Add files using upload-large-folder tool

parse/train/GvqjmSwUxkY/GvqjmSwUxkY.md

@@ -0,0 +1,359 @@
+# RETHINKING THE TRULY UNSUPERVISED IMAGE-TOIMAGE TRANSLATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Every recent image-to-image translation model uses either image-level (i.e. inputoutput pairs) or set-level (i.e. domain labels) supervision at a minimum. However, even the set-level supervision can be a serious bottleneck for data collection in practice. In this paper, we tackle image-to-image translation in a fully unsupervised setting, i.e., neither paired images nor domain labels. To this end, we propose a truly unsupervised image-to-image translation model (TUNIT) that simultaneously learns to separate image domains and translate input images into the estimated domains. Experimental results show that our model achieves comparable or even better performance than the set-level supervised model trained with full labels, generalizes well on various datasets, and is robust against the choice of hyperparameters (e.g. the preset number of pseudo domains). In addition, TUNIT extends well to the semi-supervised scenario with various amount of labels provided.
+
+# 1 INTRODUCTION
+
+Given an image of one domain, image-to-image translation is a task to generate the plausible images of the other domains. Based on the success of conditional generative models (Mirza & Osindero, 2014; Sohn et al., 2015), many image translation methods have been proposed either using imagelevel supervision (e.g. paired data) (Isola et al., 2017; Hoffman et al., 2018; Zhu et al., 2017b; Wang et al., 2018; Park et al., 2019) or using set-level supervision (e.g. domain labels) (Zhu et al., 2017a; Kim et al., 2017; Liu et al., 2017; Huang et al., 2018; Liu et al., 2019; Lee et al., 2020). Though the latter approach is generally called unsupervised as a counterpart of the former, it actually assumes that the domain labels are given a priori. This assumption can be a serious bottleneck in practice as the number of domains and samples increases. For example, labeling individual samples of a large dataset, such as FFHQ, is expensive, and the distinction across domains can be ambiguous.
+
+Here, we first clarify that unsupervised image-to-image translation should strictly denote the task without any supervision neither paired images nor domain labels. Under this definition, our goal is to develop an unsupervised translation model given a mixed set of images of many domains (Figure 1). We tackle this problem by formulating three sub-problems: 1) clustering the images by approximating the set-level characteristics (i.e. domains), 2) encoding the individual content and style of an input image, and 3) learning a mapping function among the estimated domains.
+
+To this end, we introduce a guiding network that simultaneously solves 1) unsupervised domain classification and 2) style encoding. It has two branches of providing pseudo domain labels and encoding style features, which are later used in the discriminator and the generator training, respectively. We employ a differentiable clustering method based on mutual information maximization for estimating domain labels. This helps the guiding network group similar images together while evenly separate their categories. For embedding style codes, we adopt a contrastive loss (Hadsell et al., 2006; He et al., 2020; Chen et al., 2020a), which leads the model to further understand the dissimilarity between images, resulting in better representation learning. Finally, conditioned on the style features and domain labels from the guiding network, we use generative adversarial networks (GAN) to learn the image translation functions across various domains.
+
+Although GAN and the guiding network play different roles, we do not separate their training process– our guiding network participates in the translation process. By doing so, the guiding network can exploit gradients from GAN training. The guiding network now understands the recipes of domain-separating attributes because the generator wants the style code to contain sufficient information to fool the domain-specific discriminator, and vice versa. Thanks to this interaction between the guiding network and GAN, our model successfully separates domains and translates images.
+
+![](images/726797f2ef17218ef12f9d9ccbbc5c6eb69b4f4ab8fe6e7c14af7f9a836ef628.jpg)  
+Figure 1: Levels of supervision. To perform image-to-image translation, existing methods need either (a) a dataset with input-output pairs or, (b) a dataset with domain information. Our method is capable of learning mappings among multiple domains using (c) a dataset without any supervision.
+
+We quantitatively and qualitatively compare our model with the existing set-level supervised method under unsupervised and semi-supervised setting. The experiments on various datasets show that the proposed model outperforms the previous method over all different levels of supervision. Our experimental results show that, by exploiting the synergy between two tasks, the guiding network helps the image translation model to largely improve the generation performance.
+
+Our contributions are summarized as follows:
+
+• We clarify the definition of unsupervised image-to-image translation and to the best of our knowledge, our model is the first to succeed in this task in an end-to-end manner.   
+• We propose the guiding network to handle the unsupervised translation task and show that the interaction between translation and clustering is helpful for the task.   
+• We show the effectiveness of our model through the extensive experiments on various datasets.   
+• We confirm that our model is applicable to various numbers of clusters and the practical case, where ground truth labels of several samples are available.
+
+# 2 TRULY UNSUPERVISED IMAGE-TO-IMAGE TRANSLATION (TUNIT)
+
+We consider the unsupervised image-to-image translation problem, where we have images $\chi$ from $K$ domains $K \geq 2 )$ ) without domain labels $y$ . Here, $K$ is an unknown property of the dataset. Throughout the paper, we denote $K$ as the actual number of domains in a dataset and $\hat { K }$ as the arbitrarily chosen number of domains to train models. We design a module that integrates both a domain classifier and a style encoder, which we call guiding network. It guides the translation by feeding reference images as the style code to the generator and as the pseudo domain labels to the discriminator. Using the feedback from the discriminator regarding the pseudo labels, the generator synthesizes images of the target domains (e.g. breeds) while respecting styles (e.g. fur patterns) of the reference images and maintaining the content (e.g. pose) of source images (Figure 2).
+
+# 2.1 LEARNING TO PRODUCE DOMAIN LABELS AND ENCODE STYLE FEATURES
+
+In our framework, the guiding network $E$ plays a central role as an unsupervised domain classifier as well as a style encoder. Our guiding network $E$ consists of two branches, $E _ { C }$ and $E _ { S }$ , each of which learns to provide domain labels and style codes, respectively. In experiments, we compare our guiding network against straightforward approaches, i.e.., K-means on image or feature space.
+
+Unsupervised domain classification. The discriminator requires target domain labels to provide useful gradients for image translation into the target domain. $E _ { C }$ adopts a differentiable clustering technique to provide pseudo domain labels of reference images, maximizing the mutual information (MI) between an image $\mathbf { x }$ and its randomly augmented version $\mathbf { x } ^ { + }$ (Ji et al., 2019). The optimum of the mutual information $I ( \mathbf { p } , \mathbf { p } ^ { + } )$ is reached as the entropy $H ( \mathbf { p } )$ is maximum and the conditional entropy $H ( \mathbf { p } | \mathbf { p } ^ { + } )$ is minimum, where $\mathbf { p } = E _ { C } ( \mathbf { x } )$ represents the softmax output from $E _ { C }$ , indicating a probability vector of $\mathbf { x }$ over $\cdot$ domains. Please refer to Section 3.3 for more details about $\cdot$ . Maximizing MI encourages $E _ { C }$ to assign the same domain label to the pair $\mathbf { \bar { x } }$ and $\mathbf { x } ^ { + }$ ) while evenly distributing entire samples to all domains.
+
+![](images/d06a3a92ca3377b8950e5dca64ab7215c6897ba6bc71cdd37600a52b869f55a1.jpg)  
+Figure 2: Overview of our proposed method. The figure illustrates how our model changes the breed of the cat. (a) An estimated domain from our guiding network $E$ is used to train the multi-task discriminator $D$ . (b) $E$ provides the generator $G$ with the style code of a reference image and the estimated domain is again used for GAN training.
+
+Formally, $E _ { C }$ maximizes the mutual information:
+
+$$
+\mathcal { L } _ { M I } = I ( \mathbf { p } , \mathbf { p } ^ { + } ) = I ( \mathbf { P } ) = \sum _ { i = 1 } ^ { \hat { K } } \sum _ { j = 1 } ^ { \hat { K } } \mathbf { P } _ { i j } \ln \frac { \mathbf { P } _ { i j } } { \mathbf { P } _ { i } \mathbf { P } _ { j } } ,
+$$
+
+where $f$ is a composition of random augmentations such as random cropping and affine transformation. $\mathbf { P } _ { i } = \mathbf { P } ( \mathbf { p } = i )$ denotes the $\hat { K }$ -dimensional marginal probability vector, and $\mathbf { P } _ { i j } = \mathbf { P } ( \mathbf { p } =$ $i , \mathbf { p } ^ { + } = j$ ) denotes the joint probability. To provide a deterministic one-hot label to the discriminator, we use the argmax operation (i.e. $y = \tt a r g m a x ( { E _ { C } ( x ) } ) )$ . We note that the mutual information is one way to implement TUNIT, therefore, any differentiable clustering methods can be adopted such as SCAN (Van Gansbeke et al., 2020).
+
+Style encoding and improving domain classification. $E _ { S }$ encodes an image into a style code s which provides translation guide for the generator. In addition to the style guide to the generator, $E _ { S }$ is beneficial in improving unsupervised domain classification, where pseudo labels from $E _ { C }$ fail to scale up when samples are complex and diverse (e.g., AnimalFaces (Liu et al., 2019)). Since $E _ { S }$ is an another branch of the guiding network, imposing the contrastive loss (He et al., 2020) on the style codes improves representation of the shared embeddings:
+
+$$
+\mathcal { L } _ { s t y l e } ^ { E } = - \log \frac { \exp ( \mathbf { s } \cdot \mathbf { s } ^ { + } / \tau ) } { \sum _ { i = 0 } ^ { N } \exp ( \mathbf { s } \cdot \mathbf { s } _ { i } ^ { - } / \tau ) } ,
+$$
+
+where ${ \bf s } = E _ { S } ( { \bf x } )$ . $\mathbf { x }$ and $\mathbf { x } ^ { + }$ denote an image and randomly augmented version of $\mathbf { x }$ , respectively. This $( N + 1 )$ -way classification enables $E$ to utilize not only the similarity of the positive pair (s, $\mathbf { s } ^ { + }$ ) but also the dissimilarity of the negative pairs (s, $\mathbf { s } _ { i } ^ { - }$ ). We adopt a queue to store the negative codes $\cdot$ of the previously sampled images as MoCo (He et al., 2020). By doing so, we can conduct the contrastive learning efficiently without large batch sizes (Saunshi et al., 2019). We observe that adding this objective significantly improves unsupervised classification accuracy on AnimalFaces from $6 8 . 0 \%$ to $8 4 . 1 \%$ compared to the previous approach (Ji et al., 2019).
+
+In this subsection, we describe how to perform the unsupervised image-to-image translation under the guidance of our guiding network. For successful translation, the model should provide the realistic images containing the visual feature of the target domain. To this end, we adopt three losses: 1) adversarial loss to produce realistic images, 2) style contrastive loss that encourages the model not to ignore the style codes, 3) image reconstruction loss for preserving the domain-invariant features. We explain each loss and the overall objective for each network.
+
+Adversarial loss. For adversarial training, we adopt a variant of conditional discriminator, the multitask discriminator (Mescheder et al., 2018). It is designed to conduct discrimination for each domain simultaneously. However, its gradient is calculated only with the loss for estimating the domain of the input image. For the domain label of the input image, we utilize the pseudo label from the guiding network. Formally, given the pseudo label $\tilde { y }$ for a reference image $\tilde { \mathbf { x } }$ , we train our generator $G$ and multi-task discriminator $D$ via the adversarial loss:
+
+$$
+\mathcal { L } _ { a d v } = \mathbb { E } _ { \tilde { \mathbf { x } } \sim p _ { d a t a } ( \mathbf { x } ) } [ \log D _ { \tilde { y } } ( \tilde { \mathbf { x } } ) ] + \mathbb { E } _ { \mathbf { x } , \tilde { \mathbf { x } } \sim p _ { d a t a } ( \mathbf { x } ) } [ \log ( 1 - D _ { \tilde { y } } ( G ( \mathbf { x } , \tilde { \mathbf { s } } ) ) ) ] ,
+$$
+
+where $D _ { \tilde { y } } ( \cdot )$ denotes the logit from the domain-specific $( \tilde { y } )$ discriminator, and $\tilde { \mathbf { s } } = E _ { S } ( \tilde { \mathbf { x } } )$ denotes a target style code of the reference image $\tilde { \bf x }$ . The generator $G$ learns to translate $\mathbf { x }$ to the target domain $\tilde { y }$ while reflecting the style code ˜s.
+
+Style constrastive loss. In order to prevent a degenerate case where the generator ignores the given style code ˜s and synthesize a random image of the domain $\tilde { y }$ , we impose a style contrastive loss:
+
+$$
+\mathcal { L } _ { s t y l e } ^ { G } = \mathbb { E } _ { \mathbf { x } , \tilde { \mathbf { x } } \sim p _ { d a t a } ( \mathbf { x } ) } \left[ - \log \frac { \exp ( \mathbf { s } ^ { \prime } \cdot \tilde { \mathbf { s } } ) } { \sum _ { i = 0 } ^ { N } \exp ( \mathbf { s } ^ { \prime } \cdot \mathbf { s } _ { i } ^ { - } / \tau ) } \right] .
+$$
+
+Here, $\mathbf { s } ^ { \prime } = E _ { S } ( G ( \mathbf { x } , \tilde { \mathbf { s } } ) )$ denotes the style code of the translated image $G ( \mathbf { x } , \tilde { \mathbf { s } } )$ and $\mathbf { s } _ { i } ^ { - }$ denotes the negative style codes, which are from the same queue used in equation (2). And we follow the training scheme of MoCo (He et al., 2020) as equation (2). The above loss guides the generated image $\cdot$ to have a style similar to the reference image $\cdot$ and dissimilar to negative (other) samples. By doing so, we also avoid the degenerated solution where the encoder maps all the images to the same style code of the reconstruction loss (Choi et al., 2020) based on L1 or L2 norm. Equation (2) and (4) are based on contrastive loss, but they are used for different purposes. Please refer to Appendix H for more discussion.
+
+Image reconstruction loss. To ensure that the generator $G$ can reconstruct the source image $\mathbf { x }$ when given with its original style ${ \bf s } = E _ { S } ( { \bf x } )$ , we impose an image reconstruction loss:
+
+$$
+\mathcal { L } _ { r e c } = \mathbb { E } _ { { \mathbf { x } } \sim p _ { d a t a } ( { \mathbf { x } } ) } [ | | { \mathbf { x } } - G ( { \mathbf { x } } , { \mathbf { s } } ) | | _ { 1 } ] .
+$$
+
+This objective not only ensures the generator $G$ to preserve domain-invariant characteristics (e.g., pose) of its input image $\mathbf { x }$ , but also helps to learn the style representation of the guiding network $E$ by extracting the original style s of the source image $\mathbf { x }$ .
+
+Overall objective. Finally, we train the three networks jointly as follows:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { D } = - \mathcal { L } _ { a d v } , } \\ & { \mathcal { L } _ { G } = \mathcal { L } _ { a d v } + \lambda _ { s t y l e } ^ { G } \mathcal { L } _ { s t y l e } ^ { G } + \lambda _ { r e c } \mathcal { L } _ { r e c } , } \\ & { \mathcal { L } _ { E } = \mathcal { L } _ { G } - \lambda _ { M I } \mathcal { L } _ { M I } + \lambda _ { s t y l e } ^ { E } \mathcal { L } _ { s t y l e } ^ { E } } \end{array}
+$$
+
+where $\lambda$ ’s are hyperparameters. Note that our guiding network $E$ receives feedback from $L _ { G }$ , which is essential for our method. We discuss the effect of feedback to $E$ on performance in Section 3.1.
+
+# 3 EXPERIMENTS
+
+We first evaluate TUNIT on labeled datasets by treating them as unlabeled because the desired behaviours of the translation models in labeled datasets are well defined (Section 3.1). Here, we provide an ablation study to analyze the effect of each component and compare the models both quantitatively and qualitatively. We then move on to unlabeled datasets to validate our model in the unsupervised scenario in the wild (Section 3.2). Lastly, we show that TUNIT is robust against the choice
+
+<table><tr><td rowspan="2">Configuration</td><td colspan="3">AnimalFaces-10</td><td colspan="3">Food-10</td></tr><tr><td>mFID</td><td>D&amp;C</td><td>Acc.</td><td>mFID</td><td>D&amp;C</td><td>Acc.</td></tr><tr><td>A Baseline FUNIT (supervised)</td><td>74.0</td><td>0.749 / 0.671</td><td>1.000</td><td>68.4</td><td>0.989 / 0.782</td><td>1.000</td></tr><tr><td>B (A)+Improved G&amp;D (supervised)</td><td>46.2</td><td>0.896 / 0.732</td><td>1.000</td><td>57.6</td><td>1.284 / 0.857</td><td>1.000</td></tr><tr><td>C (B)+ K-means on image space</td><td>110.7</td><td>0.822 / 0.615</td><td>0.215</td><td>90.7</td><td>0.849 / 0.648</td><td>0.201</td></tr><tr><td>D (B) + K-means on feature space</td><td>76.2</td><td>0.770 / 0.597</td><td>0.428</td><td>64.6</td><td>0.968 / 0.808</td><td>0.331</td></tr><tr><td>E (B) + Differentiable clustering</td><td>73.5</td><td>0.940 / 0.588</td><td>0.680</td><td>64.2</td><td>1.038 / 0.819</td><td>0.542</td></tr><tr><td>F TUNIT w/ sequential training</td><td>46.0</td><td>1.060 / 0.789</td><td>0.850</td><td>61.1</td><td>0.908 / 0.777</td><td>0.860</td></tr><tr><td>G TUNIT w/ joint training</td><td>47.7</td><td>1.039 / 0.805</td><td>0.841</td><td>52.2</td><td>1.079 / 0.875</td><td>0.848</td></tr></table>
+
+Table 1: Main results. mFID, Density / Coverage (D & C), and classification accuracy (Acc) of each training configuration. Note that the configurations (A) - (B) use ground-truth class labels, while (C) - (G) use pseudo-labels. We bold the best results separately for supervised and unsupervised settings.
+
+![](images/88571bb423f86a5b771181cfa37f3796a8c5c8b5c7ffc1981e9edec274f61969.jpg)  
+Figure 3: Qualitative comparison of translation results using each configuration in Table 1. Here, B reflects the style feature (e.g. species or type of food) of the reference images while A does not. The model C performs much worse than A and B in that it overly adopts the source image, not adequately merging styles and contents from both sides. The model D generates more plausible images than C but fails to reflect the characteristics of the reference images. For example, D on fifth row does not look like several pieces of dumpling due to its shape and dish color, meaning that the reference styles are not properly reflected. Similarly, E also fails to generate the dumpling in the fifth row. TUNIT with sequential training F reflects the visual features of each reference on both datasets. However, in terms of visual fidelity, we observe that G consistently outperforms F. Akin to the quantitative results, TUNIT achieves equivalent or even better visual quality than the set-level supervised model A and B.
+
+of hyperparameters (e.g. the preset number of clusters, $K .$ ) and extends well to the semi-supervised scenario (Section 3.3). In all experiments, we use FUNIT (Liu et al., 2019) as our baseline.
+
+Datasets. For the labeled datasets, we randomly select ten classes among 149 classes of AnimalFaces and 101 classes of Food-101, which we call AnimalFaces-10 and Food-10, respectively. Here, the labels are used only for the evaluation purpose. For the unlabeled datasets, we use AFHQ, FFHQ, and LSUN Car (Choi et al., 2020; Karras et al., 2019; Yu et al., 2015), which do not have any or are missing with fine-grained labels. Specifically, AFHQ roughly has three groups (i.e., dog, cat and wild), but each group contains diverse species and these species labels are not provided. FFHQ and LSUN Car contain various human faces and cars without any labels, respectively.
+
+![](images/55a9488623cb69c72d4de1bca06515d3d37bc916780476c730bb5325666af15c.jpg)  
+Figure 4: Reference-guided image translation results on unlabeled datasets.
+
+![](images/214c164faf6bdbca092f21b1ca1527fbc7947db60f659fec53878ba855201c0d.jpg)  
+Figure 5: t-SNE visualization of the style space of our guiding network trained on AFHQ Wild. Since AFHQ Wild does not have ground-truth labels, each point is colored with the guiding network’s prediction. Although we set the number of domains to be quite large $\hat { K } = 1 0 $ ), the network separates one species into two domains, which are so closely located that the model creates six clusters.
+
+Evaluation Metrics. We report two scores to assess the generated images. First, to provide a general sense of image quality, we use the mean of class-wise Frenchet Inception Distance (mFID) (Heusel ´ et al., 2017). It can avoid the degenerate case of the original FID, which assigns a good score when the model conveying the source image as is. Additionally, to provide a finer assessment of the generated images, we report Density and Coverage (D&C) (Naeem et al., 2020). D&C separately evaluates the fidelity and the diversity of the model outputs, which is also known to be robust against outliers and model hyperparameters (e.g. the number of samples used for evaluation). Denote that a lower mFID score means better image quality, and D&C scores that are bigger or closer to 1.0 indicate the better fidelity and diversity, respectively. Please refer to Appendix C for the detailed information.
+
+# 3.1 COMPARATIVE EVALUATION ON LABELED DATASETS
+
+Table 1 summarizes the effect of each component of TUNIT and rigorous comparisons with the state-of-the-art supervised method, FUNIT. First, we report the set-level supervised performance
+
+![](images/75d51c13a67bf1ecc74547dc5c8277f2332d6b5792826479497837a54811f2b0.jpg)
+
+<table><tr><td>K</td><td>AnimalFaces-10 mFID D&amp;C</td><td>Food-10 mFID</td><td>D&amp;C</td></tr><tr><td>1 4 7 10 13 16 20 50 500</td><td>129.6 0.561 /0.512 77.7 0.879 /0.738 62.7 1.016 /0.729 47.7 1.039 /0.805 56.8 0.993/ 0.805 54.1 1.093 / 0.782 55.4 1.019 / 0.778 63.8 0.858 /0.701 67.2</td><td>95.1 67.4 52.7 52.2 54.8 54.8 57.7 60.8 0.921/ 0.694 63.2</td><td>1.113 / 0.771 0.851/0.785 1.079 /0.875 1.079 /0.875 0.970/0.845 1.029 /0.857 0.937 /0.846 1.067 /0.837</td></tr></table>
+
+Table 2: t-SNE visualization of the model with (a) $\scriptstyle { \hat { K } } = 1 0$ and (b) $\scriptstyle { \hat { K } } = 2 0$ trained on AnimalFaces-10 and quantitative evaluation of our method by varying the number of pseudo domains $\cdot$ . Each point is colored with the ground-truth labels. As shown in t-SNE visualizations, even if $\cdot$ is set to overly larger than the actual number of domains, the guiding network clusters the domains reasonably well.
+
+![](images/12399874c7d44a8da845f303377a9173ddc43c29caa472ab0b6d48283c524b3f.jpg)  
+Figure 6: Qualitative comparison on the number of pseudo domains $\cdot$ . The performance varies along with $\cdot$ . When we set $\hat { K }$ large enough, the results are reasonable.
+
+of FUNIT and its variant (Table 1). Here, A is the original FUNIT and B denotes the modified FUNIT using our architecture (e.g. We do not use PatchGAN discriminator), which brings a large improvement over every score on both datasets. One simple way to extend B to the unsupervised scenario is to add an off-the-shelf clustering method and use its estimated labels instead of the ground truth. We employ K-means clustering on the image space for C, and the pretrained feature space for D. Here, we use ResNet-50 (He et al., 2016) features trained with MoCo v2 (Chen et al., 2020b) on ImageNet. Not surprisingly, because the estimated labels are inaccurate, the overall performance significantly drops. Although using the pretrained features helps a little, not only is it far from the setlevel supervised performance but it requires three steps to train the entire model, which complicates the application. This can be partially addressed by employing the differentiable clustering method (Ji et al., 2019), which trains VGG-11BN (Simonyan & Zisserman, 2015) with mutual information maximization that makes E. This reduces the number of training steps from three to two and provides better label estimation, which enables the model to approach the performance of original FUNIT A. However, as seen in the coverage score, the sample diversity is unsatisfactory.
+
+Finally, we build TUNIT by introducing the guiding network and the new objective functions described in Section 2. The changes significantly improve the accuracy on both datasets, particularly achieving similar mFID of the improved set-level supervised model B. Our final model, G matches or outperforms mFID and D&C of B. This is impressive because B utilizes oracles for training while G has no labels. Notably, TUNIT can cover $7 \% \mathrm { p }$ wider support of the data on AnimalFaces-10 than B. We conjecture that TUNIT benefits from the style codes that represent meaningful domain features learned by clustering. By comparing $\mathbb { E }$ and G, we confirm that they are comparable in terms of clustering and G is more stable in terms of inter-dataset performance. Therefore, we adopt the joint training of style encoder and clustering as our final model (G). In addition, we remove the adversarial loss for training the guiding network. It directly degrades the performance; mFID changes from 47.7 to 63.0 on AnimalFaces-10. It indicates that our training scheme takes an important portion of performance gains. Qualitative results also show superiority of TUNIT over competitors (Figure 3).
+
+Table 3: Quantitative evaluation (mFID) when few labels are available during training.   
