encoder and the actor are trained to maximize the shared goal of all agents, they can achieve higher performance with increasing $l$ . In Figure 8b, we compare the cases where $k = 1 , 2 , 3$ , and FC in which all agents can access the medium, with $l = 1$ . As we can expect, the general tendency is that the performance grows as $k$ increases.
+
+Table 2: Performance with/without encoder   
+
+<table><tr><td>FC</td><td>SchedNet -Top(1)</td><td>Schedule w/ auto-encoder</td></tr><tr><td>1</td><td>2.030</td><td>3.408</td></tr></table>
+
+Impact of joint scheduling and encoding To study the effect of jointly coupling scheduling and encoding, we devise a comparison against a pre-trained auto-encoder (Bourlard & Kamp, 1988; Hinton & Zemel, 1994). An auto-encoder was trained ahead of time, and the encoder part of this auto-encoder was placed in the Actor’s ENC module in Figure 1. The encoder part is not trained further while training the other parts of network. Henceforth, we name this modified Actor “AE”. Figure 8c shows the learning curve of AE and other baselines. Table 2 highlights the impact of joint scheduling and encoding. The numbers shown are the performance metric normalized to the FC case in the PP environment. While SchedNet-Top(1) took only 2.030 times as long as FC to finish the PP task, the AE-equipped actor took 3.408 times as long as FC. This lets us ascertain that utilizing a pre-trained auto-encoder deprives the agent of the benefit of joint the scheduler and encoder neural network in SchedNet.
+
+What messages agents broadcast In Section 4.1, we attempted to understand what the predator agents communicate when performing PP task where $k = 1$ and $l = 2$ . In this section, we look into the message in detail. Figure 9 shows the projections of the messages generated by the scheduled agent based on its own observation. In the PP task, the most important information is the location of the prey, and this can be estimated from the observation of other agents. Thus, we are interested in the location information of the prey and other agents. We classify the message into four classes based on which quadrant the prey and the predator are included, and mark each class with different colors. Figure 9a shows the messages for different relative location of prey for agents’ observation, and Figure 9b shows the messages for different locations of the agent who sends the message. We can observe that there is some general trend in the message according to the class. We thus conclude that if the agents observe the prey, they encode into the message the relevant information that is helpful to estimate the location of the prey. The agents who receive this message interpret the message to select action.
+
+![](images/43538fe12aecd8b68a5ab3741ec2cebbbd9826febe138e719171e823a127d3ab.jpg)  
+Figure 9: Projection of encoded messages into 2D plane in PP.
+
+# C.2 PARTIAL OBSERVABILITY ISSUE IN SCHEDNET
+
+In MARL, partial observability issue is one of the major problems, and there are two typical ways to tackle this issue. First, using RNN structure to indirectly remember the history can alleviate the partial observability issues. Another way is to use the observations of other agents through communication among them. In this paper, we focused more on the latter because the goal of this paper is to show the importance of learning to schedule in a practical communication environment in which the shared medium contention is inevitable.
+
+Enlarging the observation through communication is somewhat orthogonal to considering temporal correlation. Thus, we can easily merge SchedNet with RNN which can be appropriate to some partially observable environments. We add one GRU layer into each of individual encoder, action selector, and weight generator of each agent, where each GRU cell has 64 hidden nodes.
+
+Figure 10 shows the result of applying RNN. We implement IDQN with RNN, and the results show that the average steps to complete tasks of IDQN with RNN is slightly smaller than that of IDQN with feed-forward network. In this case, RNN helps to improve the performance by tackling the partial observable issue. On the other hand, SchedNet-RNN and SchedNet achieve similar performance. We think that the communication in SchedNet somewhat resolves the partial observable issues, so the impact of considering temporal correlation with RNN is relatively small. Although applying RNN to SchedNet is not really that helpful in this simple environment, we expect that in a more complex environment, using the recurrent connection is more helpful.
+
+![](images/959c23c61980c5faeb8ec4e9b2916345afd797c70441c2b8132388d179758f3f.jpg)  
+Figure 10: Impact of applying RNN $k =$ 1 and $l = 2$ )
+
+# C.3 COOPERATIVE COMMUNICATION AND NAVIGATION
+
+Result in CCN Figure 11 illustrates the learning curve of 200,000 steps in CCN. In FC, since all agents can broadcast their message during execution, they achieve the best performance. IDQN and COMA in which no communication is allowed, take a longer time to complete the task compared to other baselines. The performances of both are similar because no cooperation can be achieved without the exchange of observations in this environment. As expected, SchedNet and DIAL outperform IDQN and COMA. Although DIAL works well when there is no contention constraint, under the contention constraint, the average number of steps to complete the task in DIAL(1) is larger than that of SchedNet-Top(1). This result shows the same tendency with the result in PP environment.
+
+![](images/975df2cf0801f9d4ac1f2f8188e6bd06b5915ab12a1a7cd4ae2e61fdfa6859ad.jpg)  
+Figure 11: Comparison with other baselines
+
+# D SCHEDULER FOR DISTRIBUTED EXECUTION
+
+Issues. The role of the scheduler is to consider the constraint due to accessing a shared medium, so that only $k < n$ agents may broadcast their encoded messages. $k$ is determined by the wireless communication environment. For example, under a single wireless channel environment where each agent is located in other agents’ interference range, $k = 1$ . Although the number of agents that can be simultaneously scheduled is somewhat more complex, we abstract it with a single number $k$ because the goal of this paper lies in studying the importance of considering scheduling constraints.
+
+There are two key challenges in designing the scheduler: $( i )$ how to schedule agents in a distributed manner for decentralized execution, and $( i i )$ how to strike a good balance between simplicity in implementation and training, and the integrity of reflecting the current practice of MAC (Medium Access Control) protocols.
+
+![](images/76e58b66862e0e6034d6d3244bfc54efafc14af427479450703eba92c6492305.jpg)  
+Figure 12: Proposed scheduling architecture. Each agent $i$ calculates its scheduling weight $w _ { i }$ from weight generator (WG), and the corresponding scheduling profile $c \in \{ 0 , 1 \} ^ { n }$ is determined by the scheduling algorithm (k) (WSA(k)), satisfying the condition $| | \boldsymbol { c } | | _ { 1 } = k$ .
+
+Weight-based scheduling To tackle the challenges addressed in the previous paragraph, we propose a scheduler, called weight-based scheduler (WSA), that works based on each agent’s individual weight coming from its observation. As shown in Figure 12, the role of WSA is to map from ${ \pmb w } = [ \bar { w _ { i } } ] _ { n }$ to $^ c$ . This scheduling is extremely simple, but more importantly, highly amenable to the philosophy of distributed execution. The remaining checkpoint is whether this principle is capable of efficiently approximating practical wireless scheduling protocols. To this end, we consider the following two weight-based scheduling algorithms among many different protocols that could be devised:
+
+$\circ \ T o p ( k )$ . Selecting top $k$ agents in terms of their weight values. ◦ Softmax $( k )$ . Computing softmax values $\begin{array} { r } { \sigma ( \boldsymbol { w } ) _ { i } = \frac { \boldsymbol { e ^ { w _ { i } } } } { \sum _ { j = 1 } ^ { n } e ^ { w _ { j } } } } \end{array}$ for each agent $i$ , and then randomly selecting $k$ agents with probability in proportion to their softmax values.
+
+$\mathrm { T o p } ( k )$ can be a nice abstraction of the MaxWeight (Tassiulas & Ephremides, 1992) scheduling principle or its distributed approximation (Yi et al., 2008), in which case it is known that different choices of weight values result in achieving different performance metrics, e.g., using the amount of messages queued for being transmitted as weight. Softmax $( k )$ can be a simplified model of CSMA (Carrier Sense Multiple Access), which forms a basis of 802.11 Wi-Fi. Due to space limitation, we refer the reader to Jiang & Walrand (2010) for detail. We now present how $T o p ( k )$ and Softmax(k) work.
+
+# D.1 CARRIER SENSE MULTIPLE ACCESS (CSMA)
+
+CSMA is the one of typical distributed MAC scheduling in wireless communication system. To show the feasibility of scheduling $T o p ( k )$ and Softmax $( k )$ in a distributed manner, we will explain the variant of CSMA. In this section, we first present the concept of CSMA.
+
+How does CSMA work? The key idea of CSMA is “listen before transmit”. Under a CSMA algorithm, prior to trying to transmit a packet, senders first check whether the medium is busy or idle, and then transmit the packet only when the medium is sensed as idle, i.e., no one is using the channel. To control the aggressiveness of such medium access, each sender maintains a backoff timer, which is set to a certain value based on a pre-defined rule. The timer runs only when the medium is idle, and stops otherwise. With the backoff timer, links try to avoid collisions by the following procedure:
+
+• Each sender does not start transmission immediately when the medium is sensed idle, but keeps silent until its backoff timer expires.   
+• After a sender grabs the channel, it holds the channel for some duration, called the holding time.
+
+Depending on how to choose the backoff and holding times, there can be many variants of CSMA that work for various purposes such as fairness and throughput. Two examples of these, $T o p ( k )$ and Softmax $( k )$ , are introduced in the following sections.
+
+# D.2 A VERSION OF Distributed Top(k)
+
+In this subsection, we introduce a simple distributed scheduling algorithm, called Distributed $T o p ( k )$ , which can work with SchedNet-Top $( k )$ . It is based on CSMA where each sender determines backoff and holding times as follows. In SchedNet, each agent generates the scheduling weight $w$ based on its own observation. The agent sets its backoff time as $1 - w$ where $w$ is its schedule weight, and it waits for backoff time before it tries to broadcast its message. Once it successfully broadcasts the message, it immediately releases the channel. Thus, the agent with the highest $w$ can grab the channel in a decentralized manner without any message passing. By repeating this for $k$ times, we can realize decentralized $\mathrm { T o p } ( k )$ scheduling.
+
+To show the feasibility of distributed scheduling, we implemented the Distributed $\mathrm { T o p } ( k )$ on Contiki network simulator (Dunkels et al., 2004) and run the trained agents for the PP task. In our experiment, $\mathrm { T o p } ( k )$ agents are successfully scheduled $98 \%$ of the time, and the $2 \%$ failures are due to probabilistic collisions in which one of the colliding agents is randomly scheduled by the default collision avoidance mechanism implemented in Contiki. In this case, agents achieve $9 8 . 9 \%$ performance compared to the case where $\mathrm { T o p } ( k )$ agents are ideally scheduled.
+
+# D.3 OCSMA ALGORITHM AND Softmax(k)
+
+In this section, we explain the relation between Softmax $( k )$ and the existing CSMA-based wireless MAC protocols, called oCSMA. When we use Softmax $( k )$ in the case of $k = 1$ , the scheduling algorithm directly relates to the channel selection probability of oCSMA algorithms. First, we explain how it works and show that the resulting channel access probability has a same form with Softmax $( k )$ .
+
+How does oCSMA work? It is also based on the basic CSMA algorithm. Once each agent generates its scheduling weight $w _ { i }$ , it sets $b _ { i }$ and $h _ { i }$ to satisfy $w _ { i } = \log ( b _ { i } h _ { i } )$ . It sets its backoff and holding times following exponential distributions with means $1 / b _ { i }$ and $h _ { i }$ , respectively. Based on these backoff and holding times, each agent runs the oCSMA algorithm. In this case, if all agents are in the communication range, the probability that agent $i$ is scheduled over time is as follows:
+
+$$
+s _ { i } ( \pmb { w } ) = \frac { \exp ( w _ { i } ) } { \sum _ { j = 1 } ^ { n } \exp ( w _ { j } ) } .
+$$
+
+We refer the readers to Jang et al. (2014) for detail.

parse/train/XwATtbX3oCz/XwATtbX3oCz.md

@@ -0,0 +1,296 @@
+# REVISITING POINT CLOUD CLASSIFICATION WITH ASIMPLE AND EFFECTIVE BASELINE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Processing point cloud data is an important component of many real-world systems. As such, a wide variety of point-based approaches have been proposed, reporting steady benchmark improvements over time. We study the key ingredients of this progress and uncover two critical results. First, we find that auxiliary factors like different evaluation schemes, data augmentation strategies, and loss functions, which are independent of the model architecture, make a large difference in performance. The differences are large enough that they obscure the effect of architecture. When these factors are controlled for, PointNet++, a relatively older network, performs competitively with recent methods. Second, a very simple projection-based method, which we refer to as SimpleView, performs surprisingly well. It achieves on par or better results than sophisticated state-ofthe-art methods on ModelNet40 while being half the size of PointNet $^ { + + }$ . It also outperforms state-of-the-art methods on ScanObjectNN, a real-world point cloud benchmark, and demonstrates better cross-dataset generalization.
+
+# 1 INTRODUCTION
+
+Processing 3D point cloud data accurately is crucial in many applications including autonomous driving (Navarro-Serment et al., 2010) and robotics (Rusu et al., 2009). In these settings, sensors like LIDAR produce unordered sets of points that correspond to object surfaces. Correctly classifying objects from this data is important for 3D scene understanding (Uy et al., 2019). While classical approaches for this problem have relied on hand-crafted features (Arras et al., 2007), recent efforts have focused on the design of deep neural networks (DNNs) to learn features directly from raw point cloud data (Qi et al., 2017a). Deep learning-based methods have proven effective in aggregating information across a set of 3D points to accurately classify objects.
+
+The most widely adopted benchmark for comparing methods for point cloud classification has been ModelNet40 (Wu et al., 2015b). The accuracy on ModelNet40 has steadily improved over the last few years from $8 9 . 2 \%$ by PointNet (Qi et al., 2017a) to $9 3 . 6 \%$ by RSCNN (Liu et al., 2019c) (Fig. 1). This progress is commonly perceived to be a result of better designs of network architectures. However, after performing a careful analysis of recent works we find two surprising results. First, we find that auxiliary factors including differing evaluation schemes, data augmentation strategies, and loss functions affect performance to such a degree that it can be difficult to disentangle improvements due to the network architecture. Second, we find that a very simple projection-based architecture works surprisingly well, outperforming state-of-the-art point-based architectures.
+
+In deep learning, as results improve on a benchmark, attention is generally focused on the novel architectures used to achieve those results. However, there are many factors beyond architecture design that influence performance including data augmentation and evaluation procedure. We refer to these additional factors as a method’s protocol. A protocol defines all details orthogonal to the network architecture that can be controlled to compare differing architectures. Note that it is possible for some specific form of loss or data augmentation to be tied to a specific architecture and inapplicable to other architectures. In these cases, it would be inappropriate to treat them as part of the protocol. However, for all the methods we consider in this paper, their losses and augmentation schemes are fully compatible with each other and can be considered independently.
+
+We do experiments to study the effect of protocol and discover that it accounts for a large difference in performance, so large as to obscure the contribution of a novel architecture. For example, the performance of the PointNe $^ { + + }$ architecture (Qi et al., 2017b) jumps from $9 0 . 0 { \pm } 0 . 3 $ to $9 3 . 3 { \pm } 0 . 3 $ , when switching from its original protocol to RSCNN’s protocol (Liu et al., 2019c). We further find that the protocols that lead to the strongest performance rely on feedback from the test set, which differs from conventional evaluation setups. We re-evaluate prior architectures using the best augmentation and loss functions, while not using any feedback from the test set. We find that by taking protocol into account, the PointNet+ $^ { - + }$ architecture performs competitively with more recent ones in various settings.
+
+![](images/66a78d68df7e5c9b9390ab71fe25ac55b4de36a395b6b5ee0ae24744c87118db.jpg)  
+Figure 1: Performance of point-based models on ModelNet40. Those using $> 1 0 2 4$ points or normals are marked with triangle. Line joins the topperforming models across times.
+
+![](images/fffc0bda7909a68a6101500cd83fbf982bc671ddf80ebc3aa5cfdd2c8ab5b377.jpg)  
+Figure 2: The SimpleView Architecture. The depth images are colored only for illustration. SimpleView takes in single channel depth images as input.
+
+In addition to the surprising importance of protocol, in reviewing past approaches, another surprising discovery is that a very simple projection based baseline works very well. One needs to simply project the points to depth maps along the orthogonal views, pass them through a light-weight CNN and fuse the features. We refer to this baseline as SimpleView.
+
+Compared to previous projection-based method (Roveri et al., 2018; Sarkar et al., 2018) for pointcloud classification, SimpleView is very simple. Prior methods have developed special modules for view selection, rendering, and feature merging, as well as use larger CNN backbones that are pretrained on ImageNet (refer to Sec. 2 for more details). In contrast, SimpleView has no such special operations, and only requires simple point projections, a much smaller CNN backbone, and no ImageNet pretraining.
+
+The discovery of SimpleView is surprising because recent state-of-the-art results have all been achieved by point-based architectures of increasing sophistication. In recent literature, it is often assumed that point-based methods are the superior choice for point-cloud processing as they “do not introduce explicit information loss” (Guo et al., 2020). Prior work has stated that “convolution operation of these methods lacks the ability to capture nonlocally geometric features” (Yan et al., 2020), that a projection-base method “often demands a huge number of views for decent performance” (Liu et al., 2019c), and that projection-based methods often “fine-tune a pre-trained image-based architecture for accurate recognition” (Liu et al., 2019c). It is thus surprising that a projection-based method could achieve state-of-the-art results with a simple architecture, only a few views, and no pretraining.
+
+On ModelNet40, SimpleView performs on par or better than more sophisticated state-of-the-art networks across various protocols, which includes the ones used by prior methods (Table. 3) as well as our protocol (Table. 5). At the same time, SimpleView outperforms state-of-the-art architectures on ScanObjectNN (Uy et al., 2019), a real-world dataset where point clouds are noisy (background points, occlusions, holes in objects) and are not axis-aligned. SimpleView also demonstrates better cross-dataset generalization than prior works. Furthermore, SimpleView uses less parameters than state-of-the-art networks (Table. 5).
+
+Note that we are not proposing a new architecture or method, but simply evaluating a simple and strong projection-based baseline for point-cloud classification that is largely ignored in the literature. We do not claim any novelty in the design of SimpleView because all of its components have appeared in the literature. Our contribution is showing that such a simple baseline works surprisingly well, which is a result absent in existing literature.
+
+It is worth noting that one might think that projection-based methods are not directly comparable with point-based methods because projection-based methods may have the full mesh as input, as opposed to just a point cloud. While this is true for existing results in the literature, it is not the case with SimpleView, whose input is the exact same point cloud given to a point-based method. In other words, SimpleView is directly comparable to a point-based method because they solve the exact same task.
+
+In summary, our contributions are threefold:
+
+• We show that training and evaluation factors independent of network architecture have a large impact on point-cloud classification performance. With these factors controlled for, PointNet++ performs as well as more recent architectures. We demonstrate how SimpleView, a very simple projection based baseline performs surprisingly well on point-cloud classification. It performs on par with or better than prior networks on ModelNet40 while using fewer parameters. It also outperforms state-of-the-art methods on real-world point-cloud classification and achieves better cross-dataset generalization.
+
+# 2 RELATED WORK
+
+Point-Based Methods for Point-Cloud Analysis: A broad class of DNNs have emerged to process 3D points directly (Simonovsky & Komodakis, 2017; Zaheer et al., 2017; Klokov & Lempitsky, 2017; Xu et al., 2018; Atzmon et al., 2018; Wang et al., 2018a; Li et al., 2018a; Groh et al., 2018; Ben-Shabat et al., 2018; Xie et al., 2018; Li et al., 2018b; Liu et al., 2019a; Thomas et al., 2019; Komarichev et al., 2019; Liu et al., 2019b; Yan et al., 2020; Su et al., 2018; Zhang et al., 2019; Liu et al., 2019a; Atzmon et al., 2018). PointNet (Qi et al., 2017a) proposed one of the first strategies, where features are updated for each point with MLP layers, and aggregated with global max pooling. However, no local comparisons are performed in PointNet, which motivates PointNet+ $^ +$ (Qi et al., 2017b). PointNet $^ { - + }$ breaks subsets of points into local regions that are processed first. More explicit modeling of the spatial relations between points is performed with more recent methods (Li et al., 2018b; Liu et al., 2019c; Wu et al., 2019). For example, PointConv learns functions to define continuous 3D convolutions that can be applied to arbitrary sets of points in a neighborhood (Wu et al., 2019). RSCNN uses MLPs conditioned on the spatial relationship of two points to update and aggregate features around an individual sampled point (Liu et al., 2019c). There exist many variations to these methods, but the emerging trend is an increase in sophistication.
+
+Projection-Based Methods for Point-Cloud Classification: Projection-based methods for point cloud classification have been proposed in the literature. Notably, Roveri et al. (2018) learn to predict viewing angles and classify images in an end-to-end differentiable way. They use the ResNet50 model, pretrained on ImageNet as their backbone and a depth-image generation pipeline. Sarkar et al. (2018) propose a special multi-height rendering and feature merging scheme, and use a larger backbone network pretrained on ImageNet. Ahmed et al. (2019) manually define important views for each object category, create binary edge maps, and train an ensemble of PointNet+ $^ +$ and CNN. However, numbers in Ahmed et al. (2019) are not directly comparable to other approaches as there is a manual alignment of objects in the test set which is different from the standard ModelNet40 test set. This was confirmed with the authors. It is worth noting that even though prior work has shown sophisticated operations to be useful for achieving good results, we find that when controlling for method protocols, strong performance can be achieved with fixed orthogonal views, a smaller network, no ImageNet pretraining, and simpler rendering of points.
+
+Projection-Based Methods for Other Point-Cloud Analysis Tasks: There is a rich literature for using projection-based methods on various point-cloud analysis problems like segmentation (Ladicky et al., 2010; Tighe & Lazebnik, 2010; Riemenschneider et al., 2014; Qin et al., 2018; \` Dai & Nießner, 2018; Kalogerakis et al., 2017; Tatarchenko et al., 2018), reconstruction (Pittaluga et al., 2019) and rendering (Aliev et al., 2019). Notably, Boulch et al. (2017) use point cloud density to create scene meshes, which are then put into a mesh renderer to generate many image views at different scales. Lawin et al. (2017) render a scene point cloud from 120 views for different modalities like color, depth, and surface normal. Information from multiple modalities is then fused to generate point-wise predictions. For a detailed survey of various projection approaches on different point-cloud processing tasks, we encourage readers to check the recent survey paper by (Guo et al., 2020). In this work, SimpleView serves as a stripped-down projection-based baseline for point-cloud classification that uses a few orthogonal views and simple point projections.
+
+Table 1: Summary of various protocols.   
+
+<table><tr><td>Protocol</td><td>Data Augmentation</td><td>Model Selection</td><td>Loss</td><td>Ensemble</td><td>Training Points</td></tr><tr><td>PointNet++</td><td>jitter, random rotation, random scaling and trans.</td><td>final model</td><td>cross-entropy</td><td>Rotation Vote</td><td>fixed</td></tr><tr><td>DGCNN</td><td>random scaling and trans.</td><td>best test model</td><td>smooth-loss</td><td>No vote</td><td>fixed</td></tr><tr><td>RSCNN</td><td>random scaling and trans.</td><td>best test model</td><td>cross-entropy</td><td>Repeated Scaling Vote</td><td>resampled</td></tr><tr><td>SimpleView</td><td>random scaling and trans.</td><td>final model</td><td>smooth-loss</td><td>No vote</td><td>fixed</td></tr></table>
+
+3D shape Analysis using Rendered Images and Voxels: Many works use images rendered from object meshes for 3D shape analysis (Maturana & Scherer, 2015; Wu et al., 2015b; Yu et al., 2018; Guo et al., 2016; Shi et al., 2015; Hackel et al., 2017; Song & Xiao, 2016; 2014; Huang & You, 2016; Tchapmi et al., 2017). MVCNN exemplifies this strategy by applying a shared CNN to many rendered views and max-pooling to aggregate features (Su et al., 2015). Subsequent approaches include RotationNet which trains the network to also predict the viewpoint for each image (Kanezaki et al., 2018), GVCNN which groups features from subsets of views together before aggregating into a final prediction (Feng et al., 2018), and hypergraph methods that consider the correlation across training samples (Zhang et al., 2018; Feng et al., 2019). One notable exception is Qi et al. (2016), who use a multi-resolution variant of MVCNN, but instead of object meshes, use a voxelized version of the object for rendering. In contrast to the prior view-based methods that use object meshes with point connectivity information, and render images using basic shading and/or depth; SimpleView takes as input raw point clouds.
+
+Another class of methods is voxel-based methods that convert points to a fixed 3D grid instead, which enables the use of 3D CNNs (Qi et al., 2016; Wu et al., 2015a; Maturana & Scherer, 2015). Given the added dimension, such methods are usually restricted to a much lower resolution to represent objects. Though some strategies such as octrees have been used to address those limitations (Wang et al., 2017), the advantages to processing 3D data directly in this manner do not yet appear to outweigh the additional overhead introduced.
+
+# 3 METHOD OVERVIEW
+
+# 3.1 VARIATIONS IN EXISTING PROTOCOLS
+
+We analyze the key ingredients in the progress in point-cloud classification. Critical to our study is controlling for factors which are independent of network architecture. We refer to the factors as a method’s protocol. A protocol used by one method can be transferred to another. For our study, we analyze a subset of the highest performing methods over the past few years. This choice was further based on availability and usability of official source-code. Specifically, we choose PointNet (Qi et al., 2017a), PointNet+ $^ +$ (Qi et al., 2017b), DGCNN (Wang et al., 2018b) and RSCNN (Liu et al., 2019c). Note that we also do direct comparisons to networks apart from the ones mentioned here (Table 4).
+
+For our purposes, we do not consider any variations in input, namely the use of surface normals or more than 1024 points. Using normals or more points have been shown to improve performance in the literature. Our objective is to study factors that are not commonly perceived as a major source of performance increase. So we scope our analysis to the most widely adopted input scheme which uses 1024 points with only $x , y , z$ coordinates.
+
+Data Augmentation: Various data augmentation strategies like jittering, random rotation along y-axis, random scaling and random translation. Different methods use different combinations of these augmentations. PointNet and PointNet+ $^ +$ use all the above augmentations. However, as objects in ModelNet40 are aligned, random rotation along y-axis adversely affects the performance of a model. Hence recent methods, including RSCNN and DGCNN, do not use it. They use only random translation and random scaling. Some methods including PointCNN make a distinction between whether or not random rotation along y-axis is used, but it is not a common practice.
+
+Input Points: PointNet and PointNet+ $^ +$ use a fixed set of 1024 points per object to train the network. We refer to it as the fixed points strategy. RSCNN and PointCNN randomly sample points during each epoch, effectively exposing the model to more than 1024 points per object during the training process. We refer to this as the resampled points strategy.
+
+Table 2: DGCNN aug. and smooth loss improve the performance of all architectures   
+
+<table><tr><td colspan="2">Data Augmentation</td><td colspan="2">Model Selection</td><td colspan="2">Loss</td><td colspan="4">Architecture</td></tr><tr><td>PN++</td><td>DGCNN</td><td>Final</td><td>Best Test</td><td>C.E.</td><td>Smooth</td><td>PointNet</td><td>PN++</td><td>DGCNN</td><td>RSCNN</td></tr><tr><td></td><td></td><td></td><td>√</td><td>?</td><td></td><td>89.7±0.3</td><td>91.0±0.3</td><td>90.5±0.2</td><td>90.4±0.3</td></tr><tr><td>√</td><td></td><td>&gt;&gt;&gt;</td><td></td><td></td><td></td><td>89.0±0.2</td><td>89.8±0.2</td><td>90.0±0.4</td><td>89.4±0.1</td></tr><tr><td></td><td>√</td><td></td><td></td><td>√</td><td></td><td>89.1±0.2</td><td>92.1±0.1</td><td>91.1±0.3</td><td>91.1±0.3</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>√</td><td>89.2±0.9</td><td>92.7±0.1</td><td>91.9±0.2</td><td>91.7±0.3</td></tr></table>
+
+Loss Function: cross-entropy (CE) is used by most of the methods. However, DGCNN uses smooth-loss, where the ground-truth labels are smoothed out before calculating cross-entropy. We observe that smooth-loss improves the performance of all network architectures.
+
+Selecting Model for Testing: PointNet and PointNet+ $^ { - + }$ use the final converged model to evaluate on the test set. Since the number of epochs is a hyper-parameter that depends on factors like data, model, optimizer, and loss, in our experiments, we create a validation set from the training set to tune the number of epochs. We then retrain the model with the complete training set to the tuned number of epochs. We refer to this strategy as final model selection. We find that some methods including DGCNN and RSCNN evaluate the model on the test set after every epoch and use the best test performance as the final performance. We refer to this strategy as best test model selection.
+
+Ensemble Scheme: Some methods use an ensemble to further improve the performance. PointNet and PointNet+ $^ +$ apply the final network to multiple rotated and shuffled versions of the point cloud, and average the predictions to make the final prediction. We refer to this strategy as Rotation Vote. The shuffling operation induces randomness in prediction for PointNet $^ { + + }$ and RSCNN as they are not strictly invariant to the order of the points (Sec. 3.3 in Qi et al. (2017b)). Hence, while evaluating Rotation Vote, we do the inference 10 times per run for PointNet $^ { + + }$ and RSCNN, and report mean and standard deviation. SimpleView and PointNet are invariant to the order of the points and hence are not affected by shuffling. Some methods, including RSCNN and DensePoint, create multiple randomly scaled and randomly sampled versions of a test object. They then evaluate the final network on these multiple versions of the object and average the prediction. Since the scaling is random, it makes the test set performance random as well. RSCNN and DensePoint repeat this procedure 300 times on the test set and report the best accuracy. We refer to it as Repeated Scaling Vote. DGCNN does not use any ensemble.
+
+Table 1 summarizes the PointNet $^ { + + }$ , DGCNN and RSCNN protocols. Besides these three protocols, we also include variants of these protocols in table 3 and table 4, such as PointNet $^ { + + }$ no Vote (i.e. PointNe $^ { + + }$ but without the Rotation Vote), DGCNN CE (i.e. DGCNN but with CE loss intead of smooth loss), DGCNN CE Final (i.e. DGCNN CE but with final model selection instead of best test model selection) and RSCNN no Vote (i.e. RSCNN but without the Rotation Vote). These protocols represent prototypical settings and have been used with slight modifications in many other prior works.
+
+For example, DeepSets (Zaheer et al., 2017) used the PointNet $^ { + + }$ no Vote protocol without jittering and translation; SO-Net (Li et al., 2018a) used the DGCNN CE protocol with jittering and random scaling instead of random scaling and translation; 3DmFV (Ben-Shabat et al., 2018) used the DGCNN $C E$ protocol with additional jittering; PCNN (Atzmon et al., 2018) used the DGCNN CE Final protcol1; PointCNN (Li et al., 2018b) used the DGCNN CE protocol with randomly sampled points and small $( 1 0 ^ { \circ } )$ rotation augmentation; DensePoint (Liu et al., 2019b) used the RSCNN protocol; PointASL (Yan et al., 2020) used the DGCNN CE protocol but with additional point jittering augmentation and voting.
+
+Our Protocol: Based on our findings, we define our SimpleView protocol, which uses the best augmentation and loss functions while not using any information from the test set. Table 2 shows that
+
+Table 3: Performance of various architectures on ModelNet40. Protocol affects performance by a large amount. SimpleView performs on par or better than prior architectures across protocols.   
+
+<table><tr><td rowspan="2">Protocol→ Architecture ↓</td><td colspan="2">PointNet++</td><td colspan="2">RSCNN</td><td colspan="2">DGCNN</td></tr><tr><td>no Vote</td><td>Vote</td><td>no Vote</td><td>Vote</td><td>CE</td><td>Smooth</td></tr><tr><td>PointNet</td><td>89.0 ± 0.2</td><td>89.1 ± 0.2</td><td>90.0 ± 0.3</td><td>90.1 ± 0.2</td><td>90.1 ± 0.2</td><td>90.5 ± 0.1</td></tr><tr><td>PointNet++</td><td>89.8 ± 0.2</td><td>90.0 ± 0.3</td><td>92.7 ± 0.1</td><td>93.3 ± 0.3</td><td>92.6 ± 0.2</td><td>93.1 ± 0.2</td></tr><tr><td>DGCNN</td><td>90.0 ± 0.4</td><td>90.5 ± 0.4</td><td>92.2 ± 0.1</td><td>92.8 ± 0.5</td><td>91.9 ± 0.2</td><td>92.7 ± 0.1</td></tr><tr><td>RSCNN</td><td>89.4 ± 0.1</td><td>90.2 ± 0.2</td><td>92.1 ± 0.1</td><td>92.5 ± 0.2</td><td>91.9 ± 0.2</td><td>92.5 ± 0.1</td></tr><tr><td>SimpleView</td><td>90.7 ± 0.3</td><td>91.0 ± 0.2</td><td>92.9 ± 0.2</td><td>93.2 ± 0.1</td><td>93.1 ± 0.1</td><td>93.6 ± 0.3</td></tr></table>
+
+Table 4: Performance of various architectures on ModelNet40. Includes prior works not in Table 3. \* indicates small differences in protocol as identified in Sec. 3.1   
+
+<table><tr><td>Architecture</td><td>#Points</td><td>Closest Protocol</td><td>Acc.</td><td>PN++ Acc.</td><td>SimpleView Acc.</td></tr><tr><td>DeepSets (Zaheer et al.)</td><td>5000</td><td>PointNet++ no Vote*</td><td>90.0 ± 0.3</td><td>89.8 ± 0.2</td><td>90.7 ± 0.3</td></tr><tr><td>SO-Net (Li et al.)</td><td>2048</td><td>DGCNN CE*</td><td>90.9</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr><tr><td>3DmFV (Ben-Shabat et al.)</td><td>1024</td><td>DGCNN CE*</td><td>91.4</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr><tr><td>PCNN (Atzmon et al.)</td><td>1024</td><td>DCNN CE Final*</td><td>92.3</td><td>92.1 ± 0.1</td><td>92.5 ± 0.3</td></tr><tr><td>PointCNN (Li et al.)</td><td>1024</td><td>DGCNN CE*</td><td>92.5</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr><tr><td>DensePoint (Liu et al.)</td><td>1024</td><td>RSCNN no Vote</td><td>92.8</td><td>92.7 ± 0.1</td><td>92.9 ± 0.2</td></tr><tr><td>RSCNN-Multi (Liu et al.)</td><td>1024</td><td>RSCNN no Vote</td><td>92.9</td><td>92.7 ± 0.1</td><td>92.9 ± 0.2</td></tr><tr><td>PointANSL (Yan et al.)</td><td>1024</td><td>DGCNN CE*</td><td>92.9</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr></table>
+
+DGCNN’s augmentation (i.e random translation and scaling) and smooth-loss improve performance of all prior networks, so we use them in the SimpleView protocol. Further, similar to PointNet, PointNet $^ { + + }$ and DGCNN, we use the fixed dataset of 1024 points instead of re-sampling different points at each epoch. Re-sampling points for each epoch effectively increases the training dataset of points, making numbers incomparable to methods using a fixed dataset. We avoid any feedback from the test set and use the final model selection, where we first tune the number of epochs on the validation set then retrain the model on the entire train set. Lastly, similar to DGCNN, we do not use ensemble as it is more standard in Machine Learning to compare models without ensemble.
+
+# 3.2 SIMPLEVIEW
+
+Given a set of points SimpleView, projects them onto the six orthogonal planes to create sparse depth images. It then extracts features from the depth images using a CNN and fuses, which is then used to classify the point-cloud as shown in Fig. 1.
+
+Generating Depth Images from Point Cloud: Let $( x , y , z )$ be the coordinates of a point in the point cloud with respect to the camera. We apply perspective projection to get the 2D coordinate $( \tilde { x } \ = \ x / z , \tilde { y } \ = \ y / z )$ of $\mathbf { p }$ at depth $z$ . We also do ablations with orthographic projection and found perspective projection to work better (Table. 7). Since coordinates on image plane have to be discrete, we use $( \bar { \lceil x \rceil } , \bar { \lceil y \rceil } )$ to be the final coordinate of $\mathbf { p }$ on the image plane. Multiple points may be projected to the same discrete location on the image plane. To produce depth value at an image location, we do ablations on two choices, one the minimum depth of all points, and other weighted average depth with more weight $\textstyle { \binom { 1 } { z } }$ given to closer points (Table. 7). Empirically, we find both perform similar with the later performing slightly better. This could be because of reduction in noise due to the averaging of nearby pixels on the surface. The depth images are of resolution $1 2 8 \mathrm { ~ X ~ } 1 2 8$ .
+
+SimpleView Architecture: To make the number of parameters comparable to point-based methods, we use ResNet18 with one-fourth filters (ResNet18/4) as the backbone. For fusing features, we do ablation with two choices, pooling and concatenation. Empirically, we find concatenation to work better than pooling them (Table. 7). This could be because pooling features throws away the view information like which views are adjacent to one another. One concern could be that concatenation could make features sensitive to viewpoint, and hence the network could fail on rotated objects.
+
+Table 5: Performance of various architectures on ModelNet40 when using the best dataaugmentation and loss function; and not using any feedback from test set. SimpleView outperforms prior architectures, while having fewest parameters and comparable inference time.   
+
+<table><tr><td>Architecture ↓</td><td>Acc.</td><td>Class Acc.</td><td>Para. (M)</td><td>Time (ms)</td></tr><tr><td>PointNet</td><td>89.2 ± 0.9</td><td>85.1 ± 0.6</td><td>3.5</td><td>3.0</td></tr><tr><td>PointNet++</td><td>92.7 ± 0.1</td><td>90.0± 0.3</td><td>1.7</td><td>20.7</td></tr><tr><td>DGCNN</td><td>92.3 ± 0.3</td><td>89.1 ± 0.3</td><td>1.8</td><td>7.3</td></tr><tr><td>RSCNN</td><td>91.7 ± 0.3</td><td>88.5 ± 0.4</td><td>1.3</td><td>4.3</td></tr><tr><td>SimpleView</td><td>93.0 ± 0.4</td><td>90.5 ± 0.8</td><td>0.8</td><td>5.0</td></tr></table>
+
+Table 6: Performance of various architectures on ScanObjectNN, and cross-dataset generalization. SimpleView achieves state-of-the-art results and shows better cross dataset generalization. Numbers for prior works are from (Uy et al., 2019).   