+
+<table><tr><td rowspan="2">Configuration</td><td rowspan="2">20%</td><td colspan="3">AnimalFaces-10</td><td rowspan="2"></td><td colspan="3">Food-10</td></tr><tr><td>40%</td><td>60%</td><td>80%</td><td>20%</td><td>40% 60%</td><td>80%</td></tr><tr><td>A FUNIT</td><td></td><td>124.4</td><td>106.4</td><td>96.0</td><td>79.6</td><td>111.4</td><td>85.8</td><td>74.8</td><td>70.3</td></tr><tr><td>G</td><td>TUNIT (ours)</td><td>42.0</td><td>42.6</td><td>43.9</td><td>46.2</td><td>53.6</td><td>56.2</td><td>52.8</td><td>53.4</td></tr></table>
+
+<table><tr><td rowspan="2">Configuration</td><td colspan="4">AnimalFaces-10</td><td colspan="4">Food-10</td></tr><tr><td>1%</td><td>2%</td><td>4%</td><td>8%</td><td>1%</td><td>2%</td><td>4%</td><td>8%</td></tr><tr><td>A FUNIT</td><td>107.8</td><td>104.7</td><td>90.3</td><td>93.9</td><td>71.9</td><td>71.5</td><td>71.6</td><td>69.0</td></tr><tr><td>G TUNIT (ours)</td><td>47.9</td><td>44.8</td><td>42.7</td><td>42.4</td><td>54.5</td><td>56.1</td><td>55.9</td><td>55.8</td></tr></table>
+
+Table 4: Quantitative evaluation (mFID) when few labels are available during training. Here, an auxliary classifier is adopted to improve the FUNIT baseline by giving pseudo-labels to $\mathcal { D } _ { u n }$ .
+
+# 3.2 VALIDATION ON UNLABELED DATASET
+
+We evaluate TUNIT on the unlabeled datasets having no clear separations of the domains, which are AFHQ, FFHQ and LSUN Cars. For AFHQ, we train three individual models for dog, cat and wild. For all experiments, FUNIT is used as a baseline. We train it by presuming all labels to be the same as one. We set $\scriptstyle { \hat { K } } = 1 0$ for all the TUNIT models. More discussions on $\cdot$ will be in Section 3.3.
+
+Figure 4 demonstrates the results. We observe that the results of TUNIT adequately reflect the style feature of the references such as the textures of cats or cars and the species of the wilds. Although FFHQ has no clear domain distinctions, TUNIT captures the existence of glasses or smile as domains, and then add or remove glasses or smile. However, FUNIT performs much worse than TUNIT in this truly unsupervised scenario. For example, FUNIT outputs the inputs as is (cats and wilds) or insufficiently reflects the species (third row of AFHQ Wild). For FFHQ, despite that FUNIT makes some changes, the changes are not interpreted as meaningful domain translations. For LSUN Car, FUNIT fails to keep the fidelity.
+
+We also visualize the style space of both models to qualitatively assess the quality of the representation. Figure 5 shows the t-SNE maps trained on AFHQ Wild and the examples of each cluster. Surprisingly, TUNIT organizes the samples according to the species where it roughly separates the images into six species. Although we set $\cdot$ to be overly large, the model represents one species into two domains where those two domains position much closely. The highly disentangled, meaningful style features can be an important factor in the success of our model. On the other hand, the style features of FUNIT hardly learn meaningful domains so that the model cannot conduct the translation properly as shown in Figure 4. Because of the page limit, we include more results in Appendix E, G.
+
+# 3.3 ANALYSIS ON GENERALIZABILITY
+
+Robustness to various $\hat { K }$ ’s. When TUNIT conducts clustering for estimating domain labels, the number of clusters $\cdot$ can affect the overall performances. Here, we study the effects on different $\cdot$ on the labeled datasets and report them in Figure 6 and Table 2. As expected, the model performs best in terms of mFID when $\cdot$ equals to the ground truth $K$ (i.e. $\hat { K } { = } 1 0 $ ). Additionally, TUNIT performs reasonably well for sufficiently large $\cdot$ $\geq 7 )$ , and even with 100 times larger $\hat { K }$ than the actual number of the domains, TUNIT still works well on both datasets. From this study, we conclude that TUNIT is relatively robust against $\hat { K }$ as long as it is sufficiently large. We suggest using a sufficiently large number of $\hat { K }$ or studying different $\hat { K }$ ’s in log scale to find the optimal model.
+
+With Few labels. We investigate whether or not TUNIT is effective for the partially labeled dataset that corresponds to a more practical scenario. To this end, we use AnimalFaces-10 and Food-10. We partition the dataset $\mathcal { D }$ into the labeled set $\mathcal { D } _ { s u p }$ and the unlabeled set $\mathcal { D } _ { u n }$ with varying ratio $\gamma = \vert \mathcal { D } _ { s u p } \vert / \vert \mathcal { D } \vert$ . Like before, we choose FUNIT as a competitor. We first train the networks while changing $\gamma$ from 0.2 to 1.0 and report the result Table 3. As $\gamma$ decreases, the performance of FUNIT significantly degrades whereas our model maintains mFID around 45 and 55 on both datasets. We also train an auxiliary classifier (VGG-11BN) with $\mathcal { D } _ { s u p }$ then provide pseudo-labels for $\mathcal { D } _ { u n }$ to FUNIT. As shown in Table 4, the performance of FUNIT is no longer sensitive to the changes in $\gamma$ but still much worse than TUNIT. Although semi-supervised learning schemes can further improve FUNIT, TUNIT outperforms FUNIT using all labels as seen in Table 1. Under the empirical results in the semi-supervised scenario, we verify that TUNIT is also effective for semi-supervised setting.
+
+# 4 CONCLUSION
+
+We argue that the unsupervised image-to-image translation should not utilize any supervision, such as image-level (i.e. paired) or set-level (i.e. unpaired) supervision. Under this rigorous regime, many previous studies fall into the set-level supervised framework that uses the domain information at a minimum. In this paper, for the first time, we proposed an effective model to handle the truly unsupervised image-to-image translation. To this end, we suggested the guiding network that performs unsupervised representation learning for providing pseudo labels and the image translation tasks. The experimental results showed that guiding network indeed exploits the synergy between two tasks, and the proposed model successfully conducts the unsupervised-image-to-image translation. We also showed the generalizability on the value of $\hat { K }$ and the partially labeled scenario.
+
+# REFERENCES
+
+H. Bahng, S. Chung, S. Yoo, and J. Choo. Exploring unlabeled faces for novel attribute discovery. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5820– 5829, 2020.
+
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020a.
+
+Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020b.
+
+Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8789–8797, 2018.
+
+Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. Stargan v2: Diverse image synthesis for multiple domains. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2020.
+
+Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. In International Conference on Learning Representations, 2018.
+
+Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pp. 1735–1742. IEEE, 2006.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pp. 1026–1034, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020.
+
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in neural information processing systems, pp. 6626–6637, 2017.
+
+Geoffrey Hinton, Nitish Srivastava, and Kevin Swersky. Neural networks for machine learning lecture 6a overview of mini-batch gradient descent. 2012.
+
+R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. In International Conference on Learning Representations, 2019.
+
+Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. Cycada: Cycle-consistent adversarial domain adaptation. In International conference on machine learning, pp. 1989–1998, 2018.
+
+Xun Huang, Ming-Yu Liu, Serge Belongie, and Jan Kautz. Multimodal unsupervised image-toimage translation. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 172–189, 2018.
+
+Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1125–1134, 2017.
+
+Xu Ji, Joao F Henriques, and Andrea Vedaldi. Invariant information clustering for unsupervised˜ image classification and segmentation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 9865–9874, 2019.
+
+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of GANs for improved quality, stability, and variation. In International Conference on Learning Representations, 2018.
+
+Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4401–4410, 2019.
+
+Taeksoo Kim, Moonsu Cha, Hyunsoo Kim, Jung Kwon Lee, and Jiwon Kim. Learning to discover cross-domain relations with generative adversarial networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1857–1865. JMLR. org, 2017.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Orest Kupyn, Volodymyr Budzan, Mykola Mykhailych, Dmytro Mishkin, and Jiˇr´ı Matas. Deblurgan: Blind motion deblurring using conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8183–8192, 2018.
+
+Hsin-Ying Lee, Hung-Yu Tseng, Qi Mao, Jia-Bin Huang, Yu-Ding Lu, Maneesh Singh, and MingHsuan Yang. Dri $^ { + + }$ : Diverse image-to-image translation via disentangled representations. International Journal of Computer Vision, pp. 1–16, 2020.
+
+Jae Hyun Lim and Jong Chul Ye. Geometric gan. arXiv preprint arXiv:1705.02894, 2017.
+
+Min Lin, Qiang Chen, and Shuicheng Yan. Network in network. arXiv preprint arXiv:1312.4400, 2013.
+
+Ming-Yu Liu, Thomas Breuel, and Jan Kautz. Unsupervised image-to-image translation networks. In Advances in neural information processing systems, pp. 700–708, 2017.
+
+Ming-Yu Liu, Xun Huang, Arun Mallya, Tero Karras, Timo Aila, Jaakko Lehtinen, and Jan Kautz. Few-shot unsupervised image-to-image translation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 10551–10560, 2019.
+
+Lars Mescheder, Sebastian Nowozin, and Andreas Geiger. Which training methods for gans do actually converge? In ICML, 2018.
+
+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014.
+
+Muhammad Ferjad Naeem, Seong Joon Oh, Youngjung Uh, Yunjey Choi, and Jaejun Yoo. Reliable fidelity and diversity metrics for generative models. 2020.
+
+Taesung Park, Ming-Yu Liu, Ting-Chun Wang, and Jun-Yan Zhu. Semantic image synthesis with spatially-adaptive normalization. In CVPR, 2019.
+
+Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017.
+
+Nikunj Saunshi, Orestis Plevrakis, Sanjeev Arora, Mikhail Khodak, and Hrishikesh Khandeparkar. A theoretical analysis of contrastive unsupervised representation learning. In International Conference on Machine Learning, pp. 5628–5637, 2019.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In International Conference on Learning Representations, 2015.
+
+Saurabh Singh and Shankar Krishnan. Filter response normalization layer: Eliminating batch dependence in the training of deep neural networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11237–11246, 2020.
+
+Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. In Advances in neural information processing systems, pp. 3483–3491, 2015.
+
+Dustin Tran, Rajesh Ranganath, and David Blei. Hierarchical implicit models and likelihood-free variational inference. In Advances in Neural Information Processing Systems, pp. 5523–5533, 2017.
+
+Wouter Van Gansbeke, Simon Vandenhende, Stamatios Georgoulis, Marc Proesmans, and Luc Van Gool. Scan: Learning to classify images without labels. In European Conference on Computer Vision (ECCV), 2020.
+
+Ting-Chun Wang, Ming-Yu Liu, Jun-Yan Zhu, Andrew Tao, Jan Kautz, and Bryan Catanzaro. Highresolution image synthesis and semantic manipulation with conditional gans. In CVPR, 2018.
+
+I Zeki Yalniz, Herve J ´ egou, Kan Chen, Manohar Paluri, and Dhruv Mahajan. Billion-scale semi- ´ supervised learning for image classification. arXiv preprint arXiv:1905.00546, 2019.
+
+Dingdong Yang, Seunghoon Hong, Yunseok Jang, Tiangchen Zhao, and Honglak Lee. Diversitysensitive conditional generative adversarial networks. In International Conference on Learning Representations, 2019.
+
+Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015.
+
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223–2232, 2017a.
+
+Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward multimodal image-to-image translation. In Advances in neural information processing systems, pp. 465–476, 2017b.
+
+# APPENDIX
+
+# A RELATED WORK
+
+Image-to-image translation. Since the seminal work of Pix2Pix (Isola et al., 2017), image-to-image translation models have shown impressive results (Zhu et al., 2017a; Liu et al., 2017; Kim et al., 2017; Kupyn et al., 2018; Choi et al., 2018; Huang et al., 2018; Liu et al., 2019; Yang et al., 2019). Exploiting the cycle consistency constraint, these methods were able to train the model with a setlevel supervision (domains) solely. However, acquiring domain information can be a huge burden in practical applications where a large amount of data are gathered from several mixed domains, e.g., web images (Yalniz et al., 2019). Not only does this complicates the data collection, but it restricts the methods only applicable to the existing dataset and domains. Inspired from few shot learning, Liu et al. (2019) proposed FUNIT that works on previously unseen target classes. However, FUNIT still requires the labels for training. Recently, Bahng et al. (Bahng et al., 2020) has partially addressed this by adopting the ImageNet pre-trained classifier for extracting domain information. Unlike the previous methods, we aim to design an image-to-image translation model that can be applied without any supervision such as a pre-trained network or supervision on both the train and the test datasets.
+
+Unsupervised representation learning and clustering. Unsupervised representation learning aims to extract meaningful features for downstream tasks without any human supervision. To this end, many researchers have proposed to utilize the information that can be acquired from the data itself (Gidaris et al., 2018; Hjelm et al., 2019; Ji et al., 2019; He et al., 2020; Van Gansbeke et al., 2020). Recently, by incorporating the contrastive learning into a dictionary learning framework, MoCo (He et al., 2020) has achieved outstanding performance in various downstream tasks under reasonable mini-batch size. On the other hand, IIC (Ji et al., 2019) have utilized the mutual information maximization in a unsupervised way so that the network clusters images while assigning the images evenly. Though IIC provided a principled way to perform unsupervised clustering, it fails to scale up when combined with a difficult downstream task such as image-to-image translation. By taking the best of both worlds, we aim to solve unsupervised image-to-image translation.
+
+# B TRAINING DETAILS
+
+We train the guiding network for the first 65K iterations while freezing the update from both the generator and the discriminator. Then, we train the whole framework 100K more iterations for training all the networks. The batch size is set to 32 and 16 for $1 2 8 \times 1 2 8$ and $2 5 6 \times 2 5 6$ images, respectively. Training takes about 36 hours on a single Tesla V100 GPU with our implementation using PyTorch(Paszke et al., 2017). We use Adam (Kingma & Ba, 2014) optimizer with $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . { \dot { 9 } } 9$ for the guiding network, and RMSprop (Hinton et al., 2012) optimizer with $\alpha = 0 . 9 9$ for the generator and the discriminator. All learning rates are set to 0.0001 with a weight decay 0.0001. We adopt hinge version adversarial loss (Lim & Ye, 2017; Tran et al., 2017) with $R _ { 1 }$ regularization (Mescheder et al., 2018) using $\gamma = 1 0$ (Eq. 5). We set $\lambda _ { \mathrm { { r e c } } } = 0 . 1 , \lambda _ { \mathrm { { s t y l e } } } ^ { G } = 0 . 0 1 , \lambda _ { \mathrm { { s t y l e } } } ^ { E } = \bar { 1 }$ , and $\lambda _ { \mathrm { M I } } = 5$ in equation. 6 for all experiments. When the guiding network is simultaneously trained with the generator, we decrease $\lambda _ { \mathrm { s t y l e } } ^ { E ^ { \ast } }$ and $\lambda _ { \mathrm { M I } }$ to 0.1 and 0.5, respectively. For evaluation, we use the exponential moving average over the parameters (Karras et al., 2018) of the guiding network and the generator. We initialize the weights of convolution layers with He initialization (He et al., 2015), all biases to zero, and weights of linear layers from $N ( 0 , 0 . 0 1 )$ with zero biases. The source code will be available publicly.
+
+# C EVALUATION PROTOCOL
+
+For evaluation, we use class-wise Frechet Inception Distance (FID) (Heusel et al., 2017), which´ is often called mFID in literatures and D&C (Naeem et al., 2020). FID measures Frechet distance ´ between real and fake samples embedded by the last average pooling layer of Inception-V3 pretrained on ImageNet. Class-wise FID is obtained by averaging the FIDs of individual classes. In the experiments with fewer labels, we report the mean value of best five mFID’s over 100K iterations. For example, we use entire real images of each class and generate 810 fake images where $1 8 ~ \times$ $( K - 1 )$ source images $K = 1 0$ for AnimalFaces-10) and five reference images of AnimalFaces10 are used to produce those fake images. We choose the source images from all classes except for the target class. For each source image, the five references are selected arbitrarily. For D&C, we generate fake images the similar number of training images with randomly selected source and reference images. Then, we use Inception-V3 pre-trained on ImageNet for extracting feature vectors and measure D&C by using the feature vectors.
+
+# D ARCHITECTURE DETAILS
+
+For the guiding network, we use VGG11 before the linear layers followed by the average pooling operation as the shared part and append two branches $E _ { \mathrm { c l a s s } }$ and $E _ { \mathrm { s t y l e } }$ . The branches are one linear layer with $\hat { K }$ and 128 dimensional outputs, respectively. The detailed information of the generator, the guiding network and the discriminator architectures are provided in Table 5, Table 6 and Table 7.
+
+Table 5: Generator architecture. “ch” represents the channel multiplier that is set to 64. IN and AdaIN indicate instance normalization and adaptive instance normalization, respectively.   
+
+<table><tr><td>LAYER</td><td>RESAMPLE</td><td>NORM</td><td>OUTPUT SHAPE</td></tr><tr><td>Image x</td><td></td><td>1</td><td>128×128×3</td></tr><tr><td>Conv7×7</td><td></td><td>IN</td><td>128 ×128 × ch</td></tr><tr><td>Conv4×4</td><td>Stride 2</td><td>IN</td><td>64 × 64× 2ch</td></tr><tr><td>Conv4×4</td><td>Stride 2</td><td>IN</td><td>32 × 32×4ch</td></tr><tr><td>Conv4×4</td><td>Stride 2</td><td>IN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td></td><td>IN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td></td><td>IN</td><td>16 × 16 × 8ch</td></tr><tr><td>ResBlk</td><td></td><td>AdaIN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td></td><td>AdaIN</td><td>16 ×16× 8ch</td></tr><tr><td>Conv5×5</td><td>Upsample</td><td>AdaIN</td><td>32 × 32 × 4ch</td></tr><tr><td>Conv5×5</td><td>Upsample</td><td>AdaIN</td><td>64 ×64× 2ch</td></tr><tr><td>Conv5×5</td><td>Upsample</td><td>AdaIN</td><td>128 ×128 × ch</td></tr><tr><td>Conv7×7</td><td></td><td>1</td><td>128 ×128×3</td></tr></table>
+
+<table><tr><td>LAYER</td><td>RESAMPLE</td><td>NORM</td><td>OUTPUT SHAPE</td></tr><tr><td>Image X</td><td></td><td>■</td><td>128 ×128×3</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>64 × 64× ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>32 × 32× 2ch</td></tr><tr><td>Conv3×3</td><td>=</td><td>BN</td><td>32 × 32 ×4ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>16 ×16 ×4ch</td></tr><tr><td>Conv3×3</td><td>=</td><td>BN</td><td>16 ×16 ×8ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>8×8×8ch</td></tr><tr><td>Conv3×3</td><td>=</td><td>BN</td><td>8×8×8ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>4×4×8ch</td></tr><tr><td>GAP</td><td></td><td></td><td>1×1×8ch</td></tr><tr><td>FC</td><td></td><td></td><td>128</td></tr><tr><td>FC</td><td></td><td></td><td>K</td></tr></table>
+
+Table 6: Guiding network architecture. “ch” represents the channel multiplier that is set to 64. The architecture is based on VGG11-BN. GAP and FC denote global average polling (Lin et al., 2013) and fully connected layer, respectively.
+
+Table 7: Discriminator architecture. “ch” and $\hat { K }$ represent the channel multiplier that is set to 64 and the number of clusters, respectively. FRN indicates filter response normalization (Singh & Krishnan, 2020).   
+
+<table><tr><td>LAYER</td><td>RESAMPLE</td><td>NORM</td><td>OUTPUT SHAPE</td></tr><tr><td>Image x</td><td></td><td></td><td>128 × 128 × 3</td></tr><tr><td>Conv3×3</td><td></td><td>1</td><td>128 ×128×ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>128 × 128× ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>64×64×2ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>64 × 64 × 2ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>32 × 32 × 4ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>32 ×32×4ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>16 ×16× 8ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>8×8×16ch</td></tr><tr><td>ResBlk</td><td>=</td><td>FRN</td><td>8×8×16ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>4×4×16ch</td></tr><tr><td>LReLU</td><td></td><td>-</td><td>4×4×16ch</td></tr><tr><td>Conv4×4</td><td></td><td></td><td>1 ×1×16ch</td></tr><tr><td>LReLU</td><td></td><td></td><td>1×1×16ch</td></tr><tr><td>Conv1 ×1</td><td></td><td></td><td>K</td></tr></table>
+
+# E T-SNE VISUALIZATION & CLUSTER EXAMPLE IMAGES
+
+# E.1 AFHQ CAT
+
+![](images/fc348f4607da18baa172df674d4f49f61120f9d2e8826232de5da073fd1a8cf4.jpg)  
+Figure 7: t-SNE visualization and representative images of each domain.
+
+# E.2 AFHQ DOG
+
+![](images/2c33a719764bd4c2605c0f8b74bde3f74980d4147e314e1af19f6680f91ba8d1.jpg)  
+Figure 8: t-SNE visualization and representative images of each domain.
+
+# E.3 FFHQ
+
+![](images/8291e471dc5724d37120bc58abe1bff180aa37d5fe815b57b5511f6aa0b1f7c0.jpg)  
+Figure 9: t-SNE visualization and representative images of each domain.
+
+![](images/a326590eebf7a605dd8b25613cefa15e4c1fc8dc45a97bb723d090384a36f254.jpg)  
+Figure 10: t-SNE visualization and representative images of each domain.
+
+# F ADDITIONAL COMPARISON WITH FUNIT: AFHQ, LSUN CAR AND FFHQ
+
+![](images/07f155349578d6c592d3bda8fa9a778649e764bca7098c1ae8aa2e7d2b19cf3a.jpg)  
+Figure 11: AFHQ Cat, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. The content and the style are from the source and the reference, respectively. While FUNIT usually fails to reflect the style of the reference image, TUNIT generates the fake images with the style – color, fur texture.
+
+![](images/b3bace1042798af3e0fb5a2795170a9a5009ac7d71692429a07f1c6c18b72238.jpg)  
+Figure 12: AFHQ Wild, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. FUNIT rarely reflects the correct style of the reference image – the species, on the other hand, TUNIT translates the source image to the correct species.
+
+![](images/e084a58d925cf91bb1fa567c3d631805ab09fb971ca71f7075df97d44e937ebb.jpg)  
+Figure 13: LSUN Car, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. While TUNIT generates plausible and changes the color of the source image to that of the reference image, FUNIT not also generates unrealistic image but also fails to changes the color.
+
+![](images/5874faa938abe04badb873854ec73b30399a316537234f2f48efb68402327600.jpg)  
+Figure 14: FFHQ, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. Our model, TUNIT can remove or add the glasses to the source while preserving the identity better than FUNIT. In addition, TUNIT can change the hair color (last column) and the hair style – especially, bang (fifth column). It is hard to specify the definition of domains in the results of FUNIT while domains of TUNIT are more interpretable.
+
+# G ADDITIONAL RESULTS OF TUNIT: AFHQ, LSUN CAR, FFHQ, ANIMALFACES-10, AND S2W
+
+# G.1 ANIMALFACES-10
+
+![](images/88b0b3eeee8ffcbc01a5759b9788c45391fc79eb37935d71124a22ebc32b48c2.jpg)  
+Figure 15: AnimalFaces-10, unsupervised image-to-image translation results.
+
+# G.2 AFHQ CAT
+
+![](images/9cb4d71aa0710ef8dd9714741e5f8d79c1b0fac3feaaa2ccacb5815b7301cc21.jpg)  
+Figure 16: AFHQ Cat, unsupervised image-to-image translation results.
+
+# G.3 AFHQ DOG
+
+![](images/6b8adf8c098cb154222098a401cdbfa102bd9031f81841dad4da280e3f7123d4.jpg)  
+Figure 17: AFHQ Dogs, unsupervised image-to-image translation results.
+
+![](images/80b9f2e057c5c37a520f5b27b98152ce046e400208859aae223622c8eeb5e271.jpg)  
+Figure 18: AFHQ Wild, unsupervised image-to-image translation results.
+
+# G.5 FFHQ
+
+![](images/e9b3985e1717ce957b749fafa8203b287216137b9fb976fb00cb7af88299df7f.jpg)  
+Figure 19: FFHQ, unsupervised image-to-image translation results.
+
+![](images/532ae5574b3192ebc35d51df8e9ee4181978770b03f90bdb42b7823829f2993b.jpg)  
+Figure 20: LSUN Car, unsupervised image-to-image translation results.
+
+# G.7 SUMMER2WINTER (S2W)
+
+![](images/ddaa5e2107475c064cc2c585a0b55ec9428d95f68325596665efa96acfd37489.jpg)  
+(b) Results guided by reference images   
+Figure 21: Summer2Winter (S2W), unsupervised image-to-image translation results.
+
+# H DIFFERENCE BETWEEN EQUATION (2) AND EQUATION (4)
+
+Equation (2) and (4) have similar forms – contrastive loss, but they are used for different purposes. We use equation (2) to improve the representation power of the guiding network, which affects the performance of the generator and the discriminator. On the other hand, equation (4) is used to enforce the generator to reflect the style of a reference image when translating a source image. To examine the effect of each loss, we train models without either equation (2) or (4) on AnimalFaces10. The mFID score without equation (2) or (4) is 86.8 and 93.3, respectively. Both models are significantly worse than the original setting (47.7). It means that both equation (2) and (4) should be considered during training. In addition to the purpose, they are different in terms of the way to choose positive pairs. We use a real image and its randomly augmented version as a positive pair in equation (2) while we use the translated image and reference image as a positive pair. In summary, the role of equation (2) is to enhance the representation power of the guiding network and lead the guiding network to learn how to encode the style vector in terms of a style encoder while the role of equation (4) is to guide the generator to learn how to interpret the provided style vector as a form of the output image.
+
+# I FID AND LPIPS ON UNLABELED DATASET
+
+![](images/7a8c6b6e2634ff1d75b3020ef9c1db65c6f1bf71cd84e0c87f1e28de3dba5251.jpg)  
+Figure 22: LPIPS and FID of models and their status.