+
+<table><tr><td>Architecture ↓</td><td>TR: SONN TE: SONN</td><td>TR:MN40 TE: SONN</td><td>TR: SONN TE: MN40</td></tr><tr><td>3DmFV (Shabat et al.)</td><td>63.0</td><td>24.9</td><td>51.5</td></tr><tr><td>PointNet (Qi et al.)</td><td>68.2</td><td>31.1</td><td>50.9</td></tr><tr><td>SpiderCNN (Xu et al.)</td><td>73.7</td><td>30.9</td><td>46.6</td></tr><tr><td>PointNet++ (Qi et al.)</td><td>77.9</td><td>32.0</td><td>47.4</td></tr><tr><td>DGCNN (Wang et al.)</td><td>78.1</td><td>36.8</td><td>54.7</td></tr><tr><td>PointCNN (Li et al.)</td><td>78.5</td><td>24.6</td><td>49.2</td></tr><tr><td>SimpleView</td><td>79.5±0.5</td><td>40.5±1.4</td><td>57.9±2.1</td></tr></table>
+
+Table 7: Ablation of various choices for SimpleView on ModelNet40. The performance is evaluatated on the validation set.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Number of Views</td><td colspan="2">Image Projection</td><td colspan="2">Feature Fusion</td><td colspan="2">Image Depth</td></tr><tr><td>1</td><td>3</td><td>6</td><td>Orthographic</td><td>Perspective</td><td>Pool</td><td>Concat</td><td>Minimum</td><td>Weighted Avg.</td></tr><tr><td>Accuracy</td><td>90.7±0.1</td><td>92.1 ± 0.2</td><td>92.9±0.3</td><td>92.7 ± 0.3</td><td>92.9±0.3</td><td>91.8± 0.3</td><td>92.9± 0.3</td><td>92.8± 0.4</td><td>92.9± 0.3</td></tr></table>
+
+However, empirically, we observe that this issue is largely mitigated by rotation augmentation and SimpleView is able to achieve state-of-the-art performance on ScanObjectNN where objects are rotated. The point-clouds are scaled to be in $[ \hat { 1 } , - 1 ] ^ { 3 }$ , we keep the cameras at a distance of 1.4 units from the center with $9 0 °$ fov. We also do ablations with different number of views, comparing only front views, three orthogonal views and six orthogonal views. We find that using all six views performs the best (Table 7). We do not use ImageNet pretraining, thus making the comparison with point-based methods strictly fair, without any additional data.
+
+# 4 EXPERIMENTS
+
+ModelNet40: ModelNet40 is a the most widely adopted benchmark for point-cloud classification. It contains objects from 40 common categories. There are 9840 objects in the training set and 2468 in the test set. Objects are aligned to a common up and front direction.
+
+ScanObjectNN: ScanObjectNN is a recent real-world point cloud classification dataset. It consists of 15 classes, 11 of which are also in ModelNet40. There are a total of $1 5 \mathrm { k }$ objects in the dataset. Unlike ModelNet40, the objects in ScanObjectNN are obtained from real-world 3D scans. Hence, point clouds are noisy (occlusions, background points) and have geometric distortions such as holes. Also, unlike ModelNet40, the objects are not axis-aligned.
+
+# 4.1 EXPERIMENTS ON MODELNET40
+
+Implementation Details: We use PyTorch (Paszke et al., 2019) to implement all models and protocols while reusing the official code wherever possible. We use the official version of DGCNN and RSCNN. We confirm with the authors that the code for RSCNN-Multi, another version of RSCNN, is yet to be released. Hence we use the reported numbers of RSCNN-Multi in Table 4. PointNet and PointNet $^ { + + }$ are officially released in TensorFlow (Abadi et al., 2015). For PointNet, we adapt our code from PointNet.pytorch (Xia, accessed June, 2020) as recommended in the official repository. For PointNet++, we adapt the model code from Pointnet2 PyTorch (Wijmans, accessed June, 2020). We further make sure that the third party PyTorch code closely matches the official TensorFlow code.
+
+We use Adam (Kingma & Ba, 2014) with an initial learning rate of 1e-3 and a decay-on-plateau learning rate scheduler. The batch size and weight decay for each model are kept the same as the official version in Table 3. We use a batch size of 18 and no weight decay for SimpleView. To give the prior models the best chance on our protocol (Table 5), we additionally tune their hyperparameters on the validation set. We find that the official hyper-parameters already perform close to optimal. We train each model for 1000 epochs. Since there are small variations in final performance across different runs, we do 4 runs and report the mean and standard deviation.
+
+![](images/8b40ab61d4067d6d54c88f0a10b5099143d8f8d1f0873543ec9e9d09ffed3114.jpg)  
+Figure 3: Failure Cases for SimpleView and PointNet++. The first row shows cases where both SimpleView and PointNe $^ { + + }$ fail; the second row shows cases where SimpleView succeeds but PointNet+ $^ +$ fails; the third row shows cases where SimpleView fails but PointNet++ succeeds.
+
+Performance under various Prior Protocols: Table 3 shows the performance different architectures under various protocols. The mean performance of PointNet $^ { + + }$ improves from $8 9 . 8 \%$ to $9 3 . 3 \%$ when we switch from the PointNet $^ { + + }$ no Vote to the RSCNN Vote protocol. Similarly the performance of SimpleView improves from $9 0 . 7 \%$ to $9 3 . 6 \%$ when we switch from PointNet $^ { + + }$ no Vote to DGCNN Smooth. Since there is variance in performance across runs, we refrain from making any claims about absolute ordering between prior works. However, we do observe that in terms of mean performance, SimpleView performs on par or better than other methods under all protocols. Note that in RSCNN Vote, voting on the test set is done 300 times with reshuffled and randomly augmented points, from which the highest accuracy is selected. Hence models that have the largest variance in prediction, i.e. PointNet++ and RSCNN gain the most from it, as they are not strictly invariant to the order of points (Sec. 3.3 in Qi et al. (2017b)).
+
+Performance under the SimpleView Protocol: Table 5 shows that SimpleView outperforms prior architectures on our controlled protocol in terms of mean performance. SimpleView has the fewest number of parameters and a competitive inference speed. Inference speed is measured on an NVIDIA 2080Ti averaged across 100 runs.
+
+Fig. 3 show examples where both SimpleView and PointNet $^ { + + }$ fail, as well as examples where one of them fails and the other succeeds. Qualitatively, we find that the failure modes of SimpleView and PointNet+ $^ { \cdot + }$ are similar. We also find that a major failure mode in both SimpleView and PointNet+ $^ +$ is the confusion between the ‘flower pot’ and ‘plant’ category (see Sec. A Fig. I and Fig. II). This could be because of the lack of color information.
+
+Comparison with More Methods: In Table 4, we do one-on-one comparison between SimpleView and recent state-of-the-art methods, other than PointNet, PointNet++, RSCNN and DGCNN. We identify the closest protocol to the one used in the paper from the ones we evaluate. Table 4 shows the competitiveness of PointNet+ $^ +$ and SimpleView with other recent state-of-the-art methods.
+
+# 4.2 EXPERIMENTS ON SCANOBJECTNN
+
+Implementation Details: ScanObjectNN’s official repository trains and evaluates the state-of-theart models under the same protocol. We implement SimpleView in TensorFlow and use the official ScanObjectNN protocol for fairness. This protocol is different from the SimpleView protocol as it normalizes the point clouds and randomly samples points. We optimize our model with Adam. We use a batch size of 20 and no weight decay to train SimpleView for 300 epochs with an initial learning rate 0.001, and use the final model for testing. We use standard image-based cropping and scaling augmentation to prevent over-fitting. The hyper-parameter for cropping and scaling is found on a validation set made from ScanObjecNN’s train set. We conduct 4 runs for SimpleView. ScanObjectNN does not use a fixed set of points during test time. It instead randomly samples points from the point cloud, which adds randomness to test set performance. Hence, we evaluate each run 10 times. We report the final performance as the mean and variance of the 40 evaluations (4 runs $\times$ 10 evaluations per run).
+
+Performance on ScanObjectNN: As shown in Table 6, SimpleView outperforms prior networks on ScanObjectNN. This shows the SimpleView is effective in real world settings, with noisy and misaligned point clouds. We also perform transfer experiments to test generalizability of SimpleView. We train on ScanObjectNN and test on ModelNet40 and vice versa. Table 6 shows that SimpleView transfers across datasets better than prior methods.
+
+# 5 DISCUSSION
+
+In this work, we demonstrate how auxiliary factors orthogonal to the network architecture have a large effect on performance for point-cloud classification. When controlling for these factors, we find that a relatively older method, PointNet $^ { - + }$ (Qi et al., 2017b), performs competitively with more recent ones. Furthermore, we show that a simple baseline performs on par or better than state-ofthe-art architectures.
+
+Our results show that for future progress we should control for protocols while comparing network architectures. Our code base could serve as a useful resource for developing new models and comparing them with prior works. Our results show that the existing evidence for point-based methods is not as strong when auxiliary factors are properly controlled for, and that SimpleView is a strong baseline. But our results are not meant to discourage future research on point-based methods. It is still entirely possible that point-based methods come out ahead with additional innovations. We believe it is beneficial to explore competing approaches, including the ones that are underperforming at a particular time, as long as the results are compared in a controlled manner.
+
+Our analysis in this work was limited to point cloud classification, which is an important problem in 3D scene understanding and forms a critical part of object detection and retrieval systems. An exciting future direction would be to expand this analysis to other problems that involve point cloud data such as scene and part segmentation.
+
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+
+# A APPENDIX
+
+![](images/5f0f28b05536c705d795036eb13399fa271ef793a5d576ab9384fb248a23d99e.jpg)  
+Figure I: Confusion matrix for SimpleView when trained under our protocol
+
+![](images/74d59f4eb55d04b38c5570c36a9863bc1a14d979aff009a5cca933d3e66ad187.jpg)  
+Figure II: Confusion matrix for PointNet++ when trained under our protocol
+
+Table I: Performance of various architectures on ModelNet40 when using different amount of training data.   
+
+<table><tr><td>Per. of Training Data</td><td>RSCNN</td><td>DGCNN</td><td>PointNet</td><td>PointNet++</td><td>SimpleView</td></tr><tr><td>25%</td><td>88.2 ± 0.4</td><td>89.1 ± 0.2</td><td>86.3 ± 0.4</td><td>89.6 ± 0.4</td><td>89.7 ± 0.3</td></tr><tr><td>50%</td><td>90.4 ± 0.4</td><td>91.0 ± 0.3</td><td>88.2 ± 0.3</td><td>91.5 ± 0.2</td><td>92.1 ± 0.3</td></tr><tr><td>100 %</td><td>91.7 ± 0.3</td><td>92.3 ± 0.3</td><td>89.2 ± 0.9</td><td>92.7 ± 0.3</td><td>93.0 ± 0.4</td></tr></table>

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+        "text": "REVISITING POINT CLOUD CLASSIFICATION WITH ASIMPLE AND EFFECTIVE BASELINE",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
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+        "text": "Processing point cloud data is an important component of many real-world systems. As such, a wide variety of point-based approaches have been proposed, reporting steady benchmark improvements over time. We study the key ingredients of this progress and uncover two critical results. First, we find that auxiliary factors like different evaluation schemes, data augmentation strategies, and loss functions, which are independent of the model architecture, make a large difference in performance. The differences are large enough that they obscure the effect of architecture. When these factors are controlled for, PointNet++, a relatively older network, performs competitively with recent methods. Second, a very simple projection-based method, which we refer to as SimpleView, performs surprisingly well. It achieves on par or better results than sophisticated state-ofthe-art methods on ModelNet40 while being half the size of PointNet $^ { + + }$ . It also outperforms state-of-the-art methods on ScanObjectNN, a real-world point cloud benchmark, and demonstrates better cross-dataset generalization. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Processing 3D point cloud data accurately is crucial in many applications including autonomous driving (Navarro-Serment et al., 2010) and robotics (Rusu et al., 2009). In these settings, sensors like LIDAR produce unordered sets of points that correspond to object surfaces. Correctly classifying objects from this data is important for 3D scene understanding (Uy et al., 2019). While classical approaches for this problem have relied on hand-crafted features (Arras et al., 2007), recent efforts have focused on the design of deep neural networks (DNNs) to learn features directly from raw point cloud data (Qi et al., 2017a). Deep learning-based methods have proven effective in aggregating information across a set of 3D points to accurately classify objects. ",
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+        "text": "The most widely adopted benchmark for comparing methods for point cloud classification has been ModelNet40 (Wu et al., 2015b). The accuracy on ModelNet40 has steadily improved over the last few years from $8 9 . 2 \\%$ by PointNet (Qi et al., 2017a) to $9 3 . 6 \\%$ by RSCNN (Liu et al., 2019c) (Fig. 1). This progress is commonly perceived to be a result of better designs of network architectures. However, after performing a careful analysis of recent works we find two surprising results. First, we find that auxiliary factors including differing evaluation schemes, data augmentation strategies, and loss functions affect performance to such a degree that it can be difficult to disentangle improvements due to the network architecture. Second, we find that a very simple projection-based architecture works surprisingly well, outperforming state-of-the-art point-based architectures. ",
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+        "text": "In deep learning, as results improve on a benchmark, attention is generally focused on the novel architectures used to achieve those results. However, there are many factors beyond architecture design that influence performance including data augmentation and evaluation procedure. We refer to these additional factors as a method’s protocol. A protocol defines all details orthogonal to the network architecture that can be controlled to compare differing architectures. Note that it is possible for some specific form of loss or data augmentation to be tied to a specific architecture and inapplicable to other architectures. In these cases, it would be inappropriate to treat them as part of the protocol. However, for all the methods we consider in this paper, their losses and augmentation schemes are fully compatible with each other and can be considered independently. ",
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+        "text": "We do experiments to study the effect of protocol and discover that it accounts for a large difference in performance, so large as to obscure the contribution of a novel architecture. For example, the performance of the PointNe $^ { + + }$ architecture (Qi et al., 2017b) jumps from $9 0 . 0 { \\pm } 0 . 3 $ to $9 3 . 3 { \\pm } 0 . 3 $ , when switching from its original protocol to RSCNN’s protocol (Liu et al., 2019c). We further find that the protocols that lead to the strongest performance rely on feedback from the test set, which differs from conventional evaluation setups. We re-evaluate prior architectures using the best augmentation and loss functions, while not using any feedback from the test set. We find that by taking protocol into account, the PointNet+ $^ { - + }$ architecture performs competitively with more recent ones in various settings. ",
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+        "img_path": "images/66a78d68df7e5c9b9390ab71fe25ac55b4de36a395b6b5ee0ae24744c87118db.jpg",
+        "image_caption": [
+            "Figure 1: Performance of point-based models on ModelNet40. Those using $> 1 0 2 4$ points or normals are marked with triangle. Line joins the topperforming models across times. "
+        ],
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+        "image_caption": [
+            "Figure 2: The SimpleView Architecture. The depth images are colored only for illustration. SimpleView takes in single channel depth images as input. "
+        ],
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+        "text": "In addition to the surprising importance of protocol, in reviewing past approaches, another surprising discovery is that a very simple projection based baseline works very well. One needs to simply project the points to depth maps along the orthogonal views, pass them through a light-weight CNN and fuse the features. We refer to this baseline as SimpleView. ",
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+        "text": "Compared to previous projection-based method (Roveri et al., 2018; Sarkar et al., 2018) for pointcloud classification, SimpleView is very simple. Prior methods have developed special modules for view selection, rendering, and feature merging, as well as use larger CNN backbones that are pretrained on ImageNet (refer to Sec. 2 for more details). In contrast, SimpleView has no such special operations, and only requires simple point projections, a much smaller CNN backbone, and no ImageNet pretraining. ",
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+        "text": "The discovery of SimpleView is surprising because recent state-of-the-art results have all been achieved by point-based architectures of increasing sophistication. In recent literature, it is often assumed that point-based methods are the superior choice for point-cloud processing as they “do not introduce explicit information loss” (Guo et al., 2020). Prior work has stated that “convolution operation of these methods lacks the ability to capture nonlocally geometric features” (Yan et al., 2020), that a projection-base method “often demands a huge number of views for decent performance” (Liu et al., 2019c), and that projection-based methods often “fine-tune a pre-trained image-based architecture for accurate recognition” (Liu et al., 2019c). It is thus surprising that a projection-based method could achieve state-of-the-art results with a simple architecture, only a few views, and no pretraining. ",
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+        "text": "On ModelNet40, SimpleView performs on par or better than more sophisticated state-of-the-art networks across various protocols, which includes the ones used by prior methods (Table. 3) as well as our protocol (Table. 5). At the same time, SimpleView outperforms state-of-the-art architectures on ScanObjectNN (Uy et al., 2019), a real-world dataset where point clouds are noisy (background points, occlusions, holes in objects) and are not axis-aligned. SimpleView also demonstrates better cross-dataset generalization than prior works. Furthermore, SimpleView uses less parameters than state-of-the-art networks (Table. 5). ",
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+        "type": "text",
+        "text": "Note that we are not proposing a new architecture or method, but simply evaluating a simple and strong projection-based baseline for point-cloud classification that is largely ignored in the literature. We do not claim any novelty in the design of SimpleView because all of its components have appeared in the literature. Our contribution is showing that such a simple baseline works surprisingly well, which is a result absent in existing literature. ",
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+        "text": "It is worth noting that one might think that projection-based methods are not directly comparable with point-based methods because projection-based methods may have the full mesh as input, as opposed to just a point cloud. While this is true for existing results in the literature, it is not the case with SimpleView, whose input is the exact same point cloud given to a point-based method. In other words, SimpleView is directly comparable to a point-based method because they solve the exact same task. ",
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+        "type": "text",
+        "text": "In summary, our contributions are threefold: ",
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+        "text": "• We show that training and evaluation factors independent of network architecture have a large impact on point-cloud classification performance. With these factors controlled for, PointNet++ performs as well as more recent architectures. We demonstrate how SimpleView, a very simple projection based baseline performs surprisingly well on point-cloud classification. It performs on par with or better than prior networks on ModelNet40 while using fewer parameters. It also outperforms state-of-the-art methods on real-world point-cloud classification and achieves better cross-dataset generalization. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Point-Based Methods for Point-Cloud Analysis: A broad class of DNNs have emerged to process 3D points directly (Simonovsky & Komodakis, 2017; Zaheer et al., 2017; Klokov & Lempitsky, 2017; Xu et al., 2018; Atzmon et al., 2018; Wang et al., 2018a; Li et al., 2018a; Groh et al., 2018; Ben-Shabat et al., 2018; Xie et al., 2018; Li et al., 2018b; Liu et al., 2019a; Thomas et al., 2019; Komarichev et al., 2019; Liu et al., 2019b; Yan et al., 2020; Su et al., 2018; Zhang et al., 2019; Liu et al., 2019a; Atzmon et al., 2018). PointNet (Qi et al., 2017a) proposed one of the first strategies, where features are updated for each point with MLP layers, and aggregated with global max pooling. However, no local comparisons are performed in PointNet, which motivates PointNet+ $^ +$ (Qi et al., 2017b). PointNet $^ { - + }$ breaks subsets of points into local regions that are processed first. More explicit modeling of the spatial relations between points is performed with more recent methods (Li et al., 2018b; Liu et al., 2019c; Wu et al., 2019). For example, PointConv learns functions to define continuous 3D convolutions that can be applied to arbitrary sets of points in a neighborhood (Wu et al., 2019). RSCNN uses MLPs conditioned on the spatial relationship of two points to update and aggregate features around an individual sampled point (Liu et al., 2019c). There exist many variations to these methods, but the emerging trend is an increase in sophistication. ",
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+    {
+        "type": "text",
+        "text": "Projection-Based Methods for Point-Cloud Classification: Projection-based methods for point cloud classification have been proposed in the literature. Notably, Roveri et al. (2018) learn to predict viewing angles and classify images in an end-to-end differentiable way. They use the ResNet50 model, pretrained on ImageNet as their backbone and a depth-image generation pipeline. Sarkar et al. (2018) propose a special multi-height rendering and feature merging scheme, and use a larger backbone network pretrained on ImageNet. Ahmed et al. (2019) manually define important views for each object category, create binary edge maps, and train an ensemble of PointNet+ $^ +$ and CNN. However, numbers in Ahmed et al. (2019) are not directly comparable to other approaches as there is a manual alignment of objects in the test set which is different from the standard ModelNet40 test set. This was confirmed with the authors. It is worth noting that even though prior work has shown sophisticated operations to be useful for achieving good results, we find that when controlling for method protocols, strong performance can be achieved with fixed orthogonal views, a smaller network, no ImageNet pretraining, and simpler rendering of points. ",
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+    {
+        "type": "text",
+        "text": "Projection-Based Methods for Other Point-Cloud Analysis Tasks: There is a rich literature for using projection-based methods on various point-cloud analysis problems like segmentation (Ladicky et al., 2010; Tighe & Lazebnik, 2010; Riemenschneider et al., 2014; Qin et al., 2018; \\` Dai & Nießner, 2018; Kalogerakis et al., 2017; Tatarchenko et al., 2018), reconstruction (Pittaluga et al., 2019) and rendering (Aliev et al., 2019). Notably, Boulch et al. (2017) use point cloud density to create scene meshes, which are then put into a mesh renderer to generate many image views at different scales. Lawin et al. (2017) render a scene point cloud from 120 views for different modalities like color, depth, and surface normal. Information from multiple modalities is then fused to generate point-wise predictions. For a detailed survey of various projection approaches on different point-cloud processing tasks, we encourage readers to check the recent survey paper by (Guo et al., 2020). In this work, SimpleView serves as a stripped-down projection-based baseline for point-cloud classification that uses a few orthogonal views and simple point projections. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5da264f3aa4417efcada4433a79a46282e57b420c20638b9c276950a334af201.jpg",
+        "table_caption": [
+            "Table 1: Summary of various protocols. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Protocol</td><td>Data Augmentation</td><td>Model Selection</td><td>Loss</td><td>Ensemble</td><td>Training Points</td></tr><tr><td>PointNet++</td><td>jitter, random rotation, random scaling and trans.</td><td>final model</td><td>cross-entropy</td><td>Rotation Vote</td><td>fixed</td></tr><tr><td>DGCNN</td><td>random scaling and trans.</td><td>best test model</td><td>smooth-loss</td><td>No vote</td><td>fixed</td></tr><tr><td>RSCNN</td><td>random scaling and trans.</td><td>best test model</td><td>cross-entropy</td><td>Repeated Scaling Vote</td><td>resampled</td></tr><tr><td>SimpleView</td><td>random scaling and trans.</td><td>final model</td><td>smooth-loss</td><td>No vote</td><td>fixed</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3D shape Analysis using Rendered Images and Voxels: Many works use images rendered from object meshes for 3D shape analysis (Maturana & Scherer, 2015; Wu et al., 2015b; Yu et al., 2018; Guo et al., 2016; Shi et al., 2015; Hackel et al., 2017; Song & Xiao, 2016; 2014; Huang & You, 2016; Tchapmi et al., 2017). MVCNN exemplifies this strategy by applying a shared CNN to many rendered views and max-pooling to aggregate features (Su et al., 2015). Subsequent approaches include RotationNet which trains the network to also predict the viewpoint for each image (Kanezaki et al., 2018), GVCNN which groups features from subsets of views together before aggregating into a final prediction (Feng et al., 2018), and hypergraph methods that consider the correlation across training samples (Zhang et al., 2018; Feng et al., 2019). One notable exception is Qi et al. (2016), who use a multi-resolution variant of MVCNN, but instead of object meshes, use a voxelized version of the object for rendering. In contrast to the prior view-based methods that use object meshes with point connectivity information, and render images using basic shading and/or depth; SimpleView takes as input raw point clouds. ",
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+    {
+        "type": "text",
+        "text": "Another class of methods is voxel-based methods that convert points to a fixed 3D grid instead, which enables the use of 3D CNNs (Qi et al., 2016; Wu et al., 2015a; Maturana & Scherer, 2015). Given the added dimension, such methods are usually restricted to a much lower resolution to represent objects. Though some strategies such as octrees have been used to address those limitations (Wang et al., 2017), the advantages to processing 3D data directly in this manner do not yet appear to outweigh the additional overhead introduced. ",
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+    {
+        "type": "text",
+        "text": "3 METHOD OVERVIEW ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 VARIATIONS IN EXISTING PROTOCOLS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We analyze the key ingredients in the progress in point-cloud classification. Critical to our study is controlling for factors which are independent of network architecture. We refer to the factors as a method’s protocol. A protocol used by one method can be transferred to another. For our study, we analyze a subset of the highest performing methods over the past few years. This choice was further based on availability and usability of official source-code. Specifically, we choose PointNet (Qi et al., 2017a), PointNet+ $^ +$ (Qi et al., 2017b), DGCNN (Wang et al., 2018b) and RSCNN (Liu et al., 2019c). Note that we also do direct comparisons to networks apart from the ones mentioned here (Table 4). ",
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+        "type": "text",
+        "text": "For our purposes, we do not consider any variations in input, namely the use of surface normals or more than 1024 points. Using normals or more points have been shown to improve performance in the literature. Our objective is to study factors that are not commonly perceived as a major source of performance increase. So we scope our analysis to the most widely adopted input scheme which uses 1024 points with only $x , y , z$ coordinates. ",
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+        "type": "text",
+        "text": "Data Augmentation: Various data augmentation strategies like jittering, random rotation along y-axis, random scaling and random translation. Different methods use different combinations of these augmentations. PointNet and PointNet+ $^ +$ use all the above augmentations. However, as objects in ModelNet40 are aligned, random rotation along y-axis adversely affects the performance of a model. Hence recent methods, including RSCNN and DGCNN, do not use it. They use only random translation and random scaling. Some methods including PointCNN make a distinction between whether or not random rotation along y-axis is used, but it is not a common practice. ",
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+    },
+    {
+        "type": "text",
+        "text": "Input Points: PointNet and PointNet+ $^ +$ use a fixed set of 1024 points per object to train the network. We refer to it as the fixed points strategy. RSCNN and PointCNN randomly sample points during each epoch, effectively exposing the model to more than 1024 points per object during the training process. We refer to this as the resampled points strategy. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/35ab2b5ea16bef0ac8cab9ed9abd8e9abbae99f4da4423b3162f1ebbb063629c.jpg",
+        "table_caption": [
+            "Table 2: DGCNN aug. and smooth loss improve the performance of all architectures "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Data Augmentation</td><td colspan=\"2\">Model Selection</td><td colspan=\"2\">Loss</td><td colspan=\"4\">Architecture</td></tr><tr><td>PN++</td><td>DGCNN</td><td>Final</td><td>Best Test</td><td>C.E.</td><td>Smooth</td><td>PointNet</td><td>PN++</td><td>DGCNN</td><td>RSCNN</td></tr><tr><td></td><td></td><td></td><td>√</td><td>?</td><td></td><td>89.7±0.3</td><td>91.0±0.3</td><td>90.5±0.2</td><td>90.4±0.3</td></tr><tr><td>√</td><td></td><td>&gt;&gt;&gt;</td><td></td><td></td><td></td><td>89.0±0.2</td><td>89.8±0.2</td><td>90.0±0.4</td><td>89.4±0.1</td></tr><tr><td></td><td>√</td><td></td><td></td><td>√</td><td></td><td>89.1±0.2</td><td>92.1±0.1</td><td>91.1±0.3</td><td>91.1±0.3</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>√</td><td>89.2±0.9</td><td>92.7±0.1</td><td>91.9±0.2</td><td>91.7±0.3</td></tr></table>",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Loss Function: cross-entropy (CE) is used by most of the methods. However, DGCNN uses smooth-loss, where the ground-truth labels are smoothed out before calculating cross-entropy. We observe that smooth-loss improves the performance of all network architectures. ",
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+    },
+    {
+        "type": "text",
+        "text": "Selecting Model for Testing: PointNet and PointNet+ $^ { - + }$ use the final converged model to evaluate on the test set. Since the number of epochs is a hyper-parameter that depends on factors like data, model, optimizer, and loss, in our experiments, we create a validation set from the training set to tune the number of epochs. We then retrain the model with the complete training set to the tuned number of epochs. We refer to this strategy as final model selection. We find that some methods including DGCNN and RSCNN evaluate the model on the test set after every epoch and use the best test performance as the final performance. We refer to this strategy as best test model selection. ",
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+    },
+    {
+        "type": "text",
+        "text": "Ensemble Scheme: Some methods use an ensemble to further improve the performance. PointNet and PointNet+ $^ +$ apply the final network to multiple rotated and shuffled versions of the point cloud, and average the predictions to make the final prediction. We refer to this strategy as Rotation Vote. The shuffling operation induces randomness in prediction for PointNet $^ { + + }$ and RSCNN as they are not strictly invariant to the order of the points (Sec. 3.3 in Qi et al. (2017b)). Hence, while evaluating Rotation Vote, we do the inference 10 times per run for PointNet $^ { + + }$ and RSCNN, and report mean and standard deviation. SimpleView and PointNet are invariant to the order of the points and hence are not affected by shuffling. Some methods, including RSCNN and DensePoint, create multiple randomly scaled and randomly sampled versions of a test object. They then evaluate the final network on these multiple versions of the object and average the prediction. Since the scaling is random, it makes the test set performance random as well. RSCNN and DensePoint repeat this procedure 300 times on the test set and report the best accuracy. We refer to it as Repeated Scaling Vote. DGCNN does not use any ensemble. ",
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+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the PointNet $^ { + + }$ , DGCNN and RSCNN protocols. Besides these three protocols, we also include variants of these protocols in table 3 and table 4, such as PointNet $^ { + + }$ no Vote (i.e. PointNe $^ { + + }$ but without the Rotation Vote), DGCNN CE (i.e. DGCNN but with CE loss intead of smooth loss), DGCNN CE Final (i.e. DGCNN CE but with final model selection instead of best test model selection) and RSCNN no Vote (i.e. RSCNN but without the Rotation Vote). These protocols represent prototypical settings and have been used with slight modifications in many other prior works. ",
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+    {
+        "type": "text",
+        "text": "For example, DeepSets (Zaheer et al., 2017) used the PointNet $^ { + + }$ no Vote protocol without jittering and translation; SO-Net (Li et al., 2018a) used the DGCNN CE protocol with jittering and random scaling instead of random scaling and translation; 3DmFV (Ben-Shabat et al., 2018) used the DGCNN $C E$ protocol with additional jittering; PCNN (Atzmon et al., 2018) used the DGCNN CE Final protcol1; PointCNN (Li et al., 2018b) used the DGCNN CE protocol with randomly sampled points and small $( 1 0 ^ { \\circ } )$ rotation augmentation; DensePoint (Liu et al., 2019b) used the RSCNN protocol; PointASL (Yan et al., 2020) used the DGCNN CE protocol but with additional point jittering augmentation and voting. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our Protocol: Based on our findings, we define our SimpleView protocol, which uses the best augmentation and loss functions while not using any information from the test set. Table 2 shows that ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/22d8c94af4c62a2b3bb85af40d250701527c94e89bc1ebf73966c03dc0f3f2cb.jpg",
+        "table_caption": [
+            "Table 3: Performance of various architectures on ModelNet40. Protocol affects performance by a large amount. SimpleView performs on par or better than prior architectures across protocols. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Protocol→ Architecture ↓</td><td colspan=\"2\">PointNet++</td><td colspan=\"2\">RSCNN</td><td colspan=\"2\">DGCNN</td></tr><tr><td>no Vote</td><td>Vote</td><td>no Vote</td><td>Vote</td><td>CE</td><td>Smooth</td></tr><tr><td>PointNet</td><td>89.0 ± 0.2</td><td>89.1 ± 0.2</td><td>90.0 ± 0.3</td><td>90.1 ± 0.2</td><td>90.1 ± 0.2</td><td>90.5 ± 0.1</td></tr><tr><td>PointNet++</td><td>89.8 ± 0.2</td><td>90.0 ± 0.3</td><td>92.7 ± 0.1</td><td>93.3 ± 0.3</td><td>92.6 ± 0.2</td><td>93.1 ± 0.2</td></tr><tr><td>DGCNN</td><td>90.0 ± 0.4</td><td>90.5 ± 0.4</td><td>92.2 ± 0.1</td><td>92.8 ± 0.5</td><td>91.9 ± 0.2</td><td>92.7 ± 0.1</td></tr><tr><td>RSCNN</td><td>89.4 ± 0.1</td><td>90.2 ± 0.2</td><td>92.1 ± 0.1</td><td>92.5 ± 0.2</td><td>91.9 ± 0.2</td><td>92.5 ± 0.1</td></tr><tr><td>SimpleView</td><td>90.7 ± 0.3</td><td>91.0 ± 0.2</td><td>92.9 ± 0.2</td><td>93.2 ± 0.1</td><td>93.1 ± 0.1</td><td>93.6 ± 0.3</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/4c039c152428d524c219eea1e4aa48cf6356af06c61066ee85caf706cb137f16.jpg",
+        "table_caption": [
+            "Table 4: Performance of various architectures on ModelNet40. Includes prior works not in Table 3. \\* indicates small differences in protocol as identified in Sec. 3.1 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>#Points</td><td>Closest Protocol</td><td>Acc.</td><td>PN++ Acc.</td><td>SimpleView Acc.</td></tr><tr><td>DeepSets (Zaheer et al.)</td><td>5000</td><td>PointNet++ no Vote*</td><td>90.0 ± 0.3</td><td>89.8 ± 0.2</td><td>90.7 ± 0.3</td></tr><tr><td>SO-Net (Li et al.)</td><td>2048</td><td>DGCNN CE*</td><td>90.9</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr><tr><td>3DmFV (Ben-Shabat et al.)</td><td>1024</td><td>DGCNN CE*</td><td>91.4</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr><tr><td>PCNN (Atzmon et al.)</td><td>1024</td><td>DCNN CE Final*</td><td>92.3</td><td>92.1 ± 0.1</td><td>92.5 ± 0.3</td></tr><tr><td>PointCNN (Li et al.)</td><td>1024</td><td>DGCNN CE*</td><td>92.5</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr><tr><td>DensePoint (Liu et al.)</td><td>1024</td><td>RSCNN no Vote</td><td>92.8</td><td>92.7 ± 0.1</td><td>92.9 ± 0.2</td></tr><tr><td>RSCNN-Multi (Liu et al.)</td><td>1024</td><td>RSCNN no Vote</td><td>92.9</td><td>92.7 ± 0.1</td><td>92.9 ± 0.2</td></tr><tr><td>PointANSL (Yan et al.)</td><td>1024</td><td>DGCNN CE*</td><td>92.9</td><td>92.6 ± 0.2</td><td>93.1 ± 0.1</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DGCNN’s augmentation (i.e random translation and scaling) and smooth-loss improve performance of all prior networks, so we use them in the SimpleView protocol. Further, similar to PointNet, PointNet $^ { + + }$ and DGCNN, we use the fixed dataset of 1024 points instead of re-sampling different points at each epoch. Re-sampling points for each epoch effectively increases the training dataset of points, making numbers incomparable to methods using a fixed dataset. We avoid any feedback from the test set and use the final model selection, where we first tune the number of epochs on the validation set then retrain the model on the entire train set. Lastly, similar to DGCNN, we do not use ensemble as it is more standard in Machine Learning to compare models without ensemble. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.2 SIMPLEVIEW ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given a set of points SimpleView, projects them onto the six orthogonal planes to create sparse depth images. It then extracts features from the depth images using a CNN and fuses, which is then used to classify the point-cloud as shown in Fig. 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Generating Depth Images from Point Cloud: Let $( x , y , z )$ be the coordinates of a point in the point cloud with respect to the camera. We apply perspective projection to get the 2D coordinate $( \\tilde { x } \\ = \\ x / z , \\tilde { y } \\ = \\ y / z )$ of $\\mathbf { p }$ at depth $z$ . We also do ablations with orthographic projection and found perspective projection to work better (Table. 7). Since coordinates on image plane have to be discrete, we use $( \\bar { \\lceil x \\rceil } , \\bar { \\lceil y \\rceil } )$ to be the final coordinate of $\\mathbf { p }$ on the image plane. Multiple points may be projected to the same discrete location on the image plane. To produce depth value at an image location, we do ablations on two choices, one the minimum depth of all points, and other weighted average depth with more weight $\\textstyle { \\binom { 1 } { z } }$ given to closer points (Table. 7). Empirically, we find both perform similar with the later performing slightly better. This could be because of reduction in noise due to the averaging of nearby pixels on the surface. The depth images are of resolution $1 2 8 \\mathrm { ~ X ~ } 1 2 8$ . ",