+
+We also utilize LPIPS to evaluate the models in addition to FID and D&C. However, LPIPS is not proper to evaluate the loyalty for reflecting the reference image and the fidelity of images, we use LPIPS with FID. Figure 22 shows the result. It is clear that a model with high FID and LPIPS generates a noise-like image. Even if FID is low, a model with high LPIPS also fails to conduct the reference-guided image translation, because it does not preserve the structure of the source image. The model with low LPIPS and high FID might be an adversarial example of LPIPS. We generate the image via optimization on LPIPS. If a model exhibits low FID and LPIPS, it might not reflect the visual feature of the reference image enough. The simple combination of LPIPS and FID can detect several failed models but can not evaluate the loyalty for the reference image. We suggest that the rigorous way to combine several metrics for the quantitative evaluation of the reference-guided translation might be a interesting future work.

parse/train/GvqjmSwUxkY/GvqjmSwUxkY_content_list.json

@@ -0,0 +1,1945 @@
+[
+    {
+        "type": "text",
+        "text": "RETHINKING THE TRULY UNSUPERVISED IMAGE-TOIMAGE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Every recent image-to-image translation model uses either image-level (i.e. inputoutput pairs) or set-level (i.e. domain labels) supervision at a minimum. However, even the set-level supervision can be a serious bottleneck for data collection in practice. In this paper, we tackle image-to-image translation in a fully unsupervised setting, i.e., neither paired images nor domain labels. To this end, we propose a truly unsupervised image-to-image translation model (TUNIT) that simultaneously learns to separate image domains and translate input images into the estimated domains. Experimental results show that our model achieves comparable or even better performance than the set-level supervised model trained with full labels, generalizes well on various datasets, and is robust against the choice of hyperparameters (e.g. the preset number of pseudo domains). In addition, TUNIT extends well to the semi-supervised scenario with various amount of labels provided. ",
+        "bbox": [
+            233,
+            263,
+            764,
+            444
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            468,
+            336,
+            484
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given an image of one domain, image-to-image translation is a task to generate the plausible images of the other domains. Based on the success of conditional generative models (Mirza & Osindero, 2014; Sohn et al., 2015), many image translation methods have been proposed either using imagelevel supervision (e.g. paired data) (Isola et al., 2017; Hoffman et al., 2018; Zhu et al., 2017b; Wang et al., 2018; Park et al., 2019) or using set-level supervision (e.g. domain labels) (Zhu et al., 2017a; Kim et al., 2017; Liu et al., 2017; Huang et al., 2018; Liu et al., 2019; Lee et al., 2020). Though the latter approach is generally called unsupervised as a counterpart of the former, it actually assumes that the domain labels are given a priori. This assumption can be a serious bottleneck in practice as the number of domains and samples increases. For example, labeling individual samples of a large dataset, such as FFHQ, is expensive, and the distinction across domains can be ambiguous. ",
+        "bbox": [
+            174,
+            498,
+            825,
+            637
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Here, we first clarify that unsupervised image-to-image translation should strictly denote the task without any supervision neither paired images nor domain labels. Under this definition, our goal is to develop an unsupervised translation model given a mixed set of images of many domains (Figure 1). We tackle this problem by formulating three sub-problems: 1) clustering the images by approximating the set-level characteristics (i.e. domains), 2) encoding the individual content and style of an input image, and 3) learning a mapping function among the estimated domains. ",
+        "bbox": [
+            174,
+            645,
+            823,
+            728
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To this end, we introduce a guiding network that simultaneously solves 1) unsupervised domain classification and 2) style encoding. It has two branches of providing pseudo domain labels and encoding style features, which are later used in the discriminator and the generator training, respectively. We employ a differentiable clustering method based on mutual information maximization for estimating domain labels. This helps the guiding network group similar images together while evenly separate their categories. For embedding style codes, we adopt a contrastive loss (Hadsell et al., 2006; He et al., 2020; Chen et al., 2020a), which leads the model to further understand the dissimilarity between images, resulting in better representation learning. Finally, conditioned on the style features and domain labels from the guiding network, we use generative adversarial networks (GAN) to learn the image translation functions across various domains. ",
+        "bbox": [
+            174,
+            736,
+            825,
+            875
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although GAN and the guiding network play different roles, we do not separate their training process– our guiding network participates in the translation process. By doing so, the guiding network can exploit gradients from GAN training. The guiding network now understands the recipes of domain-separating attributes because the generator wants the style code to contain sufficient information to fool the domain-specific discriminator, and vice versa. Thanks to this interaction between the guiding network and GAN, our model successfully separates domains and translates images. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/726797f2ef17218ef12f9d9ccbbc5c6eb69b4f4ab8fe6e7c14af7f9a836ef628.jpg",
+        "image_caption": [
+            "Figure 1: Levels of supervision. To perform image-to-image translation, existing methods need either (a) a dataset with input-output pairs or, (b) a dataset with domain information. Our method is capable of learning mappings among multiple domains using (c) a dataset without any supervision. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            102,
+            756,
+            273
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            340,
+            821,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We quantitatively and qualitatively compare our model with the existing set-level supervised method under unsupervised and semi-supervised setting. The experiments on various datasets show that the proposed model outperforms the previous method over all different levels of supervision. Our experimental results show that, by exploiting the synergy between two tasks, the guiding network helps the image translation model to largely improve the generation performance. ",
+        "bbox": [
+            174,
+            390,
+            825,
+            459
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are summarized as follows: ",
+        "bbox": [
+            176,
+            465,
+            472,
+            479
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We clarify the definition of unsupervised image-to-image translation and to the best of our knowledge, our model is the first to succeed in this task in an end-to-end manner.   \n• We propose the guiding network to handle the unsupervised translation task and show that the interaction between translation and clustering is helpful for the task.   \n• We show the effectiveness of our model through the extensive experiments on various datasets.   \n• We confirm that our model is applicable to various numbers of clusters and the practical case, where ground truth labels of several samples are available. ",
+        "bbox": [
+            173,
+            492,
+            825,
+            608
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 TRULY UNSUPERVISED IMAGE-TO-IMAGE TRANSLATION (TUNIT) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            628,
+            764,
+            645
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider the unsupervised image-to-image translation problem, where we have images $\\chi$ from $K$ domains $K \\geq 2 )$ ) without domain labels $y$ . Here, $K$ is an unknown property of the dataset. Throughout the paper, we denote $K$ as the actual number of domains in a dataset and $\\hat { K }$ as the arbitrarily chosen number of domains to train models. We design a module that integrates both a domain classifier and a style encoder, which we call guiding network. It guides the translation by feeding reference images as the style code to the generator and as the pseudo domain labels to the discriminator. Using the feedback from the discriminator regarding the pseudo labels, the generator synthesizes images of the target domains (e.g. breeds) while respecting styles (e.g. fur patterns) of the reference images and maintaining the content (e.g. pose) of source images (Figure 2). ",
+        "bbox": [
+            173,
+            660,
+            825,
+            789
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 LEARNING TO PRODUCE DOMAIN LABELS AND ENCODE STYLE FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            806,
+            722,
+            820
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In our framework, the guiding network $E$ plays a central role as an unsupervised domain classifier as well as a style encoder. Our guiding network $E$ consists of two branches, $E _ { C }$ and $E _ { S }$ , each of which learns to provide domain labels and style codes, respectively. In experiments, we compare our guiding network against straightforward approaches, i.e.., K-means on image or feature space. ",
+        "bbox": [
+            174,
+            832,
+            825,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised domain classification. The discriminator requires target domain labels to provide useful gradients for image translation into the target domain. $E _ { C }$ adopts a differentiable clustering technique to provide pseudo domain labels of reference images, maximizing the mutual information (MI) between an image $\\mathbf { x }$ and its randomly augmented version $\\mathbf { x } ^ { + }$ (Ji et al., 2019). The optimum of the mutual information $I ( \\mathbf { p } , \\mathbf { p } ^ { + } )$ is reached as the entropy $H ( \\mathbf { p } )$ is maximum and the conditional entropy $H ( \\mathbf { p } | \\mathbf { p } ^ { + } )$ is minimum, where $\\mathbf { p } = E _ { C } ( \\mathbf { x } )$ represents the softmax output from $E _ { C }$ , indicating a probability vector of $\\mathbf { x }$ over $\\cdot$ domains. Please refer to Section 3.3 for more details about $\\cdot$ . Maximizing MI encourages $E _ { C }$ to assign the same domain label to the pair $\\mathbf { \\bar { x } }$ and $\\mathbf { x } ^ { + }$ ) while evenly distributing entire samples to all domains. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/d06a3a92ca3377b8950e5dca64ab7215c6897ba6bc71cdd37600a52b869f55a1.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our proposed method. The figure illustrates how our model changes the breed of the cat. (a) An estimated domain from our guiding network $E$ is used to train the multi-task discriminator $D$ . (b) $E$ provides the generator $G$ with the style code of a reference image and the estimated domain is again used for GAN training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            228,
+            98,
+            774,
+            306
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            387,
+            825,
+            488
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, $E _ { C }$ maximizes the mutual information: ",
+        "bbox": [
+            174,
+            494,
+            500,
+            510
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a302c3d2d0e4bbaee34fb6d7cba9a8921d6e58525c88b090d3f17eea148e2535.jpg",
+        "text": "$$\n\\mathcal { L } _ { M I } = I ( \\mathbf { p } , \\mathbf { p } ^ { + } ) = I ( \\mathbf { P } ) = \\sum _ { i = 1 } ^ { \\hat { K } } \\sum _ { j = 1 } ^ { \\hat { K } } \\mathbf { P } _ { i j } \\ln \\frac { \\mathbf { P } _ { i j } } { \\mathbf { P } _ { i } \\mathbf { P } _ { j } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            517,
+            679,
+            564
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f$ is a composition of random augmentations such as random cropping and affine transformation. $\\mathbf { P } _ { i } = \\mathbf { P } ( \\mathbf { p } = i )$ denotes the $\\hat { K }$ -dimensional marginal probability vector, and $\\mathbf { P } _ { i j } = \\mathbf { P } ( \\mathbf { p } =$ $i , \\mathbf { p } ^ { + } = j$ ) denotes the joint probability. To provide a deterministic one-hot label to the discriminator, we use the argmax operation (i.e. $y = \\tt a r g m a x ( { E _ { C } ( x ) } ) )$ . We note that the mutual information is one way to implement TUNIT, therefore, any differentiable clustering methods can be adopted such as SCAN (Van Gansbeke et al., 2020). ",
+        "bbox": [
+            173,
+            592,
+            826,
+            679
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Style encoding and improving domain classification. $E _ { S }$ encodes an image into a style code s which provides translation guide for the generator. In addition to the style guide to the generator, $E _ { S }$ is beneficial in improving unsupervised domain classification, where pseudo labels from $E _ { C }$ fail to scale up when samples are complex and diverse (e.g., AnimalFaces (Liu et al., 2019)). Since $E _ { S }$ is an another branch of the guiding network, imposing the contrastive loss (He et al., 2020) on the style codes improves representation of the shared embeddings: ",
+        "bbox": [
+            173,
+            685,
+            825,
+            770
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0afc1df22c52bf5a28429a81cb504bf03448707a73925e7f91d265d786d4cf75.jpg",
+        "text": "$$\n\\mathcal { L } _ { s t y l e } ^ { E } = - \\log \\frac { \\exp ( \\mathbf { s } \\cdot \\mathbf { s } ^ { + } / \\tau ) } { \\sum _ { i = 0 } ^ { N } \\exp ( \\mathbf { s } \\cdot \\mathbf { s } _ { i } ^ { - } / \\tau ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            776,
+            622,
+            815
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\bf s } = E _ { S } ( { \\bf x } )$ . $\\mathbf { x }$ and $\\mathbf { x } ^ { + }$ denote an image and randomly augmented version of $\\mathbf { x }$ , respectively. This $( N + 1 )$ -way classification enables $E$ to utilize not only the similarity of the positive pair (s, $\\mathbf { s } ^ { + }$ ) but also the dissimilarity of the negative pairs (s, $\\mathbf { s } _ { i } ^ { - }$ ). We adopt a queue to store the negative codes $\\cdot$ of the previously sampled images as MoCo (He et al., 2020). By doing so, we can conduct the contrastive learning efficiently without large batch sizes (Saunshi et al., 2019). We observe that adding this objective significantly improves unsupervised classification accuracy on AnimalFaces from $6 8 . 0 \\%$ to $8 4 . 1 \\%$ compared to the previous approach (Ji et al., 2019). ",
+        "bbox": [
+            173,
+            823,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we describe how to perform the unsupervised image-to-image translation under the guidance of our guiding network. For successful translation, the model should provide the realistic images containing the visual feature of the target domain. To this end, we adopt three losses: 1) adversarial loss to produce realistic images, 2) style contrastive loss that encourages the model not to ignore the style codes, 3) image reconstruction loss for preserving the domain-invariant features. We explain each loss and the overall objective for each network. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            213
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Adversarial loss. For adversarial training, we adopt a variant of conditional discriminator, the multitask discriminator (Mescheder et al., 2018). It is designed to conduct discrimination for each domain simultaneously. However, its gradient is calculated only with the loss for estimating the domain of the input image. For the domain label of the input image, we utilize the pseudo label from the guiding network. Formally, given the pseudo label $\\tilde { y }$ for a reference image $\\tilde { \\mathbf { x } }$ , we train our generator $G$ and multi-task discriminator $D$ via the adversarial loss: ",
+        "bbox": [
+            173,
+            219,
+            825,
+            304
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9e8a335f2af1c5ef994f3a68c46f46e95a63f13fd41f9013992ab424da092e2e.jpg",
+        "text": "$$\n\\mathcal { L } _ { a d v } = \\mathbb { E } _ { \\tilde { \\mathbf { x } } \\sim p _ { d a t a } ( \\mathbf { x } ) } [ \\log D _ { \\tilde { y } } ( \\tilde { \\mathbf { x } } ) ] + \\mathbb { E } _ { \\mathbf { x } , \\tilde { \\mathbf { x } } \\sim p _ { d a t a } ( \\mathbf { x } ) } [ \\log ( 1 - D _ { \\tilde { y } } ( G ( \\mathbf { x } , \\tilde { \\mathbf { s } } ) ) ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            309,
+            740,
+            328
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { \\tilde { y } } ( \\cdot )$ denotes the logit from the domain-specific $( \\tilde { y } )$ discriminator, and $\\tilde { \\mathbf { s } } = E _ { S } ( \\tilde { \\mathbf { x } } )$ denotes a target style code of the reference image $\\tilde { \\bf x }$ . The generator $G$ learns to translate $\\mathbf { x }$ to the target domain $\\tilde { y }$ while reflecting the style code ˜s. ",
+        "bbox": [
+            174,
+            333,
+            825,
+            376
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Style constrastive loss. In order to prevent a degenerate case where the generator ignores the given style code ˜s and synthesize a random image of the domain $\\tilde { y }$ , we impose a style contrastive loss: ",
+        "bbox": [
+            174,
+            382,
+            823,
+            411
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb7f827a7eaa0d135182e08b195352c9e4a569aeab20afddcdee90134e2ae989.jpg",
+        "text": "$$\n\\mathcal { L } _ { s t y l e } ^ { G } = \\mathbb { E } _ { \\mathbf { x } , \\tilde { \\mathbf { x } } \\sim p _ { d a t a } ( \\mathbf { x } ) } \\left[ - \\log \\frac { \\exp ( \\mathbf { s } ^ { \\prime } \\cdot \\tilde { \\mathbf { s } } ) } { \\sum _ { i = 0 } ^ { N } \\exp ( \\mathbf { s } ^ { \\prime } \\cdot \\mathbf { s } _ { i } ^ { - } / \\tau ) } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            416,
+            681,
+            458
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\mathbf { s } ^ { \\prime } = E _ { S } ( G ( \\mathbf { x } , \\tilde { \\mathbf { s } } ) )$ denotes the style code of the translated image $G ( \\mathbf { x } , \\tilde { \\mathbf { s } } )$ and $\\mathbf { s } _ { i } ^ { - }$ denotes the negative style codes, which are from the same queue used in equation (2). And we follow the training scheme of MoCo (He et al., 2020) as equation (2). The above loss guides the generated image $\\cdot$ to have a style similar to the reference image $\\cdot$ and dissimilar to negative (other) samples. By doing so, we also avoid the degenerated solution where the encoder maps all the images to the same style code of the reconstruction loss (Choi et al., 2020) based on L1 or L2 norm. Equation (2) and (4) are based on contrastive loss, but they are used for different purposes. Please refer to Appendix H for more discussion. ",
+        "bbox": [
+            173,
+            464,
+            825,
+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Image reconstruction loss. To ensure that the generator $G$ can reconstruct the source image $\\mathbf { x }$ when given with its original style ${ \\bf s } = E _ { S } ( { \\bf x } )$ , we impose an image reconstruction loss: ",
+        "bbox": [
+            173,
+            583,
+            820,
+            612
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f5ee4293712cdea444597302fde558508dd7acffc8db7529f75d4248a6066cde.jpg",
+        "text": "$$\n\\mathcal { L } _ { r e c } = \\mathbb { E } _ { { \\mathbf { x } } \\sim p _ { d a t a } ( { \\mathbf { x } } ) } [ | | { \\mathbf { x } } - G ( { \\mathbf { x } } , { \\mathbf { s } } ) | | _ { 1 } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            617,
+            625,
+            636
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This objective not only ensures the generator $G$ to preserve domain-invariant characteristics (e.g., pose) of its input image $\\mathbf { x }$ , but also helps to learn the style representation of the guiding network $E$ by extracting the original style s of the source image $\\mathbf { x }$ . ",
+        "bbox": [
+            174,
+            640,
+            825,
+            683
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Overall objective. Finally, we train the three networks jointly as follows: ",
+        "bbox": [
+            173,
+            689,
+            655,
+            704
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b9ee62ec37c6f4a7c956eb2e1fd9c61439662c07dc38bd0ed3e03825a4c6b5b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { D } = - \\mathcal { L } _ { a d v } , } \\\\ & { \\mathcal { L } _ { G } = \\mathcal { L } _ { a d v } + \\lambda _ { s t y l e } ^ { G } \\mathcal { L } _ { s t y l e } ^ { G } + \\lambda _ { r e c } \\mathcal { L } _ { r e c } , } \\\\ & { \\mathcal { L } _ { E } = \\mathcal { L } _ { G } - \\lambda _ { M I } \\mathcal { L } _ { M I } + \\lambda _ { s t y l e } ^ { E } \\mathcal { L } _ { s t y l e } ^ { E } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            708,
+            629,
+            771
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda$ ’s are hyperparameters. Note that our guiding network $E$ receives feedback from $L _ { G }$ , which is essential for our method. We discuss the effect of feedback to $E$ on performance in Section 3.1. ",
+        "bbox": [
+            173,
+            773,
+            826,
+            803
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            821,
+            326,
+            838
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We first evaluate TUNIT on labeled datasets by treating them as unlabeled because the desired behaviours of the translation models in labeled datasets are well defined (Section 3.1). Here, we provide an ablation study to analyze the effect of each component and compare the models both quantitatively and qualitatively. We then move on to unlabeled datasets to validate our model in the unsupervised scenario in the wild (Section 3.2). Lastly, we show that TUNIT is robust against the choice ",
+        "bbox": [
+            174,
+            853,
+            823,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/37862f5c879a6da80ccca8c6de523a0f2c7b9cb38b2c03ae9f05dd511cdb6493.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Configuration</td><td colspan=\"3\">AnimalFaces-10</td><td colspan=\"3\">Food-10</td></tr><tr><td>mFID</td><td>D&amp;C</td><td>Acc.</td><td>mFID</td><td>D&amp;C</td><td>Acc.</td></tr><tr><td>A Baseline FUNIT (supervised)</td><td>74.0</td><td>0.749 / 0.671</td><td>1.000</td><td>68.4</td><td>0.989 / 0.782</td><td>1.000</td></tr><tr><td>B (A)+Improved G&amp;D (supervised)</td><td>46.2</td><td>0.896 / 0.732</td><td>1.000</td><td>57.6</td><td>1.284 / 0.857</td><td>1.000</td></tr><tr><td>C (B)+ K-means on image space</td><td>110.7</td><td>0.822 / 0.615</td><td>0.215</td><td>90.7</td><td>0.849 / 0.648</td><td>0.201</td></tr><tr><td>D (B) + K-means on feature space</td><td>76.2</td><td>0.770 / 0.597</td><td>0.428</td><td>64.6</td><td>0.968 / 0.808</td><td>0.331</td></tr><tr><td>E (B) + Differentiable clustering</td><td>73.5</td><td>0.940 / 0.588</td><td>0.680</td><td>64.2</td><td>1.038 / 0.819</td><td>0.542</td></tr><tr><td>F TUNIT w/ sequential training</td><td>46.0</td><td>1.060 / 0.789</td><td>0.850</td><td>61.1</td><td>0.908 / 0.777</td><td>0.860</td></tr><tr><td>G TUNIT w/ joint training</td><td>47.7</td><td>1.039 / 0.805</td><td>0.841</td><td>52.2</td><td>1.079 / 0.875</td><td>0.848</td></tr></table>",
+        "bbox": [
+            174,
+            101,
+            825,
+            233
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Main results. mFID, Density / Coverage (D & C), and classification accuracy (Acc) of each training configuration. Note that the configurations (A) - (B) use ground-truth class labels, while (C) - (G) use pseudo-labels. We bold the best results separately for supervised and unsupervised settings. ",
+        "bbox": [
+            174,
+            243,
+            825,
+            286
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/88571bb423f86a5b771181cfa37f3796a8c5c8b5c7ffc1981e9edec274f61969.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative comparison of translation results using each configuration in Table 1. Here, B reflects the style feature (e.g. species or type of food) of the reference images while A does not. The model C performs much worse than A and B in that it overly adopts the source image, not adequately merging styles and contents from both sides. The model D generates more plausible images than C but fails to reflect the characteristics of the reference images. For example, D on fifth row does not look like several pieces of dumpling due to its shape and dish color, meaning that the reference styles are not properly reflected. Similarly, E also fails to generate the dumpling in the fifth row. TUNIT with sequential training F reflects the visual features of each reference on both datasets. However, in terms of visual fidelity, we observe that G consistently outperforms F. Akin to the quantitative results, TUNIT achieves equivalent or even better visual quality than the set-level supervised model A and B. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            296,
+            802,
+            612
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "of hyperparameters (e.g. the preset number of clusters, $K .$ ) and extends well to the semi-supervised scenario (Section 3.3). In all experiments, we use FUNIT (Liu et al., 2019) as our baseline. ",
+        "bbox": [
+            173,
+            791,
+            823,
+            819
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. For the labeled datasets, we randomly select ten classes among 149 classes of AnimalFaces and 101 classes of Food-101, which we call AnimalFaces-10 and Food-10, respectively. Here, the labels are used only for the evaluation purpose. For the unlabeled datasets, we use AFHQ, FFHQ, and LSUN Car (Choi et al., 2020; Karras et al., 2019; Yu et al., 2015), which do not have any or are missing with fine-grained labels. Specifically, AFHQ roughly has three groups (i.e., dog, cat and wild), but each group contains diverse species and these species labels are not provided. FFHQ and LSUN Car contain various human faces and cars without any labels, respectively. ",
+        "bbox": [
+            174,
+            827,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/55a9488623cb69c72d4de1bca06515d3d37bc916780476c730bb5325666af15c.jpg",
+        "image_caption": [
+            "Figure 4: Reference-guided image translation results on unlabeled datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            98,
+            789,
+            417
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/214c164faf6bdbca092f21b1ca1527fbc7947db60f659fec53878ba855201c0d.jpg",
+        "image_caption": [
+            "Figure 5: t-SNE visualization of the style space of our guiding network trained on AFHQ Wild. Since AFHQ Wild does not have ground-truth labels, each point is colored with the guiding network’s prediction. Although we set the number of domains to be quite large $\\hat { K } = 1 0 $ ), the network separates one species into two domains, which are so closely located that the model creates six clusters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            449,
+            815,
+            637
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics. We report two scores to assess the generated images. First, to provide a general sense of image quality, we use the mean of class-wise Frenchet Inception Distance (mFID) (Heusel ´ et al., 2017). It can avoid the degenerate case of the original FID, which assigns a good score when the model conveying the source image as is. Additionally, to provide a finer assessment of the generated images, we report Density and Coverage (D&C) (Naeem et al., 2020). D&C separately evaluates the fidelity and the diversity of the model outputs, which is also known to be robust against outliers and model hyperparameters (e.g. the number of samples used for evaluation). Denote that a lower mFID score means better image quality, and D&C scores that are bigger or closer to 1.0 indicate the better fidelity and diversity, respectively. Please refer to Appendix C for the detailed information. ",
+        "bbox": [
+            173,
+            722,
+            825,
+            847
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 COMPARATIVE EVALUATION ON LABELED DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            868,
+            583,
+            882