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+    {
+        "type": "text",
+        "text": "SimpleView Architecture: To make the number of parameters comparable to point-based methods, we use ResNet18 with one-fourth filters (ResNet18/4) as the backbone. For fusing features, we do ablation with two choices, pooling and concatenation. Empirically, we find concatenation to work better than pooling them (Table. 7). This could be because pooling features throws away the view information like which views are adjacent to one another. One concern could be that concatenation could make features sensitive to viewpoint, and hence the network could fail on rotated objects. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 5: Performance of various architectures on ModelNet40 when using the best dataaugmentation and loss function; and not using any feedback from test set. SimpleView outperforms prior architectures, while having fewest parameters and comparable inference time. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture ↓</td><td>Acc.</td><td>Class Acc.</td><td>Para. (M)</td><td>Time (ms)</td></tr><tr><td>PointNet</td><td>89.2 ± 0.9</td><td>85.1 ± 0.6</td><td>3.5</td><td>3.0</td></tr><tr><td>PointNet++</td><td>92.7 ± 0.1</td><td>90.0± 0.3</td><td>1.7</td><td>20.7</td></tr><tr><td>DGCNN</td><td>92.3 ± 0.3</td><td>89.1 ± 0.3</td><td>1.8</td><td>7.3</td></tr><tr><td>RSCNN</td><td>91.7 ± 0.3</td><td>88.5 ± 0.4</td><td>1.3</td><td>4.3</td></tr><tr><td>SimpleView</td><td>93.0 ± 0.4</td><td>90.5 ± 0.8</td><td>0.8</td><td>5.0</td></tr></table>",
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+        "type": "table",
+        "img_path": "images/add2a7673441e4264cd37481ef995c80d11108425bdfc8312a73296c90cb238c.jpg",
+        "table_caption": [
+            "Table 6: Performance of various architectures on ScanObjectNN, and cross-dataset generalization. SimpleView achieves state-of-the-art results and shows better cross dataset generalization. Numbers for prior works are from (Uy et al., 2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture ↓</td><td>TR: SONN TE: SONN</td><td>TR:MN40 TE: SONN</td><td>TR: SONN TE: MN40</td></tr><tr><td>3DmFV (Shabat et al.)</td><td>63.0</td><td>24.9</td><td>51.5</td></tr><tr><td>PointNet (Qi et al.)</td><td>68.2</td><td>31.1</td><td>50.9</td></tr><tr><td>SpiderCNN (Xu et al.)</td><td>73.7</td><td>30.9</td><td>46.6</td></tr><tr><td>PointNet++ (Qi et al.)</td><td>77.9</td><td>32.0</td><td>47.4</td></tr><tr><td>DGCNN (Wang et al.)</td><td>78.1</td><td>36.8</td><td>54.7</td></tr><tr><td>PointCNN (Li et al.)</td><td>78.5</td><td>24.6</td><td>49.2</td></tr><tr><td>SimpleView</td><td>79.5±0.5</td><td>40.5±1.4</td><td>57.9±2.1</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 7: Ablation of various choices for SimpleView on ModelNet40. The performance is evaluatated on the validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Number of Views</td><td colspan=\"2\">Image Projection</td><td colspan=\"2\">Feature Fusion</td><td colspan=\"2\">Image Depth</td></tr><tr><td>1</td><td>3</td><td>6</td><td>Orthographic</td><td>Perspective</td><td>Pool</td><td>Concat</td><td>Minimum</td><td>Weighted Avg.</td></tr><tr><td>Accuracy</td><td>90.7±0.1</td><td>92.1 ± 0.2</td><td>92.9±0.3</td><td>92.7 ± 0.3</td><td>92.9±0.3</td><td>91.8± 0.3</td><td>92.9± 0.3</td><td>92.8± 0.4</td><td>92.9± 0.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "However, empirically, we observe that this issue is largely mitigated by rotation augmentation and SimpleView is able to achieve state-of-the-art performance on ScanObjectNN where objects are rotated. The point-clouds are scaled to be in $[ \\hat { 1 } , - 1 ] ^ { 3 }$ , we keep the cameras at a distance of 1.4 units from the center with $9 0 °$ fov. We also do ablations with different number of views, comparing only front views, three orthogonal views and six orthogonal views. We find that using all six views performs the best (Table 7). We do not use ImageNet pretraining, thus making the comparison with point-based methods strictly fair, without any additional data. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ModelNet40: ModelNet40 is a the most widely adopted benchmark for point-cloud classification. It contains objects from 40 common categories. There are 9840 objects in the training set and 2468 in the test set. Objects are aligned to a common up and front direction. ",
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+    {
+        "type": "text",
+        "text": "ScanObjectNN: ScanObjectNN is a recent real-world point cloud classification dataset. It consists of 15 classes, 11 of which are also in ModelNet40. There are a total of $1 5 \\mathrm { k }$ objects in the dataset. Unlike ModelNet40, the objects in ScanObjectNN are obtained from real-world 3D scans. Hence, point clouds are noisy (occlusions, background points) and have geometric distortions such as holes. Also, unlike ModelNet40, the objects are not axis-aligned. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTS ON MODELNET40 ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation Details: We use PyTorch (Paszke et al., 2019) to implement all models and protocols while reusing the official code wherever possible. We use the official version of DGCNN and RSCNN. We confirm with the authors that the code for RSCNN-Multi, another version of RSCNN, is yet to be released. Hence we use the reported numbers of RSCNN-Multi in Table 4. PointNet and PointNet $^ { + + }$ are officially released in TensorFlow (Abadi et al., 2015). For PointNet, we adapt our code from PointNet.pytorch (Xia, accessed June, 2020) as recommended in the official repository. For PointNet++, we adapt the model code from Pointnet2 PyTorch (Wijmans, accessed June, 2020). We further make sure that the third party PyTorch code closely matches the official TensorFlow code. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use Adam (Kingma & Ba, 2014) with an initial learning rate of 1e-3 and a decay-on-plateau learning rate scheduler. The batch size and weight decay for each model are kept the same as the official version in Table 3. We use a batch size of 18 and no weight decay for SimpleView. To give the prior models the best chance on our protocol (Table 5), we additionally tune their hyperparameters on the validation set. We find that the official hyper-parameters already perform close to optimal. We train each model for 1000 epochs. Since there are small variations in final performance across different runs, we do 4 runs and report the mean and standard deviation. ",
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+    {
+        "type": "image",
+        "img_path": "images/8b40ab61d4067d6d54c88f0a10b5099143d8f8d1f0873543ec9e9d09ffed3114.jpg",
+        "image_caption": [
+            "Figure 3: Failure Cases for SimpleView and PointNet++. The first row shows cases where both SimpleView and PointNe $^ { + + }$ fail; the second row shows cases where SimpleView succeeds but PointNet+ $^ +$ fails; the third row shows cases where SimpleView fails but PointNet++ succeeds. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Performance under various Prior Protocols: Table 3 shows the performance different architectures under various protocols. The mean performance of PointNet $^ { + + }$ improves from $8 9 . 8 \\%$ to $9 3 . 3 \\%$ when we switch from the PointNet $^ { + + }$ no Vote to the RSCNN Vote protocol. Similarly the performance of SimpleView improves from $9 0 . 7 \\%$ to $9 3 . 6 \\%$ when we switch from PointNet $^ { + + }$ no Vote to DGCNN Smooth. Since there is variance in performance across runs, we refrain from making any claims about absolute ordering between prior works. However, we do observe that in terms of mean performance, SimpleView performs on par or better than other methods under all protocols. Note that in RSCNN Vote, voting on the test set is done 300 times with reshuffled and randomly augmented points, from which the highest accuracy is selected. Hence models that have the largest variance in prediction, i.e. PointNet++ and RSCNN gain the most from it, as they are not strictly invariant to the order of points (Sec. 3.3 in Qi et al. (2017b)). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Performance under the SimpleView Protocol: Table 5 shows that SimpleView outperforms prior architectures on our controlled protocol in terms of mean performance. SimpleView has the fewest number of parameters and a competitive inference speed. Inference speed is measured on an NVIDIA 2080Ti averaged across 100 runs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fig. 3 show examples where both SimpleView and PointNet $^ { + + }$ fail, as well as examples where one of them fails and the other succeeds. Qualitatively, we find that the failure modes of SimpleView and PointNet+ $^ { \\cdot + }$ are similar. We also find that a major failure mode in both SimpleView and PointNet+ $^ +$ is the confusion between the ‘flower pot’ and ‘plant’ category (see Sec. A Fig. I and Fig. II). This could be because of the lack of color information. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Comparison with More Methods: In Table 4, we do one-on-one comparison between SimpleView and recent state-of-the-art methods, other than PointNet, PointNet++, RSCNN and DGCNN. We identify the closest protocol to the one used in the paper from the ones we evaluate. Table 4 shows the competitiveness of PointNet+ $^ +$ and SimpleView with other recent state-of-the-art methods. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS ON SCANOBJECTNN ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details: ScanObjectNN’s official repository trains and evaluates the state-of-theart models under the same protocol. We implement SimpleView in TensorFlow and use the official ScanObjectNN protocol for fairness. This protocol is different from the SimpleView protocol as it normalizes the point clouds and randomly samples points. We optimize our model with Adam. We use a batch size of 20 and no weight decay to train SimpleView for 300 epochs with an initial learning rate 0.001, and use the final model for testing. We use standard image-based cropping and scaling augmentation to prevent over-fitting. The hyper-parameter for cropping and scaling is found on a validation set made from ScanObjecNN’s train set. We conduct 4 runs for SimpleView. ScanObjectNN does not use a fixed set of points during test time. It instead randomly samples points from the point cloud, which adds randomness to test set performance. Hence, we evaluate each run 10 times. We report the final performance as the mean and variance of the 40 evaluations (4 runs $\\times$ 10 evaluations per run). ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Performance on ScanObjectNN: As shown in Table 6, SimpleView outperforms prior networks on ScanObjectNN. This shows the SimpleView is effective in real world settings, with noisy and misaligned point clouds. We also perform transfer experiments to test generalizability of SimpleView. We train on ScanObjectNN and test on ModelNet40 and vice versa. Table 6 shows that SimpleView transfers across datasets better than prior methods. ",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we demonstrate how auxiliary factors orthogonal to the network architecture have a large effect on performance for point-cloud classification. When controlling for these factors, we find that a relatively older method, PointNet $^ { - + }$ (Qi et al., 2017b), performs competitively with more recent ones. Furthermore, we show that a simple baseline performs on par or better than state-ofthe-art architectures. ",
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+    {
+        "type": "text",
+        "text": "Our results show that for future progress we should control for protocols while comparing network architectures. Our code base could serve as a useful resource for developing new models and comparing them with prior works. Our results show that the existing evidence for point-based methods is not as strong when auxiliary factors are properly controlled for, and that SimpleView is a strong baseline. But our results are not meant to discourage future research on point-based methods. It is still entirely possible that point-based methods come out ahead with additional innovations. We believe it is beneficial to explore competing approaches, including the ones that are underperforming at a particular time, as long as the results are compared in a controlled manner. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our analysis in this work was limited to point cloud classification, which is an important problem in 3D scene understanding and forms a critical part of object detection and retrieval systems. An exciting future direction would be to expand this analysis to other problems that involve point cloud data such as scene and part segmentation. ",
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+    {
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+        "text": "A APPENDIX ",
+        "text_level": 1,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f0f28b05536c705d795036eb13399fa271ef793a5d576ab9384fb248a23d99e.jpg",
+        "image_caption": [
+            "Figure I: Confusion matrix for SimpleView when trained under our protocol "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            136,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/74d59f4eb55d04b38c5570c36a9863bc1a14d979aff009a5cca933d3e66ad187.jpg",
+        "image_caption": [
+            "Figure II: Confusion matrix for PointNet++ when trained under our protocol "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/c06bbc72e6afb3031a7639c7a00b80b0fe3064b269c002ee09f23e669eb0b1d4.jpg",
+        "table_caption": [
+            "Table I: Performance of various architectures on ModelNet40 when using different amount of training data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Per. of Training Data</td><td>RSCNN</td><td>DGCNN</td><td>PointNet</td><td>PointNet++</td><td>SimpleView</td></tr><tr><td>25%</td><td>88.2 ± 0.4</td><td>89.1 ± 0.2</td><td>86.3 ± 0.4</td><td>89.6 ± 0.4</td><td>89.7 ± 0.3</td></tr><tr><td>50%</td><td>90.4 ± 0.4</td><td>91.0 ± 0.3</td><td>88.2 ± 0.3</td><td>91.5 ± 0.2</td><td>92.1 ± 0.3</td></tr><tr><td>100 %</td><td>91.7 ± 0.3</td><td>92.3 ± 0.3</td><td>89.2 ± 0.9</td><td>92.7 ± 0.3</td><td>93.0 ± 0.4</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    }
+]

parse/train/h8flNv9x8v-/h8flNv9x8v-.md

@@ -0,0 +1,422 @@
+# Gradient Starvation: A Learning Proclivity in Neural Networks
+
+Mohammad Pezeshki1,2 Sékou-Oumar Kaba1,3 Yoshua Bengio1,2 Aaron Courville1,2 Doina Precup1,3,4 Guillaume Lajoie1,2   
+1Mila 2Université de Montréal 3McGill University 4Google DeepMind   
+corresponding authors:{pezeshki, guillaume.lajoie}@mila.quebec
+
+# Abstract
+
+We identify and formalize a fundamental gradient descent phenomenon leading to a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalances in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel but simple regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and realworld out-of-distribution (OOD) generalization experiments.
+
+# 1 Introduction
+
+In 1904, a horse named Hans attracted worldwide attention due to the belief that it was capable of doing arithmetic calculations [81]. Its trainer would ask Hans a question, and Hans would reply by tapping on the ground with its hoof. However, it was later revealed that the horse was only noticing subtle but distinctive signals in its trainer’s unconscious behavior, unbeknown to him, and not actually performing arithmetic. An analogous phenomenon has been noticed when training neural networks [e.g. 85, 109, 54, 39, 17, 14, 37, 51, 107, 76, 48, 19, 61, 77]. In many cases, state-of-the-art neural networks appear to focus on low-level superficial correlations, rather than more abstract and robustly informative features of interest [16, 88, 40, 68, 30].
+
+The rationale behind this phenomenon is well known by practitioners: given strongly-correlated and fast-to-learn features in training data, gradient descent is biased towards learning them first. However, the precise conditions leading to such learning dynamics, and how one might intervene to control this feature imbalance are not entirely understood. Recent work aims at identifying the reasons behind this phenomenon [97, 70, 22, 73, 51, 76, 100, 92, 83, 105, 42, 79, 4], while complementary work quantifies resulting shortcomings, including poor generalization to out-of-distribution (OOD) test data, reliance upon spurious correlations, and lack of robustness [30, 68, 77, 41, 63, 64, 9]. However most established work focuses on squared-error loss and its particularities, where results do not readily generalize to other objective forms. This is especially problematic since for several classification applications, cross-entropy is the loss function of choice, yielding very distinct learning dynamics. In this paper, we argue that Gradient Starvation, first coined in [26], is a leading cause for this feature imbalance in neural networks trained with cross-entropy, and propose a simple approach to mitigate it.
+
+![](images/9f945eaedc28dc850b70ed0f5a00c0d50de39ab2ea581ed85082d9da4e26613b.jpg)  
+Figure 1: Diagram illustrating the effect of gradient starvation in a simple 2-D classification task. (a) Data is not linearly separable and the learned decision boundary is curved. (b) Data is linearly separable by a small margin $\Delta = \mathrm { 0 . 1 }$ ). This small margin allows the network to discriminate confidently only along the horizontal axis and ignore the vertical axis. (c) Data is linearly separable as in (b). However, with the proposed Spectral decoupling (SD), a curved decision boundary with a large margin is learned. (d) Diagram shows the evolution of two of the features (Eq. 4) of the dynamics in three cases shown as dotted, dashed and solid lines. Analysis: (dotted) vs (dashed): Linear separability of the data results in an increase in $z _ { 1 }$ and a decrease (starvation) of $z _ { 2 }$ . (dashed) vs (solid): SD suppresses $z _ { 1 }$ and hence allows $z _ { 2 }$ to grow. Decision boundaries are averaged over ten runs. More experiments with common regularization methods are provided in App. B.
+
+Here we summarize our contributions:
+
+We provide a theoretical framework to study the learning dynamics of linearized neural networks trained with cross-entropy loss in a dual space. Using perturbation analysis, we formalize Gradient Starvation (GS) in view of the coupling between the dynamics of orthogonal directions in the feature space (Thm. 2). We leverage our theory to introduce Spectral Decoupling (SD) (Eq. 17) and prove this simple regularizer helps to decouple learning dynamics, mitigating GS. We support our findings with extensive empirical results on a variety of classification and adversarial attack tasks. All code and experiment details available at GitHub repository.
+
+In the rest of the paper, we first present a simple example to outline the consequences of GS. We then present our theoretical results before outlining a number of numerical experiments. We close with a review of related work followed by a discussion.
+
+# 2 Gradient Starvation: A simple example
+
+Consider a 2-D classification task with a training set consisting of two classes, as shown in Figure 1. A two-layer ReLU network with 500 hidden units is trained with cross-entropy loss for two different arrangements of the training points. The difference between the two arrangements is that, in one setting, the data is not linearly separable, but a slight shift makes it linearly separable in the other setting. This small shift allows the network to achieve a negligible loss by only learning to discriminate along the horizontal axis, ignoring the other. This contrasts with the other case, where both features contribute to the learned classification boundary, which arguably matches the data structure better. We observe that training longer or using different regularizers, including weight decay [58], dropout [95], batch normalization [49], as well as changing the optimization algorithm to Adam [56] or changing the network architecture or the coordinate system, do not encourage the network to learn a curved decision boundary. (See App. B for more details.)
+
+We argue that this occurs because cross-entropy loss leads to gradients “starved” of information from vertical features. Simply put, when one feature is learned faster than the others, the gradient contribution of examples containing that feature is diminished (i.e., they are correctly processed based on that feature alone). This results in a lack of sufficient gradient signal, and hence prevents any remaining features from being learned. This simple mechanism has potential consequences, which we outline below.
+
+# 2.1 Consequences of Gradient Starvation
+
+Lack of robustness. In the example above, even in the right plot, the training loss is nearly zero, and the network is very confident in its predictions. However, the decision boundary is located very close to the data points. This could lead to adversarial vulnerability as well as lack of robustness when generalizing to out-of-distribution data.
+
+Excessive invariance. GS could also result in neural networks that are invariant to task-relevant changes in the input. In the example above, it is possible to obtain a data point with low probability under the data distribution, but that would still be classified with high confidence.
+
+Implicit regularization. One might argue that according to Occam’s razor, a simpler decision boundary should generalize better. In fact, if both training and test sets share the same dominant feature (in this example, the feature along the horizontal axis), GS naturally prevents the learning of less dominant features that could otherwise result in overfitting. Therefore, depending on our assumptions on the training and test distributions, GS could also act as an implicit regularizer. We provide further discussion on the implicit regularization aspect of GS in Section 5.
+
+# 3 Theoretical Results
+
+In this section, we study the learning dynamics of neural networks trained with cross-entropy loss. Particularly, we seek to decompose the learning dynamics along orthogonal directions in the feature space of neural networks, to provide a formal definition of GS, and to derive a simple regularization method to mitigate it. For analytical tractability, we make three key assumptions: (1) we study deep networks in the Neural Tangent Kernel (NTK) regime, (2) we treat a binary classification task, (3) we decompose the interaction between two features. In Section 4, we demonstrate our results hold beyond these simplifying assumptions, for a wide range of practical settings. All derivation details can be found in $\mathbf { S M C }$ .
+
+# 3.1 Problem Setup and Gradient Starvation Definition
+
+Let ${ \mathcal { D } } = \{ { \bf X } , { \bf y } \}$ denote a training set containing $n$ datapoints with $d$ dimensions, where, $\mathbf { X } =$ $[ \mathbf { x } _ { 1 } , . . . , \mathbf { x } _ { n } ] \in \mathbb { R } ^ { n \times d }$ and their corresponding class label $\mathbf { y } \in \{ - 1 , + 1 \} ^ { n }$ . Also let $\hat { \mathbf { y } } ( \mathbf { X } ) : = f ^ { ( L ) } ( \mathbf { X } ) :$ $\mathbb { R } ^ { n \times d } \to \mathbb { R } ^ { n }$ represent the logits of an $\mathrm { L }$ -layer fully-connected neural network where each hidden layer $h ^ { ( l ) } ( x ) \in \mathbf { \bar { \mathbb { R } } } ^ { d _ { l } }$ is defined as follows,
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \boldsymbol { f } ^ { ( l ) } ( \mathbf { x } _ { i } ) = \mathbf { W } ^ { ( l ) } h ^ { ( l - 1 ) } ( \mathbf { x } _ { i } ) } \\ { h ^ { ( l ) } ( \mathbf { x } _ { i } ) = \sqrt { \frac { \gamma } { d _ { l } } } \boldsymbol { \xi } ( \boldsymbol { f } ^ { ( l ) } ( \mathbf { x } _ { i } ) ) } \end{array} \right. , l \in \{ 0 , 1 , . . . , L \} , } \end{array}
+$$
+
+in which $\mathbf { W } ^ { ( l ) } \in \mathbb { R } ^ { d _ { l } \times d _ { l - 1 } }$ is a weight matrix drawn from $\mathcal { N } ( 0 , \bf { I } )$ and $\gamma$ is a scaling factor to ensure that norm of each $\boldsymbol { h } ^ { ( l - 1 ) }$ is preserved at initialization (See [28] for a formal treatment). The function $\xi ( . )$ is also an element-wise non-linear activation function.
+
+Let $\pmb \theta = \mathrm { c o n c a t } \big ( \cup _ { l = 1 } ^ { L } \mathrm { v e c } ( \mathbf W ^ { ( l ) } ) \big ) \in \mathbb { R } ^ { m }$ be the concatenation of all vectorized weight matrices with $m$ as the total number of parameters. In the NTK regime [52], in the limit of infinite width, the output of the neural network can be approximated as a linear function of its parameters governed by the neural tangent random feature (NTRF) matrix [23],
+
+$$
+\Phi \left( \mathbf { X } , \theta \right) = \frac { \partial \hat { \mathbf { y } } \left( \mathbf { X } , \theta \right) } { \partial \theta } \in \mathbb { R } ^ { n \times m } .
+$$
+
+In the wide-width regime, the NTRF changes very little during training [62], and the output of the neural network can be approximated by a first order Taylor expansion around the initialization parameters $\pmb { \theta } _ { 0 }$ . Setting $\Phi _ { 0 } \equiv \Phi \left( \mathbf { X } , \pmb { \theta } _ { 0 } \right)$ and then, without loss of generality, centering parameters and the output coordinates to their value at the initialization ${ \bf \delta _ { \theta } }$ and $\hat { \mathbf { y } } _ { 0 , }$ ), we get
+
+$$
+\hat { \mathbf { y } } \left( \mathbf { X } , \pmb { \theta } \right) = \Phi _ { 0 } \pmb { \theta } .
+$$
+
+Dominant directions in the feature space as well as the parameter space are given by principal components of the NTRF matrix $\Phi _ { 0 }$ , which are the same as those of the NTK Gram matrix [106]. We therefore introduce the following definition.
+
+Definition 1 (Features and Responses). Consider the singular value decomposition (SVD) of the matrix $\mathbf { Y } \Phi _ { 0 } = \mathbf { U } \mathbf { S } \mathbf { V } ^ { T }$ , where $\mathbf { Y } = d i a g \left( \mathbf { y } \right)$ . The jth feature is given by $( \mathbf { \dot { V } } ^ { T } ) _ { j . }$ .. The strength of jth feature is represented by $s _ { j } = ( \mathbf { S } ) _ { j j }$ . Also, $( \mathbf { U } ) _ { \cdot j }$ contains the weights of this feature in all examples. A neural network’s response to a feature $j$ is given by $z _ { j }$ where,
+
+$$
+\mathbf { z } : = \mathbf { U } ^ { T } \mathbf { Y } \hat { \mathbf { y } } = \mathbf { S } \mathbf { V } ^ { T } \pmb { \theta } .
+$$
+
+In Eq. 4, the response to feature $j$ is the sum of the responses to every example in $( \mathbf { Y } \hat { \mathbf { y } } )$ multiplied by the weight of the feature in that example $( \mathbf { U } ^ { T } )$ . For example, if all elements of $( \mathbf { U } ) _ { \cdot j }$ are positive, it indicates a perfect correlation between this feature and class labels. We are now equipped to formally define GS.
+
+Definition 2 (Gradient Starvation). Recall the the model prescribed by Eq. 3. Let $z _ { j } ^ { \ast }$ denote the model’s response to feature $j$ at training optimum $\pmb { \theta } ^ { * 1 }$ . Feature $i$ starves the gradient for feature $j$ $i f d z _ { j } ^ { * } / d ( s _ { i } ^ { 2 } ) < 0$ .
+
+This definition of GS implies that an increase in the strength of feature $i$ has a detrimental effect on the learning of feature $j$ . We now derive conditions for which learning dynamics of system 3 suffer from GS.
+
+# 3.2 Training Dynamics
+
+We consider the widely used ridge-regularized cross-entropy loss function,
+
+$$
+\mathcal { L } \left( \pmb { \theta } \right) = \mathbf { 1 } \cdot \log \left[ 1 + \exp \left( - \mathbf { Y } \hat { \mathbf { y } } \right) \right] + \frac { \lambda } { 2 } \| \pmb { \theta } \| ^ { 2 } ,
+$$
+
+where 1 is a vector of size $n$ with all its elements equal to 1. This vector form simply represents a summation over all the elements of the vector it is multiplied to. $\lambda \in [ 0 , \infty )$ denotes the weight decay coefficient.
+
+Direct minimization of this loss function using the gradient descent obeys coupled dynamics and is difficult to treat directly [26]. To overcome this problem, we call on a variational approach that leverages the Legendre transformation of the loss function. This allows tractable dynamics that can directly incorporate rates of learning in different feature directions. Following [50], we note the following inequality,
+
+$$
+\begin{array} { r } { \log \left[ 1 + \exp \left( - \mathbf { Y } \hat { \mathbf { y } } \right) \right] \geq H ( \alpha ) - \alpha \odot \mathbf { Y } \hat { \mathbf { y } } , } \end{array}
+$$
+
+where $H ( \pmb { \alpha } ) = - \left[ \pmb { \alpha } \log \pmb { \alpha } + ( 1 - \pmb { \alpha } ) \log \left( 1 - \pmb { \alpha } \right) \right]$ is Shannon’s binary entropy function, $\alpha \in$ $( 0 , 1 ) ^ { n }$ is a variational parameter defined for each training example, and $\odot$ denotes the element-wise vector product. Crucially, the equality holds when the maximum of r.h.s. w.r.t $_ \alpha$ is achieved at $\begin{array} { r } { \pmb { \alpha } ^ { * } = \frac { \bar { { \partial \mathcal { L } } } } { \partial ( \mathbf { Y } \hat { \mathbf { y } } ) ^ { T } } } \end{array}$ , which leads to the following optimization problem,
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \mathcal { L } \left( \pmb { \theta } \right) = \operatorname* { m i n } _ { \pmb { \theta } } \operatorname* { m a x } _ { \pmb { \alpha } } \left( \mathbf { 1 } \cdot H ( \pmb { \alpha } ) - \pmb { \alpha } \mathbf { Y } \hat { \mathbf { y } } + \frac { \lambda } { 2 } \| \pmb { \theta } \| ^ { 2 } \right) ,
+$$
+
+where the order of min and max can be swapped (see Lemma 3 of [50]). Since the neural network’s output is approximated by a linear function of $\pmb \theta$ , the minimization can be performed analytically with an critical value $\pmb { \theta } _ { . } ^ { \ast T } = \triangleq \frac { 1 } { \lambda } \pmb { \alpha } \mathbf { Y } \pmb { \Phi } _ { 0 }$ , given by a weighted sum of the training examples. This results in the following maximization problem on the dual variable, i.e., $\operatorname* { m i n } _ { \theta } { \mathcal { L } } \left( \theta \right)$ is equivalent to,
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \mathcal { L } \left( \pmb { \theta } \right) = \operatorname* { m a x } _ { \pmb { \alpha } } \left( \mathbf { 1 } \cdot H ( \pmb { \alpha } ) - \frac { 1 } { 2 \lambda } \pmb { \alpha } \mathbf { Y } \pmb { \Phi } _ { 0 } \pmb { \Phi } _ { 0 } ^ { T } \mathbf { Y } ^ { T } \pmb { \alpha } ^ { T } \right) .
+$$
+
+By applying continuous-time gradient ascent on this optimization problem, we derive an autonomous differential equation for the evolution of $_ { \pmb { \alpha } }$ , which can be written in terms of features (see Definition 1),
+
+$$
+\dot { \pmb { \alpha } } = \eta \left( - \log \pmb { \alpha } + \log \left( \mathbf { 1 } - \pmb { \alpha } \right) - \frac { 1 } { \lambda } \pmb { \alpha } \mathbf { U } \mathbf { S } ^ { 2 } \mathbf { U } ^ { T } \right) ,
+$$
+
+where $\eta$ is the learning rate (see $\mathrm { S M } \mathrm { C } . 1$ for more details). For this dynamical system, we see that the logarithm term acts as barriers that keep $\alpha _ { i } \in ( 0 , 1 )$ . The other term depends on the matrix $\mathbf { U S ^ { 2 } U } ^ { T }$ , which is positive definite, and thus pushes the system towards the origin and therefore drives learning.
+
+When $\lambda \ll s _ { k } ^ { 2 }$ , where $k$ is an index over the singular values, the linear term dominates Eq. 9, and the fixed point is drawn closer towards the origin. Approximating dynamics with a first order Taylor expansion around the origin of the second term in Eq. 9, we get
+
+$$
+\dot { \boldsymbol { \alpha } } \approx \eta \left( - \log \boldsymbol { \alpha } - \frac { 1 } { \lambda } \boldsymbol { \alpha } \mathbf { U } \left( \mathbf { S } ^ { 2 } + \lambda \mathbf { I } \right) \mathbf { U } ^ { T } \right) ,
+$$
+
+with stability given by the following theorem with proof in $\mathbf { S M C }$ .
+
+Theorem 1. Any fixed points of the system in Eq. 10 is attractive in the domain $\alpha _ { i } \in ( 0 , 1 )$ .
+
+At the fixed point $\ b { \alpha } ^ { * }$ , corresponding to the optimum of Eq. 8, the feature response of the neural network is given by,
+
+$$
+\mathbf { z } ^ { \ast } = \frac { 1 } { \lambda } \mathbf { S } ^ { 2 } \mathbf { U } ^ { T } \pmb { \alpha } ^ { \ast T } .
+$$
+
+See App. A for further discussions on the distinction between "feature space" and "parameter space". Below, we study how the strength of one feature could impact the response of the network to another feature which leads to GS.
+
+# 3.3 Gradient Starvation Regime
+
+In general, we do not expect to find an analytical solution for the dynamics of the coupled non-linear dynamical system of Eq. 10. However, there are at least two cases where a decoupled form for the dynamics allows to find an exact solution. We first introduce these cases and then study their perturbation to outline general lessons.
+
+1. If the matrix of singular values $\mathbf { S } ^ { 2 }$ is proportional to the identity: This is the case where all the features have the same strength $s ^ { 2 }$ . The fixed points are then given by,
+
+$$
+\alpha _ { i } ^ { * } = \frac { \lambda \mathcal { W } ( \lambda ^ { - 1 } s ^ { 2 } + 1 ) } { s ^ { 2 } + \lambda } , \qquad z _ { j } ^ { * } = \frac { s ^ { 2 } \mathcal { W } ( \lambda ^ { - 1 } s ^ { 2 } + 1 ) } { s ^ { 2 } + \lambda } \sum _ { i } u _ { i j } ,
+$$
+
+where $\mathcal { W }$ is the Lambert W function.
+
+2. If the matrix $\mathbf { U }$ is a permutation matrix: This is the case in which each feature is associated with a single example only. The fixed points are then given by,
+
+$$
+\alpha _ { i } ^ { * } = \frac { \lambda \mathcal { W } ( \lambda ^ { - 1 } s _ { i } ^ { 2 } + 1 ) } { s _ { i } ^ { 2 } + \lambda } , \ ~ \ z _ { j } ^ { * } = \frac { s _ { i } ^ { 2 } \mathcal { W } ( \lambda ^ { - 1 } s _ { i } ^ { 2 } + 1 ) } { s _ { i } ^ { 2 } + \lambda } .
+$$
+
+To study a minimal case of starvation, we consider a variation of case 2 with the following assumption which implies that each feature is not associated with a single example anymore.
+
+Lemma 1. Assume $\mathbf { U }$ is a perturbed identity matrix (a special case of a permutation matrix) in which the off-diagonal elements are proportional to a small parameter $\delta > 0$ . Then, the fixed point of the dynamical system in Eq. 10 can be approximated by,
+
+$$
+\begin{array} { r } { { \pmb \alpha } ^ { * } = ( 1 - \log \left( { \pmb \alpha } _ { 0 } ^ { * } \right) ) \left[ { \pmb A } + d i a g \left( { \pmb \alpha } _ { 0 } ^ { * } ^ { - 1 } \right) \right] ^ { - 1 } , } \end{array}
+$$
+
+where $\mathbf { A } = \lambda ^ { - 1 } \mathbf { U } ( \mathbf { S } ^ { 2 } + \lambda \mathbf { I } ) \mathbf { U } ^ { T }$ and $\alpha _ { 0 } ^ { * }$ is the fixed point of the uncoupled system with $\delta = 0$
+
+For sake of ease of derivations, we consider the two dimensional case where,
+
+$$
+\mathbf { U } = \left( \begin{array} { c c } { \sqrt { 1 - \delta ^ { 2 } } } & { - \delta } \\ { \delta } & { \sqrt { 1 - \delta ^ { 2 } } } \end{array} \right) ,
+$$
+
+which is equivalent to a $U$ matrix with two blocks of features with no intra-block coupling and $\delta$ amount of inter-block coupling.
+
+Theorem 2 (Gradient Starvation Regime). Consider a neural network in the linear regime, trained under cross-entropy loss for a binary classification task. With definition $^ { l }$ , assuming coupling between features 1 and 2 as in Eq. 15 and $s _ { 1 } ^ { 2 } > s _ { 2 } ^ { 2 }$ , we have,
+
+$$
+\frac { \mathrm { d } z _ { 2 } ^ { * } } { \mathrm { d } s _ { 1 } ^ { 2 } } < 0 ,
+$$
+
+which implies GS.
+
+While Thm. 2 outlines conditions for GS in two dimensional feature space, we note that the same rationale naturally extends to higher dimensions, where GS is defined pairwise over feature directions. For a classification task, Thm. 2 indicates that gradient starvation occurs when the data admits different feature strengths, and coupled learning dynamics. GS is thus naturally expected with cross-entropy loss. Its detrimental effects however (as outlined in Sect. 2) arise in settings with large discrepancies between feature strengths, along with network connectivity that couples these features’ directions. This phenomenon readily extends to multi-class settings, and we validate this case with experiments in Sect. 4. Next, we introduce a simple regularizer that encourages feature decoupling, thus mitigating GS by insulating strong features from weaker ones.
+
+# 3.4 Spectral Decoupling
+
+By tracing back the equations of the previous section, one may realize that the term $U ^ { T } S ^ { 2 } U$ in Eq. 9 is not diagonal in the general case, and consequently introduces coupling between $\alpha _ { i }$ ’s and hence, between the features $z _ { i }$ ’s. We would like to discourage solutions that couple features in this way. To that end, we introduce a simple regularizer: Spectral Decoupling (SD). SD replaces the general L2 weight decay term in Eq. 5 with an L2 penalty exclusively on the network’s logits, yielding
+
+$$
+\mathcal { L } \left( \pmb { \theta } \right) = \mathbf { 1 } \cdot \log \left[ 1 + \exp \left( - \mathbf { Y } \hat { \mathbf { y } } \right) \right] + \frac { \lambda } { 2 } \| \hat { \mathbf { y } } \| ^ { 2 } .
+$$
+
+Repeating the same analysis steps taken above, but with SD instead of general L2 penalty, the critical value for $\pmb { \theta } ^ { * }$ becomes $\begin{array} { r } { \pmb { \theta } ^ { \ast } = \frac { 1 } { \lambda } \pmb { \alpha } \pmb { Y } \pmb { \Phi } _ { 0 } V \mathbf { S } ^ { - 2 } V ^ { T } } \end{array}$ . This new expression for $\pmb { \theta } ^ { * }$ results in the following modification of Eq. 9,
+
+$$
+\dot { \alpha } = \eta \left( \log \frac { \mathbf { 1 } - \alpha } { \alpha } - \frac { 1 } { \lambda } \alpha \mathbf { U } \mathbf { S } ^ { 2 } \mathbf { S } ^ { - 2 } \mathbf { U } ^ { T } \right) = \eta \left( \log \frac { \mathbf { 1 } - \alpha } { \alpha } - \frac { 1 } { \lambda } \alpha \right) ,
+$$
+
+where as earlier, log and division are taken element-wise on the coordinates of $_ { \pmb { \alpha } }$
+
+Note that in contrast to Eq. 9 the matrix multiplication involving $U$ and $S$ in Eq. 18 cancels out, leaving $\alpha _ { i }$ independent of other $\alpha _ { j \neq i }$ ’s. We point out this is true for any initial coupling, without simplifying assumptions. Thus, a simple penalty on output weights promotes decoupled dynamics across the dual parameter $\alpha _ { i }$ ’s, which track learning dynamics of feature responses (see Eq. 7). Together with Thm. 2, Eq. 18 suggests SD should mitigate GS and promote balanced learning dynamics across features. We now verify this in numerical experiments. For further intuition, we provide a simple experiment, summarized in Fig. 5, where directly visualizes the primal vs. the dual dynamics as well as the effect of the proposed spectral decoupling method.
+
+# 4 Experiments
+
+The experiments presented here are designed to outline the presence of GS and its consequences, as well as the efficacy of our proposed regularization method to alleviate them. Consequently, we highlight that achieving state-of-the-art results is not the objective. For more details including the scheme for hyper-parameter tuning, see App. B.
+
+# 4.1 Two-Moon classification and the margin
+
+Recall the simple 2-D classification task between red and blue data points in Fig. 1. Fig. 1 (c) demonstrates the learned decision boundary when SD is used. SD leads to learning a curved decision boundary with a larger margin in the input space. See App. B for additional details and experiments.
+
+# 4.2 CIFAR classification and adversarial robustness
+
+To study the classification margin in deeper networks, we conduct a classification experiment on CIFAR-10, CIFAR-100, and CIFAR-2 (cats vs dogs of CIFAR-10) [57] using a convolutional network with ReLU non-linearity. Unlike linear models, the margin to a non-linear decision boundary cannot be computed analytically. Therefore, following the approach in [72], we use "the norm of inputdisturbance required to cross the decision boundary" as a proxy for the margin. The disturbance on the input is computed by projected gradient descent (PGD) [84], a well-known adversarial attack.
+
+Table 1: Table compares adversarial robustness of ERM (vanilla cross-entropy) vs SD with a CNN trained on CIFAR-2, 10, and 100 (setup of [72]). SD consistently achieves a better OOD performance.   