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the effect of each component of TUNIT and rigorous comparisons with the state-of-the-art supervised method, FUNIT. First, we report the set-level supervised performance ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/75d51c13a67bf1ecc74547dc5c8277f2332d6b5792826479497837a54811f2b0.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            114,
+            455,
+            257
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/a6b6842c2fbddf85138b432d74facb68f0220274d008e836b48df02ed9a8fd53.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>K</td><td>AnimalFaces-10 mFID D&amp;C</td><td>Food-10 mFID</td><td>D&amp;C</td></tr><tr><td>1 4 7 10 13 16 20 50 500</td><td>129.6 0.561 /0.512 77.7 0.879 /0.738 62.7 1.016 /0.729 47.7 1.039 /0.805 56.8 0.993/ 0.805 54.1 1.093 / 0.782 55.4 1.019 / 0.778 63.8 0.858 /0.701 67.2</td><td>95.1 67.4 52.7 52.2 54.8 54.8 57.7 60.8 0.921/ 0.694 63.2</td><td>1.113 / 0.771 0.851/0.785 1.079 /0.875 1.079 /0.875 0.970/0.845 1.029 /0.857 0.937 /0.846 1.067 /0.837</td></tr></table>",
+        "bbox": [
+            470,
+            101,
+            813,
+            270
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: t-SNE visualization of the model with (a) $\\scriptstyle { \\hat { K } } = 1 0$ and (b) $\\scriptstyle { \\hat { K } } = 2 0$ trained on AnimalFaces-10 and quantitative evaluation of our method by varying the number of pseudo domains $\\cdot$ . Each point is colored with the ground-truth labels. As shown in t-SNE visualizations, even if $\\cdot$ is set to overly larger than the actual number of domains, the guiding network clusters the domains reasonably well. ",
+        "bbox": [
+            174,
+            281,
+            825,
+            342
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/12399874c7d44a8da845f303377a9173ddc43c29caa472ab0b6d48283c524b3f.jpg",
+        "image_caption": [
+            "Figure 6: Qualitative comparison on the number of pseudo domains $\\cdot$ . The performance varies along with $\\cdot$ . When we set $\\hat { K }$ large enough, the results are reasonable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            347,
+            800,
+            568
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "of FUNIT and its variant (Table 1). Here, A is the original FUNIT and B denotes the modified FUNIT using our architecture (e.g. We do not use PatchGAN discriminator), which brings a large improvement over every score on both datasets. One simple way to extend B to the unsupervised scenario is to add an off-the-shelf clustering method and use its estimated labels instead of the ground truth. We employ K-means clustering on the image space for C, and the pretrained feature space for D. Here, we use ResNet-50 (He et al., 2016) features trained with MoCo v2 (Chen et al., 2020b) on ImageNet. Not surprisingly, because the estimated labels are inaccurate, the overall performance significantly drops. Although using the pretrained features helps a little, not only is it far from the setlevel supervised performance but it requires three steps to train the entire model, which complicates the application. This can be partially addressed by employing the differentiable clustering method (Ji et al., 2019), which trains VGG-11BN (Simonyan & Zisserman, 2015) with mutual information maximization that makes E. This reduces the number of training steps from three to two and provides better label estimation, which enables the model to approach the performance of original FUNIT A. However, as seen in the coverage score, the sample diversity is unsatisfactory. ",
+        "bbox": [
+            173,
+            611,
+            825,
+            805
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we build TUNIT by introducing the guiding network and the new objective functions described in Section 2. The changes significantly improve the accuracy on both datasets, particularly achieving similar mFID of the improved set-level supervised model B. Our final model, G matches or outperforms mFID and D&C of B. This is impressive because B utilizes oracles for training while G has no labels. Notably, TUNIT can cover $7 \\% \\mathrm { p }$ wider support of the data on AnimalFaces-10 than B. We conjecture that TUNIT benefits from the style codes that represent meaningful domain features learned by clustering. By comparing $\\mathbb { E }$ and G, we confirm that they are comparable in terms of clustering and G is more stable in terms of inter-dataset performance. Therefore, we adopt the joint training of style encoder and clustering as our final model (G). In addition, we remove the adversarial loss for training the guiding network. It directly degrades the performance; mFID changes from 47.7 to 63.0 on AnimalFaces-10. It indicates that our training scheme takes an important portion of performance gains. Qualitative results also show superiority of TUNIT over competitors (Figure 3). ",
+        "bbox": [
+            173,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/90adc3545bfc1e17aa7100b3835533627951c16e27c71fb96e65a4995fef933e.jpg",
+        "table_caption": [
+            "Table 3: Quantitative evaluation (mFID) when few labels are available during training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Configuration</td><td rowspan=\"2\">20%</td><td colspan=\"3\">AnimalFaces-10</td><td rowspan=\"2\"></td><td colspan=\"3\">Food-10</td></tr><tr><td>40%</td><td>60%</td><td>80%</td><td>20%</td><td>40% 60%</td><td>80%</td></tr><tr><td>A FUNIT</td><td></td><td>124.4</td><td>106.4</td><td>96.0</td><td>79.6</td><td>111.4</td><td>85.8</td><td>74.8</td><td>70.3</td></tr><tr><td>G</td><td>TUNIT (ours)</td><td>42.0</td><td>42.6</td><td>43.9</td><td>46.2</td><td>53.6</td><td>56.2</td><td>52.8</td><td>53.4</td></tr></table>",
+        "bbox": [
+            176,
+            101,
+            821,
+            160
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/e878e3b70cd4a419d94c4351a51d9775856052f2fb2a48e9391e7d55a95d8c87.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Quantitative evaluation (mFID) when few labels are available during training. Here, an auxliary classifier is adopted to improve the FUNIT baseline by giving pseudo-labels to $\\mathcal { D } _ { u n }$ . "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Configuration</td><td colspan=\"4\">AnimalFaces-10</td><td colspan=\"4\">Food-10</td></tr><tr><td>1%</td><td>2%</td><td>4%</td><td>8%</td><td>1%</td><td>2%</td><td>4%</td><td>8%</td></tr><tr><td>A FUNIT</td><td>107.8</td><td>104.7</td><td>90.3</td><td>93.9</td><td>71.9</td><td>71.5</td><td>71.6</td><td>69.0</td></tr><tr><td>G TUNIT (ours)</td><td>47.9</td><td>44.8</td><td>42.7</td><td>42.4</td><td>54.5</td><td>56.1</td><td>55.9</td><td>55.8</td></tr></table>",
+        "bbox": [
+            178,
+            181,
+            823,
+            242
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            280,
+            825,
+            337
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.2 VALIDATION ON UNLABELED DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            353,
+            486,
+            367
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We evaluate TUNIT on the unlabeled datasets having no clear separations of the domains, which are AFHQ, FFHQ and LSUN Cars. For AFHQ, we train three individual models for dog, cat and wild. For all experiments, FUNIT is used as a baseline. We train it by presuming all labels to be the same as one. We set $\\scriptstyle { \\hat { K } } = 1 0$ for all the TUNIT models. More discussions on $\\cdot$ will be in Section 3.3. ",
+        "bbox": [
+            174,
+            378,
+            825,
+            438
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 demonstrates the results. We observe that the results of TUNIT adequately reflect the style feature of the references such as the textures of cats or cars and the species of the wilds. Although FFHQ has no clear domain distinctions, TUNIT captures the existence of glasses or smile as domains, and then add or remove glasses or smile. However, FUNIT performs much worse than TUNIT in this truly unsupervised scenario. For example, FUNIT outputs the inputs as is (cats and wilds) or insufficiently reflects the species (third row of AFHQ Wild). For FFHQ, despite that FUNIT makes some changes, the changes are not interpreted as meaningful domain translations. For LSUN Car, FUNIT fails to keep the fidelity. ",
+        "bbox": [
+            173,
+            444,
+            825,
+            556
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also visualize the style space of both models to qualitatively assess the quality of the representation. Figure 5 shows the t-SNE maps trained on AFHQ Wild and the examples of each cluster. Surprisingly, TUNIT organizes the samples according to the species where it roughly separates the images into six species. Although we set $\\cdot$ to be overly large, the model represents one species into two domains where those two domains position much closely. The highly disentangled, meaningful style features can be an important factor in the success of our model. On the other hand, the style features of FUNIT hardly learn meaningful domains so that the model cannot conduct the translation properly as shown in Figure 4. Because of the page limit, we include more results in Appendix E, G. ",
+        "bbox": [
+            173,
+            563,
+            825,
+            689
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.3 ANALYSIS ON GENERALIZABILITY ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            707,
+            454,
+            720
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Robustness to various $\\hat { K }$ ’s. When TUNIT conducts clustering for estimating domain labels, the number of clusters $\\cdot$ can affect the overall performances. Here, we study the effects on different $\\cdot$ on the labeled datasets and report them in Figure 6 and Table 2. As expected, the model performs best in terms of mFID when $\\cdot$ equals to the ground truth $K$ (i.e. $\\hat { K } { = } 1 0 $ ). Additionally, TUNIT performs reasonably well for sufficiently large $\\cdot$ $\\geq 7 )$ , and even with 100 times larger $\\hat { K }$ than the actual number of the domains, TUNIT still works well on both datasets. From this study, we conclude that TUNIT is relatively robust against $\\hat { K }$ as long as it is sufficiently large. We suggest using a sufficiently large number of $\\hat { K }$ or studying different $\\hat { K }$ ’s in log scale to find the optimal model. ",
+        "bbox": [
+            173,
+            731,
+            825,
+            867
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "With Few labels. We investigate whether or not TUNIT is effective for the partially labeled dataset that corresponds to a more practical scenario. To this end, we use AnimalFaces-10 and Food-10. We partition the dataset $\\mathcal { D }$ into the labeled set $\\mathcal { D } _ { s u p }$ and the unlabeled set $\\mathcal { D } _ { u n }$ with varying ratio $\\gamma = \\vert \\mathcal { D } _ { s u p } \\vert / \\vert \\mathcal { D } \\vert$ . Like before, we choose FUNIT as a competitor. We first train the networks while changing $\\gamma$ from 0.2 to 1.0 and report the result Table 3. As $\\gamma$ decreases, the performance of FUNIT significantly degrades whereas our model maintains mFID around 45 and 55 on both datasets. We also train an auxiliary classifier (VGG-11BN) with $\\mathcal { D } _ { s u p }$ then provide pseudo-labels for $\\mathcal { D } _ { u n }$ to FUNIT. As shown in Table 4, the performance of FUNIT is no longer sensitive to the changes in $\\gamma$ but still much worse than TUNIT. Although semi-supervised learning schemes can further improve FUNIT, TUNIT outperforms FUNIT using all labels as seen in Table 1. Under the empirical results in the semi-supervised scenario, we verify that TUNIT is also effective for semi-supervised setting. ",
+        "bbox": [
+            174,
+            868,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            222,
+            318,
+            238
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We argue that the unsupervised image-to-image translation should not utilize any supervision, such as image-level (i.e. paired) or set-level (i.e. unpaired) supervision. Under this rigorous regime, many previous studies fall into the set-level supervised framework that uses the domain information at a minimum. In this paper, for the first time, we proposed an effective model to handle the truly unsupervised image-to-image translation. To this end, we suggested the guiding network that performs unsupervised representation learning for providing pseudo labels and the image translation tasks. The experimental results showed that guiding network indeed exploits the synergy between two tasks, and the proposed model successfully conducts the unsupervised-image-to-image translation. We also showed the generalizability on the value of $\\hat { K }$ and the partially labeled scenario. ",
+        "bbox": [
+            174,
+            255,
+            825,
+            382
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            404,
+            285,
+            419
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "H. Bahng, S. Chung, S. Yoo, and J. Choo. Exploring unlabeled faces for novel attribute discovery. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5820– 5829, 2020. ",
+        "bbox": [
+            176,
+            428,
+            823,
+            469
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020a. ",
+        "bbox": [
+            173,
+            481,
+            823,
+            510
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020b. ",
+        "bbox": [
+            173,
+            520,
+            821,
+            550
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8789–8797, 2018. ",
+        "bbox": [
+            174,
+            560,
+            825,
+            616
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. Stargan v2: Diverse image synthesis for multiple domains. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2020. ",
+        "bbox": [
+            174,
+            627,
+            823,
+            670
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. In International Conference on Learning Representations, 2018. ",
+        "bbox": [
+            173,
+            680,
+            823,
+            710
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pp. 1735–1742. IEEE, 2006. ",
+        "bbox": [
+            176,
+            720,
+            821,
+            763
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pp. 1026–1034, 2015. ",
+        "bbox": [
+            174,
+            773,
+            821,
+            818
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. ",
+        "bbox": [
+            174,
+            828,
+            821,
+            871
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in neural information processing systems, pp. 6626–6637, 2017. ",
+        "bbox": [
+            176,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey Hinton, Nitish Srivastava, and Kevin Swersky. Neural networks for machine learning lecture 6a overview of mini-batch gradient descent. 2012. ",
+        "bbox": [
+            171,
+            154,
+            825,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. In International Conference on Learning Representations, 2019. ",
+        "bbox": [
+            178,
+            191,
+            823,
+            236
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. Cycada: Cycle-consistent adversarial domain adaptation. In International conference on machine learning, pp. 1989–1998, 2018. ",
+        "bbox": [
+            173,
+            243,
+            823,
+            287
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xun Huang, Ming-Yu Liu, Serge Belongie, and Jan Kautz. Multimodal unsupervised image-toimage translation. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 172–189, 2018. ",
+        "bbox": [
+            173,
+            295,
+            826,
+            338
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1125–1134, 2017. ",
+        "bbox": [
+            174,
+            347,
+            825,
+            390
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xu Ji, Joao F Henriques, and Andrea Vedaldi. Invariant information clustering for unsupervised˜ image classification and segmentation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 9865–9874, 2019. ",
+        "bbox": [
+            173,
+            397,
+            825,
+            441
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of GANs for improved quality, stability, and variation. In International Conference on Learning Representations, 2018. ",
+        "bbox": [
+            173,
+            449,
+            823,
+            492
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4401–4410, 2019. ",
+        "bbox": [
+            173,
+            501,
+            821,
+            544
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Taeksoo Kim, Moonsu Cha, Hyunsoo Kim, Jung Kwon Lee, and Jiwon Kim. Learning to discover cross-domain relations with generative adversarial networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1857–1865. JMLR. org, 2017. ",
+        "bbox": [
+            173,
+            551,
+            823,
+            595
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            171,
+            604,
+            823,
+            632
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Orest Kupyn, Volodymyr Budzan, Mykola Mykhailych, Dmytro Mishkin, and Jiˇr´ı Matas. Deblurgan: Blind motion deblurring using conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8183–8192, 2018. ",
+        "bbox": [
+            174,
+            641,
+            823,
+            685
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hsin-Ying Lee, Hung-Yu Tseng, Qi Mao, Jia-Bin Huang, Yu-Ding Lu, Maneesh Singh, and MingHsuan Yang. Dri $^ { + + }$ : Diverse image-to-image translation via disentangled representations. International Journal of Computer Vision, pp. 1–16, 2020. ",
+        "bbox": [
+            174,
+            693,
+            823,
+            737
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jae Hyun Lim and Jong Chul Ye. Geometric gan. arXiv preprint arXiv:1705.02894, 2017. ",
+        "bbox": [
+            168,
+            744,
+            766,
+            761
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Min Lin, Qiang Chen, and Shuicheng Yan. Network in network. arXiv preprint arXiv:1312.4400, 2013. ",
+        "bbox": [
+            174,
+            767,
+            821,
+            797
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ming-Yu Liu, Thomas Breuel, and Jan Kautz. Unsupervised image-to-image translation networks. In Advances in neural information processing systems, pp. 700–708, 2017. ",
+        "bbox": [
+            171,
+            806,
+            821,
+            835
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ming-Yu Liu, Xun Huang, Arun Mallya, Tero Karras, Timo Aila, Jaakko Lehtinen, and Jan Kautz. Few-shot unsupervised image-to-image translation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 10551–10560, 2019. ",
+        "bbox": [
+            176,
+            843,
+            821,
+            887
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lars Mescheder, Sebastian Nowozin, and Andreas Geiger. Which training methods for gans do actually converge? In ICML, 2018. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Muhammad Ferjad Naeem, Seong Joon Oh, Youngjung Uh, Yunjey Choi, and Jaejun Yoo. Reliable fidelity and diversity metrics for generative models. 2020. ",
+        "bbox": [
+            173,
+            140,
+            823,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Taesung Park, Ming-Yu Liu, Ting-Chun Wang, and Jun-Yan Zhu. Semantic image synthesis with spatially-adaptive normalization. In CVPR, 2019. ",
+        "bbox": [
+            173,
+            178,
+            823,
+            208
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017. ",
+        "bbox": [
+            174,
+            215,
+            823,
+            258
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nikunj Saunshi, Orestis Plevrakis, Sanjeev Arora, Mikhail Khodak, and Hrishikesh Khandeparkar. A theoretical analysis of contrastive unsupervised representation learning. In International Conference on Machine Learning, pp. 5628–5637, 2019. ",
+        "bbox": [
+            176,
+            267,
+            823,
+            311
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In International Conference on Learning Representations, 2015. ",
+        "bbox": [
+            174,
+            319,
+            821,
+            349
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Saurabh Singh and Shankar Krishnan. Filter response normalization layer: Eliminating batch dependence in the training of deep neural networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11237–11246, 2020. ",
+        "bbox": [
+            176,
+            357,
+            823,
+            401
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. In Advances in neural information processing systems, pp. 3483–3491, 2015. ",
+        "bbox": [
+            173,
+            409,
+            825,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dustin Tran, Rajesh Ranganath, and David Blei. Hierarchical implicit models and likelihood-free variational inference. In Advances in Neural Information Processing Systems, pp. 5523–5533, 2017. ",
+        "bbox": [
+            174,
+            460,
+            825,
+            502
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wouter Van Gansbeke, Simon Vandenhende, Stamatios Georgoulis, Marc Proesmans, and Luc Van Gool. Scan: Learning to classify images without labels. In European Conference on Computer Vision (ECCV), 2020. ",
+        "bbox": [
+            173,
+            511,
+            826,
+            555
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ting-Chun Wang, Ming-Yu Liu, Jun-Yan Zhu, Andrew Tao, Jan Kautz, and Bryan Catanzaro. Highresolution image synthesis and semantic manipulation with conditional gans. In CVPR, 2018. ",
+        "bbox": [
+            171,
+            563,
+            823,
+            593
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "I Zeki Yalniz, Herve J ´ egou, Kan Chen, Manohar Paluri, and Dhruv Mahajan. Billion-scale semi- ´ supervised learning for image classification. arXiv preprint arXiv:1905.00546, 2019. ",
+        "bbox": [
+            169,
+            601,
+            821,
+            631
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dingdong Yang, Seunghoon Hong, Yunseok Jang, Tiangchen Zhao, and Honglak Lee. Diversitysensitive conditional generative adversarial networks. In International Conference on Learning Representations, 2019. ",
+        "bbox": [
+            173,
+            638,
+            823,
+            681
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015. ",
+        "bbox": [
+            173,
+            690,
+            825,
+            733
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223–2232, 2017a. ",
+        "bbox": [
+            171,
+            742,
+            823,
+            785
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward multimodal image-to-image translation. In Advances in neural information processing systems, pp. 465–476, 2017b. ",
+        "bbox": [
+            173,
+            794,
+            826,
+            837
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            263,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            136,
+            344,
+            152
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Image-to-image translation. Since the seminal work of Pix2Pix (Isola et al., 2017), image-to-image translation models have shown impressive results (Zhu et al., 2017a; Liu et al., 2017; Kim et al., 2017; Kupyn et al., 2018; Choi et al., 2018; Huang et al., 2018; Liu et al., 2019; Yang et al., 2019). Exploiting the cycle consistency constraint, these methods were able to train the model with a setlevel supervision (domains) solely. However, acquiring domain information can be a huge burden in practical applications where a large amount of data are gathered from several mixed domains, e.g., web images (Yalniz et al., 2019). Not only does this complicates the data collection, but it restricts the methods only applicable to the existing dataset and domains. Inspired from few shot learning, Liu et al. (2019) proposed FUNIT that works on previously unseen target classes. However, FUNIT still requires the labels for training. Recently, Bahng et al. (Bahng et al., 2020) has partially addressed this by adopting the ImageNet pre-trained classifier for extracting domain information. Unlike the previous methods, we aim to design an image-to-image translation model that can be applied without any supervision such as a pre-trained network or supervision on both the train and the test datasets. ",
+        "bbox": [
+            174,
+            169,
+            825,
+            363
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised representation learning and clustering. Unsupervised representation learning aims to extract meaningful features for downstream tasks without any human supervision. To this end, many researchers have proposed to utilize the information that can be acquired from the data itself (Gidaris et al., 2018; Hjelm et al., 2019; Ji et al., 2019; He et al., 2020; Van Gansbeke et al., 2020). Recently, by incorporating the contrastive learning into a dictionary learning framework, MoCo (He et al., 2020) has achieved outstanding performance in various downstream tasks under reasonable mini-batch size. On the other hand, IIC (Ji et al., 2019) have utilized the mutual information maximization in a unsupervised way so that the network clusters images while assigning the images evenly. Though IIC provided a principled way to perform unsupervised clustering, it fails to scale up when combined with a difficult downstream task such as image-to-image translation. By taking the best of both worlds, we aim to solve unsupervised image-to-image translation. ",
+        "bbox": [
+            174,
+            367,
+            825,
+            520
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            542,
+            370,
+            558
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We train the guiding network for the first 65K iterations while freezing the update from both the generator and the discriminator. Then, we train the whole framework 100K more iterations for training all the networks. The batch size is set to 32 and 16 for $1 2 8 \\times 1 2 8$ and $2 5 6 \\times 2 5 6$ images, respectively. Training takes about 36 hours on a single Tesla V100 GPU with our implementation using PyTorch(Paszke et al., 2017). We use Adam (Kingma & Ba, 2014) optimizer with $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . { \\dot { 9 } } 9$ for the guiding network, and RMSprop (Hinton et al., 2012) optimizer with $\\alpha = 0 . 9 9$ for the generator and the discriminator. All learning rates are set to 0.0001 with a weight decay 0.0001. We adopt hinge version adversarial loss (Lim & Ye, 2017; Tran et al., 2017) with $R _ { 1 }$ regularization (Mescheder et al., 2018) using $\\gamma = 1 0$ (Eq. 5). We set $\\lambda _ { \\mathrm { { r e c } } } = 0 . 1 , \\lambda _ { \\mathrm { { s t y l e } } } ^ { G } = 0 . 0 1 , \\lambda _ { \\mathrm { { s t y l e } } } ^ { E } = \\bar { 1 }$ , and $\\lambda _ { \\mathrm { M I } } = 5$ in equation. 6 for all experiments. When the guiding network is simultaneously trained with the generator, we decrease $\\lambda _ { \\mathrm { s t y l e } } ^ { E ^ { \\ast } }$ and $\\lambda _ { \\mathrm { M I } }$ to 0.1 and 0.5, respectively. For evaluation, we use the exponential moving average over the parameters (Karras et al., 2018) of the guiding network and the generator. We initialize the weights of convolution layers with He initialization (He et al., 2015), all biases to zero, and weights of linear layers from $N ( 0 , 0 . 0 1 )$ with zero biases. The source code will be available publicly. ",
+        "bbox": [
+            174,
+            574,
+            825,
+            786
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C EVALUATION PROTOCOL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            411,
+            823
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For evaluation, we use class-wise Frechet Inception Distance (FID) (Heusel et al., 2017), which´ is often called mFID in literatures and D&C (Naeem et al., 2020). FID measures Frechet distance ´ between real and fake samples embedded by the last average pooling layer of Inception-V3 pretrained on ImageNet. Class-wise FID is obtained by averaging the FIDs of individual classes. In the experiments with fewer labels, we report the mean value of best five mFID’s over 100K iterations. For example, we use entire real images of each class and generate 810 fake images where $1 8 ~ \\times$ $( K - 1 )$ source images $K = 1 0$ for AnimalFaces-10) and five reference images of AnimalFaces10 are used to produce those fake images. We choose the source images from all classes except for the target class. For each source image, the five references are selected arbitrarily. For D&C, we generate fake images the similar number of training images with randomly selected source and reference images. Then, we use Inception-V3 pre-trained on ImageNet for extracting feature vectors and measure D&C by using the feature vectors. ",