+
+<table><tr><td>Dataset</td><td>Method</td><td>Train*</td><td>Test IID</td><td>Test OOD†</td></tr><tr><td rowspan="2">Cifar-2</td><td>w/o SD</td><td></td><td>100.0% ±0.0 95.2%±0.12</td><td>42.3% ±3.0</td></tr><tr><td></td><td></td><td>w/ SD(a=0.01)100.0%±0.0 95.3%±0.17</td><td>69.7% ± 2.9</td></tr><tr><td rowspan="2">Cifar-10</td><td>w/o SD</td><td>99.9% ± 0.01</td><td>92.8%±0.15</td><td>30.1% ± 2.1</td></tr><tr><td></td><td></td><td>w/ SD(=0.01)99.9%±0.01 92.9% ±0.16</td><td>67.7% ± 1.5</td></tr><tr><td rowspan="2">Cifar-100</td><td>w/o SD</td><td></td><td>99.7% ± 0.01 69.2% ±0.29</td><td>14.3% ± 2.0</td></tr><tr><td></td><td></td><td>w/SD(a=0.05)99.7%±0.0270.5%±0.26</td><td>24.9% ± 1.9</td></tr></table>
+
+† Accuracy (± std) for 10 runs.
+
+![](images/bf195fd7cc5efdd1fa50f433f2b3afcfca326e622b21c20b2e591316e226f770.jpg)  
+Figure 2: The plot shows the cumulative distribution function (CDF) of the margin for the CIFAR-2 binary classification. SD appears to improve the margin considerably.
+
+Table 1 includes the results for IID (original test set) and OOD (perturbed test set by $\epsilon _ { \mathrm { P G D } } = 0 . 0 5 )$ . Fig. 2 shows the percentange of mis-classifications as the norm of disturbance is increased for the Cifar-2 dataset. This plot can be interpreted as the cumulative distribution function (CDF) of the margin and hence a lower curve reads as a more robust network with a larger margin. This experiment suggests that when trained with vanilla cross-entropy, even slight disturbances in the input deteriorates the network’s classification accuracy. That is while spectral decoupling (SD) improves the margin considerably. Importantly, this improvement in robustness does not seem to compromise the noise-free test performance. It should also be highlighted that SD does not explicitly aim at maximizing the margin and the observed improvement is in fact a by-product of decoupled learning of latent features. See Section 5 for a discussion on why cross-entropy results in a poor margin while being considered a max-margin classifier in the literature [94].
+
+# 4.3 Colored MNIST with color bias
+
+We conduct experiments on the Colored MNIST Dataset, proposed in [9]. The task is to predict binary labels $y = - 1$ for digits 0 to 4 and $y = + 1$ for digits 5 to 9. A color channel (red, green) is artificially added to each example to deliberately impose a spurious correlation between the color and the label. The task has three environments:
+
+• Training env. 1: Color is correlated with the labels with 0.9 probability. • Training env. 2: Color is correlated with the labels with 0.8 probability. • Testing env.: Color is correlated with the labels with 0.1 probability (0.9 reversely corre
+
+Because of the opposite correlation between the color and the label in the test set, only learning to classify based on color would be disastrous at testing. For this reason, Empirical Risk Minimization (ERM) performs very poorly on the test set $( 2 3 . 7 \%$ accuracy) as shown in Tab. 2.
+
+<table><tr><td>Method</td><td>Train Accuracy</td><td>Test Accuracy</td></tr><tr><td>ERM(Vanilla Cross-Entropy)</td><td>91.1 % (±0.4)</td><td>23.7 % (±0.8)</td></tr><tr><td>REx[59]</td><td>71.5 % (±1.0)</td><td>68.7% (±0.9)</td></tr><tr><td>IRM[9]</td><td>70.5% (±0.6)</td><td>67.1 % (±1.4)</td></tr><tr><td>SD (this work)</td><td>70.0 % (±0.9)</td><td>68.4% (±1.2)</td></tr><tr><td>Oracle - (grayscale images)</td><td>73.5 % (±0.2)</td><td>73.0 % (±0.4)</td></tr><tr><td>RandomGuess</td><td>50%</td><td>50%</td></tr></table>
+
+Table 2: Test accuracy on test examples of the Colored MNIST after training for 1k epochs. The standard deviation over 10 runs is reported in parenthesis. ERM stands for the empirical risk minimization. Oracle is an ERM trained on grayscale images. Note that due to $25 \%$ label noise, a hypothetical optimum achieves $75 \%$ accuracy (the upper bound).
+
+Invariant Risk Minimization (IRM) [9] on the other hand, performs well on the test set with $( 6 7 . 1 \%$ accuracy). However, IRM requires access to multiple (two in this case) separate training environments with varying amount of spurious correlations. IRM uses the variance between environments as a signal for learning to be “invariant” to spurious correlations. Risk Extrapolation (REx) [59] is a related training method that encourages learning invariant representations. Similar to IRM, it requires access to multiple training environments in order to quantify the concept of “invariance”.
+
+SD achieves an accuracy of $6 8 . 4 \%$ . Its performance is remarkable because unlike IRM and REx, SD does not require access to multiple environments and yet performs well when trained on a single environment (in this case the aggregation of both of the training environments).
+
+A natural question that arises is “How does SD learn to ignore the color feature without having access to multiple environments?” The short answer is that it does not! In fact, we argue that SD learns the color feature but it also learns other predictive features, i.e., the digit shape features. At test time, the predictions resulting from the shape features prevail over the color feature. To validate this hypothesis, we study a trained model with each of these methods (ERM, IRM, SD) on four variants of the test environment: 1) grayscale-digits: No color channel is provided and the network should rely on shape features only. 2) colored-digits: Both color and digit are provided however the color is negatively correlated (opposite of the training set) with the label. 3) grayscaleblank: All images are grayscale and blank and hence do not provide any information. 4) colored-blank: Digit features are removed and only the color feature is kept, also with reverse label compared to training. Fig. 3 summarizes the results. For more discussions see SM B.
+
+As a final remark, we should highlight that, by design, this task assumes access to the test environment for hyperparameter tuning for all the reported methods. This is not a valid assumption in general, and hence the results should be only interpreted as a probe that shows that SD could provide an important level of control over what features are learned.
+
+![](images/d0fa5d396174a94eb275513da009f1f33178a40f2dbc241ce7dfa391f7e566f2.jpg)  
+Figure 3: Diagram comparing ERM, SD, and IRM on four different test environments on which we evaluate a pre-trained model. Top and bottom rows show the accuracy and the entropy (inverse of confidence), respectively. Analysis: Compare three values of $\mathfrak { C } \sharp \sharp \sharp \sharp \sharp \cdot$ , 9.4 % , and $4 9 . 6 \%$ : Both ERM and SD have learned the color feature but since it is inversely correlated with the label, when only the color feature is provided, as expected both ERM and SD performs poorly. Now compare $\mathbf { G } \mathbf { M } \mathbf { 0 }$ and $0 . 4 1$ : Although both ERM and SD have learned the color feature, ERM is much more confident on its predictions (zero entropy). As a consequence, when digit features are provided along with the color feature (colored-digit environment), ERM still performs poorly $( \overbrace { 2 3 . 9 \ \% } )$ but SD achieves significantly better results $( 6 7 . 2 \% )$ ). IRM ignores the color feature altogether but it requires access to multiple training environments.
+
+![](images/78aab33fb5d05d81db549ed8560e9bd995cca1540e7ae465d6c4ea573f439933.jpg)  
+Figure 4: CelebA: blond vs dark hair classification. The HairColor and the Gender are spuriously correlated which leads to poor OOD performance with ERM, however SD significantly improves performance. ERM’s worst group accuracy is significantly lower than SD.
+
+Table 3: CelebA: blond vs dark hair classification with spurious correlation. We report test performance over ten runs. SD significantly improves upon ERM. ∗Group DRO [89] requires explicit information about the spurious correlation. LfF [71] requires simultaneous training of two networks.   
+
+<table><tr><td>Method</td><td>Average Acc.</td><td>Worst Group Acc.</td></tr><tr><td>ERM</td><td>94.61 % (±0.67)</td><td>40.35 % (±1.68)</td></tr><tr><td>SD (this work)</td><td>91.64 % (±0.61)</td><td>83.24 % (±2.01)</td></tr><tr><td>LfF</td><td>N/A</td><td>81.24 % (±1.38)</td></tr><tr><td>Group DRO*</td><td>91.76 % (±0.28)</td><td>87.78 % (±0.96)</td></tr></table>
+
+# 4.4 CelebA with gender bias
+
+The CelebA dataset [65] contains $1 6 2 \mathrm { k }$ celebrity faces with binary attributes associated with each image. Following the setup of [89], the task is to classify images with respect to their hair color into two classes of blond or dark hair. However, the Gender $\in$ {Male, Female} is spuriously correlated with the $\mathtt { H a i r C o l o r } \in \{ \mathtt { B l o n d } , \mathtt { D a r k } \}$ in the training data. The rarest group which is blond males represents only $0 . 8 5 \%$ of the training data (1387 out of $1 6 2 \mathrm { k }$ examples). We train a ResNet-50 model [38] on this task. Tab. 3 summarizes the results and compares the performance of several methods. A model with vanilla cross-entropy (ERM) appears to generalize well on average but fails to generalize to the rarest group (blond males) which can be considered as “weakly" out-of-distribution (OOD). Our proposed SD improves the performance more than twofold. It should be highlighted that for this task, we use a variant of SD in which, $\frac { \lambda } { 2 } | | \hat { y } - \gamma | | _ { 2 } ^ { 2 }$ is added to the original cross-entropy loss. The hyper-parameters $\lambda$ and $\gamma$ are tuned separately for each class (a total of four hyper-parameters). This variant of SD does provably decouple the dynamics too but appears to perform better than the original SD in Eq. 17 in this task.
+
+Other proposed methods presented in Tab. 3 also show significant improvements on the performance of the worst group accuracy. The recently proposed “Learning from failure” (LfF) [71] achieves comparable results to SD, but it requires simultaneous training of two networks. Group DRO [89] is another successful method for this task. However, unlike SD, Group DRO requires explicit information about the spuriously correlated attributes. In most practical tasks, information about the spurious correlations is not provided and, dependence on the spurious correlation goes unrecognized.2
+
+# 5 Related Work and Discussion
+
+Here, we discuss the related work. Due to space constraints, further discussions are in App. A.
+
+On learning dynamics and Loss Choice. Several works including [90, 91, 1, 60] investigate the dynamics of deep linear networks trained with squared-error loss. Different decompositions of the learning process for neural networks have been used: [83, 104, 87, 105] study the learning in the Fourier domain and show that low-frequency functions are learned earlier than high-frequency ones. [90, 2, 32] provide closed-form equations for the dynamics of linear networks in terms of the principal components of the input covariance matrix. More recently, with the introduction of neural tangent kernel (NTK) [52, 62], a new line of research is to study the convergence properties of gradient descent [e.g. 8, 69, 25, 29, 7, 44, 33, 110, 11, 99]. Among them, [12, 106, 18, 22] decompose the learning process along the principal components of the NTK. The message in these works is that the training process can be decomposed into independent learning dynamics along the orthogonal directions.
+
+Most of the studies mentioned above focus on the particular squared-error loss. For a linearized network, the squared-error loss results in linear learning dynamics, which often admit an analytical solution. However, the de-facto loss function for many of the practical applications of neural networks is the cross-entropy. Using the cross-entropy as the loss function leads to significantly more complicated and non-linear dynamics, even for a linear neural network. In this work, our focus was the cross-entropy loss.
+
+On reliance upon spurious correlations and robustness. In the context of robustness in neural networks, state-of-the-art neural networks appear to naturally focus on low-level superficial correlations rather than more abstract and robustly informative features of interest (e.g. [30]). As we argue in this work, Gradient Starvation is likely an important factor contributing to this phenomenon and can result in adversarial vulnerability. There is a rich research literature on adversarial attacks and neural networks’ vulnerability [96, 34, 48, 67, 5, 47]. Interestingly, [73], [72] and [51] draw a similar conclusion and argue that “an insufficiency of the cross-entropy loss” causes excessive invariances to predictive features. Perhaps [92] is the closest to our work in which authors study the simplicity bias (SB) in stochastic gradient descent. They demonstrate that neural networks exhibit extreme bias that could lead to adversarial vulnerability.
+
+On implicit bias. Despite being highly-overparameterized, modern neural networks seem to generalize very well [108]. Modern neural networks generalize surprisingly well in numerous machine tasks. This is despite the fact that neural networks typically contain orders of magnitude more parameters than the number of examples in a training set and have sufficient capacity to fit a totally randomized dataset perfectly [108]. The widespread explanation is that the gradient descent has a form of implicit bias towards learning simpler functions that generalize better according to Occam’s razor. Our exposition of GS reinforces this explanation. In essence, when training and test data points are drawn from the same distribution, the top salient features are predictive in both sets. We conjecture that in such a scenario, by not learning the less salient features, GS naturally protects the network from overfitting.
+
+The same phenomenon is referred to as implicit bias, implicit regularization, simplicity bias and spectral bias in several works [83, 75, 36, 74, 70, 53, 94, 10, 13, 35, 82, 66].
+
+As an active line of research, numerous studies have provided different explanations for this phenomenon. For example, [70] justifies the implicit bias of neural networks by showing that stochastic gradient descent learns simpler functions first. [15, 78] suggests that a form of implicit regularization is induced by an alignment between NTK’s principal components and only a few task-relevant directions. Several other works such as [20, 35, 94, 25] recognize the convergence of gradient descent to maximum-margin solution as the essential factor for the generalizability of neural networks. It should be stressed that these work refer to the margin in the hidden space and not in the input space as pointed out in [55]. Indeed, as observed in our experiments, the maximum-margin classifier in the hidden space can be achieved at the expense of a small margin in the input space.
+
+On Gradient Starvation and no free lunch theorem. The no free lunch theorem [93, 102] states that “learning is impossible without making assumptions about training and test distributions”. Perhaps, the most commonly used assumption of machine learning is the i.i.d. assumption [98], which assumes that training and test data are identically distributed. However, in general, this assumption might not hold, and in many practical applications, there are predictive features in the training set that do not generalize to the test set. A natural question that arises is how to favor generalizable features over spurious features? The most common approaches include data augmentation, controlling the inductive biases, using regularizations, and more recently training using multiple environments.
+
+Here, we would like to elaborate on an interesting thought experiment of [79]: Suppose a neural network is provided with a chess book containing examples of chess games with the best movements indicated by a red arrow. The network can take two approaches: 1) learn how to play chess, or 2) learn just the red arrows. Either of these solutions results in zero training loss on the games in the book while only the former is generalizable to new games. With no external knowledge, the network typically learns the simpler solution.
+
+Recent work aims to leverage the invariance principle across several environments to improve robust learning. This is akin to present several chess books to a network, each with markings indicating the best moves for different sets of games. In several studies [9, 59, 79, 3], methods are developed to aggregate information from multiple training environments in a way that favors the generalizable / domain-agnostic / invariant solution. We argue that even with having access to only one training environment, there is useful information in the training set that fails to be discovered due to Gradient Starvation. The information on how to actually play chess is already available in any of the chess books. Still, as soon as the network learns the red arrows, the network has no incentive for further learning. Therefore, learning the red arrows is not an issue per se, but not learning to play chess is.
+
+Gradient Starvation: friend or foe? Here, we would like to remind the reader that GS can have both adverse and beneficial consequences. If the learned features are sufficient to generalize to the test data, gradient starvation can be viewed as an implicit regularizer. Otherwise, Gradient Starvation could have an unfavorable effect, which we observe empirically when some predictive features fail to be learned. A better understanding and control of Gradient Starvation and its impact on generalization offers promising avenues to address this issue with minimal assumptions. Indeed, our Spectral Decoupling method requires an assumption about feature imbalance but not to pinpoint them exactly, relying on modulated learning dynamics to achieve balance.
+
+GS social impact Modern neural networks are being deployed extensively in numerous machine learning tasks. Our models are used in critical applications such as autonomous driving, medical prediction, and even justice system where human lives are at stake. However, neural networks appear to base their predictions on superficial biases in the dataset. Unfortunately, biases in datasets could be neglected and pose negative impacts on our society. In fact, our Celeb-A experiment is an example of the existence of such a bias in the data. As shown in the paper, the gender-specific bias could lead to a superficial high performance and is indeed very hard to detect. Our analysis, although mostly on the theory side, could pave the path for researchers to build machine learning systems that are robust to biases and helps towards fairness in our predictions.
+
+# 6 Conclusion
+
+In this paper, we formalized Gradient Starvation (GS) as a phenomenon that emerges when training with cross-entropy loss in neural networks. By analyzing the dynamical system corresponding to the learning process in a dual space, we showed that GS could slow down the learning of certain features, even if they are present in the training set. We derived spectral decoupling (SD) regularization as a possible remedy to GS.
+
+# Acknowledgments and Disclosure of Funding
+
+The authors are grateful to Samsung Electronics Co., Ldt., CIFAR, and IVADO for their funding and Calcul Québec and Compute Canada for providing us with the computing resources. We would further like to acknowledge the significance of discussions and supports from Reyhane Askari Hemmat and Faruk Ahmed. MP would like to thank Aristide Baratin, Kostiantyn Lapchevskyi, Seyed Mohammad Mehdi Ahmadpanah, Milad Aghajohari, Kartik Ahuja, Shagun Sodhani, and Emmanuel Bengio for their invaluable help.
+
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parse/train/kFJoj7zuDVi/kFJoj7zuDVi.md

@@ -0,0 +1,361 @@
+# Towards a Theoretical Framework of Out-of-Distribution Generalization
+
+Chuanlong XieHuawei Noah’s Ark Labxie.chuanlong@huawei.com
+
+Haotian Ye   
+Peking University   
+Pazhou Lab   
+haotianye@pku.edu.cn
+
+Tianle Cai Peking University caitianle1998@pku.edu.cn
+
+# Ruichen Li
+
+# Zhenguo Li
+
+# Liwei Wang
+
+Peking University xk-lrc@pku.edu.cn
+
+Huawei Noah’s Ark Lab Li.Zhenguo@huawei.com
+
+Key Laboratory of Machine Perception, MOE, School of EECS, Institute for Artificial Intelligence, Peking University wanglw@cis.pku.edu.cn
+
+# Abstract
+
+Generalization to out-of-distribution (OOD) data is one of the central problems in modern machine learning. Recently, there is a surge of attempts to propose algorithms that mainly build upon the idea of extracting invariant features. Although intuitively reasonable, theoretical understanding of what kind of invariance can guarantee OOD generalization is still limited, and generalization to arbitrary out-of-distribution is clearly impossible. In this work, we take the first step towards rigorous and quantitative definitions of 1) what is OOD; and 2) what does it mean by saying an OOD problem is learnable. We also introduce a new concept of expansion function, which characterizes to what extent the variance is amplified in the test domains over the training domains, and therefore give a quantitative meaning of invariant features. Based on these, we prove OOD generalization error bounds. It turns out that OOD generalization largely depends on the expansion function. As recently pointed out by [21], any OOD learning algorithm without a model selection module is incomplete. Our theory naturally induces a model selection criterion. Extensive experiments on benchmark OOD datasets demonstrate that our model selection criterion has a significant advantage over baselines.
+
+# 1 Introduction
+
+One of the most fundamental assumptions of classic supervised learning is the “i.i.d. assumption”, which states that the training and the test data are independent and identically distributed. However, this assumption can be easily violated in a reality [8, 10, 11, 17, 38, 48, 56] where the test data usually have a different distribution than the training data. This motivates the research on the out-ofdistribution (OOD) generalization, or domain generalization problem, which assumes access only to data drawn from a set $\mathcal { E } _ { a v a i l }$ of available domains during training, and the goal is to generalize to a larger domain set ${ \mathcal { E } } _ { a l l }$ including unseen domains.
+
+To generalize to OOD data, most existing algorithms attempt to learn features that are invariant to a certain extent across training domains in the hope that such invariance also holds in unseen domains. For example, distributional matching-based methods [20, 35, 55] seek to learn features that have the same distribution across different domains; IRM [5] and its variants [1, 32, 33] learn feature representations such that the optimal linear classifier on top of the representation matches across domains.
+
+35th Conference on Neural Information Processing Systems (NeurIPS 2021).
+
+Though the idea of learning invariant features is intuitively reasonable, there is only limited theoretical understanding of what kind of invariance can guarantee OOD generalization. Clearly, generalization to an arbitrary out-of-distribution domain is impossible and in practice, the features can hardly be absolutely invariant from $\mathcal { E } _ { a v a i l }$ to ${ \mathcal { E } } _ { a l l }$ unless all the domains are identical. So it is necessary to first formulate what OOD data can be generalized to, or, what is the relation between the available training domain set $\mathcal { E } _ { a v a i l }$ and the entire domain set ${ \mathcal { E } } _ { a l l }$ . Meanwhile, to what extent the invariance of features on $\mathcal { E } _ { a v a i l }$ can be preserved in ${ \mathcal { E } } _ { a l l }$ should be rigorously characterized.
+
+In this paper, we take the first step towards a general OOD framework by quantitatively formalizing the relationship between $\mathcal { E } _ { a v a i l }$ and ${ \mathcal { E } } _ { a l l }$ in terms of the distributions of features and provide OOD generalization guarantees based on our quantification of the difficulty of OOD generalization problem. Specifically, we first rigorously formulate the intuition of invariant features used in previous works by introducing the “variation” and “informativeness” (Definition 3.1 and 3.2) of each feature. Our theoretical insight can then be informally stated as: for learnable OOD problems, if a feature is informative for the classification task as well as invariant over $\mathcal { E } _ { a v a i l }$ , then it is still invariant over ${ \mathcal { E } } _ { a l l }$ . In other words, invariance of informative features in $\mathcal { E } _ { a v a i l }$ can be preserved in ${ \mathcal { E } } _ { a l l }$ . We further introduce a class of functions, dubbed expansion function (Definition 3.3), to quantitatively characterize to what extent the variance of features on $\mathcal { E } _ { a v a i l }$ is amplified on ${ \mathcal { E } } _ { a l l }$ .
+
+Based on our formulation, we derive theoretical guarantees on the OOD generalization error, i.e., the gap of largest error between the domain in $\mathcal { E } _ { a v a i l }$ and domain in ${ \mathcal { E } } _ { a l l }$ . Specifically, we prove the upper and lower bound of OOD generalization error in terms of the expansion function and the variation of learned features over $\mathcal { E } _ { a v a i l }$ . Our results theoretically confirm that 1) the expansion function can reflect the difficulty of OOD generalization problem, i.e., problems with more rapidly increasing expansion functions are harder and have worse generalization guarantees; 2) the generalization error gap can tend to zero when the variation of learned features tend to zero, so minimizing the variation in $\mathcal { E } _ { a v a i l }$ can reduce the generalization error.
+
+As pointed out by Gulrajani and Lopez-Paz [21], any OOD algorithm without a specified model selection criterion is not complete. Since ${ \mathcal { E } } _ { a l l }$ is unseen, hyper-parameters can only be chosen according to $\mathcal { E } _ { a v a i l }$ . Previous selection methods mainly focus on validation accuracy over $\mathcal { E } _ { a v a i l }$ which is only a biased metric of OOD performance. On the contrary, a promising model selection method should instead be predictive of OOD performance. Inspired by our bounds, we propose a model selection method to select models with high validation accuracy and low variation, which corresponds to the upper bound of OOD error. The introduction of a model’s variation relieves the problem of classic selection methods, in which models that overfit $\mathcal { E } _ { a v a i l }$ tend to be selected. Experimental results show that our method can outperform baselines and select models with higher OOD accuracy.
+
+Contributions. We summarize our major contributions here:
+
+• We introduce a quantitative and rigorous formulation of OOD generalization problem that characterizes the relation of invariance over the training domain set $\mathcal { E } _ { a v a i l }$ and test domain set ${ \mathcal { E } } _ { a l l }$ . The core quantity in our characterization, the expansion function, determines the difficulty of an OOD generalization problem.
+
+• We prove novel OOD generalization error bounds based on our formulation. The upper and lower bounds together indicate that the expansion function well characterizes the OOD generalization ability of features with different levels of variation.
+
+• We design a model selection criterion that is inspired by our generalization bounds. Our criterion takes both the performance on training domains and the variation of models into consideration and is predictive of OOD performance according to our bounds. Experimental results demonstrate our selection criterion can choose models with higher OOD accuracy.
+
+The rest of the paper is organized as follows: Section 2 is our preliminary. In Section 3, we give our theoretical formulation. Section 4 gives our generalization bound. We propose our model selection method in Section 5. In Section 6 we conduct experiments on expansion function and model selection. We review more related works in Section 7 and conclude our work in Section 8.
+
+# 2 Preliminary
+
+Throughout the paper, we consider a multi-class classification task $\mathcal { X } \to \mathcal { Y } = \{ 1 , . . . , K \}$ .1 Let ${ \mathcal { E } } _ { a l l }$ be the domain set we want to generalize to, and $\mathcal { E } _ { a v a i l } \subseteq \mathcal { E } _ { a l l }$ be the available domain set, i.e., all domains we have during the training procedure. We denote $( X ^ { e } , Y ^ { e } )$ to be the input-label pair drawn from the data distribution of domain $e$ . The OOD generalization goal is to find a classifier $f ^ { * }$ that minimizes the worst-domain loss on ${ \mathcal { E } } _ { a l l }$ :
+
+$$
+f ^ { * } = \underset { f \in \mathcal { F } } { \mathrm { a r g m i n } } \mathcal { L } ( \mathcal { E } _ { a l l } , f ) , \mathcal { L } ( \mathcal { E } , f ) \triangleq \underset { e \in \mathcal { E } } { \mathrm { m a x } } \mathbb { E } \big [ \ell \big ( f ( X ^ { e } ) , Y ^ { e } \big ) \big ]
+$$
+
+where $\mathcal { F } : \mathcal { X } \xrightarrow { } \mathbb { R } ^ { K }$ is the the hypothetical space and $\ell ( \cdot , \cdot )$ is a loss function. Similar to previous works [5, 16, 27, 33], we assume that $f$ can be decomposed into $g \circ h$ , where $g \in \mathcal { G } : \mathbb { R } ^ { d }  \mathbb { R } ^ { K }$ is the top classifier and $h : \mathcal { X }  \mathbb { R } ^ { d }$ is a $d$ -dimensional feature extractor, i.e.,
+
+$$
+h ( x ) = ( \phi _ { 1 } ( x ) , \phi _ { 2 } ( x ) , \ldots , \phi _ { d } ( x ) ) ^ { \top } , \quad \phi _ { i } \in \Phi .
+$$
+
+Here $\Phi$ is the set of scalar feature maps which map $\mathcal { X }$ to $\mathbb { R }$ and $d$ is fixed. We will call each $\phi \in \Phi$ a feature for simplicity. Given a domain $e$ , we denote the $d$ -dimensional random vector $h ( X ^ { e } )$ as $h ^ { e }$ , one-dimensional feature $\phi ( X ^ { e } )$ as $\phi ^ { e }$ , and the conditional distribution of $h ^ { e } , \phi ^ { e }$ given $Y ^ { e } = y$ as $\mathbb { P } ( h ^ { e } | y ) , \mathbb { P } ( \phi ^ { e } | y )$ . For simplicity, we assume the data distribution is balanced in every domain, i.e., $P ( Y ^ { e } = y ) = \frac { 1 } { K } , \forall y \in \mathcal { Y } , e \in \mathcal { E } _ { a l l }$ . Our framework can be easily extended to the case where the balanced assumption is removed, with an additional term corresponding to the imbalance adding to the generalization bounds.
+
+# 3 Framework of OOD Generalization Problem
+
+The main challenge of formalizing the OOD generalization problem is to mathematically describe the connection between $\mathcal { E } _ { a v a i l }$ and ${ \mathcal { E } } _ { a l l }$ and how generalization depends on this relation. Towards this goal, we introduce several quantities to characterize the relation of feature distributions over different domains and bridge $\mathcal { E } _ { a v a i l }$ and ${ \mathcal { E } } _ { a l l }$ by expansion function (Definition 3.3) over the quantities we have introduced. Our framework is motivated by the understanding that, in an OOD generalization task, certain “property” of “good” features in $\mathcal { E } _ { a v a i l }$ should be “preserved” in ${ \mathcal { E } } _ { a l l }$ (the reason is described in Section 1). In Section 3.1, we will go into details on what we mean by “property” (variation, Definition 3.1), “good” (informativeness, Definition 3.2), and “preserved” (measured by expansion function). In Section 6.2, we further illustrate the key concepts in our framework by a real-world OOD problem.
+
+# 3.1 Formalizing OOD Problem by Quantifying Feature Distribution
+
+We first introduce the concepts “variation" and “informativeness" of a feature $\phi$ . The first one is what we expect to be preserved in ${ \mathcal { E } } _ { a l l }$ and the second one characterizes what features will be considered. Specifically, let $\rho ( \mathbb { P } , \mathbb { Q } )$ be a symmetric “distance” of two distributions. Note that $\rho$ can have many choices, like $L _ { 2 }$ Distance, Total Variation and symmetric KL-divergence, etc. The variation and informativeness are defined as follows:
+
+Definition 3.1 (Variation). The variation of feature $\phi ( \cdot )$ across a domain set $\mathcal { E }$ is
+
+$$
+\mathcal { V } _ { \rho } ( \phi , \mathcal { E } ) = \operatorname* { m a x } _ { y \in \mathcal { V } } \operatorname* { s u p } _ { e , e ^ { \prime } \in \mathcal { E } } \rho \big ( \mathbb { P } ( \phi ^ { e } | y ) , \mathbb { P } ( \phi ^ { e ^ { \prime } } | y ) \big ) .
+$$
+
+A feature $\phi ( \cdot )$ is $\varepsilon$ -invariant across $\mathcal { E }$ , $i f \varepsilon \geq \mathcal { V } ( \phi , \mathcal { E } )$ (We omit the subscript $\rho$ in case of no ambiguity). Definition 3.2 (Informativeness). The informativeness of feature $\phi ( \cdot )$ across a domain set $\mathcal { E }$ is
+
+$$
+\mathcal { T } _ { \rho } ( \phi , \mathcal { E } ) = \frac { 1 } { K ( K - 1 ) } \sum _ { \stackrel { y \neq y ^ { \prime } } { y , y ^ { \prime } \in y } } \operatorname* { m i n } _ { e \in \mathcal { E } } \rho \big ( \mathbb { P } ( \phi ^ { e } | y ) , \mathbb { P } ( \phi ^ { e } | y ^ { \prime } ) \big ) .
+$$
+
+A feature $\phi ( \cdot )$ is $\delta$ -informative across $\varepsilon$ , if $\quad \delta \leq { \mathcal { I } } ( \phi , { \mathcal { E } } )$ .
+
+The variation $\mathcal { V } ( \phi , \mathcal { E } )$ measures the stability of $\phi ( \cdot )$ over the domains in $\mathcal { E }$ and the informativeness ${ \mathcal { T } } ( \phi , { \mathcal { E } } )$ captures the ability of $\phi ( \cdot )$ to distinguish different labels. We would like to highlight that the variation and informativeness are defined on each one-dimensional feature $\phi ( \cdot )$ . Unlike previous distance between distributions defined in $d$ -dimensional space, our definitions are more reasonable and practical, since it can be easily calculated and analyzed.
+
+We are now ready to introduce the core quantity for connecting $\mathcal { E } _ { a v a i l }$ and ${ \mathcal { E } } _ { a l l }$ . Our motivation, as elaborated in the introduction section, is that, if a feature is informative for the classification task and invariant over $\mathcal { E } _ { a v a i l }$ , then to enable OOD generalization from $\mathcal { E } _ { a v a i l }$ to ${ \mathcal { E } } _ { a l l }$ , it should be still invariant over ${ \mathcal { E } } _ { a l l }$ . So the relation between $\mathcal { V } ( \phi , \mathcal { E } _ { a v a i l } )$ and $\mathinner { \gamma \mathopen { \left( \phi , \mathcal { E } _ { a l l } \right) } }$ of an informative feature captures the feasibility and difficulty of OOD generalization. To quantitatively measure this relation, we define the following function class:
+
+Definition 3.3 (Expansion Function). We say a function $s : \mathbb { R } ^ { + } \cup \{ 0 \}  \mathbb { R } ^ { + } \cup \{ 0 , + \infty \}$ is an expansion function, iff the following properties hold: 1) $s ( \cdot )$ is monotonically increasing and $s ( x ) \geq x , \forall x \geq 0 ; 2 ,$ ) $\begin{array} { r } { \operatorname* { l i m } _ { x  0 ^ { + } } s ( x ) = s ( 0 ) = 0 } \end{array}$ .
+
+This function class gives a full characterization of how the variation between $\mathcal { E } _ { a v a i l }$ and ${ \mathcal { E } } _ { a l l }$ is related. Based on this function class, we can introduce our formulation of the learnability of OOD generalization as follows:
+
+Definition 3.4 (Learnability). Let $\Phi$ be the feature space and $\rho$ be a distribution distance. We say an OOD generalization problem from $\mathcal { E } _ { a v a i l }$ to ${ \mathcal { E } } _ { a l l }$ is learnable if there exists an expansion function $s ( \cdot )$ and $\delta \geq 0$ , such that: for all $\phi \in \Phi$ satisfying ${ \mathcal { T } } _ { \rho } ( \phi , { \mathcal { E } } _ { a v a i l } ) \geq \delta$ , we have $s ( \mathcal { V } _ { \rho } ( \phi , \mathcal { E } _ { a v a i l } ) ) \geq$ $\mathcal { V } _ { \rho } \big ( \phi , \mathcal { E } _ { a l l } \big )$ . If such $s ( \cdot )$ and $\delta$ exist, we further call this problem $( s ( \cdot ) , \delta )$ -learnable. If an $o o D$ generalization problem is not learnable, we call $i t$ unlearnable.
+
+To understand the intuition and rationality of our formulation, several discussions are in order.
+
+Properties of the expansion function. In Definition 3.3, we highlight two properties of the expansion function. The first property comes naturally from the monotonicity properties of variation: any $\varepsilon _ { 1 }$ -invariant feature is also $\varepsilon _ { 2 }$ -invariant for $\varepsilon _ { 2 } \geq \varepsilon _ { 1 }$ ; and $\mathcal { V } ( \phi , \mathcal { E } _ { 1 } ) \leq \mathcal { V } ( \phi , \mathcal { E } _ { 2 } )$ for any ${ \mathcal { E } } _ { 1 } \subseteq { \mathcal { E } } _ { 2 }$ . The monotonicity also implies that larger ${ \mathcal { E } } _ { a l l }$ will induce larger $s ( \cdot ) ^ { 2 }$ and it is also harder to be generalized to. From this view, we can see that the scale of $s ( \cdot )$ can reflect the difficulty of OOD generalization. The second property is more crucial since it formulates the intuition that if an informative feature is almost invariant in $\mathcal { E } _ { a v a i l }$ , it should remain invariant in ${ \mathcal { E } } _ { a l l }$ . Without this assumption, OOD generalization can never be guaranteed because we cannot predict whether an invariant and informative feature in $\mathcal { E } _ { a v a i l }$ will vary severely in unseen ${ \mathcal { E } } _ { a l l }$ .
+
+Necessity of informativeness. We include a seemingly redundant quantity informativeness in the definition of learnability. However, this term is necessary because only informative features are responsible for the performance of classification. Non-informative but invariant features over $\mathcal { E } _ { a v a i l }$ may only capture some noise that is irrelevant to the classification problem, and we shall not expect the noise to be invariant over ${ \mathcal { E } } _ { a l l }$ . Moreover, we show in Figure 1 that in practice, many invariant but useless features in $\mathcal { E } _ { a v a i l }$ vary a lot in ${ \mathcal { E } } _ { a l l }$ , and adding the constraint of informativeness makes the expansion function reasonable. In addition, there are multiple choices of $( s ( \cdot ) , \delta )$ to make an OOD generalization problem learnable: larger $\delta$ will filter out more features, and so $\dot { s } ( \cdot )$ can be smaller (flatter). This multiplicity will result in a trade-off between $s ( \cdot )$ and $\delta$ , which will be discussed in Section 6.2.
+
+Two extreme cases: i.i.d. & unlearnable. To better understand the concept of learnability, we consider two extreme cases. (1) The first example is when all data from different $\textit { e } \in \mathcal { E } _ { a l l }$ are identically distributed, i.e., the classic supervised learning setting. This problem is $( s ( \cdot ) , 0 )$ -learnable with $s ( x ) = x$ , implying no extra difficulty in OOD generalization. (2) As an example of unlearnable, consider the following case (modified from Colored MNIST [5]): For $e \in \mathcal { E } _ { a v a i l }$ , images with label 0 always has a red background while images with label 1 has a blue background. For $e \in \mathcal { E } _ { a l l } \ \backslash \ \mathcal { E } _ { a v a i l }$ , this relationship is entirely inverse. Since data from different $e \in \mathcal { E } _ { a v a i l }$ are identically distributed but different from other $e \in \mathcal { E } _ { a l l }$ , no expansion function can make it learnable, i.e., it is OOD-unlearnable.
+
+The unlearnability of this case also coincides with our intuition: Without prior knowledge, it is not clear from merely the training data, whether the task is to distinguish digit 0 from 1, or to distinguish color red from blue. As a result, generalization to ${ \mathcal { E } } _ { a l l }$ cannot be guaranteed.
+
+# 4 Generalization Bound
+
+In this section, we consider an OOD generalization problem from $\mathcal { E } _ { a v a i l }$ to ${ \mathcal { E } } _ { a l l }$ , and our goal is to analyze the OOD generalization error of classifier $f = g \circ h$ defined by
+
+$$
+\mathrm { e r r } ( f ) = \mathcal { L } ( \mathcal { E } _ { a l l } , f ) - \mathcal { L } ( \mathcal { E } _ { a v a i l } , f ) ,
+$$
+
+where we assume the loss function $l ( \cdot , \cdot )$ is bounded by $[ 0 , C ]$ . We prove two upper bounds (4.1, 4.2) as well as a lower bound (4.3) for $\operatorname { e r r } ( f )$ based on our formulation. Our bounds together provide a complete characterization of the difficulty of OOD generalization. Since we expect that an invariant classifier can generalize to unseen domains, we hope to bound $\operatorname { e r r } ( f )$ in terms of the certain variation of $f$ . To this end, we define the variation and informativeness of $f$ in terms of its features, i.e.,
+
+$$
+\begin{array} { r l r } { \mathcal { V } ^ { \operatorname* { s u p } } ( h , \mathcal { E } _ { a v a i l } ) } & { \triangleq } & { \underset { \beta \in S ^ { d - 1 } } { \operatorname* { s u p } } \mathcal { V } ( \beta ^ { \top } h , \mathcal { E } _ { a v a i l } ) , } \\ { \mathcal { T } ^ { \operatorname* { i n f } } ( h , \mathcal { E } _ { a v a i l } ) } & { \triangleq } & { \underset { \beta \in S ^ { d - 1 } } { \operatorname* { i n f } } \mathcal { T } ( \beta ^ { \top } h , \mathcal { E } _ { a v a i l } ) , } \end{array}
+$$
+
+where $( \beta ^ { \top } h ) ( x ) = \beta ^ { \top } h ( x )$ is a feature and $S ^ { d - 1 } = \{ \beta \in \mathbb { R } ^ { d } : \| \beta \| _ { 2 } = 1 \}$ is the unit $( d - 1 )$ -sphere.