+        "bbox": [
+            174,
+            840,
+            823,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D ARCHITECTURE DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            208,
+            419,
+            224
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the guiding network, we use VGG11 before the linear layers followed by the average pooling operation as the shared part and append two branches $E _ { \\mathrm { c l a s s } }$ and $E _ { \\mathrm { s t y l e } }$ . The branches are one linear layer with $\\hat { K }$ and 128 dimensional outputs, respectively. The detailed information of the generator, the guiding network and the discriminator architectures are provided in Table 5, Table 6 and Table 7. ",
+        "bbox": [
+            174,
+            238,
+            825,
+            299
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/3fa6319c0162a420f80fbff75b78d430612399e04b876812df0fd223ff6bef8e.jpg",
+        "table_caption": [
+            "Table 5: Generator architecture. “ch” represents the channel multiplier that is set to 64. IN and AdaIN indicate instance normalization and adaptive instance normalization, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LAYER</td><td>RESAMPLE</td><td>NORM</td><td>OUTPUT SHAPE</td></tr><tr><td>Image x</td><td></td><td>1</td><td>128×128×3</td></tr><tr><td>Conv7×7</td><td></td><td>IN</td><td>128 ×128 × ch</td></tr><tr><td>Conv4×4</td><td>Stride 2</td><td>IN</td><td>64 × 64× 2ch</td></tr><tr><td>Conv4×4</td><td>Stride 2</td><td>IN</td><td>32 × 32×4ch</td></tr><tr><td>Conv4×4</td><td>Stride 2</td><td>IN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td></td><td>IN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td></td><td>IN</td><td>16 × 16 × 8ch</td></tr><tr><td>ResBlk</td><td></td><td>AdaIN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td></td><td>AdaIN</td><td>16 ×16× 8ch</td></tr><tr><td>Conv5×5</td><td>Upsample</td><td>AdaIN</td><td>32 × 32 × 4ch</td></tr><tr><td>Conv5×5</td><td>Upsample</td><td>AdaIN</td><td>64 ×64× 2ch</td></tr><tr><td>Conv5×5</td><td>Upsample</td><td>AdaIN</td><td>128 ×128 × ch</td></tr><tr><td>Conv7×7</td><td></td><td>1</td><td>128 ×128×3</td></tr></table>",
+        "bbox": [
+            310,
+            313,
+            687,
+            541
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/3847c7169934371781bde2e512c97117ab8e40144ce0fc06620f813358dddde3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LAYER</td><td>RESAMPLE</td><td>NORM</td><td>OUTPUT SHAPE</td></tr><tr><td>Image X</td><td></td><td>■</td><td>128 ×128×3</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>64 × 64× ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>32 × 32× 2ch</td></tr><tr><td>Conv3×3</td><td>=</td><td>BN</td><td>32 × 32 ×4ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>16 ×16 ×4ch</td></tr><tr><td>Conv3×3</td><td>=</td><td>BN</td><td>16 ×16 ×8ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>8×8×8ch</td></tr><tr><td>Conv3×3</td><td>=</td><td>BN</td><td>8×8×8ch</td></tr><tr><td>Conv3×3</td><td>MaxPool</td><td>BN</td><td>4×4×8ch</td></tr><tr><td>GAP</td><td></td><td></td><td>1×1×8ch</td></tr><tr><td>FC</td><td></td><td></td><td>128</td></tr><tr><td>FC</td><td></td><td></td><td>K</td></tr></table>",
+        "bbox": [
+            310,
+            617,
+            687,
+            829
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Guiding network architecture. “ch” represents the channel multiplier that is set to 64. The architecture is based on VGG11-BN. GAP and FC denote global average polling (Lin et al., 2013) and fully connected layer, respectively. ",
+        "bbox": [
+            174,
+            847,
+            823,
+            888
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/54d588d072ecae59ef29936f8e1d95622dfbdff167f1dea4c9b5abbea4d4ce19.jpg",
+        "table_caption": [
+            "Table 7: Discriminator architecture. “ch” and $\\hat { K }$ represent the channel multiplier that is set to 64 and the number of clusters, respectively. FRN indicates filter response normalization (Singh & Krishnan, 2020). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LAYER</td><td>RESAMPLE</td><td>NORM</td><td>OUTPUT SHAPE</td></tr><tr><td>Image x</td><td></td><td></td><td>128 × 128 × 3</td></tr><tr><td>Conv3×3</td><td></td><td>1</td><td>128 ×128×ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>128 × 128× ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>64×64×2ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>64 × 64 × 2ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>32 × 32 × 4ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>32 ×32×4ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>16 ×16× 8ch</td></tr><tr><td>ResBlk</td><td></td><td>FRN</td><td>16 ×16 ×8ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>8×8×16ch</td></tr><tr><td>ResBlk</td><td>=</td><td>FRN</td><td>8×8×16ch</td></tr><tr><td>ResBlk</td><td>AvgPool</td><td>FRN</td><td>4×4×16ch</td></tr><tr><td>LReLU</td><td></td><td>-</td><td>4×4×16ch</td></tr><tr><td>Conv4×4</td><td></td><td></td><td>1 ×1×16ch</td></tr><tr><td>LReLU</td><td></td><td></td><td>1×1×16ch</td></tr><tr><td>Conv1 ×1</td><td></td><td></td><td>K</td></tr></table>",
+        "bbox": [
+            308,
+            342,
+            687,
+            616
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E T-SNE VISUALIZATION & CLUSTER EXAMPLE IMAGES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            658,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E.1 AFHQ CAT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            132,
+            297,
+            148
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/fc348f4607da18baa172df674d4f49f61120f9d2e8826232de5da073fd1a8cf4.jpg",
+        "image_caption": [
+            "Figure 7: t-SNE visualization and representative images of each domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            174,
+            795,
+            823
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E.2 AFHQ DOG ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            302,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c33a719764bd4c2605c0f8b74bde3f74980d4147e314e1af19f6680f91ba8d1.jpg",
+        "image_caption": [
+            "Figure 8: t-SNE visualization and representative images of each domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            145,
+            795,
+            792
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E.3 FFHQ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            263,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/8291e471dc5724d37120bc58abe1bff180aa37d5fe815b57b5511f6aa0b1f7c0.jpg",
+        "image_caption": [
+            "Figure 9: t-SNE visualization and representative images of each domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            145,
+            795,
+            791
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/a326590eebf7a605dd8b25613cefa15e4c1fc8dc45a97bb723d090384a36f254.jpg",
+        "image_caption": [
+            "Figure 10: t-SNE visualization and representative images of each domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            145,
+            795,
+            787
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "F ADDITIONAL COMPARISON WITH FUNIT: AFHQ, LSUN CAR AND FFHQ ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            769,
+            136
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/07f155349578d6c592d3bda8fa9a778649e764bca7098c1ae8aa2e7d2b19cf3a.jpg",
+        "image_caption": [
+            "Figure 11: AFHQ Cat, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. The content and the style are from the source and the reference, respectively. While FUNIT usually fails to reflect the style of the reference image, TUNIT generates the fake images with the style – color, fur texture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            159,
+            797,
+            443
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/b3bace1042798af3e0fb5a2795170a9a5009ac7d71692429a07f1c6c18b72238.jpg",
+        "image_caption": [
+            "Figure 12: AFHQ Wild, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. FUNIT rarely reflects the correct style of the reference image – the species, on the other hand, TUNIT translates the source image to the correct species. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            534,
+            797,
+            818
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/e084a58d925cf91bb1fa567c3d631805ab09fb971ca71f7075df97d44e937ebb.jpg",
+        "image_caption": [
+            "Figure 13: LSUN Car, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. While TUNIT generates plausible and changes the color of the source image to that of the reference image, FUNIT not also generates unrealistic image but also fails to changes the color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            128,
+            797,
+            411
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/5874faa938abe04badb873854ec73b30399a316537234f2f48efb68402327600.jpg",
+        "image_caption": [
+            "Figure 14: FFHQ, unsupervised reference-guided image-to-image translation results of FUNIT and TUNIT. Our model, TUNIT can remove or add the glasses to the source while preserving the identity better than FUNIT. In addition, TUNIT can change the hair color (last column) and the hair style – especially, bang (fifth column). It is hard to specify the definition of domains in the results of FUNIT while domains of TUNIT are more interpretable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            529,
+            797,
+            813
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "G ADDITIONAL RESULTS OF TUNIT: AFHQ, LSUN CAR, FFHQ, ANIMALFACES-10, AND S2W ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            740,
+            136
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "G.1 ANIMALFACES-10 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            151,
+            348,
+            165
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/88b0b3eeee8ffcbc01a5759b9788c45391fc79eb37935d71124a22ebc32b48c2.jpg",
+        "image_caption": [
+            "Figure 15: AnimalFaces-10, unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            169,
+            787,
+            872
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "G.2 AFHQ CAT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            299,
+            117
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/9cb4d71aa0710ef8dd9714741e5f8d79c1b0fac3feaaa2ccacb5815b7301cc21.jpg",
+        "image_caption": [
+            "Figure 16: AFHQ Cat, unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            126,
+            795,
+            847
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "G.3 AFHQ DOG ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            303,
+            117
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b8adf8c098cb154222098a401cdbfa102bd9031f81841dad4da280e3f7123d4.jpg",
+        "image_caption": [
+            "Figure 17: AFHQ Dogs, unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            126,
+            795,
+            845
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/80b9f2e057c5c37a520f5b27b98152ce046e400208859aae223622c8eeb5e271.jpg",
+        "image_caption": [
+            "Figure 18: AFHQ Wild, unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            125,
+            795,
+            847
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "G.5 FFHQ ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            264,
+            118
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "image",
+        "img_path": "images/e9b3985e1717ce957b749fafa8203b287216137b9fb976fb00cb7af88299df7f.jpg",
+        "image_caption": [
+            "Figure 19: FFHQ, unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            126,
+            795,
+            849
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "image",
+        "img_path": "images/532ae5574b3192ebc35d51df8e9ee4181978770b03f90bdb42b7823829f2993b.jpg",
+        "image_caption": [
+            "Figure 20: LSUN Car, unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            117,
+            795,
+            849
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "G.7 SUMMER2WINTER (S2W) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            400,
+            117
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "image",
+        "img_path": "images/ddaa5e2107475c064cc2c585a0b55ec9428d95f68325596665efa96acfd37489.jpg",
+        "image_caption": [
+            "(b) Results guided by reference images ",
+            "Figure 21: Summer2Winter (S2W), unsupervised image-to-image translation results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            126,
+            823,
+            804
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "H DIFFERENCE BETWEEN EQUATION (2) AND EQUATION (4) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            687,
+            118
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Equation (2) and (4) have similar forms – contrastive loss, but they are used for different purposes. We use equation (2) to improve the representation power of the guiding network, which affects the performance of the generator and the discriminator. On the other hand, equation (4) is used to enforce the generator to reflect the style of a reference image when translating a source image. To examine the effect of each loss, we train models without either equation (2) or (4) on AnimalFaces10. The mFID score without equation (2) or (4) is 86.8 and 93.3, respectively. Both models are significantly worse than the original setting (47.7). It means that both equation (2) and (4) should be considered during training. In addition to the purpose, they are different in terms of the way to choose positive pairs. We use a real image and its randomly augmented version as a positive pair in equation (2) while we use the translated image and reference image as a positive pair. In summary, the role of equation (2) is to enhance the representation power of the guiding network and lead the guiding network to learn how to encode the style vector in terms of a style encoder while the role of equation (4) is to guide the generator to learn how to interpret the provided style vector as a form of the output image. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            328
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "I FID AND LPIPS ON UNLABELED DATASET ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            348,
+            555,
+            364
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a8c6b6e2634ff1d75b3020ef9c1db65c6f1bf71cd84e0c87f1e28de3dba5251.jpg",
+        "image_caption": [
+            "Figure 22: LPIPS and FID of models and their status. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            396,
+            679,
+            645
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "We also utilize LPIPS to evaluate the models in addition to FID and D&C. However, LPIPS is not proper to evaluate the loyalty for reflecting the reference image and the fidelity of images, we use LPIPS with FID. Figure 22 shows the result. It is clear that a model with high FID and LPIPS generates a noise-like image. Even if FID is low, a model with high LPIPS also fails to conduct the reference-guided image translation, because it does not preserve the structure of the source image. The model with low LPIPS and high FID might be an adversarial example of LPIPS. We generate the image via optimization on LPIPS. If a model exhibits low FID and LPIPS, it might not reflect the visual feature of the reference image enough. The simple combination of LPIPS and FID can detect several failed models but can not evaluate the loyalty for the reference image. We suggest that the rigorous way to combine several metrics for the quantitative evaluation of the reference-guided translation might be a interesting future work. ",
+        "bbox": [
+            173,
+            691,
+            825,
+            845
+        ],
+        "page_idx": 27
+    }
+]

parse/train/NPOWF_ZLfC5/NPOWF_ZLfC5.md

@@ -0,0 +1,403 @@
+# Automatic Symmetry Discovery with Lie Algebra Convolutional Network
+
+Nima Dehmamy Northwestern University nimadt@bu.edu
+
+Robin Walters Northeastern University rwalters@northeastern.edu
+
+Yanchen Liu Northeastern University liu.yanc@northeastern.edu
+
+Dashun Wang Northwestern University dashun.wang@kellogg.northwestern.edu
+
+Rose Yu University of California San Diego roseyu@ucsd.edu
+
+# Abstract
+
+Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current. These connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences.1
+
+# 1 Introduction
+
+Incorporating symmetries into a deep learning architecture can reduce sample complexity, improve generalization, while significantly decreasing the number of model parameters (Cohen et al., 2019b; Cohen & Welling, 2016b; Ravanbakhsh et al., 2017; Ravanbakhsh, 2020; Wang et al., 2020). For instance, Convolutional Neural Networks (CNN) (LeCun et al., 1989, 1998) implement translation symmetry through weight sharing. General principles for constructing symmetry-aware group equivariant neural networks were introduced in Cohen & Welling (2016b), Kondor & Trivedi (2018), and Cohen et al. (2019b).
+
+However, most work on equivariant networks requires knowing the symmetry group a priori. A different equivariant model needs to be re-designed for each symmetry group. In practice, we may not have a good inductive bias and such knowledge of the symmetries may not be available. Constructing and selecting the equivariant network with the appropriate symmetry group becomes quite tedious. Furthermore, many existing works are limited to finite groups such as permutations Hartford et al. (2018); Ravanbakhsh et al. (2017); Zaheer et al. (2017), 90 degree rotations Cohen et al. (2018) or dihedral groups $\mathrm Ḋ N Ḍ$ and $E ( 2 )$ Weiler & Cesa (2019).
+
+For a continuous group, existing approaches either discretize the group Weiler et al. (2018a,b); Cohen & Welling (2016a), or use a truncated sum over irreducible representations (irreps) Weiler & Cesa (2019); Weiler et al. (2018a) via spherical harmonics in Worrall et al. (2017) or more general Clebsch-Gordon coefficients Kondor et al. (2018); Bogatskiy et al. (2020). These approaches are prone to approximation error. Recently, Finzi et al. (2020) propose to approximates the integral over the Lie group by Monte Carlo sampling. This approach requires implementing the matrix exponential and obtaining a local neighborhood for each point. Both parametrizing Lie groups for sampling and finding irreps are computationally expensive. Finzi et al. (2021) provide a general algorithm for constructing equivariant multi-layer perceptrons (MLP), but require explicit knowledge of the group to encode its irreps, and solving a set of constraints.
+
+We provide a novel framework for designing equivariant neural networks. We leverage the fact that Lie groups can be constructed from a set of infinitesimal generators, called Lie algebras. A Lie algebra has a finite basis, assuming the group is finite-dimensional. Working with the Lie algebra basis allows us to encode an infinite group without discretizing or summing over irreps. Additionally, all Lie algebras have the same general structure and hence can be implemented the same way. We propose Lie Algebra Convolutional Network (L-conv), a novel architecture that can automatically discover symmetries from data. Our main contributions can be summarized as follows:
+
+• We propose the Lie algebra convolutional network (L-conv), a building block for constructing group equivariant neural networks. We prove that multi-layer L-conv can approximate group convolutional layers, including CNNs, and find graph convolutional networks to be a special case of L-conv.   
+We can learn the Lie algebra basis in L-conv, enabling automatic symmetry discovery.   
+• L-conv also reveals interesting connections between physics and learning: equivariant loss generalizes important Lagrangians in field theory; robustness and equivariance can be expressed as Euler-Lagrange equations and Noether currents.
+
+Learning symmetries from data has been studied in limited settings for commutative Lie groups as in Cohen & Welling (2014), 2D rotations and translations in Rao & Ruderman (1999), Sohl-Dickstein et al. (2010) or permutations (Anselmi et al., 2019). In the non-commutative case, GeoManCEr (Pfau et al., 2020) uses data points related by small transformations to learn non-abelian Lie groups, but it does not introduce an equivariant layer architecture. (Zhou et al., 2020) propose a general method for symmetry discovery. Yet, their weight-sharing scheme and the symmetry generators are very different from ours. Our approach use much fewer parameters and has a direct interpretation using Lie algebras (SI B.3). Benton et al. (2020) propose Augerino to learn a distribution over data augmentations. It also involves Lie algebras, but is restricted to a subgroup of 2D affine transformations and requires matrix logarithm and sampling (SI B.3). In contrast, our approach is simpler and more general. Our approach uses composition of small transformations to achieve large transformations. In this sense bears some resemblance to symnets (Gens & Domingos, 2014), but the rest of the construction is different.
+
+# 2 Background
+
+We review the core concepts L-conv builds upon: equivariance, group convolution and Lie algebras.
+
+Notations. Unless explicitly stated, $a$ in $A ^ { a }$ is an index, not an exponent. We use the Einstein summation $\begin{array} { r } { A ^ { a } B _ { a b } = \sum _ { a } A ^ { \dot { a } } B _ { a b } = [ A B ] _ { b } } \end{array}$ , where a repeated upper and lower index are summed.
+
+Equivariance. Let $s$ be a topological space on which a Lie group $G$ (continuous group) acts from the left, meaning for all $\pmb { x } \in \ b { S }$ and $g \in G$ , $g { \pmb x } \in { \mathcal { S } }$ . We refer to $s$ as the base space. Let $\mathcal { F }$ , the “feature space”, be the vector space $\mathcal { F } = \mathbb { R } ^ { m }$ . Each data point is a feature map $f : S  { \mathcal { F } }$ . The action of $G$ on the input of $f$ induces an action on feature maps. For “scalar” features, for $u \in G$ , the transformed features $u \cdot f$ are given by
+
+$$
+u \cdot f ( \pmb { x } ) = f ( u ^ { - 1 } \pmb { x } ) .
+$$
+
+Denote the space of all functions from $s$ to $\mathcal { F }$ by $\mathcal { F } ^ { S }$ , so that $f \in \mathcal { F } ^ { s }$ . Let $F$ be a mapping to a new feature space $\mathcal { F } ^ { \prime } = \mathbb { R } ^ { m ^ { \prime } }$ , meaning $F : \mathcal { F } ^ { s } \to \mathcal { F } ^ { \prime } { } ^ { s }$ . We say $F$ is equivariant under $G$ if $G$ acts on ${ \mathcal { F } } ^ { \prime }$
+
+![](images/6c3a9c8f35d4da74c4e308612267a2b9221f619d9ec042857d2aff947d0785e5.jpg)  
+Figure 1: Lie group and Lie algebra: Illustration of the group manifold of a Lie group $G$ (left). The Lie algebra ${ \mathfrak { g } } = T _ { I } G$ is the tangent space at the identity $I$ . $L _ { i }$ are a basis for $T _ { I } G$ . If $G$ is connected, $\forall g \in G$ there exist paths like $\gamma$ from $I$ to $g$ and $g$ can be written as a path-ordered integral $\begin{array} { r } { g = P \exp [ \bar { \int _ { \gamma } d t ^ { i } } L _ { i } ] } \end{array}$ . Base space Right is a schematic of the base space $s$ as a manifold. The lift ${ \pmb x } = g { \pmb x } _ { 0 }$ takes ${ \pmb x } \in { \pmb S }$ to $g \in G$ , and maps the tangent spaces $T _ { x } S  T _ { g } G$ . Each Lie algebra basis $L _ { i } \in \mathfrak { g } = T _ { I } G$ generates a vector field $\hat { L } _ { i }$ on the tangent bundle $T G$ via the pushforward $\hat { L } _ { i } ( g ) = g L _ { i } g ^ { - 1 }$ . Via the lift, $L _ { i }$ also generates a vector field $\bar { L } _ { i } = \hat { L } _ { i } ^ { \alpha } ( { \pmb x } ) \partial _ { \alpha } = [ g L _ { i } \bar { \pmb x _ { 0 } } ] ^ { \alpha } \partial _ { \alpha }$ .
+
+and for $u \in G$ , we have
+
+$$
+u \cdot ( F ( f ) ) = F ( u \cdot f ) .
+$$
+
+Group Convolution. Kondor & Trivedi (2018) showed that $F$ is a linear equivariant map if and only if it performs a group convolution (G-conv). To define G-conv, we first lift $_ { \textbf { \em x } }$ to elements in $G$ (Kondor $\&$ Trivedi, 2018). Specifically, we pick an origin ${ \pmb x } _ { 0 } \in { \mathcal { S } }$ and replace each point ${ \pmb x } = g { \pmb x } _ { 0 }$ by $g$ . We will often drop $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ for brevity and write $f ( g ) \equiv f ( g \pmb { x } _ { 0 } )$ . Let $\kappa : G \to \mathbb { R } ^ { m ^ { \prime } } \otimes \mathbb { R } ^ { m }$ be a linear transformation from $\mathcal { F }$ to ${ \mathcal { F } } ^ { \prime }$ . G-conv is defined as
+
+$$
+[ \kappa \star f ] ( g ) = \int _ { G } \kappa ( g ^ { - 1 } v ) f ( v ) d v = \int _ { G } \kappa ( v ) f ( g v ) d v ,
+$$
+
+We denote the Haar measure on $G$ as $d v \equiv d \mu ( v )$ for brevity.
+
+Equivariance of G-conv. G-conv in equation 3 is equivariant (Kondor $\&$ Trivedi, 2018). By definition, for $w \in G$ we have
+
+$$
+\begin{array} { l } { { [ \kappa \star w \cdot f ] ( g ) = \displaystyle \int _ { G } \kappa ( v ) w \cdot f ( g v ) d v = \displaystyle \int _ { G } \kappa ( v ) f ( w ^ { - 1 } g v ) d v } } \\ { { = [ \kappa \star f ] ( w ^ { - 1 } g ) = w \cdot [ \kappa \star f ] ( g ) } } \end{array}
+$$
+
+Existing works on equivariance networks implement $\int _ { G }$ by discretizing the group or summing over irreps. We take a different approach and use the infinitesimal generators of the group. While a Lie group $G$ is infinite, usually it can be generated using a small number of infinitesimal generator, comprising its “Lie algebra”. We use the Lie algebra to introduce a building block to approximate G-conv. Figure 1 visualizes a Lie group, Lie algebra and the concept we discuss below.
+
+Lie algebra. Let $G$ be a Lie group, which includes common continuous groups. Group elements $u \in G$ infinitesimally close to the identity element $I$ can be written as $u \stackrel { - } { \approx } I + \stackrel { - } \epsilon ^ { i } L _ { i }$ (note Einstein summation), where $L _ { i } \in { \mathfrak { g } }$ with the Lie algebra ${ \mathfrak { g } } = T _ { I } G$ is the tangent space of $G$ at the identity element. The Lie algebra has the property that it is closed under a Lie bracket $[ \cdot , \cdot ] : { \mathfrak { g } } \times { \mathfrak { g } } \to { \mathfrak { g } }$
+
+$$
+[ L _ { i } , L _ { j } ] = { c _ { i j } } ^ { k } L _ { k } ,
+$$
+
+which is skew-symmetric and satisfies the Jacobi identity. Here the coefficients $c _ { i j } { } ^ { k } \in \mathbb { R }$ or $\mathbb { C }$ are called the structure constants of the Lie algebra. For matrix representations of $\mathfrak { g }$ , $[ L _ { i } , L _ { j } ] =$ $L _ { i } L _ { j } - L _ { j } L _ { i }$ is the commutator. The $L _ { i }$ are called the infinitesimal generators of the Lie group.
+
+Exponential map. If the manifold of $G$ is connected 2, an exponential map $\exp : { \mathfrak { g } } \to G$ can be defined such that $\overset { \cdot } { g } = \exp [ t ^ { i } L _ { i } ] \in G$ . For matrix groups, if $G$ is connected and compact, the matrix exponential is such a map and it is surjective. For most other groups (except ${ \mathrm { G L } } _ { d } ( \mathbb { C } )$ and nilpotent groups) it is not surjective. Nevertheless, for any connected group every $g \in G$ can be written as a product $\begin{array} { r } { g = \prod _ { a } \exp [ t _ { a } ^ { i } L _ { i } ] } \end{array}$ (Hall, 2015). Making ${ \dot { t } } _ { a } ^ { i }$ infinitesimal steps $d t ^ { i } ( s )$ tangent to a path $\gamma$ from $I$ to $g$ on $G$ yields the surjective path-ordered exponential in physics, denoted as $\begin{array} { r } { g = P \exp [ \int _ { \gamma } d t ^ { i } L _ { i } ] } \end{array}$ (SI A, and see Time-ordering in Weinberg (1995, p143)).
+
+Pushforward. $L _ { i } \in T _ { I } G$ can be pushed forward to $\hat { L } _ { i } ( g ) = g L _ { i } g ^ { - 1 } \in T _ { g } G$ to form a basis for $T _ { g } G$ , satisfying the same Lie algebra $[ \hat { L } _ { i } ( g ) , \hat { L } _ { j } ( g ) ] = { c _ { i j } } ^ { k } \hat { L } _ { k } ( g )$ . The manifold of $G$ together with the set of all $T _ { g } G$ attached to each $g$ forms the tangent bundle $T G$ , a type of fiber bundle (Lee et al., 2009). $\hat { L } _ { i }$ is a vector field on $_ { T G }$ . The lift maps $\hat { L } _ { i }$ to an equivalent vector field on $T S$ , which we will also denote by $\hat { L } _ { i }$ . Figure 1 illustrates the flow of these vector fields on $T G$ and $T S$ .
+
+# 3 Lie Algebra Convolutional Network
+
+We can use the Lie algebra basis $L _ { i } \in { \mathfrak { g } }$ to construct the Lie group $G$ with the exponential map. Similarly, we show that Lie algebras can also serve as building blocks to construct $\mathbf { G }$ -conv layers. We propose the Lie algebra convolutional network (L-conv). The key idea is to approximate the kernel $\kappa ( u )$ using localized kernels which can be constructed using the Lie algebra (Fig. 2). This is possible because the exponential map is a generalization of a Taylor expansion. We show that a G-conv whose kernel is concentrated near the identity can be expanded in the Lie algebra.