+
+Necessity of using supremum over linear combination. One seemingly plausible definition of the variation of a classifier $f$ can be the supremum over all $\mathcal { V } ( \phi _ { i } , \mathcal { E } _ { a v a i l } ) , i \in [ d ]$ . However, as is shown in Appendix 1, it is possible that two high-dimensional joint distributions have close marginal distribution in each dimension, while they do not overlap. In other words, there exist cases where $\mathcal { V } ( \phi _ { i } , \mathcal { E } _ { a l l } ) = 0 , \forall i \in [ d ]$ but after applying the top model $g$ over $\phi _ { i }$ ’s, the distribution varies a lot in $\mathcal { E } _ { a v a i l }$ . Our definition comes from the simple idea that the class of top model $\mathcal { G }$ is at least a linear space, so we should at least consider the variation of every (normalized) linear combination of $h ( \cdot )$ With this, we can guarantee the joint distribution distance is still small.
+
+Theorem 4.1 (Main Theorem). Suppose we have learned a classifier $f ( x ) = g ( h ( x ) )$ such that $\forall e \in \mathcal { E } _ { a l l }$ and $\forall y \in \mathcal { V } ;$ $, p _ { h ^ { e } | Y ^ { e } } ( h | y ) \in L ^ { 2 } ( \mathbb R ^ { d } )$ . Denote the characteristic function of random variable $h ^ { e } | Y ^ { e }$ as $\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) = \mathbb { E } [ \exp \{ i \langle t , h ^ { e } \rangle \} | Y ^ { e } = y ]$ . Assume the hypothetical space $\mathcal { F }$ satisfies the following regularity conditions that $\exists \alpha , M _ { 1 } , M _ { 2 } > 0 , \forall f \in \mathcal { F } , \forall e \in \mathcal { E } _ { a l l } , y \in \mathcal { V }$ ,
+
+$$
+\int _ { h \in \mathbb { R } ^ { d } } p _ { h ^ { e } | Y ^ { e } } ( h | y ) | h | ^ { \alpha } \mathrm { d } h \leq M _ { 1 } \quad a n d \quad \int _ { t \in \mathbb { R } ^ { d } } | \hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) | | t | ^ { \alpha } \mathrm { d } t \leq M _ { 2 } .
+$$
+
+$I f ( \mathcal { E } _ { a v a i l } , \mathcal { E } _ { a l l } )$ is $\left( s ( \cdot ) , \mathcal { T } ^ { i n f } ( h , \mathcal { E } _ { a v a i l } ) \right)$ -learnable under $\Phi$ with Total Variation $\rho ^ { 3 }$ , then we have
+
+$$
+\operatorname { e r r } ( f ) \leq O \left( s \left( \mathcal { V } _ { \rho } ^ { s u p } ( h , \mathcal { E } _ { a v a i l } ) \right) ^ { \frac { \alpha ^ { 2 } } { ( \alpha + d ) ^ { 2 } } } \right) .
+$$
+
+Here $\rho$ is total variation distance, and $O ( \cdot )$ depends on $d , C , \alpha , M _ { 1 } , M _ { 2 }$ .
+
+The above theorem holds for a general classifier learned by any algorithms. Due to its generality, we need to introduce some technical regularity conditions on the density function. The assumption (4) assume the decay rate of density and its characteristic function, which is common in the literature, e.g. [14]. This theorem demonstrates that, the generalization error can be bounded by a function of the variation of $h$ , and it converges to 0 as the variation approaches to 0. Under some special but typical case where the top model $g$ is linear, we can further show that even without the regularity conditions in Theorem 4.1, we have a much better (linear) convergence rate.
+
+Theorem 4.2 (Linear Top Model). Consider any loss satisfying $\begin{array} { r } { \ell ( \hat { y } , y ) = \sum _ { k = 1 } ^ { K } \ell _ { 0 } ( \hat { y } _ { k } , y _ { k } ) } \end{array}$ .4 For any classifier with linear top model $g$ , i.e.,
+
+$$
+f ( x ) = A h ( x ) + b w i t h A \in \mathbb { R } ^ { K \times d } , b \in \mathbb { R } ^ { K } ,
+$$
+
+$i f ( \mathcal { E } _ { a v a i l } , \mathcal { E } _ { a l l } )$ is $\left( s ( \cdot ) , \mathcal { T } ^ { i n f } ( h , \mathcal { E } _ { a v a i l } ) \right)$ -learnable under $\Phi$ with Total Variation $\rho$ , then we have
+
+$$
+\mathrm { e r r } ( f ) \leq O \Bigl ( s \bigl ( \mathcal { V } ^ { s u p } ( h , \mathcal { E } _ { a v a i l } ) \bigr ) \Bigr ) .
+$$
+
+Here $O ( \cdot )$ depends only on d and $C$ .
+
+Discussion. Theorem 4.1 shows that, for any model, the generalization gap depends largely on the model’s variation captured by $\mathcal { V } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } )$ . The result is irrelevant to the algorithm and provides a guarantee for the generalization gap from $\mathcal { E } _ { a v a i l }$ to ${ \mathcal { E } } _ { a l l }$ , so long as the learned model $f$ is invariant, i.e. $\mathcal { V } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } )$ is small. When $s ( \cdot )$ is fixed, a model with smaller $\mathcal { V } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } )$ results in a smaller gap, which matches our understanding that invariant features in $\mathcal { E } _ { a v a i l }$ are somehow invariant in ${ \mathcal { E } } _ { a l l }$ . When $\mathcal { V } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } )$ is fixed, more difficult generalization will generate a larger expansion function, which leads to a larger gap. For the Gaussian class with bounded mean and variance, $\alpha \gg d$ and the convergent rate is almost linear.
+
+However, without any constraint to $g$ , the convergent rate might be small. Theorem 4.2 then offers a generalization bound with a linear convergent rate under mild assumptions when $g$ is linear, which is common in reality. It relaxes the concentration assumption (Formula 4) and asks only for the integrability of the density. The convergent rate is identical to the convergent rate of the expansion function, showing that $s ( \cdot )$ captures the generalization quite well.
+
+Proof Sketch of Theorem 4.1. The proof of the main result, Theorem 4.1, is decomposed into the following steps. First, we transform $\operatorname { e r r } ( f )$ into the Total Variation of joint distributions of features in different domains (Step 1). To bound the Total Variation, it is sufficient to bound the distance of the corresponding Fourier transform, and further, it is equivalent to bound the Radon transform of joint distributions (Step 2). Eventually, we show that $\mathcal { V } ^ { \mathrm { s u p } } ( \beta ^ { \top } h , \mathcal { E } _ { a v a i l } )$ can be used to bound the Radon transform, which finishes the proof (Step 3).
+
+Step 1. The OOD generalization error can be bounded as:
+
+$$
+\mathrm { e r r } ( f ) \le \operatorname* { s u p } _ { ( e , e ^ { \prime } ) \in ( \mathcal { E } _ { a v a i l } , \mathcal { E } _ { a l l } ) } \frac { C } { K } \sum _ { y \in \mathcal { Y } } \int _ { h \in \mathbb { R } ^ { d } } \big | p _ { h ^ { e } | Y ^ { e } } ( h | y ) - p _ { h ^ { e ^ { \prime } } | Y ^ { e ^ { \prime } } } ( h | y ) ) \big | \mathrm { d } h .
+$$
+
+Step 2. According to the assumption (4), the dominant term in (7) is
+
+$$
+\int _ { | h | \leq r _ { 1 } } \Big | \int _ { | t | \leq r _ { 2 } } e ^ { - i \langle h , t \rangle } \big ( \hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) - \hat { p } _ { h ^ { e ^ { \prime } } | Y ^ { e ^ { \prime } } } ( t | y ) \big ) \big ) \mathrm { d } t \Big | \mathrm { d } t ,
+$$
+
+where $r _ { 1 }$ and $r _ { 2 }$ are well-selected scalars that depend on $s \big ( \mathcal { V } _ { \rho } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } ) \big )$ . By the Projection Theorem [31, 42] and the Fourier Inversion Formula, (8) is bounded above by
+
+$$
+O ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } ) \times \int _ { u \in \mathbb { R } } \big | \mathscr { R } _ { e ^ { \prime } } ( \beta , u ) - \mathscr { R } _ { e } ( \beta , u ) \big | \mathrm { d } u ,
+$$
+
+where $\mathcal { R } _ { e } ( \beta , u )$ is the Radon transform of $p _ { h ^ { e } | Y ^ { e } } ( t | y )$ .
+
+Step 3. The right-hand side of Formula 8 can be bounded by $O \big ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } s \big ( \mathcal { V } _ { \rho } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } ) \big ) \big )$ . We finish the proof by selecting appropriate $r _ { 1 }$ and $r _ { 2 }$ to balance the rate of the dominant term and other minor terms. For more details, please see Appendix 2 for the complete proofs.
+
+Now we turn to the lower bound of $\operatorname { e r r } ( f )$
+
+Theorem 4.3 (Lower Bound). Consider 0-1 loss: $\ell ( \hat { y } , y ) = \mathbb { I } ( \hat { y } \neq y )$ . For any $\delta > 0$ and any exps.t. $k x \leq s ( x ) < + \infty , x \in [ 0 , t ]$ $\begin{array} { r } { s _ { + } ^ { \prime } ( 0 ) \triangleq \operatorname* { l i m } _ { x \to 0 ^ { + } } \frac { s ( x ) - s ( 0 ) } { x } \in ( 1 , + \infty ) } \end{array}$ s(x)−s(0) ∈ (1, +∞); 2) exists k > 1, t > 0, $C _ { 0 }$ $O O D$ generaliz $( \mathcal { E } _ { a v a i l } , \mathcal { E } _ { a l l } )$ that is $( s ( \cdot ) , \delta )$ -learnable under linear feature space $\Phi$ w.r.t symmetric $K L$ -divergence $\rho$ , s.t. $\forall \varepsilon \ \in \ [ 0 , \frac { t } { 2 } ] .$ , the optimal classifier $f$ satisfying $\mathcal { V } ^ { s u p } ( h , \mathcal { E } _ { a v a i l } ) = \varepsilon$ will have the OOD generalization error lower bounded by
+
+$$
+\mathrm { e r r } ( f ) \geq C _ { 0 } \cdot s ( \mathcal { V } ^ { s u p } ( h , \mathcal { E } _ { a v a i l } ) ) .
+$$
+
+Theorem 4.3 shows that $\operatorname { e r r } ( f )$ of optimal classifier $f$ is lower bounded by its variation. Here “optimal” means the classifier that minimize $\mathcal { L } ( f , \mathcal { E } _ { a v a i l } )$ . Altogether, the above three theorems offer a bidirectional control of OOD generalization error, showing that our formulation can offer a fine-grained description of most OOD generalization problem in a theoretical way. To pursue a good OOD performance, OOD algorithm should focus on improving predictive performance on $\mathcal { E } _ { a v a i l }$ and controlling the variation $\mathcal { V } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } )$ simultaneously. Note that this bound starts from population error, and we call for future works to combine our generalization bound and traditional bound from data samples to population error, giving a more complete characterization of the problem.
+
+# 5 Variation as a Factor of Model Selection Criterion
+
+As is pointed out in [21], model selection has a significant effect on domain generalization, and any OOD algorithm without a model selection criterion is not complete. [21] trained more than 45,900 models with different algorithms, and results show that when traditional selection methods are applied, none of OOD algorithms can outperform ERM [58] by a significant margin. This result is not strange, since traditional selection methods focus mainly on (validation) accuracy, which is biased in OOD generalization [21, 63]. A very typical example is Colored MNIST [5], where the image is colored according to the label, but the relationship varies across domains. As explained in [5], ERM principle will only capture this spurious feature (color) and performs badly in ${ \mathcal { E } } _ { a l l }$ . Since ERM is exactly minimizing loss in $\mathcal { E } _ { a v a i l }$ , any model selection method using validation accuracy alone is likely to choose ERM rather than any other OOD algorithm [63]. Thus no algorithm will have a significant improvement compared to ERM.
+
+A natural question arises: what else can we use, in addition to accuracy? Theorem 4.1 points out that, learning feature with small variation across $\mathcal { E } _ { a v a i l }$ is important for decreasing OOD generalization error. Once a model $f$ achieves a small $\mathcal { V } ^ { \mathrm { s u p } } ( h , \mathcal { E } _ { a v a i l } )$ , then $\operatorname { e r r } ( f )$ will be small. If the validation accuracy is also high, we shall know that the OOD accuracy will remain high. To this end, we propose our heuristic selection criterion (Algorithm 1). Instead of considering validation accuracy alone, we combine it with feature variation and select the model with high validation accuracy as well as low variation.
+
+# Algorithm 1: Model Selection
+
+Input: available dataset $\mathcal { X } _ { a v a i l } = ( \mathcal { X } _ { t r a i n } , \mathcal { X } _ { v a l } )$ , candidate models set $\mathcal { M }$ , var_acc_rate $r _ { 0 }$ .   
+for $f = g \circ h$ in $\mathcal { M }$ do for $i$ in $[ d ]$ do $\hat { \mathcal { V } } _ { i } \gets \operatorname* { m a x } _ { y \in \mathcal { V } , \mathcal { X } ^ { e } \neq \mathcal { X } ^ { e ^ { \prime } } \in \mathcal { X } _ { a v a i l } }$ Total Variation $( \mathbb { P } ( \phi _ { i } ^ { e } | y ) , \mathbb { P } ( \phi _ { i } ^ { e ^ { \prime } } | y ) )$ ; .Use GPU KDE end $\mathcal { V } _ { f } \gets \mathrm { m e a n } _ { i \in [ d ] } \hat { \mathcal { V } } _ { i }$ $\operatorname { A c c } _ { f } $ compute validation accuracy of $f$ using $\mathcal { X } _ { v a l }$   
+end   
+Retur ${ \mathfrak { n } } \operatorname { a r g m a x } _ { f \in { \mathcal { M } } } ( \operatorname { A c c } _ { f } - r _ { 0 } \mathcal { V } _ { f } )$
+
+We briefly explain Algorithm 1 here. For each candidate model, we calculate its variation using the average of each feature’s variation, i.e., $\begin{array} { r } { \frac { 1 } { d } \sum _ { i \in [ d ] } \mathcal { V } ( \phi _ { i } , \mathcal { X } _ { a v a i l } ) } \end{array}$ . When deriving the bounds, we use $\mathcal { V } ^ { \mathrm { s u p } }$ instead of their average because we need to consider the worst case, i.e., the worst top model. In practice, we find out that the average of $\mathcal { V } ( \phi _ { i } , \mathcal { X } _ { a v a i l } )$ is enough to improve selection.
+
+Our criterion of model selection is
+
+$$
+\operatorname { A c c } _ { f } { - r _ { 0 } \gamma _ { f } } ,
+$$
+
+i.e., we select a model with high validation accuracy and low variation simultaneously. Here $r _ { 0 }$ is a hyper-parameter representing the concrete relationship between $\operatorname { e r r } ( f )$ and $\mathcal { V } _ { f }$ . Although we have already used one hyper-parameter to help select multiple hyper-parameter combinations, it is natural to ask whether we can further get rid of the selection of $r _ { 0 }$ . Since $r _ { 0 }$ represents the relationship between variation and accuracy, which is actually determined by the unknown expansion function, explicitly calculating $r _ { 0 }$ is not possible. However, we can empirically estimate $r _ { 0 }$ using $\begin{array} { r } { r _ { 0 } = \frac { \mathrm { S t d } _ { f \in \hat { \mathcal { M } } } \mathrm { A c c } _ { f } } { \mathrm { S t d } _ { f \in \hat { \mathcal { M } } } \mathcal { V } _ { f } } } \end{array}$ where $\hat { \mathcal { M } } \subset \mathcal { M }$ is the model with not bad validation accuracy. We do not use the whole set $\mathcal { M }$ because some OOD algorithms will perform extremely bad when the penalty is huge, and these models will influence our estimation of the ratio. Since high validation means large informativeness in learned features, the use of $\hat { \mathcal { M } }$ is an implicit application of informative assumption.
+
+As shown in Section 6.1, our method can select models with higher OOD accuracy in various OOD datasets. We also explain in Appendix 3 why our method can outperform the traditional method in Color MNIST, where the dataset is hand-make and simple enough to calculate the expansion function.
+
+# 6 Experiments
+
+In this section, we conduct experiments to compare our model selection criterion (Section 5) with the baseline method5 [21]. Since both the variation and informativeness in Definition 3.1 are based on one-dimensional features, we can directly estimate these quantities feature-by-feature and design model selection method based on them. To verify the existence of the expansion function and to see what it’s like in a real-world dataset, we plot nearly 2 million features trained in a common-used OOD dataset and compute their variation and informativeness. We then draw the expansion function for this problem.
+
+# 6.1 Experiments on Model Selection
+
+In this section, we conduct experiments to compare the performance of models selected by our method and by validation accuracy. We train models on different datasets, different $\mathcal { E } _ { a v a i l }$ , and select models according to a different selection criteria. We then compare the OOD accuracy of selected models.
+
+Settings We train our model on three benchmark OOD datasets (PACS [34], OfficeHome [59], VLCS [57]) and consider all possible selections of $( \mathcal { E } _ { a v a i l } , \mathcal { E } _ { a l l } )$ . We choose ResNet–50 as our network architecture. We use ERM [58] and four common-used OOD algorithms (CORAL [55], Inter-domain Mixup [62], Group DRO [51], and IRM [5]). For each environment setup, we train 200 models using different algorithms, penalties, learning rates, and epoch. After training, we employ different selection methods and compare the OOD accuracy of the selected models. As stated in Section 5, we use the standard deviation of $\nu$ and validation accuracy in $\hat { \mathcal { M } }$ to estimate $r _ { 0 }$ , where ${ \hat { \mathcal { M } } } = \{ f \in { \mathcal { M } } : \operatorname { A c c } _ { f } \geq \operatorname* { m a x } _ { \hat { f } } \operatorname { A c c } _ { \hat { f } } - 0 . 1 \}$ . Note that calculating $\mathcal { V } ( \phi _ { i } , \mathcal { X } _ { a v a i l } )$ takes calculus many times, so we design a parallel GPU kernel density estimation to speed up the whole process a hundred times and manage to finish one model in seconds. For more details about the experiments, see Appendix 4.
+
+Table 1: Model Selection Result. “Env” denotes the unseen domain during training. “Val” denotes the OOD accuracy of model selected by validation accuracy.   
+
+<table><tr><td rowspan="3">PACS</td><td>Env</td><td>A</td><td>C</td><td>P</td><td>S</td><td>avg</td><td>acc inc</td></tr><tr><td>Val</td><td>85.20%</td><td>80.42%</td><td>96.17%</td><td>77.86%</td><td>84.91%</td><td>1</td></tr><tr><td>Ours</td><td>88.72%</td><td>81.74%</td><td>96.83%</td><td>79.00%</td><td>86.57%</td><td>1.66%↑</td></tr><tr><td rowspan="3">OfficeHome</td><td>Env</td><td>A</td><td>C</td><td>P</td><td>R</td><td>avg</td><td>acc inc</td></tr><tr><td>Val</td><td>61.85%</td><td>55.56%</td><td>74.72%</td><td>76.25%</td><td>67.09%</td><td>-</td></tr><tr><td>Ours</td><td>65.76%</td><td>55.07%</td><td>75.20%</td><td>76.31%</td><td>68.09%</td><td>1.00%↑</td></tr><tr><td rowspan="3">VLCS</td><td>Env</td><td>C</td><td>L</td><td>S</td><td>V</td><td>avg</td><td>acc inc</td></tr><tr><td>Val</td><td>97.46%</td><td>64.83%</td><td>69.50%6</td><td>70.97%</td><td>75.69%</td><td>1</td></tr><tr><td>Ours</td><td>97.81%</td><td>66.98%</td><td>69.50%</td><td>70.97%</td><td>76.32%</td><td>0.63%↑</td></tr></table>
+
+Result We summarize our experimental results in Table 1. For each environment setup, we select the best model according to Algorithm 1 and validation accuracy. The results show that on all datasets, our selection criterion significantly outperforms the validation accuracy in average OOD accuracy. For a more detailed comparison, our method improves the OOD accuracy in most of the 12 setups. Our experiments demonstrate that our criterion can help select models with higher OOD accuracy.
+
+![](images/c514b7d4698237506faccba2638407d1bd68cc2a61ac79a424f03d7abc31b464.jpg)  
+Figure 1: The expansion function of the OOD generalization problem on Office-Home. The $\mathbf { X } ^ { } -$ -axis stands for $\mathcal { V } ( \phi , \mathcal { E } _ { a v a i l } )$ and the y-axis for $\mathcal { V } ( \phi , \mathcal { E } _ { a l l } )$ . There are approximately 2 million points in each image, with each point representing a feature, and its color represents its informativeness. The solid red line stands for the expansion function under the corresponding $\delta$ . When $\delta$ increases, the expansion function decreases. When $\delta = 0$ , no expansion function can make it learnable.
+
+One may wonder if the expansion function really exists and what it will look like for a real-world OOD generalization task. In this section, we consider the OOD dataset Office-Home [59]. We explicitly plot millions of features’ $\mathcal { V } _ { \rho } \big ( \phi , \mathcal { E } _ { a v a i l } \big )$ and $\mathcal { V } _ { \rho } \big ( \phi , \mathcal { E } _ { a l l } \big )$ with Total Variation $\rho$ to see what the expansion function is like in this task. We take the architecture as ResNet-50 [23], and we trained thousands of models with more than five algorithms, obtaining about 2 million features. The results are in Figure 1.
+
+Existence of $s ( \cdot )$ . When $\delta = 0$ , some non-informative features are nearly 0-invariant across $\mathcal { E } _ { a v a i l }$ but are varying across ${ \mathcal { E } } _ { a l l }$ , so no expansion function can make this task learnable, i.e., this task is NOT $( s ( \cdot ) , 0 )$ for any expansion function. But as $\delta$ increases, only informative features are left, and now we can find appropriate $s ( \cdot )$ to make it learnable. We can clearly realize from the figure that $s ( \cdot )$ do exist when $\delta \geq 0 . 1 5$ .
+
+Trade-off between $s ( \cdot )$ and $\delta$ . The second phenomenon is that the slope of $s ( \cdot )$ decreases as $\delta$ increases, showing a trade-off between $s ( \cdot )$ and $\delta$ . Although this trade-off comes naturally from the definition of learnability, it has a deep meaning. As is shown in Section 4, $\operatorname { e r r } ( f )$ is bounded by $O ( s ( \varepsilon ) )$ where $\varepsilon$ is the variation of the model. To make the bound tighter, a natural idea is to choose a flatter $s ( \cdot )$ . However, a flatter $s ( \cdot )$ corresponds to a larger $\delta$ . Typically, learning a model to meet this higher informativeness requirement is more difficult, and it is possible that the algorithm achieves this by capturing more domain-specific features, which will therefore increase the variation of the model, $\varepsilon$ . As a result, we are not sure whether $s ( \varepsilon )$ will increase or decrease. We believe this is also the essence of model selection: i.e., to trade-off between the variation and informativeness of a model, which is done in Formula 11.
+
+# 7 More Related Works
+
+Domain generalization [12, 39], or OOD generalization, has drawn much attention recently [21, 30]. The goal is to learn a model from several training domains and expect good performance on unseen test domains. [60, 64] offer a comprehensive survey. A popular solution is to extract domain-invariant feature representation. [45] and [49] proved that when the model is linear, the invariance under training domains can help discover invariant features on test domains. [5] introduces the invariant prediction into neural networks and proposes a practical objective function. After that, a lot of works arise from the view of causal discovery, distributional robustness and conditional independence [1, 7, 16, 15, 26, 32, 33, 43, 51, 61]. On the other hand, some works point out the weakness of existing methods from the theoretical and experimental perspectives [2, 21, 29, 41, 50].
+
+The OOD generalization requires restrictions on how the target domains may differ. A straightforward approach is to define a set of test domains around the training domain using some distribution distance measure [6, 13, 19, 25, 51, 53, 54, 61]. Another feasible route is the causal framework which is robust to the test distributions caused by interventions[44, 46] on variables, e.g., [5, 24, 36, 37, 40, 47, 49, 52]. The principle of these methods is that a causal model is invariant and can achieve the minimal worstcase risk [4, 22, 44, 49]. Since the test distribution is unknown, additional assumptions are required for generalization analysis. [12, 18, 39] assume that the domains are generated from a hyper-distribution and measures the average risk estimation error bound. [3] derives a risk bound for any linear combination of training domains. For more related results in domain adaptation, a closed field where the test domains can be seen but are unlabeled, please see [9, 10, 28].
+
+# 8 Conclusion
+
+In this paper, we take the first step towards a rigorous theoretical framework of OOD generalization. We propose a mathematical formulation to characterize the learnability of OOD generalization problem. Based on our framework, we prove generalization bounds and give guarantees for OOD generalization error. Inspired by our bound, we design a model selection criterion to check the model’s variation and validation accuracy simultaneously. Experiments show that our metric has a significant advantage over the traditional selection method.
+
+# Acknowledgments and Disclosure of Funding
+
+Authors are thankful to the anonymous reviewers for their helpful and constructive feedback.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 5.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In the appendix.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We use [21] and we cite it in Section 6.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/kFJoj7zuDVi/kFJoj7zuDVi_content_list.json

@@ -0,0 +1,1490 @@
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+    {
+        "type": "text",
+        "text": "Towards a Theoretical Framework of Out-of-Distribution Generalization ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chuanlong XieHuawei Noah’s Ark Labxie.chuanlong@huawei.com",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Haotian Ye   \nPeking University   \nPazhou Lab   \nhaotianye@pku.edu.cn ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tianle Cai Peking University caitianle1998@pku.edu.cn ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Ruichen Li ",
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+        "text": "Zhenguo Li ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Liwei Wang ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Peking University xk-lrc@pku.edu.cn ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Huawei Noah’s Ark Lab Li.Zhenguo@huawei.com ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Key Laboratory of Machine Perception, MOE, School of EECS, Institute for Artificial Intelligence, Peking University wanglw@cis.pku.edu.cn ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generalization to out-of-distribution (OOD) data is one of the central problems in modern machine learning. Recently, there is a surge of attempts to propose algorithms that mainly build upon the idea of extracting invariant features. Although intuitively reasonable, theoretical understanding of what kind of invariance can guarantee OOD generalization is still limited, and generalization to arbitrary out-of-distribution is clearly impossible. In this work, we take the first step towards rigorous and quantitative definitions of 1) what is OOD; and 2) what does it mean by saying an OOD problem is learnable. We also introduce a new concept of expansion function, which characterizes to what extent the variance is amplified in the test domains over the training domains, and therefore give a quantitative meaning of invariant features. Based on these, we prove OOD generalization error bounds. It turns out that OOD generalization largely depends on the expansion function. As recently pointed out by [21], any OOD learning algorithm without a model selection module is incomplete. Our theory naturally induces a model selection criterion. Extensive experiments on benchmark OOD datasets demonstrate that our model selection criterion has a significant advantage over baselines. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One of the most fundamental assumptions of classic supervised learning is the “i.i.d. assumption”, which states that the training and the test data are independent and identically distributed. However, this assumption can be easily violated in a reality [8, 10, 11, 17, 38, 48, 56] where the test data usually have a different distribution than the training data. This motivates the research on the out-ofdistribution (OOD) generalization, or domain generalization problem, which assumes access only to data drawn from a set $\\mathcal { E } _ { a v a i l }$ of available domains during training, and the goal is to generalize to a larger domain set ${ \\mathcal { E } } _ { a l l }$ including unseen domains. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To generalize to OOD data, most existing algorithms attempt to learn features that are invariant to a certain extent across training domains in the hope that such invariance also holds in unseen domains. For example, distributional matching-based methods [20, 35, 55] seek to learn features that have the same distribution across different domains; IRM [5] and its variants [1, 32, 33] learn feature representations such that the optimal linear classifier on top of the representation matches across domains. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "35th Conference on Neural Information Processing Systems (NeurIPS 2021). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Though the idea of learning invariant features is intuitively reasonable, there is only limited theoretical understanding of what kind of invariance can guarantee OOD generalization. Clearly, generalization to an arbitrary out-of-distribution domain is impossible and in practice, the features can hardly be absolutely invariant from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ unless all the domains are identical. So it is necessary to first formulate what OOD data can be generalized to, or, what is the relation between the available training domain set $\\mathcal { E } _ { a v a i l }$ and the entire domain set ${ \\mathcal { E } } _ { a l l }$ . Meanwhile, to what extent the invariance of features on $\\mathcal { E } _ { a v a i l }$ can be preserved in ${ \\mathcal { E } } _ { a l l }$ should be rigorously characterized. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we take the first step towards a general OOD framework by quantitatively formalizing the relationship between $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ in terms of the distributions of features and provide OOD generalization guarantees based on our quantification of the difficulty of OOD generalization problem. Specifically, we first rigorously formulate the intuition of invariant features used in previous works by introducing the “variation” and “informativeness” (Definition 3.1 and 3.2) of each feature. Our theoretical insight can then be informally stated as: for learnable OOD problems, if a feature is informative for the classification task as well as invariant over $\\mathcal { E } _ { a v a i l }$ , then it is still invariant over ${ \\mathcal { E } } _ { a l l }$ . In other words, invariance of informative features in $\\mathcal { E } _ { a v a i l }$ can be preserved in ${ \\mathcal { E } } _ { a l l }$ . We further introduce a class of functions, dubbed expansion function (Definition 3.3), to quantitatively characterize to what extent the variance of features on $\\mathcal { E } _ { a v a i l }$ is amplified on ${ \\mathcal { E } } _ { a l l }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on our formulation, we derive theoretical guarantees on the OOD generalization error, i.e., the gap of largest error between the domain in $\\mathcal { E } _ { a v a i l }$ and domain in ${ \\mathcal { E } } _ { a l l }$ . Specifically, we prove the upper and lower bound of OOD generalization error in terms of the expansion function and the variation of learned features over $\\mathcal { E } _ { a v a i l }$ . Our results theoretically confirm that 1) the expansion function can reflect the difficulty of OOD generalization problem, i.e., problems with more rapidly increasing expansion functions are harder and have worse generalization guarantees; 2) the generalization error gap can tend to zero when the variation of learned features tend to zero, so minimizing the variation in $\\mathcal { E } _ { a v a i l }$ can reduce the generalization error. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As pointed out by Gulrajani and Lopez-Paz [21], any OOD algorithm without a specified model selection criterion is not complete. Since ${ \\mathcal { E } } _ { a l l }$ is unseen, hyper-parameters can only be chosen according to $\\mathcal { E } _ { a v a i l }$ . Previous selection methods mainly focus on validation accuracy over $\\mathcal { E } _ { a v a i l }$ which is only a biased metric of OOD performance. On the contrary, a promising model selection method should instead be predictive of OOD performance. Inspired by our bounds, we propose a model selection method to select models with high validation accuracy and low variation, which corresponds to the upper bound of OOD error. The introduction of a model’s variation relieves the problem of classic selection methods, in which models that overfit $\\mathcal { E } _ { a v a i l }$ tend to be selected. Experimental results show that our method can outperform baselines and select models with higher OOD accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contributions. We summarize our major contributions here: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We introduce a quantitative and rigorous formulation of OOD generalization problem that characterizes the relation of invariance over the training domain set $\\mathcal { E } _ { a v a i l }$ and test domain set ${ \\mathcal { E } } _ { a l l }$ . The core quantity in our characterization, the expansion function, determines the difficulty of an OOD generalization problem. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We prove novel OOD generalization error bounds based on our formulation. The upper and lower bounds together indicate that the expansion function well characterizes the OOD generalization ability of features with different levels of variation. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We design a model selection criterion that is inspired by our generalization bounds. Our criterion takes both the performance on training domains and the variation of models into consideration and is predictive of OOD performance according to our bounds. Experimental results demonstrate our selection criterion can choose models with higher OOD accuracy. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The rest of the paper is organized as follows: Section 2 is our preliminary. In Section 3, we give our theoretical formulation. Section 4 gives our generalization bound. We propose our model selection method in Section 5. In Section 6 we conduct experiments on expansion function and model selection. We review more related works in Section 7 and conclude our work in Section 8. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 Preliminary ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Throughout the paper, we consider a multi-class classification task $\\mathcal { X } \\to \\mathcal { Y } = \\{ 1 , . . . , K \\}$ .1 Let ${ \\mathcal { E } } _ { a l l }$ be the domain set we want to generalize to, and $\\mathcal { E } _ { a v a i l } \\subseteq \\mathcal { E } _ { a l l }$ be the available domain set, i.e., all domains we have during the training procedure. We denote $( X ^ { e } , Y ^ { e } )$ to be the input-label pair drawn from the data distribution of domain $e$ . The OOD generalization goal is to find a classifier $f ^ { * }$ that minimizes the worst-domain loss on ${ \\mathcal { E } } _ { a l l }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92ece3d80a18d8b09d86d490b1aa6271dee39a3b130e884626dc355a67e41bdf.jpg",
+        "text": "$$\nf ^ { * } = \\underset { f \\in \\mathcal { F } } { \\mathrm { a r g m i n } } \\mathcal { L } ( \\mathcal { E } _ { a l l } , f ) , \\mathcal { L } ( \\mathcal { E } , f ) \\triangleq \\underset { e \\in \\mathcal { E } } { \\mathrm { m a x } } \\mathbb { E } \\big [ \\ell \\big ( f ( X ^ { e } ) , Y ^ { e } \\big ) \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            195,
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+            224
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } : \\mathcal { X } \\xrightarrow { } \\mathbb { R } ^ { K }$ is the the hypothetical space and $\\ell ( \\cdot , \\cdot )$ is a loss function. Similar to previous works [5, 16, 27, 33], we assume that $f$ can be decomposed into $g \\circ h$ , where $g \\in \\mathcal { G } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { K }$ is the top classifier and $h : \\mathcal { X }  \\mathbb { R } ^ { d }$ is a $d$ -dimensional feature extractor, i.e., ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee0a57e21bd4e230b25e434833fe9ced52367d0fd15ac040eda84f95a58b070d.jpg",
+        "text": "$$\nh ( x ) = ( \\phi _ { 1 } ( x ) , \\phi _ { 2 } ( x ) , \\ldots , \\phi _ { d } ( x ) ) ^ { \\top } , \\quad \\phi _ { i } \\in \\Phi .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            284,
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+            303
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here $\\Phi$ is the set of scalar feature maps which map $\\mathcal { X }$ to $\\mathbb { R }$ and $d$ is fixed. We will call each $\\phi \\in \\Phi$ a feature for simplicity. Given a domain $e$ , we denote the $d$ -dimensional random vector $h ( X ^ { e } )$ as $h ^ { e }$ , one-dimensional feature $\\phi ( X ^ { e } )$ as $\\phi ^ { e }$ , and the conditional distribution of $h ^ { e } , \\phi ^ { e }$ given $Y ^ { e } = y$ as $\\mathbb { P } ( h ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e } | y )$ . For simplicity, we assume the data distribution is balanced in every domain, i.e., $P ( Y ^ { e } = y ) = \\frac { 1 } { K } , \\forall y \\in \\mathcal { Y } , e \\in \\mathcal { E } _ { a l l }$ . Our framework can be easily extended to the case where the balanced assumption is removed, with an additional term corresponding to the imbalance adding to the generalization bounds. ",
+        "bbox": [
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+            407
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Framework of OOD Generalization Problem ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            426,
+            580,
+            444
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The main challenge of formalizing the OOD generalization problem is to mathematically describe the connection between $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ and how generalization depends on this relation. Towards this goal, we introduce several quantities to characterize the relation of feature distributions over different domains and bridge $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ by expansion function (Definition 3.3) over the quantities we have introduced. Our framework is motivated by the understanding that, in an OOD generalization task, certain “property” of “good” features in $\\mathcal { E } _ { a v a i l }$ should be “preserved” in ${ \\mathcal { E } } _ { a l l }$ (the reason is described in Section 1). In Section 3.1, we will go into details on what we mean by “property” (variation, Definition 3.1), “good” (informativeness, Definition 3.2), and “preserved” (measured by expansion function). In Section 6.2, we further illustrate the key concepts in our framework by a real-world OOD problem. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3.1 Formalizing OOD Problem by Quantifying Feature Distribution ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first introduce the concepts “variation\" and “informativeness\" of a feature $\\phi$ . The first one is what we expect to be preserved in ${ \\mathcal { E } } _ { a l l }$ and the second one characterizes what features will be considered. Specifically, let $\\rho ( \\mathbb { P } , \\mathbb { Q } )$ be a symmetric “distance” of two distributions. Note that $\\rho$ can have many choices, like $L _ { 2 }$ Distance, Total Variation and symmetric KL-divergence, etc. The variation and informativeness are defined as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Definition 3.1 (Variation). The variation of feature $\\phi ( \\cdot )$ across a domain set $\\mathcal { E }$ is ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/b5e6d90d4f6e707f43d0a5fd09cd50b46b59f7364145ae7e58a3a5a8d9599fbf.jpg",
+        "text": "$$\n\\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } ) = \\operatorname* { m a x } _ { y \\in \\mathcal { V } } \\operatorname* { s u p } _ { e , e ^ { \\prime } \\in \\mathcal { E } } \\rho \\big ( \\mathbb { P } ( \\phi ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e ^ { \\prime } } | y ) \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "A feature $\\phi ( \\cdot )$ is $\\varepsilon$ -invariant across $\\mathcal { E }$ , $i f \\varepsilon \\geq \\mathcal { V } ( \\phi , \\mathcal { E } )$ (We omit the subscript $\\rho$ in case of no ambiguity). Definition 3.2 (Informativeness). The informativeness of feature $\\phi ( \\cdot )$ across a domain set $\\mathcal { E }$ is ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5d2d9f29351ffd114b5530974aa3d8ba7d34f3c8275c029e6a978717a1c0731.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\rho } ( \\phi , \\mathcal { E } ) = \\frac { 1 } { K ( K - 1 ) } \\sum _ { \\stackrel { y \\neq y ^ { \\prime } } { y , y ^ { \\prime } \\in y } } \\operatorname* { m i n } _ { e \\in \\mathcal { E } } \\rho \\big ( \\mathbb { P } ( \\phi ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e } | y ^ { \\prime } ) \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A feature $\\phi ( \\cdot )$ is $\\delta$ -informative across $\\varepsilon$ , if $\\quad \\delta \\leq { \\mathcal { I } } ( \\phi , { \\mathcal { E } } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The variation $\\mathcal { V } ( \\phi , \\mathcal { E } )$ measures the stability of $\\phi ( \\cdot )$ over the domains in $\\mathcal { E }$ and the informativeness ${ \\mathcal { T } } ( \\phi , { \\mathcal { E } } )$ captures the ability of $\\phi ( \\cdot )$ to distinguish different labels. We would like to highlight that the variation and informativeness are defined on each one-dimensional feature $\\phi ( \\cdot )$ . Unlike previous distance between distributions defined in $d$ -dimensional space, our definitions are more reasonable and practical, since it can be easily calculated and analyzed. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to introduce the core quantity for connecting $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ . Our motivation, as elaborated in the introduction section, is that, if a feature is informative for the classification task and invariant over $\\mathcal { E } _ { a v a i l }$ , then to enable OOD generalization from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ , it should be still invariant over ${ \\mathcal { E } } _ { a l l }$ . So the relation between $\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )$ and $\\mathinner { \\gamma \\mathopen { \\left( \\phi , \\mathcal { E } _ { a l l } \\right) } }$ of an informative feature captures the feasibility and difficulty of OOD generalization. To quantitatively measure this relation, we define the following function class: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.3 (Expansion Function). We say a function $s : \\mathbb { R } ^ { + } \\cup \\{ 0 \\}  \\mathbb { R } ^ { + } \\cup \\{ 0 , + \\infty \\}$ is an expansion function, iff the following properties hold: 1) $s ( \\cdot )$ is monotonically increasing and $s ( x ) \\geq x , \\forall x \\geq 0 ; 2 ,$ ) $\\begin{array} { r } { \\operatorname* { l i m } _ { x  0 ^ { + } } s ( x ) = s ( 0 ) = 0 } \\end{array}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This function class gives a full characterization of how the variation between $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ is related. Based on this function class, we can introduce our formulation of the learnability of OOD generalization as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.4 (Learnability). Let $\\Phi$ be the feature space and $\\rho$ be a distribution distance. We say an OOD generalization problem from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ is learnable if there exists an expansion function $s ( \\cdot )$ and $\\delta \\geq 0$ , such that: for all $\\phi \\in \\Phi$ satisfying ${ \\mathcal { T } } _ { \\rho } ( \\phi , { \\mathcal { E } } _ { a v a i l } ) \\geq \\delta$ , we have $s ( \\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } _ { a v a i l } ) ) \\geq$ $\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )$ . If such $s ( \\cdot )$ and $\\delta$ exist, we further call this problem $( s ( \\cdot ) , \\delta )$ -learnable. If an $o o D$ generalization problem is not learnable, we call $i t$ unlearnable. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To understand the intuition and rationality of our formulation, several discussions are in order. ",
+        "bbox": [
+            171,
+            436,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Properties of the expansion function. In Definition 3.3, we highlight two properties of the expansion function. The first property comes naturally from the monotonicity properties of variation: any $\\varepsilon _ { 1 }$ -invariant feature is also $\\varepsilon _ { 2 }$ -invariant for $\\varepsilon _ { 2 } \\geq \\varepsilon _ { 1 }$ ; and $\\mathcal { V } ( \\phi , \\mathcal { E } _ { 1 } ) \\leq \\mathcal { V } ( \\phi , \\mathcal { E } _ { 2 } )$ for any ${ \\mathcal { E } } _ { 1 } \\subseteq { \\mathcal { E } } _ { 2 }$ . The monotonicity also implies that larger ${ \\mathcal { E } } _ { a l l }$ will induce larger $s ( \\cdot ) ^ { 2 }$ and it is also harder to be generalized to. From this view, we can see that the scale of $s ( \\cdot )$ can reflect the difficulty of OOD generalization. The second property is more crucial since it formulates the intuition that if an informative feature is almost invariant in $\\mathcal { E } _ { a v a i l }$ , it should remain invariant in ${ \\mathcal { E } } _ { a l l }$ . Without this assumption, OOD generalization can never be guaranteed because we cannot predict whether an invariant and informative feature in $\\mathcal { E } _ { a v a i l }$ will vary severely in unseen ${ \\mathcal { E } } _ { a l l }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Necessity of informativeness. We include a seemingly redundant quantity informativeness in the definition of learnability. However, this term is necessary because only informative features are responsible for the performance of classification. Non-informative but invariant features over $\\mathcal { E } _ { a v a i l }$ may only capture some noise that is irrelevant to the classification problem, and we shall not expect the noise to be invariant over ${ \\mathcal { E } } _ { a l l }$ . Moreover, we show in Figure 1 that in practice, many invariant but useless features in $\\mathcal { E } _ { a v a i l }$ vary a lot in ${ \\mathcal { E } } _ { a l l }$ , and adding the constraint of informativeness makes the expansion function reasonable. In addition, there are multiple choices of $( s ( \\cdot ) , \\delta )$ to make an OOD generalization problem learnable: larger $\\delta$ will filter out more features, and so $\\dot { s } ( \\cdot )$ can be smaller (flatter). This multiplicity will result in a trade-off between $s ( \\cdot )$ and $\\delta$ , which will be discussed in Section 6.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Two extreme cases: i.i.d. & unlearnable. To better understand the concept of learnability, we consider two extreme cases. (1) The first example is when all data from different $\\textit { e } \\in \\mathcal { E } _ { a l l }$ are identically distributed, i.e., the classic supervised learning setting. This problem is $( s ( \\cdot ) , 0 )$ -learnable with $s ( x ) = x$ , implying no extra difficulty in OOD generalization. (2) As an example of unlearnable, consider the following case (modified from Colored MNIST [5]): For $e \\in \\mathcal { E } _ { a v a i l }$ , images with label 0 always has a red background while images with label 1 has a blue background. For $e \\in \\mathcal { E } _ { a l l } \\ \\backslash \\ \\mathcal { E } _ { a v a i l }$ , this relationship is entirely inverse. Since data from different $e \\in \\mathcal { E } _ { a v a i l }$ are identically distributed but different from other $e \\in \\mathcal { E } _ { a l l }$ , no expansion function can make it learnable, i.e., it is OOD-unlearnable. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The unlearnability of this case also coincides with our intuition: Without prior knowledge, it is not clear from merely the training data, whether the task is to distinguish digit 0 from 1, or to distinguish color red from blue. As a result, generalization to ${ \\mathcal { E } } _ { a l l }$ cannot be guaranteed. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Generalization Bound ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            151,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider an OOD generalization problem from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ , and our goal is to analyze the OOD generalization error of classifier $f = g \\circ h$ defined by ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f61601e575911b7b38e2ff41bc91c80f1ca3bdfad17110177fbe71a235a390ce.jpg",
+        "text": "$$\n\\mathrm { e r r } ( f ) = \\mathcal { L } ( \\mathcal { E } _ { a l l } , f ) - \\mathcal { L } ( \\mathcal { E } _ { a v a i l } , f ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            217,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we assume the loss function $l ( \\cdot , \\cdot )$ is bounded by $[ 0 , C ]$ . We prove two upper bounds (4.1, 4.2) as well as a lower bound (4.3) for $\\operatorname { e r r } ( f )$ based on our formulation. Our bounds together provide a complete characterization of the difficulty of OOD generalization. Since we expect that an invariant classifier can generalize to unseen domains, we hope to bound $\\operatorname { e r r } ( f )$ in terms of the certain variation of $f$ . To this end, we define the variation and informativeness of $f$ in terms of its features, i.e., ",
+        "bbox": [
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+            238,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37ed4f7a0dc7d2319bdc5086f2478da0e7cc992b3f71f92313ccf9703cd17289.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\mathcal { V } ^ { \\operatorname* { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { s u p } } \\mathcal { V } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\\\ { \\mathcal { T } ^ { \\operatorname* { i n f } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { i n f } } \\mathcal { T } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            313,
+            647,
+            372
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $( \\beta ^ { \\top } h ) ( x ) = \\beta ^ { \\top } h ( x )$ is a feature and $S ^ { d - 1 } = \\{ \\beta \\in \\mathbb { R } ^ { d } : \\| \\beta \\| _ { 2 } = 1 \\}$ is the unit $( d - 1 )$ -sphere. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Necessity of using supremum over linear combination. One seemingly plausible definition of the variation of a classifier $f$ can be the supremum over all $\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a v a i l } ) , i \\in [ d ]$ . However, as is shown in Appendix 1, it is possible that two high-dimensional joint distributions have close marginal distribution in each dimension, while they do not overlap. In other words, there exist cases where $\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a l l } ) = 0 , \\forall i \\in [ d ]$ but after applying the top model $g$ over $\\phi _ { i }$ ’s, the distribution varies a lot in $\\mathcal { E } _ { a v a i l }$ . Our definition comes from the simple idea that the class of top model $\\mathcal { G }$ is at least a linear space, so we should at least consider the variation of every (normalized) linear combination of $h ( \\cdot )$ With this, we can guarantee the joint distribution distance is still small. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1 (Main Theorem). Suppose we have learned a classifier $f ( x ) = g ( h ( x ) )$ such that $\\forall e \\in \\mathcal { E } _ { a l l }$ and $\\forall y \\in \\mathcal { V } ;$ $, p _ { h ^ { e } | Y ^ { e } } ( h | y ) \\in L ^ { 2 } ( \\mathbb R ^ { d } )$ . Denote the characteristic function of random variable $h ^ { e } | Y ^ { e }$ as $\\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) = \\mathbb { E } [ \\exp \\{ i \\langle t , h ^ { e } \\rangle \\} | Y ^ { e } = y ]$ . Assume the hypothetical space $\\mathcal { F }$ satisfies the following regularity conditions that $\\exists \\alpha , M _ { 1 } , M _ { 2 } > 0 , \\forall f \\in \\mathcal { F } , \\forall e \\in \\mathcal { E } _ { a l l } , y \\in \\mathcal { V }$ , ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee4b4c076d84f9c894a2e146552ae306c55f38d76c671d508d171889b7b12f9f.jpg",
+        "text": "$$\n\\int _ { h \\in \\mathbb { R } ^ { d } } p _ { h ^ { e } | Y ^ { e } } ( h | y ) | h | ^ { \\alpha } \\mathrm { d } h \\leq M _ { 1 } \\quad a n d \\quad \\int _ { t \\in \\mathbb { R } ^ { d } } | \\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) | | t | ^ { \\alpha } \\mathrm { d } t \\leq M _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            585,
+            745,
+            619
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$I f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ is $\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)$ -learnable under $\\Phi$ with Total Variation $\\rho ^ { 3 }$ , then we have ",
+        "bbox": [
+            174,
+            625,
+            802,
+            641
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/47d72154cd89cf0c6271de3f68baa90e658deea6245d64bec4f750c758b38256.jpg",
+        "text": "$$\n\\operatorname { e r r } ( f ) \\leq O \\left( s \\left( \\mathcal { V } _ { \\rho } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\right) ^ { \\frac { \\alpha ^ { 2 } } { ( \\alpha + d ) ^ { 2 } } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            647,
+            616,
+            676
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here $\\rho$ is total variation distance, and $O ( \\cdot )$ depends on $d , C , \\alpha , M _ { 1 } , M _ { 2 }$ . ",
+        "bbox": [
+            176,
+            681,
+            648,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above theorem holds for a general classifier learned by any algorithms. Due to its generality, we need to introduce some technical regularity conditions on the density function. The assumption (4) assume the decay rate of density and its characteristic function, which is common in the literature, e.g. [14]. This theorem demonstrates that, the generalization error can be bounded by a function of the variation of $h$ , and it converges to 0 as the variation approaches to 0. Under some special but typical case where the top model $g$ is linear, we can further show that even without the regularity conditions in Theorem 4.1, we have a much better (linear) convergence rate. ",
+        "bbox": [
+            173,
+            705,
+            826,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.2 (Linear Top Model). Consider any loss satisfying $\\begin{array} { r } { \\ell ( \\hat { y } , y ) = \\sum _ { k = 1 } ^ { K } \\ell _ { 0 } ( \\hat { y } _ { k } , y _ { k } ) } \\end{array}$ .4 For any classifier with linear top model $g$ , i.e., ",
+        "bbox": [
+            173,
+            809,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd483c8cfb69d41c51abea8bfcbef01bb15ff1bdd62fa9a0e20893ef0103bc4c.jpg",
+        "text": "$$\nf ( x ) = A h ( x ) + b w i t h A \\in \\mathbb { R } ^ { K \\times d } , b \\in \\mathbb { R } ^ { K } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            843,
+            647,
+            862
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$i f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ is $\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)$ -learnable under $\\Phi$ with Total Variation $\\rho$ , then we have ",
+        "bbox": [
+            171,
+            90,
+            797,
+            108
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bff65b7e956cd09791c5ed2c8933cda9684cbbae3c4dc554a897bdb02f4ed4da.jpg",
+        "text": "$$\n\\mathrm { e r r } ( f ) \\leq O \\Bigl ( s \\bigl ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\bigr ) \\Bigr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            113,
+            594,
+            140
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here $O ( \\cdot )$ depends only on d and $C$ . ",
+        "bbox": [
+            174,
+            145,
+            413,
+            160
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Discussion. Theorem 4.1 shows that, for any model, the generalization gap depends largely on the model’s variation captured by $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ . The result is irrelevant to the algorithm and provides a guarantee for the generalization gap from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ , so long as the learned model $f$ is invariant, i.e. $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ is small. When $s ( \\cdot )$ is fixed, a model with smaller $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ results in a smaller gap, which matches our understanding that invariant features in $\\mathcal { E } _ { a v a i l }$ are somehow invariant in ${ \\mathcal { E } } _ { a l l }$ . When $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ is fixed, more difficult generalization will generate a larger expansion function, which leads to a larger gap. For the Gaussian class with bounded mean and variance, $\\alpha \\gg d$ and the convergent rate is almost linear. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "However, without any constraint to $g$ , the convergent rate might be small. Theorem 4.2 then offers a generalization bound with a linear convergent rate under mild assumptions when $g$ is linear, which is common in reality. It relaxes the concentration assumption (Formula 4) and asks only for the integrability of the density. The convergent rate is identical to the convergent rate of the expansion function, showing that $s ( \\cdot )$ captures the generalization quite well. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proof Sketch of Theorem 4.1. The proof of the main result, Theorem 4.1, is decomposed into the following steps. First, we transform $\\operatorname { e r r } ( f )$ into the Total Variation of joint distributions of features in different domains (Step 1). To bound the Total Variation, it is sufficient to bound the distance of the corresponding Fourier transform, and further, it is equivalent to bound the Radon transform of joint distributions (Step 2). Eventually, we show that $\\mathcal { V } ^ { \\mathrm { s u p } } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } )$ can be used to bound the Radon transform, which finishes the proof (Step 3). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Step 1. The OOD generalization error can be bounded as: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c22961cf79642649ac7052d3371c49427c7a1ab896566ae2ddd688515f2d0dd.jpg",
+        "text": "$$\n\\mathrm { e r r } ( f ) \\le \\operatorname* { s u p } _ { ( e , e ^ { \\prime } ) \\in ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } ) } \\frac { C } { K } \\sum _ { y \\in \\mathcal { Y } } \\int _ { h \\in \\mathbb { R } ^ { d } } \\big | p _ { h ^ { e } | Y ^ { e } } ( h | y ) - p _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( h | y ) ) \\big | \\mathrm { d } h .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Step 2. According to the assumption (4), the dominant term in (7) is ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/8d84abc42db5ac0bce20ba33a361ed49a07b3c2d298320a7da3642158de97201.jpg",
+        "text": "$$\n\\int _ { | h | \\leq r _ { 1 } } \\Big | \\int _ { | t | \\leq r _ { 2 } } e ^ { - i \\langle h , t \\rangle } \\big ( \\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) - \\hat { p } _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( t | y ) \\big ) \\big ) \\mathrm { d } t \\Big | \\mathrm { d } t ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            555,
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+            592
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $r _ { 1 }$ and $r _ { 2 }$ are well-selected scalars that depend on $s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big )$ . By the Projection Theorem [31, 42] and the Fourier Inversion Formula, (8) is bounded above by ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/49950396e3d8096d0baf981fde2f0be38a8c185cda445644133fe0a23d73c6a7.jpg",
+        "text": "$$\nO ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } ) \\times \\int _ { u \\in \\mathbb { R } } \\big | \\mathscr { R } _ { e ^ { \\prime } } ( \\beta , u ) - \\mathscr { R } _ { e } ( \\beta , u ) \\big | \\mathrm { d } u ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { R } _ { e } ( \\beta , u )$ is the Radon transform of $p _ { h ^ { e } | Y ^ { e } } ( t | y )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Step 3. The right-hand side of Formula 8 can be bounded by $O \\big ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big ) \\big )$ . We finish the proof by selecting appropriate $r _ { 1 }$ and $r _ { 2 }$ to balance the rate of the dominant term and other minor terms. For more details, please see Appendix 2 for the complete proofs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Now we turn to the lower bound of $\\operatorname { e r r } ( f )$ ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Theorem 4.3 (Lower Bound). Consider 0-1 loss: $\\ell ( \\hat { y } , y ) = \\mathbb { I } ( \\hat { y } \\neq y )$ . For any $\\delta > 0$ and any exps.t. $k x \\leq s ( x ) < + \\infty , x \\in [ 0 , t ]$ $\\begin{array} { r } { s _ { + } ^ { \\prime } ( 0 ) \\triangleq \\operatorname* { l i m } _ { x \\to 0 ^ { + } } \\frac { s ( x ) - s ( 0 ) } { x } \\in ( 1 , + \\infty ) } \\end{array}$ s(x)−s(0) ∈ (1, +∞); 2) exists k > 1, t > 0, $C _ { 0 }$ $O O D$ generaliz $( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ that is $( s ( \\cdot ) , \\delta )$ -learnable under linear feature space $\\Phi$ w.r.t symmetric $K L$ -divergence $\\rho$ , s.t. $\\forall \\varepsilon \\ \\in \\ [ 0 , \\frac { t } { 2 } ] .$ , the optimal classifier $f$ satisfying $\\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) = \\varepsilon$ will have the OOD generalization error lower bounded by ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b71427ee15ff225ca02625e3250ad817c12b27937efe9a03a49e7ec1eee529f3.jpg",
+        "text": "$$\n\\mathrm { e r r } ( f ) \\geq C _ { 0 } \\cdot s ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 4.3 shows that $\\operatorname { e r r } ( f )$ of optimal classifier $f$ is lower bounded by its variation. Here “optimal” means the classifier that minimize $\\mathcal { L } ( f , \\mathcal { E } _ { a v a i l } )$ . Altogether, the above three theorems offer a bidirectional control of OOD generalization error, showing that our formulation can offer a fine-grained description of most OOD generalization problem in a theoretical way. To pursue a good OOD performance, OOD algorithm should focus on improving predictive performance on $\\mathcal { E } _ { a v a i l }$ and controlling the variation $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ simultaneously. Note that this bound starts from population error, and we call for future works to combine our generalization bound and traditional bound from data samples to population error, giving a more complete characterization of the problem. ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "5 Variation as a Factor of Model Selection Criterion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As is pointed out in [21], model selection has a significant effect on domain generalization, and any OOD algorithm without a model selection criterion is not complete. [21] trained more than 45,900 models with different algorithms, and results show that when traditional selection methods are applied, none of OOD algorithms can outperform ERM [58] by a significant margin. This result is not strange, since traditional selection methods focus mainly on (validation) accuracy, which is biased in OOD generalization [21, 63]. A very typical example is Colored MNIST [5], where the image is colored according to the label, but the relationship varies across domains. As explained in [5], ERM principle will only capture this spurious feature (color) and performs badly in ${ \\mathcal { E } } _ { a l l }$ . Since ERM is exactly minimizing loss in $\\mathcal { E } _ { a v a i l }$ , any model selection method using validation accuracy alone is likely to choose ERM rather than any other OOD algorithm [63]. Thus no algorithm will have a significant improvement compared to ERM. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A natural question arises: what else can we use, in addition to accuracy? Theorem 4.1 points out that, learning feature with small variation across $\\mathcal { E } _ { a v a i l }$ is important for decreasing OOD generalization error. Once a model $f$ achieves a small $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ , then $\\operatorname { e r r } ( f )$ will be small. If the validation accuracy is also high, we shall know that the OOD accuracy will remain high. To this end, we propose our heuristic selection criterion (Algorithm 1). Instead of considering validation accuracy alone, we combine it with feature variation and select the model with high validation accuracy as well as low variation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Model Selection ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Input: available dataset $\\mathcal { X } _ { a v a i l } = ( \\mathcal { X } _ { t r a i n } , \\mathcal { X } _ { v a l } )$ , candidate models set $\\mathcal { M }$ , var_acc_rate $r _ { 0 }$ .   \nfor $f = g \\circ h$ in $\\mathcal { M }$ do for $i$ in $[ d ]$ do $\\hat { \\mathcal { V } } _ { i } \\gets \\operatorname* { m a x } _ { y \\in \\mathcal { V } , \\mathcal { X } ^ { e } \\neq \\mathcal { X } ^ { e ^ { \\prime } } \\in \\mathcal { X } _ { a v a i l } }$ Total Variation $( \\mathbb { P } ( \\phi _ { i } ^ { e } | y ) , \\mathbb { P } ( \\phi _ { i } ^ { e ^ { \\prime } } | y ) )$ ; .Use GPU KDE end $\\mathcal { V } _ { f } \\gets \\mathrm { m e a n } _ { i \\in [ d ] } \\hat { \\mathcal { V } } _ { i }$ $\\operatorname { A c c } _ { f } $ compute validation accuracy of $f$ using $\\mathcal { X } _ { v a l }$   \nend   \nRetur ${ \\mathfrak { n } } \\operatorname { a r g m a x } _ { f \\in { \\mathcal { M } } } ( \\operatorname { A c c } _ { f } - r _ { 0 } \\mathcal { V } _ { f } )$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We briefly explain Algorithm 1 here. For each candidate model, we calculate its variation using the average of each feature’s variation, i.e., $\\begin{array} { r } { \\frac { 1 } { d } \\sum _ { i \\in [ d ] } \\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } ) } \\end{array}$ . When deriving the bounds, we use $\\mathcal { V } ^ { \\mathrm { s u p } }$ instead of their average because we need to consider the worst case, i.e., the worst top model. In practice, we find out that the average of $\\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } )$ is enough to improve selection. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our criterion of model selection is ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f4a8d3348d12ad167259139dedde77e64f96549c3f99a91438c1b14e5a3f0957.jpg",
+        "text": "$$\n\\operatorname { A c c } _ { f } { - r _ { 0 } \\gamma _ { f } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "i.e., we select a model with high validation accuracy and low variation simultaneously. Here $r _ { 0 }$ is a hyper-parameter representing the concrete relationship between $\\operatorname { e r r } ( f )$ and $\\mathcal { V } _ { f }$ . Although we have already used one hyper-parameter to help select multiple hyper-parameter combinations, it is natural to ask whether we can further get rid of the selection of $r _ { 0 }$ . Since $r _ { 0 }$ represents the relationship between variation and accuracy, which is actually determined by the unknown expansion function, explicitly calculating $r _ { 0 }$ is not possible. However, we can empirically estimate $r _ { 0 }$ using $\\begin{array} { r } { r _ { 0 } = \\frac { \\mathrm { S t d } _ { f \\in \\hat { \\mathcal { M } } } \\mathrm { A c c } _ { f } } { \\mathrm { S t d } _ { f \\in \\hat { \\mathcal { M } } } \\mathcal { V } _ { f } } } \\end{array}$ where $\\hat { \\mathcal { M } } \\subset \\mathcal { M }$ is the model with not bad validation accuracy. We do not use the whole set $\\mathcal { M }$ because some OOD algorithms will perform extremely bad when the penalty is huge, and these models will influence our estimation of the ratio. Since high validation means large informativeness in learned features, the use of $\\hat { \\mathcal { M } }$ is an implicit application of informative assumption. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As shown in Section 6.1, our method can select models with higher OOD accuracy in various OOD datasets. We also explain in Appendix 3 why our method can outperform the traditional method in Color MNIST, where the dataset is hand-make and simple enough to calculate the expansion function. ",
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+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct experiments to compare our model selection criterion (Section 5) with the baseline method5 [21]. Since both the variation and informativeness in Definition 3.1 are based on one-dimensional features, we can directly estimate these quantities feature-by-feature and design model selection method based on them. To verify the existence of the expansion function and to see what it’s like in a real-world dataset, we plot nearly 2 million features trained in a common-used OOD dataset and compute their variation and informativeness. We then draw the expansion function for this problem. ",
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+    {
+        "type": "text",
+        "text": "6.1 Experiments on Model Selection ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we conduct experiments to compare the performance of models selected by our method and by validation accuracy. We train models on different datasets, different $\\mathcal { E } _ { a v a i l }$ , and select models according to a different selection criteria. We then compare the OOD accuracy of selected models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Settings We train our model on three benchmark OOD datasets (PACS [34], OfficeHome [59], VLCS [57]) and consider all possible selections of $( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ . We choose ResNet–50 as our network architecture. We use ERM [58] and four common-used OOD algorithms (CORAL [55], Inter-domain Mixup [62], Group DRO [51], and IRM [5]). For each environment setup, we train 200 models using different algorithms, penalties, learning rates, and epoch. After training, we employ different selection methods and compare the OOD accuracy of the selected models. As stated in Section 5, we use the standard deviation of $\\nu$ and validation accuracy in $\\hat { \\mathcal { M } }$ to estimate $r _ { 0 }$ , where ${ \\hat { \\mathcal { M } } } = \\{ f \\in { \\mathcal { M } } : \\operatorname { A c c } _ { f } \\geq \\operatorname* { m a x } _ { \\hat { f } } \\operatorname { A c c } _ { \\hat { f } } - 0 . 1 \\}$ . Note that calculating $\\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } )$ takes calculus many times, so we design a parallel GPU kernel density estimation to speed up the whole process a hundred times and manage to finish one model in seconds. For more details about the experiments, see Appendix 4. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/cb2470f47e4fb512f2684a9e17c4e49890f16f65e01308e130f28a69279cde8a.jpg",
+        "table_caption": [
+            "Table 1: Model Selection Result. “Env” denotes the unseen domain during training. “Val” denotes the OOD accuracy of model selected by validation accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">PACS</td><td>Env</td><td>A</td><td>C</td><td>P</td><td>S</td><td>avg</td><td>acc inc</td></tr><tr><td>Val</td><td>85.20%</td><td>80.42%</td><td>96.17%</td><td>77.86%</td><td>84.91%</td><td>1</td></tr><tr><td>Ours</td><td>88.72%</td><td>81.74%</td><td>96.83%</td><td>79.00%</td><td>86.57%</td><td>1.66%↑</td></tr><tr><td rowspan=\"3\">OfficeHome</td><td>Env</td><td>A</td><td>C</td><td>P</td><td>R</td><td>avg</td><td>acc inc</td></tr><tr><td>Val</td><td>61.85%</td><td>55.56%</td><td>74.72%</td><td>76.25%</td><td>67.09%</td><td>-</td></tr><tr><td>Ours</td><td>65.76%</td><td>55.07%</td><td>75.20%</td><td>76.31%</td><td>68.09%</td><td>1.00%↑</td></tr><tr><td rowspan=\"3\">VLCS</td><td>Env</td><td>C</td><td>L</td><td>S</td><td>V</td><td>avg</td><td>acc inc</td></tr><tr><td>Val</td><td>97.46%</td><td>64.83%</td><td>69.50%6</td><td>70.97%</td><td>75.69%</td><td>1</td></tr><tr><td>Ours</td><td>97.81%</td><td>66.98%</td><td>69.50%</td><td>70.97%</td><td>76.32%</td><td>0.63%↑</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Result We summarize our experimental results in Table 1. For each environment setup, we select the best model according to Algorithm 1 and validation accuracy. The results show that on all datasets, our selection criterion significantly outperforms the validation accuracy in average OOD accuracy. For a more detailed comparison, our method improves the OOD accuracy in most of the 12 setups. Our experiments demonstrate that our criterion can help select models with higher OOD accuracy. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c514b7d4698237506faccba2638407d1bd68cc2a61ac79a424f03d7abc31b464.jpg",
+        "image_caption": [
+            "Figure 1: The expansion function of the OOD generalization problem on Office-Home. The $\\mathbf { X } ^ { } -$ -axis stands for $\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )$ and the y-axis for $\\mathcal { V } ( \\phi , \\mathcal { E } _ { a l l } )$ . There are approximately 2 million points in each image, with each point representing a feature, and its color represents its informativeness. The solid red line stands for the expansion function under the corresponding $\\delta$ . When $\\delta$ increases, the expansion function decreases. When $\\delta = 0$ , no expansion function can make it learnable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "One may wonder if the expansion function really exists and what it will look like for a real-world OOD generalization task. In this section, we consider the OOD dataset Office-Home [59]. We explicitly plot millions of features’ $\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a v a i l } \\big )$ and $\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )$ with Total Variation $\\rho$ to see what the expansion function is like in this task. We take the architecture as ResNet-50 [23], and we trained thousands of models with more than five algorithms, obtaining about 2 million features. The results are in Figure 1. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Existence of $s ( \\cdot )$ . When $\\delta = 0$ , some non-informative features are nearly 0-invariant across $\\mathcal { E } _ { a v a i l }$ but are varying across ${ \\mathcal { E } } _ { a l l }$ , so no expansion function can make this task learnable, i.e., this task is NOT $( s ( \\cdot ) , 0 )$ for any expansion function. But as $\\delta$ increases, only informative features are left, and now we can find appropriate $s ( \\cdot )$ to make it learnable. We can clearly realize from the figure that $s ( \\cdot )$ do exist when $\\delta \\geq 0 . 1 5$ . ",
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+        "type": "text",
+        "text": "Trade-off between $s ( \\cdot )$ and $\\delta$ . The second phenomenon is that the slope of $s ( \\cdot )$ decreases as $\\delta$ increases, showing a trade-off between $s ( \\cdot )$ and $\\delta$ . Although this trade-off comes naturally from the definition of learnability, it has a deep meaning. As is shown in Section 4, $\\operatorname { e r r } ( f )$ is bounded by $O ( s ( \\varepsilon ) )$ where $\\varepsilon$ is the variation of the model. To make the bound tighter, a natural idea is to choose a flatter $s ( \\cdot )$ . However, a flatter $s ( \\cdot )$ corresponds to a larger $\\delta$ . Typically, learning a model to meet this higher informativeness requirement is more difficult, and it is possible that the algorithm achieves this by capturing more domain-specific features, which will therefore increase the variation of the model, $\\varepsilon$ . As a result, we are not sure whether $s ( \\varepsilon )$ will increase or decrease. We believe this is also the essence of model selection: i.e., to trade-off between the variation and informativeness of a model, which is done in Formula 11. ",
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+        "text": "7 More Related Works ",
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+        "text": "Domain generalization [12, 39], or OOD generalization, has drawn much attention recently [21, 30]. The goal is to learn a model from several training domains and expect good performance on unseen test domains. [60, 64] offer a comprehensive survey. A popular solution is to extract domain-invariant feature representation. [45] and [49] proved that when the model is linear, the invariance under training domains can help discover invariant features on test domains. [5] introduces the invariant prediction into neural networks and proposes a practical objective function. After that, a lot of works arise from the view of causal discovery, distributional robustness and conditional independence [1, 7, 16, 15, 26, 32, 33, 43, 51, 61]. On the other hand, some works point out the weakness of existing methods from the theoretical and experimental perspectives [2, 21, 29, 41, 50]. ",
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+        "text": "The OOD generalization requires restrictions on how the target domains may differ. A straightforward approach is to define a set of test domains around the training domain using some distribution distance measure [6, 13, 19, 25, 51, 53, 54, 61]. Another feasible route is the causal framework which is robust to the test distributions caused by interventions[44, 46] on variables, e.g., [5, 24, 36, 37, 40, 47, 49, 52]. The principle of these methods is that a causal model is invariant and can achieve the minimal worstcase risk [4, 22, 44, 49]. Since the test distribution is unknown, additional assumptions are required for generalization analysis. [12, 18, 39] assume that the domains are generated from a hyper-distribution and measures the average risk estimation error bound. [3] derives a risk bound for any linear combination of training domains. For more related results in domain adaptation, a closed field where the test domains can be seen but are unlabeled, please see [9, 10, 28]. ",
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+        "text": "8 Conclusion ",
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+        "text": "In this paper, we take the first step towards a rigorous theoretical framework of OOD generalization. We propose a mathematical formulation to characterize the learnability of OOD generalization problem. Based on our framework, we prove generalization bounds and give guarantees for OOD generalization error. Inspired by our bound, we design a model selection criterion to check the model’s variation and validation accuracy simultaneously. Experiments show that our metric has a significant advantage over the traditional selection method. ",
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+        "text": "Acknowledgments and Disclosure of Funding ",
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+        "text": "Authors are thankful to the anonymous reviewers for their helpful and constructive feedback. ",
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+        "text": "References ",
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parse/train/r1lq1hRqYQ/r1lq1hRqYQ.md

@@ -0,0 +1,249 @@
+# FROM LANGUAGE TO GOALS: INVERSE REINFORCE-MENT LEARNING FOR VISION-BASED INSTRUCTIONFOLLOWING
+
+Justin Fu ∗, Anoop Korattikara, Sergey Levine, Sergio Guadarrama Google AI {justinfu,kbanoop,slevine,sguada}@google.com
+
+# ABSTRACT
+
+Reinforcement learning is a promising framework for solving control problems, but its use in practical situations is hampered by the fact that reward functions are often difficult to engineer. Specifying goals and tasks for autonomous machines, such as robots, is a significant challenge: conventionally, reward functions and goal states have been used to communicate objectives. But people can communicate objectives to each other simply by describing or demonstrating them. How can we build learning algorithms that will allow us to tell machines what we want them to do? In this work, we investigate the problem of grounding language commands as reward functions using inverse reinforcement learning, and argue that language-conditioned rewards are more transferable than language-conditioned policies to new environments. We propose language-conditioned reward learning (LC-RL), which grounds language commands as a reward function represented by a deep neural network. We demonstrate that our model learns rewards that transfer to novel tasks and environments on realistic, high-dimensional visual environments with natural language commands, whereas directly learning a languageconditioned policy leads to poor performance.
+
+# 1 INTRODUCTION
+
+While reinforcement learning provides a powerful and flexible framework for describing and solving control tasks, it requires the practitioner to specify objectives in terms of reward functions. Engineering reward functions is often done by experienced practitioners and researchers, and even then can pose a significant challenge, such as when working with complex image-based observations. While researchers have investigated alternative means of specifying objectives, such as learning from demonstration (Argall et al., 2009), or through binary preferences (Christiano et al., 2017), language is often a more natural and desirable way for humans to communicate goals.
+
+A common approach to building natural language interfaces for reinforcement learning agents is to build language-conditioned policies
+
+![](images/5e71170ca08fb2257826210ece6f73ff54f1ace30220b36c57506856bd504733.jpg)  
+Figure 1: A task where an agent (green triangle) must execute the command “go to the fruit bowl.” This is a simple example where the reward function is easier to specify than the policy.
+
+that directly map observations and language commands to a sequence of actions that perform the desired task. However, this requires the policy to solve two challenging problems together: understanding how to plan and solve tasks in the physical world, and understanding the language command itself. The trained policy must simultaneously interpret a command and plan through possibly complicated environment dynamics. The performance of the system then hinges entirely on its ability to generalize to new environments - if either the language interpretation or the physical control fail to generalize, the entire system will fail. We can recognize instead that the role of language in such a system is to communicate the goal, and rather than mapping language directly to policies, we propose to learn how to convert language-defined goals into reward functions. In this manner, the agent can learn how to plan and perform the task on its own via reinforcement learning, directly interacting with the environment, without relying on zero-shot transfer of policies. A simple example is shown in Figure 1, where an agent is tasked with navigating through a house. If an agent is commanded “go to the fruit bowl”, a valid reward function could simply be a fruit bowl detector from first-person views of the agent. However, if we were to learn a mapping from language to actions, given the same goal description, the model would need to generate a different plan for each house.
+
+In this work, we investigate the feasibility of grounding free-form natural language commands as reward functions using inverse reinforcement learning (IRL). Learning language-conditioned rewards poses unique computational problems. IRL methods generally require solving a reinforcement learning problem as an inner-loop (Ziebart, 2010), or rely on potentially unstable adversarial optimization procedures (Finn et al., 2016; Fu et al., 2018). This is compounded by the fact that we wish to train our model across multiple tasks, meaning the IRL problem itself is an inner-loop. In order to isolate the language-learning problem from the difficulties in solving reinforcement learning and adversarial learning problems, we base our method on an exact MaxEnt IRL (Ziebart, 2010) procedure, which requires full knowledge of environment dynamics to train a language-conditioned reward function represented by a deep neural network. While using exact IRL procedures may seem limiting, in many cases (such as indoor robotic navigation) full environment dynamics are available, and this formulation allows us to remove the difficulty of using RL from the training procedure. The crucial insight is that we can use dynamic programming methods during training to learn a reward function that maps from observations, but we do not need knowledge of dynamics to use the reward function, meaning during test time we can evaluate using a reinforcement learning agent without knowledge of the underlying environment dynamics. We evaluate our method on a dataset of realistic indoor house navigation and pick-and-place tasks using the SUNCG dataset, with natural language commands. We demonstrate that our approach generalizes not only to novel tasks, but also to entirely new scenes, while directly learning a language-conditioned policy leads to poor performance and fails to generalize.