+
+Let $\delta _ { \eta } ( u ) \in \mathbb { R }$ denote a normalized localized kernel, meaning $\begin{array} { r } { \int _ { G } \delta _ { \eta } ( g ) d g = 1 } \end{array}$ , and with support on a small neighborhood of size $\eta$ centered around the identity $I$ (i.e., $\delta _ { \eta } ( I + \epsilon ^ { i } L _ { i } ) \to 0$ if $\| \epsilon \| ^ { 2 } > \eta ^ { 2 } )$ . We pick $\delta _ { \eta } ( v _ { \epsilon } ) \sim \theta ( \eta ^ { 2 } - \| \epsilon \| ^ { 2 } )$ , for $v _ { \epsilon } = I + \epsilon ^ { i } L _ { i } \in T _ { I } G$ and $\delta _ { \eta } ( v ) = 0$ for all other $v \notin T _ { I } G \left( \theta ( \cdot ) \right.$ being the Heaviside step function). Let $\kappa _ { 0 } : G \to \mathbb { R } ^ { m ^ { \prime } } \otimes \mathbb { R } ^ { m }$ be given by
+
+$$
+\left[ \kappa _ { 0 } \right] _ { a } ^ { b } ( u ) = \left[ W ^ { 0 } \right] _ { a } ^ { c } \delta _ { \eta } \left( u \left( I - \left[ \bar { \epsilon } ^ { i } \right] _ { c } ^ { b } L _ { i } \right) \right)
+$$
+
+where $W ^ { 0 } \in \mathbb { R } ^ { m ^ { \prime } } \otimes \mathbb { R } ^ { h }$ and $\overline { { \epsilon } } ^ { i } \in \mathbb { R } ^ { h } \otimes \mathbb { R } ^ { m }$ are constants, and we choose $\vert [ \overline { { { \epsilon } } } ^ { i } ] _ { b } ^ { a } \vert < \eta$ . Note that $\begin{array} { r } { ( I + \epsilon ^ { i } L _ { i } ) ( I - \overline { { \epsilon } } ^ { j } L _ { j } ) = I + [ \epsilon - \overline { { \epsilon } } ] ^ { i } L _ { i } + O ( \eta ^ { 2 } ) } \end{array}$ . Therefore,
+
+$$
+\int \epsilon ^ { i } d \epsilon \delta _ { \eta } \left( ( I + \epsilon ^ { i } L _ { i } ) ( I - \overline { { { \epsilon } } } ^ { j } L _ { j } ) \right) = \overline { { { \epsilon } } } ^ { i }
+$$
+
+The localized kernels $\kappa _ { 0 }$ can be used to approximate G-conv.
+
+Linear expansion of $\mathbf { G }$ -conv with localized kernel. We can expand a G-conv whose kernel is $\kappa _ { 0 } ( u ) = \bar { W } ^ { 0 } \delta _ { \eta } ( u )$ in the Lie algebra of $G$ to linear order. With $v _ { \epsilon } \stackrel { \textstyle \mathsf { \bar { \alpha } } } { = } I + \epsilon ^ { i } L _ { i }$ , we have (see SI A)
+
+$$
+\begin{array} { l } { { \displaystyle Q [ f ] ( g ) = [ \kappa _ { 0 } \star f ] ( g ) = \int _ { G } d v \kappa _ { 0 } ( v ) f ( g v ) = \int _ { \| \epsilon \| < \eta } d v _ { \epsilon } \kappa _ { 0 } ( v _ { \epsilon } ) f ( g v _ { \epsilon } ) } } \\ { ~ } \\ { { \displaystyle ~ = W ^ { 0 } \int d \epsilon \delta _ { \eta } ( v _ { \epsilon } ) \left[ f ( g ) + \epsilon ^ { i } g L _ { i } \cdot \frac d { d g } f ( g ) + O ( \epsilon ^ { 2 } ) \right] } } \\ { { \displaystyle ~ = W ^ { 0 } \left[ I + \bar { \epsilon } ^ { i } g L _ { i } \cdot \frac d { d g } \right] f ( g ) + O ( \eta ^ { 2 } ) } } \end{array}
+$$
+
+with $W ^ { 0 } \in \mathbb { R } ^ { m ^ { \prime } } \otimes \mathbb { R } ^ { h }$ and $\overline { { \epsilon } } ^ { i } \in \mathbb { R } ^ { h } \otimes \mathbb { R } ^ { m }$ , as before. Here $d \epsilon$ is the integration measure on the Lie algebra ${ \mathfrak { g } } = T _ { I } G$ induced by the Haar measure $d v _ { \epsilon }$ on $G$ .
+
+Interpreting the derivatives. In a matrix representation of $G$ , we have $\begin{array} { r } { g L _ { i } \cdot \frac { d f } { d g } = [ g L _ { i } ] _ { \alpha } ^ { \beta } \frac { d f } { d g _ { \alpha } ^ { \beta } } = } \end{array}$ $\operatorname { T r } \left[ [ g L _ { i } ] ^ { T } { \frac { d f } { d g } } \right]$ . This can be written in terms of partial derivatives $\partial _ { \alpha } f ( \pmb x ) = \partial f / \partial \pmb x ^ { \alpha }$ as follows. Using $\pmb { x } ^ { \rho } = g _ { \sigma } ^ { \rho } \pmb { x } _ { 0 } ^ { \sigma }$ , we have $\begin{array} { r } { \frac { d f ( g x _ { 0 } ) } { d g _ { \beta } ^ { \alpha } } = { \pmb x } _ { 0 } ^ { \beta } \partial _ { \alpha } f ( { \pmb x } ) } \end{array}$ , and so
+
+$$
+\hat { L } _ { i } f ( \pmb { x } ) \equiv g L _ { i } \cdot \frac { d f } { d g } = [ g L _ { i } ] _ { \beta } ^ { \alpha } \pmb { x } _ { 0 } ^ { \beta } \partial _ { \alpha } f ( \pmb { x } ) = [ g L _ { i } \pmb { x } _ { 0 } ] \cdot \nabla f ( \pmb { x } )
+$$
+
+Hence, for each $L _ { i }$ , the pushforward $g L _ { i } g ^ { - 1 }$ generates a flow on $s$ through the vector field $\hat { L } _ { i } \equiv { }$ $g L _ { i } \cdot d / d g = [ g L _ { i } g ^ { - 1 } { \pmb x } ] ^ { \bar { \alpha } } \partial _ { \alpha }$ (Fig. 1).
+
+![](images/66a444b0af7483094d85e1019feae8039ad1ee494275c1042f0548072df9fede.jpg)  
+Figure 2: Sketch of the procedure for approximating G-conv using L-conv. First, the kernel is written as the sum of a number of localized kernels $\kappa _ { k }$ with support around $u _ { k }$ (left). Each of the $\kappa _ { k }$ is then moved toward identity by composing multiple L-conv layers $Q _ { \epsilon ^ { \prime } } \circ Q _ { \epsilon } \dots \kappa _ { k }$ (right).
+
+Lie algebra convolutional (L-conv) layer. Equation 8 states that for a kernel localized near the identity, the effect of the kernel can be summarized in $W ^ { 0 }$ and $\overline { { \epsilon } } ^ { i } \hat { L } _ { i }$ . Note that we do not need to perform the integral over $G$ explicitly anymore. Instead of working with a kernel $\kappa _ { 0 }$ , we only need to specify $W ^ { 0 }$ and $\overline { { \epsilon } } ^ { i }$ . Hence, in general, we define the Lie algebra convolution (L-conv) as
+
+$$
+\begin{array} { l } { { Q [ f ] ( { \pmb x } ) = W ^ { 0 } \left[ I + \overline { { { \epsilon } } } ^ { i } \hat { L } _ { i } \right] f ( { \pmb x } ) } } \\ { { \ = W ^ { 0 } \left[ I + \overline { { { \epsilon } } } ^ { i } [ g L _ { i } { \pmb x } _ { 0 } ] ^ { \alpha } \partial _ { \alpha } \right] f ( { \pmb x } ) } } \end{array}
+$$
+
+Being an expansion of G-conv, L-conv inherits the equivariance of $\mathbf { G }$ -conv, as we show next.
+
+Proposition 1 (Equivariance of L-conv). With assumptions above, $L$ -conv is equivariant under $G$
+
+Proof: First, note that the components of $\hat { L } _ { i }$ transform as $[ \hat { L } _ { i } ( v \pmb { x } ) ] ^ { \alpha } = [ v g L _ { i } \pmb { x } _ { 0 } ] ^ { \alpha } = v _ { \beta } ^ { \alpha } \hat { L } _ { i } ( \pmb { x } ) ^ { \beta }$ , while the partial transforms as $\partial / \partial [ v \pmb { x } ] ^ { \alpha } = [ v ^ { - 1 } ] _ { \alpha } ^ { \gamma } \partial _ { \gamma }$ . As a result in $\hat { L } _ { i } = [ g L _ { i } { \pmb x } _ { 0 } ] ^ { \alpha } \partial _ { \alpha }$ all factors of $v$ cancel, meaning for $v \in G$ , $\hat { L } _ { i } ( v \pmb { x } ) = \hat { L } _ { i } ( \pmb { x } )$ . This is because of the fact that $\hat { L } _ { i } \in T S$ is a vector field (i.e. 1-tensor) and, thus, invariant under change of basis. Plugging into equation 10, for $w \in G$
+
+$$
+\begin{array} { l } { w \cdot Q [ f ] ( { \pmb x } ) = Q [ f ] ( w ^ { - 1 } { \pmb x } ) = W ^ { 0 } \left[ I + \bar { \epsilon } ^ { i } \hat { L } _ { i } ( w ^ { - 1 } { \pmb x } ) \right] f ( w ^ { - 1 } { \pmb x } ) } \\ { = W ^ { 0 } \left[ I + \bar { \epsilon } ^ { i } \hat { L } _ { i } ( g ) \right] f ( w ^ { - 1 } { \pmb x } ) = W ^ { 0 } \left[ I + \bar { \epsilon } ^ { i } \hat { L } _ { i } ( g ) \right] w \cdot f ( { \pmb x } ) = Q [ w \cdot f ] ( { \pmb x } ) ( \bar { \epsilon } ^ { i } - { \pmb x } ) } \end{array}
+$$
+
+which proves L-conv is equivariant.
+
+Examples. Using equation 9 we can calculate $\mathrm { L }$ -conv for specific groups (details in SI A.2). For translations $G = T _ { n } = ( \mathbb { R } ^ { n } , + )$ , we find the generators become simple partial derivatives $\hat { L } _ { i } = \partial _ { i }$ (SI A.2.2), yielding $f ( \pmb { x } ) + \epsilon ^ { \alpha } \partial _ { \alpha } f ( \pmb { x } )$ . For 2D rotations (SI A.2.1) the generator $\hat { L } \equiv ( x \partial _ { y } - y \partial _ { x } ) = \partial _ { \theta }$ , which is the angular momentum operator about the $\mathbf { Z }$ -axis in quantum mechanics and field theories. For rotations with scaling, $G = S O ( 2 ) \times \mathbb { R } ^ { + }$ , we have two $L _ { i }$ , one $\hat { L } _ { \theta } = \partial _ { \theta }$ from $s o ( 2 )$ and a scaling with $L _ { r } = I$ , yielding $\hat { L } _ { r } = x \partial _ { x } + y \partial _ { y } = r \partial _ { r }$ . Next, we discuss the form of L-conv on discrete data.
+
+# 3.1 Approximating G-conv using L-conv
+
+L-conv can be used as a basic building block to construct G-conv with more general kernels. Figure 2 sketches the argument described here (see also SI A.1).
+
+Theorem 1 (G-conv from L-convs). $G$ -conv equation 3 can be approximated using $L$ -conv layers.
+
+Proof: The procedure involves two steps, as illustrated in Fig. 2: 1) approximate the kernel using localized kernels as the $\delta _ { \eta }$ in L-conv; 2) move the kernels towards identity using multiple L-conv layers. The following lemma outline the details. 
+
+Lemma 1 (Approximating the kernel). Let the kernel $\kappa : G \to { \mathcal { F } } ^ { \prime } \otimes { \mathcal { F } }$ with $\textstyle \int _ { G } \| \kappa ( g ) \| ^ { 2 } d g < \infty$ be continuously differentiable with $\| d \kappa ( g ) / d g \| ^ { 2 } < \xi ^ { 2 }$ , and with compact support over $G _ { 0 } \subset G$ . Let The $\kappa _ { k } ( g ) = c _ { k } \delta _ { \eta } ( u _ { k } ^ { - 1 } g )$ set oand $N$ els with ssuch that $\eta$ neighborhoodapproximates f , $u _ { k } \in G$ $c _ { k } \in \mathcal { F } ^ { \prime } \otimes \mathcal { F }$ $u _ { k } \in G$ $\begin{array} { r } { \tilde { \kappa } = \sum _ { k = 1 } ^ { N } \kappa _ { k } } \end{array}$ $\kappa _ { \ast }$ $\begin{array} { r } { \int _ { G } \| \kappa ( g ) - \tilde { \kappa } ( g ) \| ^ { 2 } d g < \zeta ^ { 2 } } \end{array}$ for arbitrary small $\zeta \in \mathbb { R } _ { + }$ .
+
+Proof: See SI A.1 for details. The intuition is similar to the universal approximation theorem for neural networks (Hornik et al., 1989; Cybenko, 1989), only generalized to a group manifold instead of $\mathbb { R }$ . Let $B _ { 0 }$ be the set of $v _ { \epsilon } = I + \epsilon ^ { i } \bar { L _ { i } } \in \mathfrak { g }$ , with $\| \dot { \epsilon } \| ^ { 2 } < \dot { \eta } ^ { 2 }$ . Choose a set of $u _ { k } \in G$ such that the k  k 0   that on small enough neighborhoods neighborhoods $B _ { k } = u _ { k } B _ { 0 } \subset G$ cover the support $B _ { k } \subset G$ 0 , for any two $G _ { 0 }$ of $u , v \in B _ { k }$ $\kappa$ . The bound we have $\| d \kappa ( g ) / d g \| ^ { 2 } < \xi ^ { 2 }$ $\| \kappa ( \dot { u } ) - \ddot { \kappa } ( v ) \| ^ { 2 } \leq \eta ^ { 2 } \xi ^ { 2 }$ means where $| G _ { 0 } |$ is the volume of the support of $\kappa$ . Hence, for $g \in B _ { k }$ , $\kappa ( g )$ can be approximated with $\kappa _ { k } ( g ) = \kappa ( u _ { k } ) \delta _ { \eta } ( u _ { k } ^ { - 1 } g )$ , with normalized localized kernels $\delta _ { \eta } ( g )$ , and any element $u _ { k } ~ \in$ $B _ { k }$ . We show that the approximation error of using $\tilde { \kappa } = \sum _ { k } \kappa _ { k }$ to approximate $\kappa$ is bounded by $\begin{array} { r } { \int _ { G } d g \| \kappa ( g ) - \tilde { \kappa } ( g ) \| ^ { 2 } < | G _ { 0 } | \eta ^ { 2 } \xi ^ { 2 } } \end{array}$ . Any desired error bound $\zeta$ can then be attained by choosing small enough $\eta$ for neighborhood sizes. 
+
+Thus, we can approximate a large class of kernels as $\begin{array} { r } { \kappa ( g ) \approx \sum _ { k } \kappa _ { k } ( g ) } \end{array}$ where the local kernels $\kappa _ { k } ( g ) = c _ { k } \delta _ { \eta } ( u _ { k } ^ { - 1 } g )$ have support only on an $\eta$ neighborhood of $u _ { k } \in G$ . Here $c _ { k } \in \mathbb { R } ^ { m ^ { \prime } } \otimes \mathbb { R } ^ { m }$ are constants and $\delta _ { \eta } ( u )$ is as in equation 8. Using this, $\mathbf { G }$ -conv equation 3 becomes
+
+$$
+[ \kappa \star f ] ( g ) = \sum _ { k } c _ { k } \int d v \delta _ { \eta } ( u _ { k } ^ { - 1 } v ) f ( g v ) = \sum _ { k } c _ { k } [ \delta _ { \eta } \star f ] ( g u _ { k } ) .
+$$
+
+The kernels $\kappa _ { k }$ are localized around $u _ { k }$ , whereas in L-conv the kernel is around identity. We can compose L-conv layers to move $\kappa _ { k }$ from $u _ { k }$ to identity.
+
+Lemma 2 (Moving kernels to identity). $\kappa _ { k }$ can be moved near identity using a multilayer $L$ -conv.
+
+Proof: In equation 12, write $u _ { k } = v _ { \epsilon } u _ { k } ^ { \prime }$ , with $v _ { \epsilon } = I + \epsilon ^ { i } L _ { i } \in \mathfrak { g }$ . Using the definition equation 10 an L-conv layer $Q _ { \epsilon } = I - \epsilon ^ { i } \hat { L } _ { i }$ performs a first order Taylor expansion (SI A.1) and so $Q _ { \epsilon } [ \delta _ { \eta } ] ( u _ { \ \boldsymbol { k } } ^ { \prime - 1 } v ) =$ $\delta _ { \eta } ( u _ { k } ^ { - 1 } v ) + O ( \epsilon ^ { 2 } )$ . Thus, applying one $\mathrm { L }$ -conv layer moves the localized kernel along $v _ { \epsilon }$ on $G$ . Writing $u _ { k }$ as the product of a set of small group elements $\textstyle u _ { k } = \prod _ { a = 1 } ^ { p } v _ { a }$ , with $v _ { a } = I + \epsilon _ { a } ^ { i } L _ { i } \in \mathfrak { g }$ Defining L-conv layers $Q _ { a } = I - \epsilon _ { a } ^ { i } \hat { L } _ { i }$ , we can write
+
+$$
+\kappa _ { k } ( g ) \approx c _ { k } Q _ { p } \circ \cdot \cdot \cdot \circ Q _ { 1 } \circ \delta _ { \eta } ( g )
+$$
+
+meaning $\kappa _ { k }$ localized around $u _ { k }$ can be written as a $p$ layer L-conv acting on a kernel $\delta _ { \eta } ( g )$ , localized around the identity of the group. With $\| \epsilon _ { a } \| < \eta$ , the error in $u _ { k }$ is $O ( \eta ^ { p + 1 } )$ . 
+
+Thus, we conclude that any G-conv equation 3 can be approximated by multilayer L-conv. Furthermore, for compact $G$ , using the theorem in Kondor & Trivedi (2018), we can show that any equivariant feedforward neural network can be approximated using multilayer L-conv with nonlinearities.
+
+Equivariance of nonlinearity. Pointwise nonlinearities give equivariant maps between scalar feature maps. To see this, let $\sigma : \mathbb { R }  \mathbb { R }$ . We extend $\sigma : { \mathcal { F } }  { \mathcal { F } }$ by applying $\sigma$ component-wise. Let $f : S  { \mathcal { F } }$ be a scalar feature map (i.e., $g \cdot f ( { \pmb x } ) = f ( g ^ { - 1 } { \pmb x } ) )$ . Then
+
+$$
+g \cdot ( \sigma \circ ( f ) ) ( { \pmb x } ) = \sigma \circ ( f ) ( g ^ { - 1 } { \pmb x } ) = \sigma \circ ( g \cdot f ) ( { \pmb x } ) .
+$$
+
+Since the composition of equivariant maps is equivariant, given equivariant linear mapping $Q : { \mathcal { F } } ^ { S } $ $\mathcal { F } ^ { \prime } \mathcal { S }$ (i.e. $g \cdot { \bar { Q } } [ f ] = Q [ g \cdot { \bar { f } } ] )$ , the layer $f \mapsto \bar { \sigma } \circ Q [ f ]$ is equivariant. Hence we have the corollary:
+
+Corollary 1. Assume $G$ is compact and acts on $s$ transitively. Then any equivariant feedforward neural network (FNN) can be approximated using multilayer $L$ -conv with point-wise nonlinearities.
+
+Proof: A FNN is defined as $\sigma _ { p } \circ F _ { p } [ \cdot \cdot \cdot [ \sigma _ { 1 } \circ F _ { 1 } [ f ] ] ( \pmb { x } )$ where $F _ { k }$ are linear and $\sigma _ { k }$ are point-wise nonlinearities. By Theorem 1 of Kondor & Trivedi (2018), any linear layer in the equivariant FNN is a G-conv, which by Theorem 1 can be approximated by multilayer L-conv. Therefore, multilayer L-conv with nonlinearity can approximate any equivariant FNN. 
+
+Finally, to our knowledge it is not known whether every equivariant function can be approximated by equivariant FNN for a Lie group $G$ . Hence, the corollary above is not a universal approximation theorem for equivariant scalar functions in terms of L-conv. However, it does show that multilayer $L$ -conv is equally expressive as other equivariant networks. Next, we discuss implementation details.
+
+# 4 Discretized space and implementation: the tensor notation
+
+In many datasets, such as images, $f ( { \pmb x } )$ is not given as continuous function, but rather as a discrete array, with ${ \cal S } = \{ { \pmb x } _ { 0 } , \ldots x _ { d - 1 } \}$ containing $d$ points. Each ${ \pmb x } _ { \mu }$ represents a coordinate in higher dimensional space, e.g. on a $1 0 \times 1 0$ image, $\scriptstyle { \mathbf { { \vec { x } } } } _ { 0 }$ is $( x , y ) = ( 0 , \dot { 0 } )$ point and ${ \pmb x } _ { 9 9 }$ is $( x , y ) = ( 9 , { \bar { 9 } } )$ .
+
+Feature maps and group action In the tensor notation, we encode $\pmb { x } _ { \pmb { \mu } } \in \mathcal { S }$ as the canonical basis (one-hot) vectors in ${ \pmb x } _ { \pmb { \mu } } \in \mathbb { R } ^ { d }$ with $[ { \pmb x } _ { \mu } ] _ { \nu } = \delta _ { \mu \nu }$ (Kronecker delta), e.g. $\pmb { x } _ { 0 } = ( 1 , 0 , \dots , 0 )$ . The features become $\pmb { f } \in \mathcal { F } = \mathbb { R } ^ { d } \otimes \mathbb { R } ^ { m }$ , meaning $d \times m$ tensors, with $f ( \pmb { x } _ { \mu } ) = \pmb { x } _ { \mu } ^ { T } \pmb { f } = \pmb { f } _ { \mu }$ . Although $s$ is discrete, the group acting on $\mathcal { F }$ can be continuous (e.g. image rotations). Any $G \subseteq { \mathrm { G L } } _ { d } ( \mathbb { R } )$ of the general linear group (invertible $d \times d$ matrices) acts on $\pmb { x } _ { \pmb { \mu } } \in \mathbb { R } ^ { d }$ and $f \in { \mathcal { F } }$ . We define $f ( g \cdot { \pmb x } _ { \mu } ) = { \pmb x } _ { \mu } ^ { T } g ^ { T } { \pmb f } , \forall g \in G$ , so that for $w \in G$ we have
+
+$$
+w \cdot f ( \pmb { x } _ { \mu } ) = f ( \pmb { w } ^ { - 1 } \cdot \pmb { x } _ { \mu } ) = \pmb { x } _ { \mu } ^ { T } \pmb { w } ^ { - 1 T } \pmb { f } = [ \pmb { w } ^ { - 1 } \pmb { x } _ { \mu } ] ^ { T } \pmb { f }
+$$
+
+Dropping the position $\scriptstyle { \pmb { x } } _ { \mu }$ , the transformed features are matrix product $w \cdot f = w ^ { - 1 T } f$ . We can write $\mathbf { G }$ -conv in this notation (SI B). Similarly, we can rewrite $\mathrm { L }$ -conv equation 8 in the tensor notation. Defining $v _ { \epsilon } = I + \overline { { { \epsilon } } } ^ { i } L _ { i }$
+
+$$
+\begin{array} { r l } & { \qquad Q [ \pmb { f } ] ( g ) = W ^ { 0 } f \left( g \left( I + \overline { { \epsilon } } ^ { i } L _ { i } \right) \right) = \pmb { x } _ { 0 } ^ { T } \left( I + \overline { { \epsilon } } ^ { i } L _ { i } \right) ^ { T } g ^ { T } f W ^ { 0 T } } \\ & { \qquad = \left( \pmb { x } + \overline { { \epsilon } } ^ { i } [ g L _ { i } \pmb { x } _ { 0 } ] \right) ^ { T } \pmb { f } W ^ { 0 T } . } \end{array}
+$$
+
+Here, $\hat { L } _ { i } = g L _ { i } { \pmb x } _ { 0 }$ is exactly the matrix analogue of pushforward vector field $\hat { L } _ { i }$ in equation 9. The equivariance of L-conv in tensor notation is again evident from the $g ^ { T } f$ , resulting in
+
+$$
+Q [ w \cdot f ] ( g ) = \pmb { x } _ { 0 } ^ { T } \pmb { v } _ { \epsilon } ^ { T } g ^ { T } w ^ { - 1 T } \pmb { f } W ^ { 0 T } = Q [ \pmb { f } ] ( w ^ { - 1 } g ) = w \cdot Q [ \pmb { f } ] ( g )
+$$
+
+Tensor L-conv layer implementation The discrete space $\mathrm { L }$ -conv equation 15 can be rewritten using the global Lie algebra basis $\hat { L } _ { i }$
+
+$$
+Q [ f ] = \left( f + \hat { L } _ { i } f \bar { \epsilon } ^ { i } \right) W ^ { 0 T } , Q [ f ] _ { \mu } ^ { a } = f _ { \mu } ^ { b } [ W ^ { 0 T } ] _ { b } ^ { a } + [ \hat { L } _ { i } ] _ { \mu } ^ { \nu } f _ { \nu } ^ { c } \left[ W ^ { i } \right] _ { c } ^ { a }
+$$
+
+Where $W ^ { i } = W ^ { 0 } \overline { { { \epsilon } } } ^ { i }$ , $W ^ { 0 } \in \mathbb { R } ^ { m _ { i n } } \otimes \mathbb { R } ^ { m _ { o u t } }$ and $\overline { { \epsilon } } ^ { i } \in \mathbb { R } ^ { m _ { i n } } \otimes \mathbb { R } ^ { m _ { i n } }$ are trainable weights. The $\hat { L } _ { i }$ can be either inserted as inductive bias or they can be learned to discover symmetries.