+
+# 2 RELATED WORK
+
+A popular class of approaches to language grounding in reinforcement learning is to directly train a policy that consumes language as an input. Several works adopt a behavioral cloning approach, where the model is trained using supervised learning with language-action sequences pairs (Anderson et al., 2018; Mei et al., 2016; Sung et al., 2015). A second approach is to forego demonstrations but instead reward an agent whenever the desired task is completed (Shah et al., 2018; Misra et al., 2017; Hermann et al., 2017; Branavan et al., 2009). This approach requires reward functions (the task completion detector) to be hand-designed for the training tasks considered. Another related approach is semantic parsing, which has also been used to convert language into an executable form that corresponds to actions within an environment (Forbes et al., 2015; Misra et al., 2014; Tellex et al., 2011). In a related task to instruction following, Das et al. (2018) consider an embodied question-answering task where an agent must produce an answer to a question, where the relevant information lies within the environment. They adopt a hybrid approach, where they pretrain with supervised learning but also give the agent reward for completing intermediate tasks. Overall, our experiments show that policy-based approaches have worse generalization performance to new environments, because the policy must rely on zero-shot generalization at test time as we show in Section 6.3. While in this paper we argue for the performance benefits of a reward-based approach, a reason one may want to adopt a policy-based approach over a reward-based one is if one cannot run RL to train a new policy in a new environment, such as for time or safety reasons.
+
+A second approach to the language grounding problem is to learn a mapping from language to reward functions. There are several other works that apply IRL or IRL-like procedures to the problem of language grounding. Perhaps most closely related to our work is MacGlashan et al. (2015), which also aims to learn a language-conditioned reward function via IRL. However, this method requires an extensively hand-designed, symbolic reward function class, whereas we use generic, differentiable function approximators that can handle arbitrary observations, including raw images. Bahdanau et al. (2018); Tung et al. (2018) also learn language-conditioned reward functions, but do not perform IRL, meaning that the objective does not correspond to matching the expert’s trajectory distribution. Tung et al. (2018) train a task-completion classifier, but do not evaluate their reward on control problems. The strategy they use is similar to directly regressing onto a ground-truth reward function, which we include a comparison to in Section 6 to as an oracle baseline. Bahdanau et al. (2018) adopt an adversarial approach similar to GAIL (Ho & Ermon, 2016), and use the learned discriminator as the reward function. While this produces a reward function, it does not provide any guarantees that the resulting reward function can be reoptimized in new environments to yield behavior similar to the expert. We believe our work is the first to apply language-conditioned inverse reinforcement learning to environments with image observations and deep neural networks, and we show that our rewards generalize to novel tasks and environments.
+
+# 3 BACKGROUND
+
+We build off of the MaxEnt IRL model (Ziebart et al., 2008), which considers an entropy-regularized Markov decision process (MDP), defined by the tuple $( S , \mathcal { A } , \mathcal { T } , r , \gamma , \rho _ { 0 } )$ . $s , A$ are the state and action spaces respectively and $\gamma \in ( 0 , 1 )$ is the discount factor. $\mathcal { T } ( s ^ { \prime } | s , a )$ represents the transition distribution or dynamics. We additionally consider partially-observed environments, where each state is associated with an observation within an observations space $o \in \mathcal { O }$ .
+
+The goal of ”forward” reinforcement learning is to find the optimal policy $\pi ^ { * }$ . Let $r ( \tau ) \ =$ $\scriptstyle \sum _ { t = 0 } ^ { T } \gamma ^ { t } r ( s _ { t } , a _ { t } )$ denote the returns of a trajectory, where $\tau$ denotes a sequence of states and actions $( s _ { 0 } , a _ { 0 } , . . . s _ { T } , a _ { T } )$ . The MaxEnt RL objective is then to find $\begin{array} { r } { \pi ^ { * } = \arg \operatorname* { m i n } _ { \pi } { E _ { \tau \sim \pi } [ r ( \tau ) + H ( \tau ) ] } } \end{array}$ .
+
+Inverse reinforcement learning (IRL) seeks to infer the reward function $r ( s , a )$ given a set of expert demonstrations $\mathcal { D } = \{ \tau _ { 1 } , . . . , \tau _ { N } \} .$ ,. In IRL, we assume the demonstrations are drawn from an optimal policy $\pi ^ { * } ( a | s )$ . We can interpret the IRL problem as solving the maximum likelihood problem:
+
+$$
+\operatorname * { m a x } _ { \theta } E _ { \tau \sim \mathcal { D } } \left[ \log p _ { \theta } ( \tau ) \right] ,
+$$
+
+In the MaxEnt IRL framework, optimal trajectories are observed with probabilities proportional to the exponentiated returns, meaning $p ( \tau ) \propto \exp \{ r ( \tau ) \}$ (Ziebart et al., 2008). Thus, learning a reward function $r _ { \theta } ( \tau )$ is equivalent to fitting an energy-based model $p _ { \theta } ( \tau ) \propto \exp \{ r _ { \theta } ( \tau ) \}$ to the maximum likelihood objective in Eqn 1. The gradient to update the reward function is (Ziebart, 2010):
+
+$$
+\nabla _ { \theta } E [ \log p _ { \theta } ( \tau ) ] = \sum _ { s , a } ( \rho ^ { \mathcal { D } } ( s , a ) - \rho _ { \theta } ^ { * } ( s , a ) ) \nabla _ { \theta } r _ { \theta } ( s , a ) ,
+$$
+
+where $\rho ^ { \mathcal { D } } ( s , a )$ represents the state-action marginal of the demonstrations, and $\rho _ { \theta } ^ { * } ( s , a )$ represents the state-action marginal of the optimal policy under reward $r _ { \theta } ( s , a )$ .
+
+# 4 MULTI-TASK IRL
+
+A unique challenge of the language-conditioned IRL problem, compared to standard IRL, is that the goal is to learn a reward function that generalizes across multiple tasks. While standard IRL methods are typically trained and evaluated on the same task, we want our language-conditioned reward function to produce correct behavior when presented with new tasks. Several previous works consider a multi-task scenario, such as in a Bayesian or meta-learning setting (Li & Burdick, 2017; Dimitrakakis & Rothkopf, 2012; Choi & Kim, 2012). We adopt a similar approach adapted for the language-IRL problem, and formalize the notion of a task, denoted by $\xi$ , as an MDP, where individual tasks may not share the same state spaces, dynamics or reward functions. Each task is associated with a context $c _ { \xi }$ which is a unique identifier (i.e. an indicator vector) for that task. Thus, we wish to optimize the following multi-task objective, where $\tau _ { \xi }$ denotes expert demonstrations for that task:
+
+$$
+\operatorname* { m a x } _ { \theta } E _ { \xi } [ E _ { \tau _ { \xi } } [ \log p _ { \theta } ( \tau _ { \xi } , c _ { \xi } ) ] ]
+$$
+
+Algorithm 1 Language-Conditioned Reward Learning (LC-RL)   
+
+<table><tr><td>1: Obtain expert demonstrations and language describing the goal.</td></tr><tr><td>2: Initialize reward function rθ .</td></tr><tr><td>3: for step t in {1,...,N} do</td></tr><tr><td>4: Sample task ε, demonstrations dg ,and language Lg.</td></tr><tr><td>5: Compute optimal q*(s,a) using q-iteration and p*(s,a) using the forward algorithm</td></tr><tr><td></td></tr><tr><td>6: Update reward rθ with the gradient (ρdε(s,a) - p*(s,a)) Vre(o,a,Lg) 7: end for</td></tr></table>
+
+In order to optimize this objective, we first require that all tasks share the same observation space and action space, and the reward to be a function of the observation, rather than of the state. For example, in our experiments, all observations are in the form of $3 2 \mathrm { x } 2 4$ images taken from simulated houses, but the state space for each house is allowed to differ (i.e., the houses have different layouts). This means the same reward can be used across all MDPs even though the state spaces differ.
+
+Second, we share the reward function across all tasks, but substitute a language command $\mathcal { L } _ { \xi }$ as a proxy for the context $c _ { \xi }$ , resulting in a model $p _ { \theta } ( \tau _ { \xi } , \mathcal { L } _ { \xi } )$ that takes as input language, states, and actions. For computational efficiency we run stochastic gradient descent on the objective in Eqn. 3 by sampling over the set of environments on each iteration.
+
+# 5 LANGUAGE-CONDITIONED REWARD LEARNING (LC-RL)
+
+We learn language-conditioned reward functions using maximum causal entropy IRL, adapted for a multi-task setting and rewards represented by language-conditioned convolutional neural networks. While during training we use dynamic programming methods that require dynamics knowledge, we do not need knowledge of dynamics to evaluate the reward function. Thus, at test time we can use standard model-free RL algorithms to learn the task from the inferred reward function in new environments. Our algorithm is briefly summarized in Algorithm 1.
+
+# 5.1 COMPUTING EXACT IRL GRADIENT UPDATES
+
+In order to take gradient steps on the objective of Eqn. 3, we update our reward function in terms of the Maximum Entropy IRL gradient (Ziebart, 2010) according to Eqn. 2. The stochastic gradient update (for a single task $\xi$ ) adapted to our case is:
+
+$$
+\nabla _ { \theta } E _ { \tau } [ \log p _ { \theta } ( \tau _ { \xi } , \mathcal { L } _ { \xi } ) ] = \sum _ { s , a } ( \rho ^ { d } ( s , a ) - \rho _ { \theta } ^ { * } ( s , a ) ) \nabla _ { \theta } r _ { \theta } ( o ( s ) , a , \mathcal { L } _ { \xi } )
+$$
+
+Where $o ( s )$ denotes the observation for state $s$ . Note that during training we need access to the ground truth states $s$ . While the update depends on the underlying state, the reward itself is only a function of the observation, the action, and the language. This enables us to evaluate the reward without knowing the underlying state space and dynamics of the environment. While requiring dynamics knowledge during training may seem limiting, in practice many environments we may wish to train a robot in can easily be mapped. This training strategy is analogous to training a robot in only known environments such as a laboratory, but the resulting reward can be used in unknown environments.
+
+In order to compute $\rho _ { \theta } ^ { * } ( s , a )$ , one normally has to first compute the optimal policy with respect to reward $r _ { \theta } ( o , a )$ using reinforcement learning, and then compute the occupancy measure using the forward algorithm for Markov chains to compute the state visitation distributions at each time-step. Because this embeds a difficult RL optimization problem within nested inner-loops, this quickly becomes computationally intractable. Thus, we train in tabular environments with known dynamics, where we can compute optimal policies exactly using Q-iteration. However, we emphasize that this is only a training time restriction, and knowledge of dynamics is not required to evaluate the rewards.
+
+![](images/f3cb66f02a1107fd18e35bab67acccca5473dba6383bdc7079370bdcbbbd5bd5.jpg)  
+Figure 2: Our reward function architecture. Our network receives as input a panoramic semantic image (4 views) and a language command represented as a sequence of one-hot word vectors, and outputs a scalar reward.
+
+# 5.2 ARCHITECTURE
+
+Our network architecture is shown in Figure 2. The network has two main modalities of input: a variable-length language input represented by a sequence of one-hot vectors (one vector for each tokenized word), and a panoramic image observation.
+
+The language embedding is formed by processing the language input sequence through an LSTM network, and the final time-step of the topmost layer is used as a fixed-dimensional embedding elanguage of the input command.
+
+The agent receives image observations in the form of four $3 2 \mathrm { x } 2 4$ image observations, one for each cardinal direction view (N, S, E, W). The convolutional neural network (CNN) consists of a sequence of convolutional and max pool layers, with the final operation being a channel-wise global pooling operation that produces an image embedding of the same length as the language embedding. Each image is passed through an identical CNN with shared weights, and the outputs are summed together to form the image embedding. That is, $e _ { \mathrm { i m a g e } } = \mathbf { C N N } ( \mathrm { i m } \mathbf { g } _ { N } ) + \mathbf { C N N } ( \mathrm { i m } \mathbf { g } _ { S } ) + \mathbf { C N N } ( \mathrm { i m } \mathbf { g } _ { E } ) +$ $\mathbf { C N N } ( \mathrm { i m g } _ { W } )$ .
+
+Finally, these two embeddings are element-wise multiplied and passed through a fully-connected network (FC) to produce a reward output. Letting $\odot$ denote elementwise multiplication, we have $r = \mathrm { F C } ( e _ { \mathrm { i m a g e } } \odot e _ { \mathrm { l a n g u a g e } } )$ .
+
+We found that the max global-pooling architecture in the CNN was able to select out objects from a scene and allow the language embedding to modulate which features to attend to. We selected our architecture via a hyper-parameter search, and found that the choice of using an element-wise multiplication versus a concatenation for combining embeddings had no appreciable performance difference, but a global pooling architecture performed significantly better than using a fully connected layer at the end of the CNN.
+
+# 6 EVALUATION
+
+We evaluate our method within a collection of simulated indoor house environments, built on top of the SUNCG (Song et al., 2017) dataset. The SUNCG dataset provides a large repository of complex and realistic 3D environments which we find very suitable for our goals. An example task from our environment is shown in Figures 3 and 4. An example of a successful execution of a task is showin in Fig. 5.
+
+# 6.1 ENVIRONMENT
+
+We consider two typical kinds of tasks that an indoor robot may wish to perform:
+
+• Navigation (NAV): In the navigation task, the agent is given a location which corresponds to a room or object, and the agent must navigate through the house to reach the target location. For example, in Fig. 3, the target could be ”cup” or ”laptop” or ”living room”.
+
+![](images/cdace129ad738b81b305696edad646670941ae49887c617f761c225c048e1dc5.jpg)  
+Figure 3: An example task. The green segment corresponds to the solution of a NAV task, ”go to the cup”, where the cup is circled in green. The green plus blue segments represents a path for the PICK task, ”move the cup to the bed”, where the bed is circled in blue.
+
+![](images/28ca2602fae11df6b47ce7b36c7d39de637ba4b5a4bc8160c0e57e228da93603.jpg)  
+Figure 4: Example first-person RGB (left) and semantic (right) images from the bedroom inside the house depicted in Figure 3. We only use the semantic labels as input to our model.
+
+![](images/d1a108a7bf4b6d49ce6fd1915455ff10baca4541408a9db638895372665a8140.jpg)  
+Figure 5: Top row: First-person view of an agent executing the task: “move vase to living room”. The vase is circled in green in the 3rd image. Bottom row: A bird’s eye view of the initial (left) and final (right) positions of the agent (green triangle) and the vase (green outline).
+
+• Pick-and-place (PICK): In the pick and place task, the agent must move an object from one location to another. For example, in Fig. 1 the task could be to move the cup from to the sink to the kitchen table.
+
+Each environment corresponds to one 3D scene, which is discretized into a grid to form a tabular environment where the grid coordinates plus agent orientation (N, S, E, W) correspond to the state of the agent. The agent receives observations with two components: one is a free-form language command, and one is a first-person panoramic image of the environment. Because the agent can move objects without directly looking at them, the panoramic view gives the agent a full view of its surroundings. The panoramic image is formed from 4 semantic image observations, one for each orientation of the agent. Each semantic image observation is $3 2 \mathrm { x } 3 2 $ pixels and contains 61 channels, one per semantic image class. Each agent is equipped with 4 actions: step forward one grid tile, turn left or right, or interact with an object.
+
+We generate language commands based on a preset grammar, and using names of objects and locations associated with the task. These are of the form ”go to $X ^ { \ast }$ for NAV tasks, or ”move X to $\mathbf { Y } ^ { \prime \prime }$ for PICK tasks, where X and $\mathrm { Y }$ stand for names of locations and objects within the environment. We explicitly do not use step-by-step instruction language such as ”turn left, walk down the hallway, go through the door”, as these commands remove the planning aspect of the problem and tell the agent directly which actions to take in order to solve the problem.
+
+The interact action only has meaning within the PICK task. Executing this action will either pick up an object if the agent is within a 1 meter of an object, or drop an object if the agent is currently holding an object. To limit the size of the state space, within a single task, there is only one object an agent may interact with and two locations the object can be in. This setup only increases the size of the state-space by a factor of 3. However, different objects may be placed in different locations across environments, meaning the model still must learn to detect the object in the correct location rather than memorizing the specific object and location associated with a single task.
+
+# 6.2 EXPERIMENTAL PROCEDURE
+
+In order to evaluate how well different methods generalize, we split our dataset of tasks into three segments: a training set, and two test sets – “task” and “house”. The ”task” test set contains tasks within the same houses as training, but requires the agent to interact with novel combinations of objects and locations. The ”house” test set contains tasks on entirely new houses that were not in the training set. The purpose of this split is to investigate varying degrees of generalization: the ”task” test set requires the model to execute novel language commands, but using landmarks and objects which were seen during training. The ”house” test set adds another layer of difficulty, requiring the model to detect familiar objects situated in entirely new scenes.
+
+In total, our dataset contains 1413 tasks (716 PICK, 697 NAV). Across all tasks, there are 14 objects, and 76 different house layouts. There are 1004 tasks $( 7 1 \% )$ in the training set, 236 $( 1 7 \% )$ in the ”task” test set, and 173 $( 1 2 \% )$ in the ”house” test set.
+
+We evaluate two methods for reward learning. LC-RL refers to the language-conditioned IRL method outlined in Section 5, which takes as input demonstration and language pairs and learns a shared reward function across all tasks. In particular, we use 10 demonstrations per task, sampled from the computed optimal policy. To provide an upper bound for performance, we can also regress directly onto the ground-truth rewards, a method we label as “Reward Regression”. While this is not possible in a practical scenario, this evaluation serves to show oracle performance on our task. Note that our method does not require access to ground-truth rewards, and only uses demonstrations.
+
+Success rates for each reward-learning method are evaluated using two policy learning procedures. Q-iteration (QI) computes the optimal policy exactly using dynamic programming, which we report in Table 1. We also experiment with reoptimizing the learned reward using DQN (Mnih et al., 2015), a sample-based RL method that does not require ground-truth knowledge of the environment dynamics. We use the position, orientation, and whether an object is held as the observation (this is identical to the state representation). This experiment represents the testing use-case where we can evaluate the reward at test-time in novel, unmapped environments despite the fact that at training time we require dynamics knowledge. However, because the probability that the random policy receives rewards on our task is tiny, we found that epsilon-greedy exploration was not enough. Thus, we also report results using a reward shaping term with a state-based potential equal to the optimal value function $\mathrm { N g }$ et al., 1999). We note that this shaping term does require dynamics knowledge compute, but we include this result to highlight the difficulty of the RL problem even if reward learning is done properly.
+
+We also compare against two baselines derived from GAIL (Ho & Ermon, 2016), using the learned discriminator as the “reward” function. We first compare to AGILE (Bahdanau et al., 2018), which modifies GAIL to use a goal-based discriminator and false negative filtering, using DQN as a policy optimizer and $\rho \ : = \ : 0 . 2 5$ . We found it difficult to learn rewards using a reinforcement learningbased policy optimizer, and the model was only able to solve the simpler NAV environments. This experiment emphasizes the gap between using a sampling-based solver and an exact solver during the training of reward-based methods. To create a more fair comparison, we also compare against GAIL using a dynamic programming solver (labeled GAIL-Exact), and we see that the performance is comparable to LC-RL on training environments, but performs significantly worse on test environments. These results are in line with our intuitions - GAIL and IRL are equivalent in training scenarios (Ho & Ermon, 2016), but the discriminator of GAIL does not correspond to the true reward function, and thus performs worse when evaluated in novel environments.
+
+In order compare against a policy-learning approach, we compare against an optimal behavioral cloning baseline. We train the optimal cloning baseline by computing the exact optimal policy using Q-iteration, and perform supervised learning to regress directly on to the optimal action probabilities. To make a fair comparison, we keep the policy architecture identical to the reward architecture, except we add two additional inputs: the orientation indicator and and indicator on whether the object (during PICK tasks) is held by the agent or not. Each indicator is transformed by an embedding lookup, and all embeddings are element-wise multiplied along with with the language and image embeddings in the original architecture.
+
+Table 1: Success rates (in percentages) across task categories. Each result is averaged over 3 seeds. Test-Task refers to testing on novel tasks within the same houses as training, whereas Test-House refers to testing novel tasks in novel houses. The AGILE method is described in (Bahdanau et al., 2018)   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=9>Train                  Test-Task              Test-House</td></tr><tr><td rowspan=3 colspan=1>Optimal Policy CloningAGILE</td><td rowspan=1 colspan=1>PICK</td><td rowspan=1 colspan=1>NAV</td><td rowspan=1 colspan=1>Total</td><td rowspan=1 colspan=1>PICK</td><td rowspan=1 colspan=1>NAV</td><td rowspan=1 colspan=1>Total</td><td rowspan=1 colspan=1>PICK</td><td rowspan=1 colspan=1>NAV</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>20.7</td><td rowspan=1 colspan=1>61.6</td><td rowspan=1 colspan=1>40.3</td><td rowspan=1 colspan=1>10.1</td><td rowspan=1 colspan=1>29.4</td><td rowspan=1 colspan=1>19.6</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>17.2</td><td rowspan=1 colspan=1>8.5</td></tr><tr><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>40.9</td><td rowspan=1 colspan=1>18.0</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>34.1</td><td rowspan=1 colspan=1>16.8</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>30.6</td><td rowspan=1 colspan=1>15.1</td></tr><tr><td rowspan=1 colspan=1>GAIL-Exact</td><td rowspan=1 colspan=1>59.4</td><td rowspan=1 colspan=1>73.5</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>49.1</td><td rowspan=1 colspan=1>50.4</td><td rowspan=1 colspan=1>49.8</td><td rowspan=1 colspan=1>23.5</td><td rowspan=1 colspan=1>35.4</td><td rowspan=1 colspan=1>28.3</td></tr><tr><td rowspan=1 colspan=1>LC-RL(ours)</td><td rowspan=1 colspan=1>63.8</td><td rowspan=1 colspan=1>69.7</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>56.7</td><td rowspan=1 colspan=1>47.8</td><td rowspan=1 colspan=1>51.9</td><td rowspan=1 colspan=1>32.1</td><td rowspan=1 colspan=1>39.4</td><td rowspan=1 colspan=1>36.4</td></tr><tr><td rowspan=1 colspan=1>Reward Reg. (Oracle)</td><td rowspan=1 colspan=1>87.0</td><td rowspan=1 colspan=1>85.0</td><td rowspan=1 colspan=1>86.1</td><td rowspan=1 colspan=1>82.5</td><td rowspan=1 colspan=1>67.0</td><td rowspan=1 colspan=1>74.1</td><td rowspan=1 colspan=1>70.6</td><td rowspan=1 colspan=1>62.3</td><td rowspan=1 colspan=1>65.7</td></tr></table>
+
+Table 2: Success rates (in percentages) on using DQN to re-optimize learned rewards. For reference, we also include Q-iteration results (labeled QI) from Table 1 as an oracle comparison.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Shaping</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test-Task</td><td rowspan=1 colspan=1>Test-House</td></tr><tr><td rowspan=2 colspan=1>LC-RL (DQN)LC-RL(QI)</td><td rowspan=1 colspan=1>YesNo</td><td rowspan=1 colspan=1>12.98.0</td><td rowspan=1 colspan=1>14.17.7</td><td rowspan=1 colspan=1>14.98.0</td></tr><tr><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>51.9</td><td rowspan=1 colspan=1>36.4</td></tr><tr><td rowspan=2 colspan=1>Reward Regression (DQN)Reward Regression (QI)</td><td rowspan=1 colspan=1>YesNo</td><td rowspan=1 colspan=1>54.77.5</td><td rowspan=1 colspan=1>58.98.3</td><td rowspan=1 colspan=1>57.59.2</td></tr><tr><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>86.1</td><td rowspan=1 colspan=1>74.1</td><td rowspan=1 colspan=1>65.7</td></tr></table>
+
+# 6.3 EXPERIMENTAL RESULTS
+
+Our main experimental results on reward learning are reported in Table 1, and experiments in reoptimizing the learned reward function are reported in Table 2. Qualitative results with diagrams of learned reward functions can be found in Appendix B. Additional supplementary material can be viewed at https://sites.google.com/view/language-irl, and experiment hyperparameters are detailed in Appendix A.
+
+We found that both LC-RL and Reward Regression were able to learn reward functions which generalize to both novel tasks and novel house layouts, and both achieve significant performance over the policy-based approach. As expected, we found that Reward Regression has superior performance when compared to LC-RL, due to the fact that it uses oracle ground-truth supervision.
+
+We include examples of learned reward functions for both methods in Appendix B. We found that a common error made by the learned rewards, aside from simply misidentifying objects and locations, was rewarding the agent for reaching the goal position without placing the object down on PICK tasks. This is reflected in the results as the performance on PICK tasks is much lower than that of NAV tasks. Additionally, there is some ambiguity in the language commands, as the same environment may contain multiple copies of a single object or location, and we do not consider the case when agents can ask for additional clarification (for example, there are 2 beds in Fig. 3).
+
+We observed especially poor performance from the cloning baseline on both training as well as testing environments, even though it was trained by directly regressing onto the optimal policy. We suspect that it is significantly more difficult for the cloning baseline to learn across multiple environments. Our language is high-level and only consists of descriptions of the goal task (such as ”move the cup to the bathroom”) rather than step-by-step instructions used in other work such as Mei et al. (2016) that allow the policy to follow a sequence of instructions. This makes the task much more difficult for a policy-learning agent as it needs to learn a mapping from language to house layouts instead of blindly following the actions specified in the language command.
+
+Regarding re-optimization of the learned rewards, we found that DQN with epsilon-greedy exploration alone achieved poor performance compared to the exact solver and comparable performance to the cloning baseline (however, note that our cloning baseline was regressing onto the exact optimal actions). Adding a shaping term based on the value-function improves results, but computing this shaping term requires ground-truth knowledge of the environment dynamics. We also note that it appears that rewards learned through regression are easier to re-optimize than rewards learned through IRL. One explanation for this is that IRL rewards appear more “noisy” (for example, see reward plots in Appendix B, because small variations in the reward may not affect the trajectories taken by the optimal policy if a large reward occurs at the goal position. However, while RL is training it may never see the large reward and thus is heavily influenced by small, spurious variations in the reward. Nevertheless, with proper exploration methods, we believe that language-conditioned reward learning provides a performant and conceptually simple method for grounding language as concrete tasks an agent can perform within an interactive environment.
+
+# 7 CONCLUSION
+
+In this paper, we introduced LC-RL, an algorithm for scalable training of language-conditioned reward functions represented by neural networks. Our method restricts training to tractable domains with known dynamics, but learns a reward function which can be used with standard RL methods in environments with unknown dynamics. We demonstrate that the reward-learning approach to instruction following outperforms the policy-learning when evaluated in test environments, because the reward-learning enables an agent to learn and interact within the test environment rather than relying on zero-shot policy transfer.
+
+# REFERENCES
+
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+Brenna D. Argall, Sonia Chernova, Manuela Veloso, and Brett Browning. A survey of robot learning from demonstration. Robotics and autonomous systems, 57(5):469–483, 2009.   
+D. Bahdanau, F. Hill, J. Leike, E. Hughes, P. Kohli, and E. Grefenstette. Learning to Follow Language Instructions with Adversarial Reward Induction. ArXiv e-prints, June 2018.   
+S. R. K. Branavan, Harr Chen, Luke S. Zettlemoyer, and Regina Barzilay. Reinforcement learning for mapping instructions to actions. In Proceedings of the Joint Conference of the 47th Annual Meeting of the ACL and the 4th International Joint Conference on Natural Language Processing of the AFNLP, 2009.   
+Jaedeug Choi and Kee-eung Kim. Nonparametric bayesian inverse reinforcement learning for multiple reward functions. In Advances in Neural Information Processing Systems (NIPS). 2012.   
+P. Christiano, J. Leike, T. B. Brown, M. Martic, S. Legg, and D. Amodei. Deep reinforcement learning from human preferences. In Advances in Neural Information Processing Systems (NIPS), 2017.   
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+Chelsea Finn, Paul Christiano, Pieter Abbeel, and Sergey Levine. A connection between generative adversarial networks, inverse reinforcement learning, and energy-based models. CoRR, abs/1611.03852, 2016.
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+Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adversarial inverse reinforcement learning. In International Conference on Learning Representations (ICLR), 2018.
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+Karl Moritz Hermann, Felix Hill, Simon Green, Fumin Wang, Ryan Faulkner, Hubert Soyer, David Szepesvari, Wojciech Marian Czarnecki, Max Jaderberg, Denis Teplyashin, Marcus Wainwright, Chris Apps, Demis Hassabis, and Phil Blunsom. Grounded language learning in a simulated 3d world. CoRR, abs/1706.06551, 2017. URL http://arxiv.org/abs/1706.06551.
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+Hongyuan Mei, Mohit Bansal, and Matthew R. Walter. Listen, attend, and walk: Neural mapping of navigational instructions to action sequences. In AAAI Conference on Artificial Intelligence (AAAI), 2016.
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+Dipendra Kumar Misra, Jaeyong Sung, Kevin Lee, and Ashutosh Saxena. Tell me dave: Contextsensitive grounding of natural language to manipulation instructions. In Robotics: Science and Systems (RSS), 2014.
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+Dipendra Kumar Misra, John Langford, and Yoav Artzi. Mapping instructions and visual observations to actions with reinforcement learning. In Conference on Empirical Methods in Natural Language Processing (EMNLP), 2017.
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+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, feb 2015. ISSN 0028-0836.
+
+Andrew Ng, Daishi Harada, and Stuart Russell. Policy invariance under reward transformations: Theory and application to reward shaping. In International Conference on Machine Learning (ICML), 1999.
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+Ethan Perez, Florian Strub, Harm de Vries, Vincent Dumoulin, and Aaron C. Courville. Film: Visual reasoning with a general conditioning layer. 2018.
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+Pararth Shah, Marek Fiser, Aleksandra Faust, J. Chase Kew, and Dilek Hakkani-Tur. Follownet: ¨ Robot navigation by following natural language directions with deep reinforcement learning. CoRR, abs/1805.06150, 2018.
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+Shuran Song, Fisher Yu, Andy Zeng, Angel X Chang, Manolis Savva, and Thomas Funkhouser. Semantic scene completion from a single depth image. Proceedings of 29th IEEE Conference on Computer Vision and Pattern Recognition, 2017.
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+Jaeyong Sung, Seok Hyun Jin, and Ashutosh Saxena. Robobarista: Object part based transfer of manipulation trajectories from crowd-sourcing in 3d pointclouds. In International Symposium on Robotics Research (ISRR), 2015.
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+Stefanie Tellex, Thomas Kollar, Steven Dickerson, Matthew R. Walter, Ashis Gopal Banerjee, Seth Teller, and Nicholas Roy. Understanding natural language commands for robotic navigation and mobile manipulation. In Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence, AAAI’11, pp. 1507–1514. AAAI Press, 2011. URL http://dl.acm.org/ citation.cfm?id=2900423.2900661.
+
+Hsiao-Yu Fish Tung, Adam W. Harley, Liang-Kang Huang, and Katerina Fragkiadaki. Reward learning from narrated demonstrations. CoRR, abs/1804.10692, 2018.
+
+Brian Ziebart. Modeling purposeful adaptive behavior with the principle of maximum causal entropy. PhD thesis, Carnegie Mellon University, 2010.
+
+Brian Ziebart, Andrew Maas, Andrew Bagnell, and Anind Dey. Maximum entropy inverse reinforcement learning. In AAAI Conference on Artificial Intelligence (AAAI), 2008.
+
+# APPENDICES
+
+# A EXPERIMENT HYPERPARAMETERS
+
+For our environment, we give the agent a time limit 30 time-steps to complete a task. For the purpose of reward regression and generating demonstrations, the environment gives a reward of 10 when the agent successfully completes the task. We sample demonstrations from the optimal policy using this ground-truth reward. Our MDP solvers use a discount of $\gamma = 0 . 9 9$ .
+
+For our model, we used 10 demonstrations per environment to train IRL, and optimized with Adam using a learning rate of $5 * 1 0 ^ { - 4 }$ .
+
+For our convolutional neural network, we used a 5x5 convolution with 16 filters, followed by a $3 \mathrm { x } 3$ convolution with 32 filters. The size of each embedding was 32. The final fully connected layer had sizes of 32, and 1 (for the final output). We did not find significant performance differences increasing the number of filters or embedding sizes.
+
+We selected our architecture through a hyper-parameter sweep, with train and test accuracies presented below (averaged over 3 seeds each). The main architectures we swept through were whether to produce the image embedding via a global pooling layer (labeled Pooling) versus a single fully connected layer (labeled FC) versus FiLM (Perez et al., 2018), and whether to combine the language and image embeddings using a point-wise multiplication or a pooling operation (labeled Mult) versus concatenation (labeled Concat).
+
+<table><tr><td></td><td>Train</td><td>Test-Task</td></tr><tr><td>Pooling, Mult</td><td>67.3</td><td>49.7</td></tr><tr><td>Pooling，Add</td><td>64.1</td><td>45.5</td></tr><tr><td>Pooling,Concat</td><td>65.8</td><td>48.5</td></tr><tr><td>FiLM,Mult</td><td>60.6</td><td>43.2</td></tr><tr><td>FiLM, Add</td><td>67.0</td><td>49.1</td></tr><tr><td>FiLM, Concat</td><td>63.2</td><td>44.6</td></tr><tr><td>FC,Mult</td><td>53.1</td><td>38.4</td></tr><tr><td>FC,Add</td><td>55.2</td><td>39.1</td></tr><tr><td>FC,Concat</td><td>48.1</td><td>35.2</td></tr></table>
+
+We also conducted a single ablation study over the size of the image below, using the Pooling, Mult architecture. We found that the impact of the image size between (32 by 24) and (64 by 64) was negligible, so we selected the smaller image size for computational reasons.
+
+<table><tr><td></td><td>Train</td><td>Test-Task</td></tr><tr><td>32 by 24</td><td>67.3</td><td>49.7</td></tr><tr><td>64by 64</td><td>70.6</td><td>51.1</td></tr></table>
+
+# B LEARNED REWARDS AND QUALITATIVE RESULTS
+
+Below is an example of a learned reward (and the computed value function) from our IRL model. The task presented here is to bring the fruit bowl (green arrow) to the bathroom (red arrow). In the top row, we plot the reward function, and in the bottom row we plot the resulting value function. The left column shows the rewards/values before the object (fruit bowl) is acquired, and the right column shows the rewards/values after. Note that before the object is acquired, the value directs the agent to the fruit bowl, and once the object is found, the value directs the agent to the bathroom.
+
+In these figures, a blue shaded square means a high values and red means low. The green-outlined tiles correspond to all locations within a 1-meter radius of the fruit bowl, or all tiles an agent can pick up the bowl from. The red-outlined tile likewise represents all tiles within a 1-meter radius of the drop-off location in the bathroom. The blue-outlined square represents the starting location of the agent.
+
+![](images/a3f722a784f488063d9086586d1cbe549962be9ed605ab14a2e19cbeb2bd93e1.jpg)
+
+Next, we show rewards learned by 3 different methods: inverse reinforcement learning (IRL), GAIL, and reward regression. Again, low rewards are denoted by red squares and high rewards are denoted by blue squares. For each task, we also include a birds-eye view of task, where the object is highlighted in green and the agent is denoted by a green triangle.
+
+In general, we find that rewards learned by IRL and GAIL tend to be noisy and contain small artifacts. This is not unexpected, as both of these methods are sample-based and observe demonstrations instead of ground-truth rewards as supervision. We believe that such artifacts are detrimental when using RL to reoptimize the learned reward, as without adequate exploration RL cannot find the large reward at the true goal state, and instead ends up finding local minima.
+
+![](images/9180e6a22420e96b242aac03f10d95d6219aaf028879cf0568e02b4784659c54.jpg)  
+Figure 6: Learned rewards and a corresponding birds-eye view rollout for the task ”go to fruit bowl”.
+
+![](images/1bcfa437dfe95c994d4815721d6b7f50a47b95fd2802cc4e2dbacbc407a79f24.jpg)  
+Figure 7: Learned rewards and a corresponding birds-eye view rollout for the task ”move pan to living room”.
+
+# C OBSERVATION CACHING
+
+The running time for dynamic programming algorithms for solving MDPs (such as q-iteration) scales with the size of the state space, and in our environments we found reward evaluation becoming a major bottleneck in runtime. One major optimization we make to our algorithm is to cache computation on repeated observations. Computation-wise, the main bottleneck in Algorithm 1 is evaluating and back-propagating the reward function at all states and actions, rather than Q-iteration itself (even though it carries a cubic run-time dependency on the size of the state space). However, in many environments this is extremely wasteful. For example, in the house depicted in Figure 3, information about where the cup is located must be included in the state. However, if our observations are images of what the robot sees, whether the cup is in the kitchen or in the bathroom has no impact on the images inside the living room. This means that we should only need to evaluate our reward in each living room images once for both locations of the cup. A major factor in speeding up our computation was to cache such repeated computation. In practice we found this to be significant, resulting in 10-100 times speedups on reward computation depending on the structure of the environment.

parse/train/r1lq1hRqYQ/r1lq1hRqYQ_content_list.json

@@ -0,0 +1,1356 @@
+[
+    {
+        "type": "text",
+        "text": "FROM LANGUAGE TO GOALS: INVERSE REINFORCE-MENT LEARNING FOR VISION-BASED INSTRUCTIONFOLLOWING",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Justin Fu ∗, Anoop Korattikara, Sergey Levine, Sergio Guadarrama Google AI {justinfu,kbanoop,slevine,sguada}@google.com ",
+        "bbox": [
+            183,
+            195,
+            650,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            273,
+            544,
+            289
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning is a promising framework for solving control problems, but its use in practical situations is hampered by the fact that reward functions are often difficult to engineer. Specifying goals and tasks for autonomous machines, such as robots, is a significant challenge: conventionally, reward functions and goal states have been used to communicate objectives. But people can communicate objectives to each other simply by describing or demonstrating them. How can we build learning algorithms that will allow us to tell machines what we want them to do? In this work, we investigate the problem of grounding language commands as reward functions using inverse reinforcement learning, and argue that language-conditioned rewards are more transferable than language-conditioned policies to new environments. We propose language-conditioned reward learning (LC-RL), which grounds language commands as a reward function represented by a deep neural network. We demonstrate that our model learns rewards that transfer to novel tasks and environments on realistic, high-dimensional visual environments with natural language commands, whereas directly learning a languageconditioned policy leads to poor performance. ",
+        "bbox": [
+            233,
+            306,
+            764,
+            527
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            555,
+            334,
+            571
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While reinforcement learning provides a powerful and flexible framework for describing and solving control tasks, it requires the practitioner to specify objectives in terms of reward functions. Engineering reward functions is often done by experienced practitioners and researchers, and even then can pose a significant challenge, such as when working with complex image-based observations. While researchers have investigated alternative means of specifying objectives, such as learning from demonstration (Argall et al., 2009), or through binary preferences (Christiano et al., 2017), language is often a more natural and desirable way for humans to communicate goals. ",
+        "bbox": [
+            174,
+            588,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A common approach to building natural language interfaces for reinforcement learning agents is to build language-conditioned policies ",
+        "bbox": [
+            174,
+            803,
+            483,
+            844
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e71170ca08fb2257826210ece6f73ff54f1ace30220b36c57506856bd504733.jpg",
+        "image_caption": [
+            "Figure 1: A task where an agent (green triangle) must execute the command “go to the fruit bowl.” This is a simple example where the reward function is easier to specify than the policy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            529,
+            592,
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+            755
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "that directly map observations and language commands to a sequence of actions that perform the desired task. However, this requires the policy to solve two challenging problems together: understanding how to plan and solve tasks in the physical world, and understanding the language command itself. The trained policy must simultaneously interpret a command and plan through possibly complicated environment dynamics. The performance of the system then hinges entirely on its ability to generalize to new environments - if either the language interpretation or the physical control fail to generalize, the entire system will fail. We can recognize instead that the role of language in such a system is to communicate the goal, and rather than mapping language directly to policies, we propose to learn how to convert language-defined goals into reward functions. In this manner, the agent can learn how to plan and perform the task on its own via reinforcement learning, directly interacting with the environment, without relying on zero-shot transfer of policies. A simple example is shown in Figure 1, where an agent is tasked with navigating through a house. If an agent is commanded “go to the fruit bowl”, a valid reward function could simply be a fruit bowl detector from first-person views of the agent. However, if we were to learn a mapping from language to actions, given the same goal description, the model would need to generate a different plan for each house. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+            268
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we investigate the feasibility of grounding free-form natural language commands as reward functions using inverse reinforcement learning (IRL). Learning language-conditioned rewards poses unique computational problems. IRL methods generally require solving a reinforcement learning problem as an inner-loop (Ziebart, 2010), or rely on potentially unstable adversarial optimization procedures (Finn et al., 2016; Fu et al., 2018). This is compounded by the fact that we wish to train our model across multiple tasks, meaning the IRL problem itself is an inner-loop. In order to isolate the language-learning problem from the difficulties in solving reinforcement learning and adversarial learning problems, we base our method on an exact MaxEnt IRL (Ziebart, 2010) procedure, which requires full knowledge of environment dynamics to train a language-conditioned reward function represented by a deep neural network. While using exact IRL procedures may seem limiting, in many cases (such as indoor robotic navigation) full environment dynamics are available, and this formulation allows us to remove the difficulty of using RL from the training procedure. The crucial insight is that we can use dynamic programming methods during training to learn a reward function that maps from observations, but we do not need knowledge of dynamics to use the reward function, meaning during test time we can evaluate using a reinforcement learning agent without knowledge of the underlying environment dynamics. We evaluate our method on a dataset of realistic indoor house navigation and pick-and-place tasks using the SUNCG dataset, with natural language commands. We demonstrate that our approach generalizes not only to novel tasks, but also to entirely new scenes, while directly learning a language-conditioned policy leads to poor performance and fails to generalize. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            578,
+            344,
+            593
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A popular class of approaches to language grounding in reinforcement learning is to directly train a policy that consumes language as an input. Several works adopt a behavioral cloning approach, where the model is trained using supervised learning with language-action sequences pairs (Anderson et al., 2018; Mei et al., 2016; Sung et al., 2015). A second approach is to forego demonstrations but instead reward an agent whenever the desired task is completed (Shah et al., 2018; Misra et al., 2017; Hermann et al., 2017; Branavan et al., 2009). This approach requires reward functions (the task completion detector) to be hand-designed for the training tasks considered. Another related approach is semantic parsing, which has also been used to convert language into an executable form that corresponds to actions within an environment (Forbes et al., 2015; Misra et al., 2014; Tellex et al., 2011). In a related task to instruction following, Das et al. (2018) consider an embodied question-answering task where an agent must produce an answer to a question, where the relevant information lies within the environment. They adopt a hybrid approach, where they pretrain with supervised learning but also give the agent reward for completing intermediate tasks. Overall, our experiments show that policy-based approaches have worse generalization performance to new environments, because the policy must rely on zero-shot generalization at test time as we show in Section 6.3. While in this paper we argue for the performance benefits of a reward-based approach, a reason one may want to adopt a policy-based approach over a reward-based one is if one cannot run RL to train a new policy in a new environment, such as for time or safety reasons. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A second approach to the language grounding problem is to learn a mapping from language to reward functions. There are several other works that apply IRL or IRL-like procedures to the problem of language grounding. Perhaps most closely related to our work is MacGlashan et al. (2015), which also aims to learn a language-conditioned reward function via IRL. However, this method requires an extensively hand-designed, symbolic reward function class, whereas we use generic, differentiable function approximators that can handle arbitrary observations, including raw images. Bahdanau et al. (2018); Tung et al. (2018) also learn language-conditioned reward functions, but do not perform IRL, meaning that the objective does not correspond to matching the expert’s trajectory distribution. Tung et al. (2018) train a task-completion classifier, but do not evaluate their reward on control problems. The strategy they use is similar to directly regressing onto a ground-truth reward function, which we include a comparison to in Section 6 to as an oracle baseline. Bahdanau et al. (2018) adopt an adversarial approach similar to GAIL (Ho & Ermon, 2016), and use the learned discriminator as the reward function. While this produces a reward function, it does not provide any guarantees that the resulting reward function can be reoptimized in new environments to yield behavior similar to the expert. We believe our work is the first to apply language-conditioned inverse reinforcement learning to environments with image observations and deep neural networks, and we show that our rewards generalize to novel tasks and environments. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            304,
+            326,
+            320
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We build off of the MaxEnt IRL model (Ziebart et al., 2008), which considers an entropy-regularized Markov decision process (MDP), defined by the tuple $( S , \\mathcal { A } , \\mathcal { T } , r , \\gamma , \\rho _ { 0 } )$ . $s , A$ are the state and action spaces respectively and $\\gamma \\in ( 0 , 1 )$ is the discount factor. $\\mathcal { T } ( s ^ { \\prime } | s , a )$ represents the transition distribution or dynamics. We additionally consider partially-observed environments, where each state is associated with an observation within an observations space $o \\in \\mathcal { O }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The goal of ”forward” reinforcement learning is to find the optimal policy $\\pi ^ { * }$ . Let $r ( \\tau ) \\ =$ $\\scriptstyle \\sum _ { t = 0 } ^ { T } \\gamma ^ { t } r ( s _ { t } , a _ { t } )$ denote the returns of a trajectory, where $\\tau$ denotes a sequence of states and actions $( s _ { 0 } , a _ { 0 } , . . . s _ { T } , a _ { T } )$ . The MaxEnt RL objective is then to find $\\begin{array} { r } { \\pi ^ { * } = \\arg \\operatorname* { m i n } _ { \\pi } { E _ { \\tau \\sim \\pi } [ r ( \\tau ) + H ( \\tau ) ] } } \\end{array}$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Inverse reinforcement learning (IRL) seeks to infer the reward function $r ( s , a )$ given a set of expert demonstrations $\\mathcal { D } = \\{ \\tau _ { 1 } , . . . , \\tau _ { N } \\} .$ ,. In IRL, we assume the demonstrations are drawn from an optimal policy $\\pi ^ { * } ( a | s )$ . We can interpret the IRL problem as solving the maximum likelihood problem: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa0fd58f9222c7708294772a3a3feffa29b7f59c8f976972089defff2df0acc6.jpg",
+        "text": "$$\n\\operatorname * { m a x } _ { \\theta } E _ { \\tau \\sim \\mathcal { D } } \\left[ \\log p _ { \\theta } ( \\tau ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            510,
+            580,
+            532
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the MaxEnt IRL framework, optimal trajectories are observed with probabilities proportional to the exponentiated returns, meaning $p ( \\tau ) \\propto \\exp \\{ r ( \\tau ) \\}$ (Ziebart et al., 2008). Thus, learning a reward function $r _ { \\theta } ( \\tau )$ is equivalent to fitting an energy-based model $p _ { \\theta } ( \\tau ) \\propto \\exp \\{ r _ { \\theta } ( \\tau ) \\}$ to the maximum likelihood objective in Eqn 1. The gradient to update the reward function is (Ziebart, 2010): ",
+        "bbox": [
+            174,
+            536,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/69b2b8daed3e4674bbcf5ecaccd3a9c8d52d4bb357f36a3dce35ca8d80b853f7.jpg",
+        "text": "$$\n\\nabla _ { \\theta } E [ \\log p _ { \\theta } ( \\tau ) ] = \\sum _ { s , a } ( \\rho ^ { \\mathcal { D } } ( s , a ) - \\rho _ { \\theta } ^ { * } ( s , a ) ) \\nabla _ { \\theta } r _ { \\theta } ( s , a ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            608,
+            684,
+            643
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\rho ^ { \\mathcal { D } } ( s , a )$ represents the state-action marginal of the demonstrations, and $\\rho _ { \\theta } ^ { * } ( s , a )$ represents the state-action marginal of the optimal policy under reward $r _ { \\theta } ( s , a )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 MULTI-TASK IRL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            702,
+            352,
+            717
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A unique challenge of the language-conditioned IRL problem, compared to standard IRL, is that the goal is to learn a reward function that generalizes across multiple tasks. While standard IRL methods are typically trained and evaluated on the same task, we want our language-conditioned reward function to produce correct behavior when presented with new tasks. Several previous works consider a multi-task scenario, such as in a Bayesian or meta-learning setting (Li & Burdick, 2017; Dimitrakakis & Rothkopf, 2012; Choi & Kim, 2012). We adopt a similar approach adapted for the language-IRL problem, and formalize the notion of a task, denoted by $\\xi$ , as an MDP, where individual tasks may not share the same state spaces, dynamics or reward functions. Each task is associated with a context $c _ { \\xi }$ which is a unique identifier (i.e. an indicator vector) for that task. Thus, we wish to optimize the following multi-task objective, where $\\tau _ { \\xi }$ denotes expert demonstrations for that task: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/3db1845f532b51a7bac21922374c9151bf92a1d4c99d774b384751031e76c55d.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } E _ { \\xi } [ E _ { \\tau _ { \\xi } } [ \\log p _ { \\theta } ( \\tau _ { \\xi } , c _ { \\xi } ) ] ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "table",
+        "img_path": "images/e1e66da165ca2ad5f53a8bc6dc1d6c415b53fd6b0c954df81209f925d6738dee.jpg",
+        "table_caption": [
+            "Algorithm 1 Language-Conditioned Reward Learning (LC-RL) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>1: Obtain expert demonstrations and language describing the goal.</td></tr><tr><td>2: Initialize reward function rθ .</td></tr><tr><td>3: for step t in {1,...,N} do</td></tr><tr><td>4: Sample task ε, demonstrations dg ,and language Lg.</td></tr><tr><td>5: Compute optimal q*(s,a) using q-iteration and p*(s,a) using the forward algorithm</td></tr><tr><td></td></tr><tr><td>6: Update reward rθ with the gradient (ρdε(s,a) - p*(s,a)) Vre(o,a,Lg) 7: end for</td></tr></table>",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to optimize this objective, we first require that all tasks share the same observation space and action space, and the reward to be a function of the observation, rather than of the state. For example, in our experiments, all observations are in the form of $3 2 \\mathrm { x } 2 4$ images taken from simulated houses, but the state space for each house is allowed to differ (i.e., the houses have different layouts). This means the same reward can be used across all MDPs even though the state spaces differ. ",
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+    {
+        "type": "text",
+        "text": "Second, we share the reward function across all tasks, but substitute a language command $\\mathcal { L } _ { \\xi }$ as a proxy for the context $c _ { \\xi }$ , resulting in a model $p _ { \\theta } ( \\tau _ { \\xi } , \\mathcal { L } _ { \\xi } )$ that takes as input language, states, and actions. For computational efficiency we run stochastic gradient descent on the objective in Eqn. 3 by sampling over the set of environments on each iteration. ",
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+    {
+        "type": "text",
+        "text": "5 LANGUAGE-CONDITIONED REWARD LEARNING (LC-RL) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We learn language-conditioned reward functions using maximum causal entropy IRL, adapted for a multi-task setting and rewards represented by language-conditioned convolutional neural networks. While during training we use dynamic programming methods that require dynamics knowledge, we do not need knowledge of dynamics to evaluate the reward function. Thus, at test time we can use standard model-free RL algorithms to learn the task from the inferred reward function in new environments. Our algorithm is briefly summarized in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "5.1 COMPUTING EXACT IRL GRADIENT UPDATES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In order to take gradient steps on the objective of Eqn. 3, we update our reward function in terms of the Maximum Entropy IRL gradient (Ziebart, 2010) according to Eqn. 2. The stochastic gradient update (for a single task $\\xi$ ) adapted to our case is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e76417de215c3130d36d9269e9dcfb87bd62710f7e84259358fd0158e0d8acd4.jpg",
+        "text": "$$\n\\nabla _ { \\theta } E _ { \\tau } [ \\log p _ { \\theta } ( \\tau _ { \\xi } , \\mathcal { L } _ { \\xi } ) ] = \\sum _ { s , a } ( \\rho ^ { d } ( s , a ) - \\rho _ { \\theta } ^ { * } ( s , a ) ) \\nabla _ { \\theta } r _ { \\theta } ( o ( s ) , a , \\mathcal { L } _ { \\xi } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Where $o ( s )$ denotes the observation for state $s$ . Note that during training we need access to the ground truth states $s$ . While the update depends on the underlying state, the reward itself is only a function of the observation, the action, and the language. This enables us to evaluate the reward without knowing the underlying state space and dynamics of the environment. While requiring dynamics knowledge during training may seem limiting, in practice many environments we may wish to train a robot in can easily be mapped. This training strategy is analogous to training a robot in only known environments such as a laboratory, but the resulting reward can be used in unknown environments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to compute $\\rho _ { \\theta } ^ { * } ( s , a )$ , one normally has to first compute the optimal policy with respect to reward $r _ { \\theta } ( o , a )$ using reinforcement learning, and then compute the occupancy measure using the forward algorithm for Markov chains to compute the state visitation distributions at each time-step. Because this embeds a difficult RL optimization problem within nested inner-loops, this quickly becomes computationally intractable. Thus, we train in tabular environments with known dynamics, where we can compute optimal policies exactly using Q-iteration. However, we emphasize that this is only a training time restriction, and knowledge of dynamics is not required to evaluate the rewards. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f3cb66f02a1107fd18e35bab67acccca5473dba6383bdc7079370bdcbbbd5bd5.jpg",
+        "image_caption": [
+            "Figure 2: Our reward function architecture. Our network receives as input a panoramic semantic image (4 views) and a language command represented as a sequence of one-hot word vectors, and outputs a scalar reward. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.2 ARCHITECTURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our network architecture is shown in Figure 2. The network has two main modalities of input: a variable-length language input represented by a sequence of one-hot vectors (one vector for each tokenized word), and a panoramic image observation. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The language embedding is formed by processing the language input sequence through an LSTM network, and the final time-step of the topmost layer is used as a fixed-dimensional embedding elanguage of the input command. ",
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+        "type": "text",
+        "text": "The agent receives image observations in the form of four $3 2 \\mathrm { x } 2 4$ image observations, one for each cardinal direction view (N, S, E, W). The convolutional neural network (CNN) consists of a sequence of convolutional and max pool layers, with the final operation being a channel-wise global pooling operation that produces an image embedding of the same length as the language embedding. Each image is passed through an identical CNN with shared weights, and the outputs are summed together to form the image embedding. That is, $e _ { \\mathrm { i m a g e } } = \\mathbf { C N N } ( \\mathrm { i m } \\mathbf { g } _ { N } ) + \\mathbf { C N N } ( \\mathrm { i m } \\mathbf { g } _ { S } ) + \\mathbf { C N N } ( \\mathrm { i m } \\mathbf { g } _ { E } ) +$ $\\mathbf { C N N } ( \\mathrm { i m g } _ { W } )$ . ",
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+    {
+        "type": "text",
+        "text": "Finally, these two embeddings are element-wise multiplied and passed through a fully-connected network (FC) to produce a reward output. Letting $\\odot$ denote elementwise multiplication, we have $r = \\mathrm { F C } ( e _ { \\mathrm { i m a g e } } \\odot e _ { \\mathrm { l a n g u a g e } } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We found that the max global-pooling architecture in the CNN was able to select out objects from a scene and allow the language embedding to modulate which features to attend to. We selected our architecture via a hyper-parameter search, and found that the choice of using an element-wise multiplication versus a concatenation for combining embeddings had no appreciable performance difference, but a global pooling architecture performed significantly better than using a fully connected layer at the end of the CNN. ",
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+    {
+        "type": "text",
+        "text": "6 EVALUATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate our method within a collection of simulated indoor house environments, built on top of the SUNCG (Song et al., 2017) dataset. The SUNCG dataset provides a large repository of complex and realistic 3D environments which we find very suitable for our goals. An example task from our environment is shown in Figures 3 and 4. An example of a successful execution of a task is showin in Fig. 5. ",
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+    {
+        "type": "text",
+        "text": "6.1 ENVIRONMENT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We consider two typical kinds of tasks that an indoor robot may wish to perform: ",
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+        "type": "text",
+        "text": "• Navigation (NAV): In the navigation task, the agent is given a location which corresponds to a room or object, and the agent must navigate through the house to reach the target location. For example, in Fig. 3, the target could be ”cup” or ”laptop” or ”living room”. ",
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+    {
+        "type": "image",
+        "img_path": "images/cdace129ad738b81b305696edad646670941ae49887c617f761c225c048e1dc5.jpg",
+        "image_caption": [
+            "Figure 3: An example task. The green segment corresponds to the solution of a NAV task, ”go to the cup”, where the cup is circled in green. The green plus blue segments represents a path for the PICK task, ”move the cup to the bed”, where the bed is circled in blue. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/28ca2602fae11df6b47ce7b36c7d39de637ba4b5a4bc8160c0e57e228da93603.jpg",
+        "image_caption": [
+            "Figure 4: Example first-person RGB (left) and semantic (right) images from the bedroom inside the house depicted in Figure 3. We only use the semantic labels as input to our model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
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+        "img_path": "images/d1a108a7bf4b6d49ce6fd1915455ff10baca4541408a9db638895372665a8140.jpg",
+        "image_caption": [
+            "Figure 5: Top row: First-person view of an agent executing the task: “move vase to living room”. The vase is circled in green in the 3rd image. Bottom row: A bird’s eye view of the initial (left) and final (right) positions of the agent (green triangle) and the vase (green outline). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Pick-and-place (PICK): In the pick and place task, the agent must move an object from one location to another. For example, in Fig. 1 the task could be to move the cup from to the sink to the kitchen table. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Each environment corresponds to one 3D scene, which is discretized into a grid to form a tabular environment where the grid coordinates plus agent orientation (N, S, E, W) correspond to the state of the agent. The agent receives observations with two components: one is a free-form language command, and one is a first-person panoramic image of the environment. Because the agent can move objects without directly looking at them, the panoramic view gives the agent a full view of its surroundings. The panoramic image is formed from 4 semantic image observations, one for each orientation of the agent. Each semantic image observation is $3 2 \\mathrm { x } 3 2 $ pixels and contains 61 channels, one per semantic image class. Each agent is equipped with 4 actions: step forward one grid tile, turn left or right, or interact with an object. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We generate language commands based on a preset grammar, and using names of objects and locations associated with the task. These are of the form ”go to $X ^ { \\ast }$ for NAV tasks, or ”move X to $\\mathbf { Y } ^ { \\prime \\prime }$ for PICK tasks, where X and $\\mathrm { Y }$ stand for names of locations and objects within the environment. We explicitly do not use step-by-step instruction language such as ”turn left, walk down the hallway, go through the door”, as these commands remove the planning aspect of the problem and tell the agent directly which actions to take in order to solve the problem. ",
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+    {
+        "type": "text",
+        "text": "The interact action only has meaning within the PICK task. Executing this action will either pick up an object if the agent is within a 1 meter of an object, or drop an object if the agent is currently holding an object. To limit the size of the state space, within a single task, there is only one object an agent may interact with and two locations the object can be in. This setup only increases the size of the state-space by a factor of 3. However, different objects may be placed in different locations across environments, meaning the model still must learn to detect the object in the correct location rather than memorizing the specific object and location associated with a single task. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6.2 EXPERIMENTAL PROCEDURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to evaluate how well different methods generalize, we split our dataset of tasks into three segments: a training set, and two test sets – “task” and “house”. The ”task” test set contains tasks within the same houses as training, but requires the agent to interact with novel combinations of objects and locations. The ”house” test set contains tasks on entirely new houses that were not in the training set. The purpose of this split is to investigate varying degrees of generalization: the ”task” test set requires the model to execute novel language commands, but using landmarks and objects which were seen during training. The ”house” test set adds another layer of difficulty, requiring the model to detect familiar objects situated in entirely new scenes. ",
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+        "type": "text",
+        "text": "In total, our dataset contains 1413 tasks (716 PICK, 697 NAV). Across all tasks, there are 14 objects, and 76 different house layouts. There are 1004 tasks $( 7 1 \\% )$ in the training set, 236 $( 1 7 \\% )$ in the ”task” test set, and 173 $( 1 2 \\% )$ in the ”house” test set. ",
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+    {
+        "type": "text",
+        "text": "We evaluate two methods for reward learning. LC-RL refers to the language-conditioned IRL method outlined in Section 5, which takes as input demonstration and language pairs and learns a shared reward function across all tasks. In particular, we use 10 demonstrations per task, sampled from the computed optimal policy. To provide an upper bound for performance, we can also regress directly onto the ground-truth rewards, a method we label as “Reward Regression”. While this is not possible in a practical scenario, this evaluation serves to show oracle performance on our task. Note that our method does not require access to ground-truth rewards, and only uses demonstrations. ",
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+    {
+        "type": "text",
+        "text": "Success rates for each reward-learning method are evaluated using two policy learning procedures. Q-iteration (QI) computes the optimal policy exactly using dynamic programming, which we report in Table 1. We also experiment with reoptimizing the learned reward using DQN (Mnih et al., 2015), a sample-based RL method that does not require ground-truth knowledge of the environment dynamics. We use the position, orientation, and whether an object is held as the observation (this is identical to the state representation). This experiment represents the testing use-case where we can evaluate the reward at test-time in novel, unmapped environments despite the fact that at training time we require dynamics knowledge. However, because the probability that the random policy receives rewards on our task is tiny, we found that epsilon-greedy exploration was not enough. Thus, we also report results using a reward shaping term with a state-based potential equal to the optimal value function $\\mathrm { N g }$ et al., 1999). We note that this shaping term does require dynamics knowledge compute, but we include this result to highlight the difficulty of the RL problem even if reward learning is done properly. ",
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+    {
+        "type": "text",
+        "text": "We also compare against two baselines derived from GAIL (Ho & Ermon, 2016), using the learned discriminator as the “reward” function. We first compare to AGILE (Bahdanau et al., 2018), which modifies GAIL to use a goal-based discriminator and false negative filtering, using DQN as a policy optimizer and $\\rho \\ : = \\ : 0 . 2 5$ . We found it difficult to learn rewards using a reinforcement learningbased policy optimizer, and the model was only able to solve the simpler NAV environments. This experiment emphasizes the gap between using a sampling-based solver and an exact solver during the training of reward-based methods. To create a more fair comparison, we also compare against GAIL using a dynamic programming solver (labeled GAIL-Exact), and we see that the performance is comparable to LC-RL on training environments, but performs significantly worse on test environments. These results are in line with our intuitions - GAIL and IRL are equivalent in training scenarios (Ho & Ermon, 2016), but the discriminator of GAIL does not correspond to the true reward function, and thus performs worse when evaluated in novel environments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order compare against a policy-learning approach, we compare against an optimal behavioral cloning baseline. We train the optimal cloning baseline by computing the exact optimal policy using Q-iteration, and perform supervised learning to regress directly on to the optimal action probabilities. To make a fair comparison, we keep the policy architecture identical to the reward architecture, except we add two additional inputs: the orientation indicator and and indicator on whether the object (during PICK tasks) is held by the agent or not. Each indicator is transformed by an embedding lookup, and all embeddings are element-wise multiplied along with with the language and image embeddings in the original architecture. ",
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+    {
+        "type": "table",
+        "img_path": "images/3f7871d3428abd72d1bfaef8acebed94633fa298a124a5ce82583950dc6587e8.jpg",
+        "table_caption": [
+            "Table 1: Success rates (in percentages) across task categories. Each result is averaged over 3 seeds. Test-Task refers to testing on novel tasks within the same houses as training, whereas Test-House refers to testing novel tasks in novel houses. The AGILE method is described in (Bahdanau et al., 2018) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=9>Train                  Test-Task              Test-House</td></tr><tr><td rowspan=3 colspan=1>Optimal Policy CloningAGILE</td><td rowspan=1 colspan=1>PICK</td><td rowspan=1 colspan=1>NAV</td><td rowspan=1 colspan=1>Total</td><td rowspan=1 colspan=1>PICK</td><td rowspan=1 colspan=1>NAV</td><td rowspan=1 colspan=1>Total</td><td rowspan=1 colspan=1>PICK</td><td rowspan=1 colspan=1>NAV</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>20.7</td><td rowspan=1 colspan=1>61.6</td><td rowspan=1 colspan=1>40.3</td><td rowspan=1 colspan=1>10.1</td><td rowspan=1 colspan=1>29.4</td><td rowspan=1 colspan=1>19.6</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>17.2</td><td rowspan=1 colspan=1>8.5</td></tr><tr><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>40.9</td><td rowspan=1 colspan=1>18.0</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>34.1</td><td rowspan=1 colspan=1>16.8</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>30.6</td><td rowspan=1 colspan=1>15.1</td></tr><tr><td rowspan=1 colspan=1>GAIL-Exact</td><td rowspan=1 colspan=1>59.4</td><td rowspan=1 colspan=1>73.5</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>49.1</td><td rowspan=1 colspan=1>50.4</td><td rowspan=1 colspan=1>49.8</td><td rowspan=1 colspan=1>23.5</td><td rowspan=1 colspan=1>35.4</td><td rowspan=1 colspan=1>28.3</td></tr><tr><td rowspan=1 colspan=1>LC-RL(ours)</td><td rowspan=1 colspan=1>63.8</td><td rowspan=1 colspan=1>69.7</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>56.7</td><td rowspan=1 colspan=1>47.8</td><td rowspan=1 colspan=1>51.9</td><td rowspan=1 colspan=1>32.1</td><td rowspan=1 colspan=1>39.4</td><td rowspan=1 colspan=1>36.4</td></tr><tr><td rowspan=1 colspan=1>Reward Reg. (Oracle)</td><td rowspan=1 colspan=1>87.0</td><td rowspan=1 colspan=1>85.0</td><td rowspan=1 colspan=1>86.1</td><td rowspan=1 colspan=1>82.5</td><td rowspan=1 colspan=1>67.0</td><td rowspan=1 colspan=1>74.1</td><td rowspan=1 colspan=1>70.6</td><td rowspan=1 colspan=1>62.3</td><td rowspan=1 colspan=1>65.7</td></tr></table>",
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+        "table_caption": [
+            "Table 2: Success rates (in percentages) on using DQN to re-optimize learned rewards. For reference, we also include Q-iteration results (labeled QI) from Table 1 as an oracle comparison. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Shaping</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test-Task</td><td rowspan=1 colspan=1>Test-House</td></tr><tr><td rowspan=2 colspan=1>LC-RL (DQN)LC-RL(QI)</td><td rowspan=1 colspan=1>YesNo</td><td rowspan=1 colspan=1>12.98.0</td><td rowspan=1 colspan=1>14.17.7</td><td rowspan=1 colspan=1>14.98.0</td></tr><tr><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>51.9</td><td rowspan=1 colspan=1>36.4</td></tr><tr><td rowspan=2 colspan=1>Reward Regression (DQN)Reward Regression (QI)</td><td rowspan=1 colspan=1>YesNo</td><td rowspan=1 colspan=1>54.77.5</td><td rowspan=1 colspan=1>58.98.3</td><td rowspan=1 colspan=1>57.59.2</td></tr><tr><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>86.1</td><td rowspan=1 colspan=1>74.1</td><td rowspan=1 colspan=1>65.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6.3 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our main experimental results on reward learning are reported in Table 1, and experiments in reoptimizing the learned reward function are reported in Table 2. Qualitative results with diagrams of learned reward functions can be found in Appendix B. Additional supplementary material can be viewed at https://sites.google.com/view/language-irl, and experiment hyperparameters are detailed in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "We found that both LC-RL and Reward Regression were able to learn reward functions which generalize to both novel tasks and novel house layouts, and both achieve significant performance over the policy-based approach. As expected, we found that Reward Regression has superior performance when compared to LC-RL, due to the fact that it uses oracle ground-truth supervision. ",
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+    {
+        "type": "text",
+        "text": "We include examples of learned reward functions for both methods in Appendix B. We found that a common error made by the learned rewards, aside from simply misidentifying objects and locations, was rewarding the agent for reaching the goal position without placing the object down on PICK tasks. This is reflected in the results as the performance on PICK tasks is much lower than that of NAV tasks. Additionally, there is some ambiguity in the language commands, as the same environment may contain multiple copies of a single object or location, and we do not consider the case when agents can ask for additional clarification (for example, there are 2 beds in Fig. 3). ",
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+        "type": "text",
+        "text": "We observed especially poor performance from the cloning baseline on both training as well as testing environments, even though it was trained by directly regressing onto the optimal policy. We suspect that it is significantly more difficult for the cloning baseline to learn across multiple environments. Our language is high-level and only consists of descriptions of the goal task (such as ”move the cup to the bathroom”) rather than step-by-step instructions used in other work such as Mei et al. (2016) that allow the policy to follow a sequence of instructions. This makes the task much more difficult for a policy-learning agent as it needs to learn a mapping from language to house layouts instead of blindly following the actions specified in the language command. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Regarding re-optimization of the learned rewards, we found that DQN with epsilon-greedy exploration alone achieved poor performance compared to the exact solver and comparable performance to the cloning baseline (however, note that our cloning baseline was regressing onto the exact optimal actions). Adding a shaping term based on the value-function improves results, but computing this shaping term requires ground-truth knowledge of the environment dynamics. We also note that it appears that rewards learned through regression are easier to re-optimize than rewards learned through IRL. One explanation for this is that IRL rewards appear more “noisy” (for example, see reward plots in Appendix B, because small variations in the reward may not affect the trajectories taken by the optimal policy if a large reward occurs at the goal position. However, while RL is training it may never see the large reward and thus is heavily influenced by small, spurious variations in the reward. Nevertheless, with proper exploration methods, we believe that language-conditioned reward learning provides a performant and conceptually simple method for grounding language as concrete tasks an agent can perform within an interactive environment. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduced LC-RL, an algorithm for scalable training of language-conditioned reward functions represented by neural networks. Our method restricts training to tractable domains with known dynamics, but learns a reward function which can be used with standard RL methods in environments with unknown dynamics. We demonstrate that the reward-learning approach to instruction following outperforms the policy-learning when evaluated in test environments, because the reward-learning enables an agent to learn and interact within the test environment rather than relying on zero-shot policy transfer. ",
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+    {
+        "type": "text",
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+        "type": "text",
+        "text": "APPENDICES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A EXPERIMENT HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For our environment, we give the agent a time limit 30 time-steps to complete a task. For the purpose of reward regression and generating demonstrations, the environment gives a reward of 10 when the agent successfully completes the task. We sample demonstrations from the optimal policy using this ground-truth reward. Our MDP solvers use a discount of $\\gamma = 0 . 9 9$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For our model, we used 10 demonstrations per environment to train IRL, and optimized with Adam using a learning rate of $5 * 1 0 ^ { - 4 }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For our convolutional neural network, we used a 5x5 convolution with 16 filters, followed by a $3 \\mathrm { x } 3$ convolution with 32 filters. The size of each embedding was 32. The final fully connected layer had sizes of 32, and 1 (for the final output). We did not find significant performance differences increasing the number of filters or embedding sizes. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We selected our architecture through a hyper-parameter sweep, with train and test accuracies presented below (averaged over 3 seeds each). The main architectures we swept through were whether to produce the image embedding via a global pooling layer (labeled Pooling) versus a single fully connected layer (labeled FC) versus FiLM (Perez et al., 2018), and whether to combine the language and image embeddings using a point-wise multiplication or a pooling operation (labeled Mult) versus concatenation (labeled Concat). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/90c6698fc133027d828a9c8b185479d4f0b064539a1c10d3d61d7dbc7970b52c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Train</td><td>Test-Task</td></tr><tr><td>Pooling, Mult</td><td>67.3</td><td>49.7</td></tr><tr><td>Pooling，Add</td><td>64.1</td><td>45.5</td></tr><tr><td>Pooling,Concat</td><td>65.8</td><td>48.5</td></tr><tr><td>FiLM,Mult</td><td>60.6</td><td>43.2</td></tr><tr><td>FiLM, Add</td><td>67.0</td><td>49.1</td></tr><tr><td>FiLM, Concat</td><td>63.2</td><td>44.6</td></tr><tr><td>FC,Mult</td><td>53.1</td><td>38.4</td></tr><tr><td>FC,Add</td><td>55.2</td><td>39.1</td></tr><tr><td>FC,Concat</td><td>48.1</td><td>35.2</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We also conducted a single ablation study over the size of the image below, using the Pooling, Mult architecture. We found that the impact of the image size between (32 by 24) and (64 by 64) was negligible, so we selected the smaller image size for computational reasons. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ec2661195e908fd9964e4338cdefcd9bccfefa5025f6383c2076b17bc58708f8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Train</td><td>Test-Task</td></tr><tr><td>32 by 24</td><td>67.3</td><td>49.7</td></tr><tr><td>64by 64</td><td>70.6</td><td>51.1</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "B LEARNED REWARDS AND QUALITATIVE RESULTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Below is an example of a learned reward (and the computed value function) from our IRL model. The task presented here is to bring the fruit bowl (green arrow) to the bathroom (red arrow). In the top row, we plot the reward function, and in the bottom row we plot the resulting value function. The left column shows the rewards/values before the object (fruit bowl) is acquired, and the right column shows the rewards/values after. Note that before the object is acquired, the value directs the agent to the fruit bowl, and once the object is found, the value directs the agent to the bathroom. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In these figures, a blue shaded square means a high values and red means low. The green-outlined tiles correspond to all locations within a 1-meter radius of the fruit bowl, or all tiles an agent can pick up the bowl from. The red-outlined tile likewise represents all tiles within a 1-meter radius of the drop-off location in the bathroom. The blue-outlined square represents the starting location of the agent. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/a3f722a784f488063d9086586d1cbe549962be9ed605ab14a2e19cbeb2bd93e1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Next, we show rewards learned by 3 different methods: inverse reinforcement learning (IRL), GAIL, and reward regression. Again, low rewards are denoted by red squares and high rewards are denoted by blue squares. For each task, we also include a birds-eye view of task, where the object is highlighted in green and the agent is denoted by a green triangle. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "In general, we find that rewards learned by IRL and GAIL tend to be noisy and contain small artifacts. This is not unexpected, as both of these methods are sample-based and observe demonstrations instead of ground-truth rewards as supervision. We believe that such artifacts are detrimental when using RL to reoptimize the learned reward, as without adequate exploration RL cannot find the large reward at the true goal state, and instead ends up finding local minima. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9180e6a22420e96b242aac03f10d95d6219aaf028879cf0568e02b4784659c54.jpg",
+        "image_caption": [
+            "Figure 6: Learned rewards and a corresponding birds-eye view rollout for the task ”go to fruit bowl”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1bcfa437dfe95c994d4815721d6b7f50a47b95fd2802cc4e2dbacbc407a79f24.jpg",
+        "image_caption": [
+            "Figure 7: Learned rewards and a corresponding birds-eye view rollout for the task ”move pan to living room”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C OBSERVATION CACHING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "The running time for dynamic programming algorithms for solving MDPs (such as q-iteration) scales with the size of the state space, and in our environments we found reward evaluation becoming a major bottleneck in runtime. One major optimization we make to our algorithm is to cache computation on repeated observations. Computation-wise, the main bottleneck in Algorithm 1 is evaluating and back-propagating the reward function at all states and actions, rather than Q-iteration itself (even though it carries a cubic run-time dependency on the size of the state space). However, in many environments this is extremely wasteful. For example, in the house depicted in Figure 3, information about where the cup is located must be included in the state. However, if our observations are images of what the robot sees, whether the cup is in the kitchen or in the bathroom has no impact on the images inside the living room. This means that we should only need to evaluate our reward in each living room images once for both locations of the cup. A major factor in speeding up our computation was to cache such repeated computation. In practice we found this to be significant, resulting in 10-100 times speedups on reward computation depending on the structure of the environment. ",
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+    }
+]