+
+To implement L-conv, note that the formula of equation 17 is quite similar to a Graph Convolutional Network (GCN) (Kipf & Welling, 2016). For each $i$ , the shared convolutional weights are $\overline { { \epsilon } } ^ { i } W ^ { 0 T }$ and the aggregation function of the GCN, a function of the graph adjacency matrix, is $\hat { L } _ { i }$ in L-conv. Thus, Lconv can be implemented as GCN modules for each $\hat { L } _ { i }$ , plus a residual connection for the $f W ^ { 0 T }$ term.
+
+Figure 3 shows the schematic of the L-conv layer. In a naive implementation, $\hat { L } _ { i }$ can be general $d \times d$ matrices. However, being vector fields generated by the Lie algebra, $\hat { L } _ { i }$ has a more constrained structure which allows them to be encoded and learned using much fewer parameters than a $d \times d$ matrix. Specifically, encoding the topology of $s$ as a graph (see SI B.1), the incidence matrix replaces partial derivatives (Schaub et al., 2020) in equation 9 and the $L _ { i }$ become weighting of the edges. This weighting is similar to
+
+![](images/7da7ef66be0a2de4ec31dd46c4a64379db96e59b7bf42566f6da3a91ec8a4ec1.jpg)  
+Figure 3: L-conv layer architecture. $L _ { i }$ only act on the $d$ flattened spatial dimensions, and $W ^ { i }$ only act on the $m _ { i n }$ input features and returns $m _ { o u t }$ output features. For each $i$ , Lconv is analogous to a GCN with $d$ nodes and $m _ { i n }$ features.
+
+Gauge Equivariant Mesh (GEM) CNN (Cohen et al., 2019a). Indeed, in L-conv the lift ${ \pmb x } _ { \mu } = g _ { \mu } { \pmb x } _ { 0 }$ fixes the gauge by mapping neighbors of $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ to neighbors of ${ \pmb x } _ { \mu }$ . Changing how the discrete $s$ samples an underlying continuous space will change $g _ { \mu }$ and hence the gauge.
+
+Choosing the number of $L _ { i }$ . Beside the width of $W ^ { 0 }$ and $\bar { \epsilon } ^ { i }$ , the number $n _ { L }$ of $L _ { i }$ is a hyperparameter in L-conv. For instance, if $s$ is a discretization of $n$ dimensional space the symmetry group is likely $G \subset { \mathrm { G L } } _ { n } ( \mathbb { R } ) \ltimes T _ { n }$ , with $n _ { L } \sim O ( n ^ { 2 } )$ . Note that $n _ { L }$ is independent of the size $d$ of the discretized space (e.g. number of pixels) and generally ${ n ^ { 2 } \ll d }$ . Choosing $n _ { L }$ larger than the true number of $L _ { i }$ only results in an over-complete basis and shouldn’t be a problem. We conducted small controlled experiments to verify how multilayer L-conv approximates $\mathbf { G }$ -conv (SI C).
+
+![](images/c8c7405c79a9c40c28b49ba9a2f69384016cff3c9158d9a55a31f7886f000548.jpg)  
+Figure 4: Learning the infinitesimal generator of $S O ( 2 )$ Left shows the architecture for learning rotation angles between pairs of images (SI C.3). Next to it is the $L$ learned using recursive L-conv in this experiment. Middle $L$ is learned using a fixed small rotation angle $\theta = \pi / 1 \bar { 0 }$ , and right shows $L$ found using the numeric solution from the data.
+
+Learning symmetries using L-conv. Rao & Ruderman (1999) introduced a basic version of Lconv and showed that it can learn 1D translation and 2D rotation. We conducted experiments to learn large rotation angle between two images (SI C), shown in Fig. 4. Left shows the architecture for learning the rotation angles between a pair of $7 \times 7$ random images $f$ and $R ( \theta ) f$ with $\theta \in [ 0 , \pi / 3 )$ . Second left is the learned $L \in S O ( { \bar { 2 } } )$ using 3 recursive layer L-conv. Middle is the $L$ learned using L-conv with fixed small rotation angle $\theta = \pi / 1 0$ (SI C.2) and right is the exact solution $R = \overline { { ( Y X ^ { T } ) ( X ^ { T } X ) ^ { - 1 } } }$ . While the middle $L$ is less noisy, it does not capture weights beyond first neighbors of each pixel. (also see SI C for a discussion on symmetry discovery literature.)
+
+L-conv can potentially replace other equivariant layers in a neural network. We conducted limited experiments for this on small image datasets (SI D). L-conv allows one to look for potential symmetries in data which may have been scrambled or harbors hidden symmetries.
+
+# 5 Relation to other architectures
+
+CNN. This is a special case of expressing G-conv as $\mathrm { L }$ -conv when the group is continuous 1D translations. The arguments here generalize trivially to higher dimensions. Rao & Ruderman (1999, sec. 4) used the Shanperiodic 1D arrays as $f _ { \rho } ^ { \prime } = g ( z ) _ { \rho } ^ { \nu } f _ { \nu }$ Inter. He $\begin{array} { r } { g ( z ) _ { \rho } ^ { \nu } = \frac { 1 } { d } \sum _ { p = - d / 2 } ^ { d / 2 } \cos \left( \frac { 2 \pi p } { d } ( z + \rho - \nu ) \right) } \end{array}$ $z$ $g ( z )$ $G$ $g ( w ) g ( z ) = g ( w + z )$ with $g ( 0 ) _ { \rho } ^ { \nu } = \delta _ { \rho } ^ { \nu }$ . For any $z = \mu \in \mathbb { Z }$ , $g _ { \mu } = g ( z = \mu )$ are circulant matrices that shift by $\mu$ as $[ g _ { \mu } ] _ { \nu } ^ { \rho } = \delta _ { \nu - \mu } ^ { \dot { \rho } }$ . Thus, a 1D CNN with kernel size $k$ can be written suing $g _ { \mu }$ as
+
+$$
+F ( { \pmb f } ) _ { \nu } ^ { a } = \sigma \left( \sum _ { \mu = 0 } ^ { k } { \pmb f } _ { \nu - \mu } ^ { c } [ W ^ { \mu } ] _ { c } ^ { a } + b ^ { a } \right) = \sigma \left( \sum _ { \mu = 0 } ^ { k } [ g _ { \mu } { \pmb f } ] _ { \nu } ^ { c } [ W ^ { \mu } ] _ { c } ^ { a } + b ^ { a } \right)
+$$
+
+where $W , b$ are the filter weights and biases. $g _ { \mu }$ can be approximated using the Lie algebra and written as multi-layer $\mathrm { L }$ -conv as in sec. 3.1. Using $g ( 0 ) _ { \rho } ^ { - } \approx \delta ( \rho - \nu )$ , the single Lie algebra basis $[ \hat { L } ] _ { 0 } = \partial _ { z } g ( z ) | _ { z  0 }$ , acts as $\hat { L } f ( z ) \approx - \partial _ { z } f ( z )$ (because $\begin{array} { r } { \int \partial _ { z } \delta ( z - \nu ) f ( z ) = - \partial _ { \nu } f ( \nu ) ) } \end{array}$ . Its components are $\begin{array} { r } { \hat { L } _ { \rho } ^ { \nu } = L ( \rho - \nu ) = \sum _ { p } \frac { 2 \pi p } { d ^ { 2 } } \sin \left( \frac { 2 \pi p } { d } ( \rho - \nu ) \right) } \end{array}$ , which are also circulant due to the $( \rho - \nu )$ dependence. Hence, $\begin{array} { r } { [ \hat { L } \pmb { f } ] _ { \rho } = \sum _ { \nu } L ( \rho - \nu ) \pmb { f } _ { \nu } = [ L \star \pmb { f } ] _ { \nu } } \end{array}$ is a convolution. Rao & Ruderman (1999) already showed that this $\hat { L }$ can reproduce finite discrete shifts $g _ { \mu }$ used in CNN. They used a primitive version of L-conv with $g _ { \mu } = ( I + \epsilon \hat { L } ) ^ { N }$ . Thus, $\mathrm { L }$ -conv can approximate 1D CNN. This result generalizes easily to higher dimensions.
+
+Graph Convolutional Network (GCN). Let $\pmb { A }$ be the adjacency matrix of a graph. In equation 17 if ${ \hat { L } } _ { i } = h ( A )$ , such as $\hat { L } _ { i } = D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ , we obtain a GCN (Kipf & Welling, 2016) $( D _ { \mu \nu } =$ $\delta _ { \mu \nu } \sum _ { \rho } A _ { \mu \rho }$ being the degree matrix). So in the special case where all neighbors of each node $< \mu >$ have the same edge weight, meaning $[ \hat { L } _ { i } ] _ { \mu } ^ { \nu } = [ \hat { L } _ { i } ] _ { \mu } ^ { \rho } , \forall \nu , \rho \in < \mu >$ , equation 8 is uniformly aggregating over neighbors and $\mathrm { L }$ -conv reduces to a GCN. Note that this similarity is not just superficial. In GCN $\mathbf { \nabla } \cdot h ( \mathbf { A } ) = { \hat { L } }$ is in fact a Lie algebra basis. When $\hat { L } = h ( A )$ , the vector field is the flow of isotropic diffusion $d f / d t = h ( A ) f$ from each node to its neighbors. This vector field defines one parameter Lie group with elements $\begin{array} { r } { \dot { g } ( t ) = \exp [ h ( A ) t ] } \end{array}$ . Hence, L-conv for flow groups with a single generator are GCN. These flow groups include Hamiltonian flows and other linear dynamical systems. The main difference between L-conv and GCN is that L-conv can assign a different weight to each neighbor of the same node, similar to GEM-CNN (Cohen et al., 2019a) with a fixed gauge set by $g _ { \mu }$ . Next, we discuss the mathematical properties of the loss functions for L-conv.
+
+# 6 Group invariant loss
+
+Loss functions of equivariant networks are rarely discussed. Yet, recent work by Kunin et al. (2020) showed the existence of symmetry directions in the loss landscape. To understand how the symmetry generators in L-conv manifest themselves in the loss landscape, we work out the explicit example of a mean square error (MSE) loss. Because $G$ is the symmetry group, $f$ and $g \cdot f$ should result in the same optimal parameters. Hence, the minima of the loss function need to be group invariant. One way to satisfy this is for the loss itself to be group invariant, which can be constructed by integrating over $G$ (global pooling (Bronstein et al., 2021)). A function $\begin{array} { r } { I = \int _ { G } d g F ( g ) } \end{array}$ is $G$ -invariant (SI A.3). We can also change the integration to $\textstyle \int _ { S } d ^ { n } x$ by change of variable $d g / d x$ (see SI A.3 for discussion on stabilizers).
+
+MSE loss and Field Theory. The MSE is given by $\begin{array} { r } { I = \sum _ { n } \int _ { G } d g \| Q [ f _ { n } ] ( g ) \| ^ { 2 } } \end{array}$ , where $f _ { n }$ are data samples and $Q [ f ]$ is L-conv or another $G$ -equivariant function. In supervised learning the input is a pair $f _ { n } , y _ { n }$ . $G$ can also act on the labels $y _ { n }$ . We assme that $y _ { n }$ are either also scalar features $y _ { n } : S  \mathbb { R } ^ { m _ { y } }$ with a group action $g \cdot y _ { n } ( x ) \stackrel { \cdot } { = } y _ { n } ( g ^ { - 1 } x )$ (e.g. $f _ { n }$ and $y _ { n }$ are both images), or that $y _ { n }$ are categorical. In the latter case $g \cdot y _ { n } = y _ { n }$ because the only representations of a continuous $G$ on a discrete set are constant. We can concatenate the inputs to $\bar { \phi _ { n } } \equiv [ f _ { n } | y _ { n } ]$ with a well-defined $G$ action $g \cdot \phi _ { n } = [ g \cdot f _ { n } | g \cdot y _ { n } ]$ . The collection of combined inputs $\Phi = ( \phi _ { 1 } , \ldots , \phi _ { N } ) ^ { T }$ is an $( m + m _ { y } ) \times N$ matrix. Using equations 8 and 9, the MSE loss with parameters $W = \{ W ^ { 0 } , { \overline { { \epsilon } } } \}$ becomes (SI A.3.1)
+
+$$
+\begin{array} { l } { { \displaystyle I [ \Phi ; W ] = \int _ { G } d g { \mathcal L } [ \Phi ; W ] = \int _ { G } d g \left\| W ^ { 0 } \left[ I + \overline { { { \epsilon } } } ^ { i } [ \hat { L } _ { i } ] ^ { \alpha } \partial _ { \alpha } \right] \Phi ( g ) \right\| ^ { 2 } } } \\ { { \displaystyle ~ = \int _ { S } \frac { d ^ { n } x } { \left| \frac { \partial x } { \partial g } \right| } \left[ \Phi ^ { T } { \bf m } _ { 2 } \Phi + \partial _ { \alpha } \Phi ^ { T } { \bf h } ^ { \alpha \beta } \partial _ { \beta } \Phi + [ \hat { L } _ { i } ] ^ { \alpha } \partial _ { \alpha } \left( \Phi ^ { T } { \bf v } ^ { i } \Phi \right) \right] } } \end{array}
+$$
+
+Equation 19 generalizes the free field theories in physics (Polyakov, 2018). Here $\left| { \frac { \partial x } { \partial g } } \right|$ is the determinant of the Jacobian, $W ^ { i } = W ^ { 0 } \overline { { { \epsilon } } } ^ { i }$ and
+
+$$
+\begin{array} { r } { { \bf m } _ { 2 } = W ^ { 0 T } W ^ { 0 } , \qquad { \bf h } ^ { \alpha \beta } ( { \bf x } ) = \bar { \epsilon } ^ { i T } { \bf m } _ { 2 } \bar { \epsilon } ^ { j } [ \hat { L } _ { i } ] ^ { \alpha } [ \hat { L } _ { j } ] ^ { \beta } , \qquad { \bf v } ^ { i } = { \bf m } _ { 2 } \bar { \epsilon } ^ { i } . } \end{array}
+$$
+
+Note that $\mathbf { h }$ has feature space indices via $[ \overline { { \epsilon } } ^ { i T } \mathbf { m } _ { 2 } \overline { { \epsilon } } ^ { j } ] _ { a b }$ , with index symmetry $\mathbf { h } _ { a b } ^ { \alpha \beta } = \mathbf { h } _ { b a } ^ { \beta \alpha }$ When (i.e. is a 1D scalar), becomes a a Riemannian metric for . In general h combines a 2-tensor $\mathbf { h } _ { a b } = \mathbf { h } _ { a b } ^ { \alpha \beta } \partial _ { \alpha } \partial _ { \beta } \in T S \otimes T S$ with an inner product $h ^ { T } \mathbf { h } ^ { \alpha \beta } f$ on the feature space $\mathcal { F }$ .
+
+In field theory, the motivation is to preserve spatial symmetries for the metric $\mathbf { h }$ . In equation 19, h transforms equivariantly as a 2-tensor $v \cdot \mathbf { h } ^ { \alpha \beta ^ { \bullet } } = [ v ^ { - \bar { 1 } } ] _ { \rho } ^ { \alpha } [ v ^ { - 1 } ] _ { \gamma } ^ { \beta } \mathbf { h } ^ { \rho \gamma } ( \pmb { x } )$ for $v \in G$ (SI A.3). The last term in equation 19 vanishes for many groups (SI A.3) and it is also absent in physics.
+
+Robustness and Euler-Lagrange Equation. Equivariant neural networks are more robust. To check this, we can quantify how the network would perform for an input $\phi ^ { \prime } = \phi + \delta \phi$ which adds a small random perturbation $\delta \phi$ to a data point $\phi$ . Robustness to such perturbation would mean that, for optimal parameters $W ^ { * }$ , the loss function would not change, i.e. $I [ \phi ^ { \prime } ; W ^ { * } ] = I [ \phi ; W ^ { * } ]$ , requiring $I$ to be minimized around real data points $\phi$ .
+
+This can be cast as a variational equation $\delta I [ \phi ; W ^ { * } ] = 0$ , which yield the familiar Euler-Lagrange (EL) equation (SI A.4). Therefore, for an equivariant network to be robust, i.e. $\delta I [ \phi ; W ^ { * } ] / \bar { \delta { \phi } } = \mathbf { \bar { 0 } }$ we would require the data points $\phi$ to satisfy the EL equations for optimal parameters $W ^ { * }$ :
+
+$$
+{ \mathrm { R o b u s t n e s s ~ t o ~ r a n d o m ~ n o i s e } } \longleftrightarrow \mathrm { E L : } \quad { \frac { \partial { \mathcal { L } } } { \partial \phi ^ { b } } } - \partial _ { \alpha } { \frac { \partial { \mathcal { L } } } { \partial ( \partial _ { \alpha } \phi ^ { b } ) } } = 0
+$$
+
+where the partial derivative terms appear because of the L-conv layer.
+
+Equivariance and Conservation laws. Conserved currents, via Noether’s theorem provide a way to find hidden symmetries (see also Kunin et al. (2020)). The idea is that the equivariance condition equation 2 can be written for the integrand of the loss, $\mathcal { L } [ \phi , W ]$ . If we write the equivariance equation for infinitesimal $v _ { \epsilon }$ , we obtain a vector field which is divergence free. Since $G$ is the symmetry of the system, transforming an input $\phi  w \cdot \phi$ by $w \in G$ the integrand should change equivariantly, meaning $\mathcal { L } [ w \cdot \phi ] = w \cdot \mathcal { L } [ \phi ]$ . When robustness error is minimized as in equation 21, an infinitesimal $w \approx I + \eta ^ { i } L _ { i }$ , with $\delta \phi = \epsilon ^ { i } \hat { L } _ { i } \phi$ , results in a conserved current (SI A.4)
+
+$$
+J ^ { \alpha } = \frac { \partial \mathcal { L } } { \partial ( \partial _ { \alpha } \phi ^ { b } ) } \delta \phi ^ { b } - \frac { \partial \mathcal { L } } { \partial x ^ { \alpha } } \delta x ^ { \alpha } , \qquad \delta I [ \phi ; W ^ { * } ] = 0 \quad \Rightarrow \partial _ { \alpha } J ^ { \alpha } = 0
+$$
+
+The above equation shows that for equivariant networks with a given symmetry, the deviation in data along the symmetry direction $( \hat { L } _ { i } )$ yields a divergence free current $J ^ { \alpha }$ , known as Noether current. It also provides an alternative means to discover symmetry generators $L _ { i }$ by minimizing $\| \partial _ { \alpha } J ^ { \alpha } \|$ . Note that this Noether current is the “stress-energy” tensor, associated with space (or space-time) variations $\delta \mathbfit { x }$ (Landau, 2013) (SI A.5). We can potentially design more general equivariant networks leading to other Noether currents.
+
+# 7 Conclusion and Discussions
+
+We propose the Lie algebra convolutional neural network (L-conv), an infinitesimal version of Gconv. L-conv layers do not require encoding irreps or discretizing the group, and can be combined to approximate any feedforward equivariant networks on compact groups. Additionally, L-conv’s universal and simple structure allows us to discover symmetries from data. It is easy to implement, with a formula similar to GCN. We validated that L-conv can learn the correct Lie algebra basis in a synthetic experiment.
+
+We discover several intriguing connections between L-conv and physics. Our derivation shows that equivariant neural networks based on L-conv lead to Noether’s theorem and conservation laws. Conversely, we can also optimize Noether current to discover symmetries. Furthermore, the current equivariance formulation only pertains to “spatial symmetries” (i.e. $G$ acts on $s$ ). In physics, more general “internal symmetries” are quite common (e.g. particle physics). We can potentially design more general equivariant networks with L-conv encoding such symmetries.
+
+Our method also shed lights on scientific machine learning, especially for physical sciences. Physicists generally use simple polynomial forms for the Lagrangian, or the loss function. These “perturbative” Lagrangian lead to divergences in quantum field theory. However, it is believed the true Lagrangian is more complicated. Hence, more expressive L-conv based models can potentially provide more advanced ansatze for solving scientific problems.
+
+# Acknowledgments and Disclosure of Funding
+
+R. Walters is supported by a Postdoctoral Fellowship from the Roux Institute and NSF grants #2107256 and #2134178. This work was supported in part by the U. S. Army Research Office under Grant W911NF-20-1-0334, DOE ASCR 2493 and NSF Grant #2134274. N. Dehmamy and D. Wang were supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0354.
+
+# References
+
+Anselmi, F., Evangelopoulos, G., Rosasco, L., and Poggio, T. Symmetry-adapted representation learning. Pattern Recognition, 86:201–208, 2019.
+
+Benton, G., Finzi, M., Izmailov, P., and Wilson, A. G. Learning invariances in neural networks. arXiv preprint arXiv:2010.11882, 2020.
+
+Bogatskiy, A., Anderson, B., Offermann, J., Roussi, M., Miller, D., and Kondor, R. Lorentz group equivariant neural network for particle physics. In International Conference on Machine Learning, pp. 992–1002. PMLR, 2020.
+
+Bronstein, M. M., Bruna, J., Cohen, T., and Velickovi ˇ c, P. Geometric deep learning: Grids, groups, ´ graphs, geodesics, and gauges. arXiv preprint arXiv:2104.13478, 2021.   
+Cohen, T. and Welling, M. Learning the irreducible representations of commutative lie groups. In International Conference on Machine Learning, pp. 1755–1763, 2014.   
+Cohen, T. and Welling, M. Group equivariant convolutional networks. In International conference on machine learning, pp. 2990–2999, 2016a.   
+Cohen, T., Weiler, M., Kicanaoglu, B., and Welling, M. Gauge equivariant convolutional networks and the icosahedral cnn. In International Conference on Machine Learning, pp. 1321–1330. PMLR, 2019a.   
+Cohen, T. S. and Welling, M. Steerable cnns. 2016b.   
+Cohen, T. S., Geiger, M., Köhler, J., and Welling, M. Spherical cnns. In International Conference on Learning Representations, 2018.   
+Cohen, T. S., Geiger, M., and Weiler, M. A general theory of equivariant cnns on homogeneous spaces. In Advances in Neural Information Processing Systems, pp. 9142–9153, 2019b.   
+Cybenko, G. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems, 2(4):303–314, 1989.   
+Finzi, M., Stanton, S., Izmailov, P., and Wilson, A. G. Generalizing convolutional neural networks for equivariance to lie groups on arbitrary continuous data. arXiv preprint arXiv:2002.12880, 2020.   
+Finzi, M., Welling, M., and Wilson, A. G. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. arXiv preprint arXiv:2104.09459, 2021.   
+Gens, R. and Domingos, P. M. Deep symmetry networks. Advances in neural information processing systems, 27:2537–2545, 2014.   
+Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chemistry. In International Conference on Machine Learning, pp. 1263–1272. PMLR, 2017.   
+Hall, B. Lie groups, Lie algebras, and representations: an elementary introduction, volume 222. Springer, 2015.   
+Hartford, J., Graham, D., Leyton-Brown, K., and Ravanbakhsh, S. Deep models of interactions across sets. In International Conference on Machine Learning, pp. 1909–1918. PMLR, 2018.   
+Hornik, K., Stinchcombe, M., and White, H. Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366, 1989.   
+Kipf, T. N. and Welling, M. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.   
+Kondor, R. and Trivedi, S. On the generalization of equivariance and convolution in neural networks to the action of compact groups. arXiv preprint arXiv:1802.03690, 2018.   
+Kondor, R., Lin, Z., and Trivedi, S. Clebsch–gordan nets: a fully fourier space spherical convolutional neural network. In Advances in Neural Information Processing Systems, pp. 10117–10126, 2018.   
+Kunin, D., Sagastuy-Brena, J., Ganguli, S., Yamins, D. L., and Tanaka, H. Neural mechanics: Symmetry and broken conservation laws in deep learning dynamics. arXiv preprint arXiv:2012.04728, 2020.   
+Landau, L. D. The classical theory of fields, volume 2. Elsevier, 2013.   
+LeCun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W., and Jackel, L. D. Backpropagation applied to handwritten zip code recognition. Neural computation, 1(4):541–551, 1989.
+
+LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Lee, J. M., Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., et al. Manifolds and differential geometry. Topology, 643:658, 2009.
+
+Pfau, D., Higgins, I., Botev, A., and Racanière, S. Disentangling by subspace diffusion. arXiv preprint arXiv:2006.12982, 2020.
+
+Polyakov, A. M. Gauge fields and strings. Routledge, 2018.
+
+Rao, R. P. and Ruderman, D. L. Learning lie groups for invariant visual perception. In Advances in neural information processing systems, pp. 810–816, 1999.
+
+Ravanbakhsh, S. Universal equivariant multilayer perceptrons. arXiv preprint arXiv:2002.02912, 2020.
+
+Ravanbakhsh, S., Schneider, J., and Poczos, B. Equivariance through parameter-sharing. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2892–2901. JMLR. org, 2017.
+
+Schaub, M. T., Benson, A. R., Horn, P., Lippner, G., and Jadbabaie, A. Random walks on simplicial complexes and the normalized hodge 1-laplacian. SIAM Review, 62(2):353–391, 2020.
+
+Sohl-Dickstein, J., Wang, C. M., and Olshausen, B. A. An unsupervised algorithm for learning lie group transformations. arXiv preprint arXiv:1001.1027, 2010.
+
+Wang, R., Walters, R., and Yu, R. Incorporating symmetry into deep dynamics models for improved generalization. In International Conference on Learning Representations, 2020.
+
+Weiler, M. and Cesa, G. General e (2)-equivariant steerable cnns. In Advances in Neural Information Processing Systems, pp. 14334–14345, 2019.
+
+Weiler, M., Geiger, M., Welling, M., Boomsma, W., and Cohen, T. S. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. In Advances in Neural Information Processing Systems, pp. 10381–10392, 2018a.
+
+Weiler, M., Hamprecht, F. A., and Storath, M. Learning steerable filters for rotation equivariant cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 849–858, 2018b.
+
+Weinberg, S. The quantum theory of fields, volume 3. Cambridge university press, 1995.
+
+Whitaker, E. On the functions which are represented by the expansion of interpolating theory. In Proc. Roy. Soc. Edinburgh, volume 35, pp. 181–194, 1915.
+
+Worrall, D. E., Garbin, S. J., Turmukhambetov, D., and Brostow, G. J. Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5028–5037, 2017.
+
+Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. Deep sets. In Advances in neural information processing systems, pp. 3391–3401, 2017.
+
+Zhou, A., Knowles, T., and Finn, C. Meta-learning symmetries by reparameterization. arXiv preprint arXiv:2007.02933, 2020.
+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+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? [No]   
+(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]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(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]   
+(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? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(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/NPOWF_ZLfC5/NPOWF_ZLfC5_content_list.json

@@ -0,0 +1,1848 @@
+[
+    {
+        "type": "text",
+        "text": "Automatic Symmetry Discovery with Lie Algebra Convolutional Network ",
+        "text_level": 1,
+        "bbox": [
+            200,
+            122,
+            797,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nima Dehmamy Northwestern University nimadt@bu.edu ",
+        "bbox": [
+            261,
+            222,
+            424,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Robin Walters Northeastern University rwalters@northeastern.edu ",
+        "bbox": [
+            517,
+            222,
+            736,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yanchen Liu Northeastern University liu.yanc@northeastern.edu ",
+        "bbox": [
+            212,
+            285,
+            428,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dashun Wang Northwestern University dashun.wang@kellogg.northwestern.edu ",
+        "bbox": [
+            473,
+            285,
+            784,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Rose Yu University of California San Diego roseyu@ucsd.edu ",
+        "bbox": [
+            383,
+            348,
+            614,
+            390
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            426,
+            535,
+            443
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current. These connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences.1 ",
+        "bbox": [
+            233,
+            457,
+            766,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            646,
+            310,
+            664
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Incorporating symmetries into a deep learning architecture can reduce sample complexity, improve generalization, while significantly decreasing the number of model parameters (Cohen et al., 2019b; Cohen & Welling, 2016b; Ravanbakhsh et al., 2017; Ravanbakhsh, 2020; Wang et al., 2020). For instance, Convolutional Neural Networks (CNN) (LeCun et al., 1989, 1998) implement translation symmetry through weight sharing. General principles for constructing symmetry-aware group equivariant neural networks were introduced in Cohen & Welling (2016b), Kondor & Trivedi (2018), and Cohen et al. (2019b). ",
+        "bbox": [
+            174,
+            676,
+            825,
+            773
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, most work on equivariant networks requires knowing the symmetry group a priori. A different equivariant model needs to be re-designed for each symmetry group. In practice, we may not have a good inductive bias and such knowledge of the symmetries may not be available. Constructing and selecting the equivariant network with the appropriate symmetry group becomes quite tedious. Furthermore, many existing works are limited to finite groups such as permutations Hartford et al. (2018); Ravanbakhsh et al. (2017); Zaheer et al. (2017), 90 degree rotations Cohen et al. (2018) or dihedral groups $\\mathrm Ḋ N Ḍ$ and $E ( 2 )$ Weiler & Cesa (2019). ",
+        "bbox": [
+            173,
+            781,
+            826,
+            878
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For a continuous group, existing approaches either discretize the group Weiler et al. (2018a,b); Cohen & Welling (2016a), or use a truncated sum over irreducible representations (irreps) Weiler & Cesa (2019); Weiler et al. (2018a) via spherical harmonics in Worrall et al. (2017) or more general Clebsch-Gordon coefficients Kondor et al. (2018); Bogatskiy et al. (2020). These approaches are prone to approximation error. Recently, Finzi et al. (2020) propose to approximates the integral over the Lie group by Monte Carlo sampling. This approach requires implementing the matrix exponential and obtaining a local neighborhood for each point. Both parametrizing Lie groups for sampling and finding irreps are computationally expensive. Finzi et al. (2021) provide a general algorithm for constructing equivariant multi-layer perceptrons (MLP), but require explicit knowledge of the group to encode its irreps, and solving a set of constraints. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We provide a novel framework for designing equivariant neural networks. We leverage the fact that Lie groups can be constructed from a set of infinitesimal generators, called Lie algebras. A Lie algebra has a finite basis, assuming the group is finite-dimensional. Working with the Lie algebra basis allows us to encode an infinite group without discretizing or summing over irreps. Additionally, all Lie algebras have the same general structure and hence can be implemented the same way. We propose Lie Algebra Convolutional Network (L-conv), a novel architecture that can automatically discover symmetries from data. Our main contributions can be summarized as follows: ",
+        "bbox": [
+            173,
+            236,
+            825,
+            333
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose the Lie algebra convolutional network (L-conv), a building block for constructing group equivariant neural networks. We prove that multi-layer L-conv can approximate group convolutional layers, including CNNs, and find graph convolutional networks to be a special case of L-conv.   \nWe can learn the Lie algebra basis in L-conv, enabling automatic symmetry discovery.   \n• L-conv also reveals interesting connections between physics and learning: equivariant loss generalizes important Lagrangians in field theory; robustness and equivariance can be expressed as Euler-Lagrange equations and Noether currents. ",
+        "bbox": [
+            215,
+            344,
+            825,
+            476
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning symmetries from data has been studied in limited settings for commutative Lie groups as in Cohen & Welling (2014), 2D rotations and translations in Rao & Ruderman (1999), Sohl-Dickstein et al. (2010) or permutations (Anselmi et al., 2019). In the non-commutative case, GeoManCEr (Pfau et al., 2020) uses data points related by small transformations to learn non-abelian Lie groups, but it does not introduce an equivariant layer architecture. (Zhou et al., 2020) propose a general method for symmetry discovery. Yet, their weight-sharing scheme and the symmetry generators are very different from ours. Our approach use much fewer parameters and has a direct interpretation using Lie algebras (SI B.3). Benton et al. (2020) propose Augerino to learn a distribution over data augmentations. It also involves Lie algebras, but is restricted to a subgroup of 2D affine transformations and requires matrix logarithm and sampling (SI B.3). In contrast, our approach is simpler and more general. Our approach uses composition of small transformations to achieve large transformations. In this sense bears some resemblance to symnets (Gens & Domingos, 2014), but the rest of the construction is different. ",
+        "bbox": [
+            173,
+            487,
+            825,
+            666
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Background ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            688,
+            308,
+            705
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We review the core concepts L-conv builds upon: equivariance, group convolution and Lie algebras. ",
+        "bbox": [
+            174,
+            719,
+            823,
+            734
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Notations. Unless explicitly stated, $a$ in $A ^ { a }$ is an index, not an exponent. We use the Einstein summation $\\begin{array} { r } { A ^ { a } B _ { a b } = \\sum _ { a } A ^ { \\dot { a } } B _ { a b } = [ A B ] _ { b } } \\end{array}$ , where a repeated upper and lower index are summed. ",
+        "bbox": [
+            173,
+            741,
+            823,
+            770
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Equivariance. Let $s$ be a topological space on which a Lie group $G$ (continuous group) acts from the left, meaning for all $\\pmb { x } \\in \\ b { S }$ and $g \\in G$ , $g { \\pmb x } \\in { \\mathcal { S } }$ . We refer to $s$ as the base space. Let $\\mathcal { F }$ , the “feature space”, be the vector space $\\mathcal { F } = \\mathbb { R } ^ { m }$ . Each data point is a feature map $f : S  { \\mathcal { F } }$ . The action of $G$ on the input of $f$ induces an action on feature maps. For “scalar” features, for $u \\in G$ , the transformed features $u \\cdot f$ are given by ",
+        "bbox": [
+            173,
+            775,
+            825,
+            845
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/53252c8756ac423d54d291ca4c7484ac549ed8861e83c679e5bb465272096c89.jpg",
+        "text": "$$\nu \\cdot f ( \\pmb { x } ) = f ( u ^ { - 1 } \\pmb { x } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            852,
+            571,
+            869
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Denote the space of all functions from $s$ to $\\mathcal { F }$ by $\\mathcal { F } ^ { S }$ , so that $f \\in \\mathcal { F } ^ { s }$ . Let $F$ be a mapping to a new feature space $\\mathcal { F } ^ { \\prime } = \\mathbb { R } ^ { m ^ { \\prime } }$ , meaning $F : \\mathcal { F } ^ { s } \\to \\mathcal { F } ^ { \\prime } { } ^ { s }$ . We say $F$ is equivariant under $G$ if $G$ acts on ${ \\mathcal { F } } ^ { \\prime }$ ",
+        "bbox": [
+            174,
+            878,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/6c3a9c8f35d4da74c4e308612267a2b9221f619d9ec042857d2aff947d0785e5.jpg",
+        "image_caption": [
+            "Figure 1: Lie group and Lie algebra: Illustration of the group manifold of a Lie group $G$ (left). The Lie algebra ${ \\mathfrak { g } } = T _ { I } G$ is the tangent space at the identity $I$ . $L _ { i }$ are a basis for $T _ { I } G$ . If $G$ is connected, $\\forall g \\in G$ there exist paths like $\\gamma$ from $I$ to $g$ and $g$ can be written as a path-ordered integral $\\begin{array} { r } { g = P \\exp [ \\bar { \\int _ { \\gamma } d t ^ { i } } L _ { i } ] } \\end{array}$ . Base space Right is a schematic of the base space $s$ as a manifold. The lift ${ \\pmb x } = g { \\pmb x } _ { 0 }$ takes ${ \\pmb x } \\in { \\pmb S }$ to $g \\in G$ , and maps the tangent spaces $T _ { x } S  T _ { g } G$ . Each Lie algebra basis $L _ { i } \\in \\mathfrak { g } = T _ { I } G$ generates a vector field $\\hat { L } _ { i }$ on the tangent bundle $T G$ via the pushforward $\\hat { L } _ { i } ( g ) = g L _ { i } g ^ { - 1 }$ . Via the lift, $L _ { i }$ also generates a vector field $\\bar { L } _ { i } = \\hat { L } _ { i } ^ { \\alpha } ( { \\pmb x } ) \\partial _ { \\alpha } = [ g L _ { i } \\bar { \\pmb x _ { 0 } } ] ^ { \\alpha } \\partial _ { \\alpha }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            89,
+            748,
+            231
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and for $u \\in G$ , we have ",
+        "bbox": [
+            174,
+            376,
+            331,
+            390
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3b643ff5ceed4d5f0dfefff9b4004387d6642dae5203ae4a6583b285fbfacac.jpg",
+        "text": "$$\nu \\cdot ( F ( f ) ) = F ( u \\cdot f ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            395,
+            576,
+            412
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Group Convolution. Kondor & Trivedi (2018) showed that $F$ is a linear equivariant map if and only if it performs a group convolution (G-conv). To define G-conv, we first lift $_ { \\textbf { \\em x } }$ to elements in $G$ (Kondor $\\&$ Trivedi, 2018). Specifically, we pick an origin ${ \\pmb x } _ { 0 } \\in { \\mathcal { S } }$ and replace each point ${ \\pmb x } = g { \\pmb x } _ { 0 }$ by $g$ . We will often drop $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ for brevity and write $f ( g ) \\equiv f ( g \\pmb { x } _ { 0 } )$ . Let $\\kappa : G \\to \\mathbb { R } ^ { m ^ { \\prime } } \\otimes \\mathbb { R } ^ { m }$ be a linear transformation from $\\mathcal { F }$ to ${ \\mathcal { F } } ^ { \\prime }$ . G-conv is defined as ",
+        "bbox": [
+            173,
+            415,
+            825,
+            488
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f15948df31c6ea5df00e8dd27123468bd5662675dcb147714d4f57902acd89b.jpg",
+        "text": "$$\n[ \\kappa \\star f ] ( g ) = \\int _ { G } \\kappa ( g ^ { - 1 } v ) f ( v ) d v = \\int _ { G } \\kappa ( v ) f ( g v ) d v ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            489,
+            674,
+            522
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We denote the Haar measure on $G$ as $d v \\equiv d \\mu ( v )$ for brevity. ",
+        "bbox": [
+            174,
+            526,
+            575,
+            541
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Equivariance of G-conv. G-conv in equation 3 is equivariant (Kondor $\\&$ Trivedi, 2018). By definition, for $w \\in G$ we have ",
+        "bbox": [
+            173,
+            546,
+            825,
+            575
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e81a4d11531281edd61f4b14be2b963b8b179c9c5b2b319b2a56f3a4bb42940.jpg",
+        "text": "$$\n\\begin{array} { l } { { [ \\kappa \\star w \\cdot f ] ( g ) = \\displaystyle \\int _ { G } \\kappa ( v ) w \\cdot f ( g v ) d v = \\displaystyle \\int _ { G } \\kappa ( v ) f ( w ^ { - 1 } g v ) d v } } \\\\ { { = [ \\kappa \\star f ] ( w ^ { - 1 } g ) = w \\cdot [ \\kappa \\star f ] ( g ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            578,
+            702,
+            633
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Existing works on equivariance networks implement $\\int _ { G }$ by discretizing the group or summing over irreps. We take a different approach and use the infinitesimal generators of the group. While a Lie group $G$ is infinite, usually it can be generated using a small number of infinitesimal generator, comprising its “Lie algebra”. We use the Lie algebra to introduce a building block to approximate G-conv. Figure 1 visualizes a Lie group, Lie algebra and the concept we discuss below. ",
+        "bbox": [
+            173,
+            636,
+            826,
+            707
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lie algebra. Let $G$ be a Lie group, which includes common continuous groups. Group elements $u \\in G$ infinitesimally close to the identity element $I$ can be written as $u \\stackrel { - } { \\approx } I + \\stackrel { - } \\epsilon ^ { i } L _ { i }$ (note Einstein summation), where $L _ { i } \\in { \\mathfrak { g } }$ with the Lie algebra ${ \\mathfrak { g } } = T _ { I } G$ is the tangent space of $G$ at the identity element. The Lie algebra has the property that it is closed under a Lie bracket $[ \\cdot , \\cdot ] : { \\mathfrak { g } } \\times { \\mathfrak { g } } \\to { \\mathfrak { g } }$ ",
+        "bbox": [
+            173,
+            712,
+            825,
+            768
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db4edd97bd1535240721b6ba05ddf6d7af261107fd508217811e6eaccb2dc4fb.jpg",
+        "text": "$$\n[ L _ { i } , L _ { j } ] = { c _ { i j } } ^ { k } L _ { k } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            772,
+            562,
+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "which is skew-symmetric and satisfies the Jacobi identity. Here the coefficients $c _ { i j } { } ^ { k } \\in \\mathbb { R }$ or $\\mathbb { C }$ are called the structure constants of the Lie algebra. For matrix representations of $\\mathfrak { g }$ , $[ L _ { i } , L _ { j } ] =$ $L _ { i } L _ { j } - L _ { j } L _ { i }$ is the commutator. The $L _ { i }$ are called the infinitesimal generators of the Lie group. ",
+        "bbox": [
+            173,
+            796,
+            825,
+            840
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Exponential map. If the manifold of $G$ is connected 2, an exponential map $\\exp : { \\mathfrak { g } } \\to G$ can be defined such that $\\overset { \\cdot } { g } = \\exp [ t ^ { i } L _ { i } ] \\in G$ . For matrix groups, if $G$ is connected and compact, the matrix exponential is such a map and it is surjective. For most other groups (except ${ \\mathrm { G L } } _ { d } ( \\mathbb { C } )$ and nilpotent groups) it is not surjective. Nevertheless, for any connected group every $g \\in G$ can be written as a product $\\begin{array} { r } { g = \\prod _ { a } \\exp [ t _ { a } ^ { i } L _ { i } ] } \\end{array}$ (Hall, 2015). Making ${ \\dot { t } } _ { a } ^ { i }$ infinitesimal steps $d t ^ { i } ( s )$ tangent to a path $\\gamma$ from $I$ to $g$ on $G$ yields the surjective path-ordered exponential in physics, denoted as $\\begin{array} { r } { g = P \\exp [ \\int _ { \\gamma } d t ^ { i } L _ { i } ] } \\end{array}$ (SI A, and see Time-ordering in Weinberg (1995, p143)). ",
+        "bbox": [
+            176,
+            847,
+            823,
+            877
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            165
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Pushforward. $L _ { i } \\in T _ { I } G$ can be pushed forward to $\\hat { L } _ { i } ( g ) = g L _ { i } g ^ { - 1 } \\in T _ { g } G$ to form a basis for $T _ { g } G$ , satisfying the same Lie algebra $[ \\hat { L } _ { i } ( g ) , \\hat { L } _ { j } ( g ) ] = { c _ { i j } } ^ { k } \\hat { L } _ { k } ( g )$ . The manifold of $G$ together with the set of all $T _ { g } G$ attached to each $g$ forms the tangent bundle $T G$ , a type of fiber bundle (Lee et al., 2009). $\\hat { L } _ { i }$ is a vector field on $_ { T G }$ . The lift maps $\\hat { L } _ { i }$ to an equivalent vector field on $T S$ , which we will also denote by $\\hat { L } _ { i }$ . Figure 1 illustrates the flow of these vector fields on $T G$ and $T S$ . ",
+        "bbox": [
+            173,
+            171,
+            826,
+            252
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Lie Algebra Convolutional Network ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            268,
+            504,
+            287
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can use the Lie algebra basis $L _ { i } \\in { \\mathfrak { g } }$ to construct the Lie group $G$ with the exponential map. Similarly, we show that Lie algebras can also serve as building blocks to construct $\\mathbf { G }$ -conv layers. We propose the Lie algebra convolutional network (L-conv). The key idea is to approximate the kernel $\\kappa ( u )$ using localized kernels which can be constructed using the Lie algebra (Fig. 2). This is possible because the exponential map is a generalization of a Taylor expansion. We show that a G-conv whose kernel is concentrated near the identity can be expanded in the Lie algebra. ",
+        "bbox": [
+            173,
+            300,
+            826,
+            385
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $\\delta _ { \\eta } ( u ) \\in \\mathbb { R }$ denote a normalized localized kernel, meaning $\\begin{array} { r } { \\int _ { G } \\delta _ { \\eta } ( g ) d g = 1 } \\end{array}$ , and with support on a small neighborhood of size $\\eta$ centered around the identity $I$ (i.e., $\\delta _ { \\eta } ( I + \\epsilon ^ { i } L _ { i } ) \\to 0$ if $\\| \\epsilon \\| ^ { 2 } > \\eta ^ { 2 } )$ . We pick $\\delta _ { \\eta } ( v _ { \\epsilon } ) \\sim \\theta ( \\eta ^ { 2 } - \\| \\epsilon \\| ^ { 2 } )$ , for $v _ { \\epsilon } = I + \\epsilon ^ { i } L _ { i } \\in T _ { I } G$ and $\\delta _ { \\eta } ( v ) = 0$ for all other $v \\notin T _ { I } G \\left( \\theta ( \\cdot ) \\right.$ being the Heaviside step function). Let $\\kappa _ { 0 } : G \\to \\mathbb { R } ^ { m ^ { \\prime } } \\otimes \\mathbb { R } ^ { m }$ be given by ",
+        "bbox": [
+            173,
+            390,
+            826,
+            454
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cbb59ba3ac440b30f93c9cf114d50c2a0296a8be1eb67414edf14fda4c9ead32.jpg",
+        "text": "$$\n\\left[ \\kappa _ { 0 } \\right] _ { a } ^ { b } ( u ) = \\left[ W ^ { 0 } \\right] _ { a } ^ { c } \\delta _ { \\eta } \\left( u \\left( I - \\left[ \\bar { \\epsilon } ^ { i } \\right] _ { c } ^ { b } L _ { i } \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            458,
+            640,
+            484
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $W ^ { 0 } \\in \\mathbb { R } ^ { m ^ { \\prime } } \\otimes \\mathbb { R } ^ { h }$ and $\\overline { { \\epsilon } } ^ { i } \\in \\mathbb { R } ^ { h } \\otimes \\mathbb { R } ^ { m }$ are constants, and we choose $\\vert [ \\overline { { { \\epsilon } } } ^ { i } ] _ { b } ^ { a } \\vert < \\eta$ . Note that $\\begin{array} { r } { ( I + \\epsilon ^ { i } L _ { i } ) ( I - \\overline { { \\epsilon } } ^ { j } L _ { j } ) = I + [ \\epsilon - \\overline { { \\epsilon } } ] ^ { i } L _ { i } + O ( \\eta ^ { 2 } ) } \\end{array}$ . Therefore, ",
+        "bbox": [
+            173,
+            491,
+            823,
+            525
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3bcd9110716995c1f77772c3935ba98c968c7cd829be7d41831680f97c7c7fcd.jpg",
+        "text": "$$\n\\int \\epsilon ^ { i } d \\epsilon \\delta _ { \\eta } \\left( ( I + \\epsilon ^ { i } L _ { i } ) ( I - \\overline { { { \\epsilon } } } ^ { j } L _ { j } ) \\right) = \\overline { { { \\epsilon } } } ^ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            530,
+            627,
+            563
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The localized kernels $\\kappa _ { 0 }$ can be used to approximate G-conv. ",
+        "bbox": [
+            173,
+            566,
+            578,
+            582
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Linear expansion of $\\mathbf { G }$ -conv with localized kernel. We can expand a G-conv whose kernel is $\\kappa _ { 0 } ( u ) = \\bar { W } ^ { 0 } \\delta _ { \\eta } ( u )$ in the Lie algebra of $G$ to linear order. With $v _ { \\epsilon } \\stackrel { \\textstyle \\mathsf { \\bar { \\alpha } } } { = } I + \\epsilon ^ { i } L _ { i }$ , we have (see SI A) ",
+        "bbox": [
+            174,
+            587,
+            825,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab8c701544dec16a18899e6e07b53ade55d5f18993996ff8b4900ab28287aac7.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle Q [ f ] ( g ) = [ \\kappa _ { 0 } \\star f ] ( g ) = \\int _ { G } d v \\kappa _ { 0 } ( v ) f ( g v ) = \\int _ { \\| \\epsilon \\| < \\eta } d v _ { \\epsilon } \\kappa _ { 0 } ( v _ { \\epsilon } ) f ( g v _ { \\epsilon } ) } } \\\\ { ~ } \\\\ { { \\displaystyle ~ = W ^ { 0 } \\int d \\epsilon \\delta _ { \\eta } ( v _ { \\epsilon } ) \\left[ f ( g ) + \\epsilon ^ { i } g L _ { i } \\cdot \\frac d { d g } f ( g ) + O ( \\epsilon ^ { 2 } ) \\right] } } \\\\ { { \\displaystyle ~ = W ^ { 0 } \\left[ I + \\bar { \\epsilon } ^ { i } g L _ { i } \\cdot \\frac d { d g } \\right] f ( g ) + O ( \\eta ^ { 2 } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            621,
+            735,
+            729
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with $W ^ { 0 } \\in \\mathbb { R } ^ { m ^ { \\prime } } \\otimes \\mathbb { R } ^ { h }$ and $\\overline { { \\epsilon } } ^ { i } \\in \\mathbb { R } ^ { h } \\otimes \\mathbb { R } ^ { m }$ , as before. Here $d \\epsilon$ is the integration measure on the Lie algebra ${ \\mathfrak { g } } = T _ { I } G$ induced by the Haar measure $d v _ { \\epsilon }$ on $G$ . ",
+        "bbox": [
+            173,
+            733,
+            823,
+            763
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Interpreting the derivatives. In a matrix representation of $G$ , we have $\\begin{array} { r } { g L _ { i } \\cdot \\frac { d f } { d g } = [ g L _ { i } ] _ { \\alpha } ^ { \\beta } \\frac { d f } { d g _ { \\alpha } ^ { \\beta } } = } \\end{array}$ $\\operatorname { T r } \\left[ [ g L _ { i } ] ^ { T } { \\frac { d f } { d g } } \\right]$ . This can be written in terms of partial derivatives $\\partial _ { \\alpha } f ( \\pmb x ) = \\partial f / \\partial \\pmb x ^ { \\alpha }$ as follows. Using $\\pmb { x } ^ { \\rho } = g _ { \\sigma } ^ { \\rho } \\pmb { x } _ { 0 } ^ { \\sigma }$ , we have $\\begin{array} { r } { \\frac { d f ( g x _ { 0 } ) } { d g _ { \\beta } ^ { \\alpha } } = { \\pmb x } _ { 0 } ^ { \\beta } \\partial _ { \\alpha } f ( { \\pmb x } ) } \\end{array}$ , and so ",
+        "bbox": [
+            173,
+            770,
+            826,
+            837
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14d479da8bc3c0231806fa01842bd7738803393e97af38cf54e4abf8c26570cf.jpg",
+        "text": "$$\n\\hat { L } _ { i } f ( \\pmb { x } ) \\equiv g L _ { i } \\cdot \\frac { d f } { d g } = [ g L _ { i } ] _ { \\beta } ^ { \\alpha } \\pmb { x } _ { 0 } ^ { \\beta } \\partial _ { \\alpha } f ( \\pmb { x } ) = [ g L _ { i } \\pmb { x } _ { 0 } ] \\cdot \\nabla f ( \\pmb { x } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            842,
+            699,
+            876
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Hence, for each $L _ { i }$ , the pushforward $g L _ { i } g ^ { - 1 }$ generates a flow on $s$ through the vector field $\\hat { L } _ { i } \\equiv { }$ $g L _ { i } \\cdot d / d g = [ g L _ { i } g ^ { - 1 } { \\pmb x } ] ^ { \\bar { \\alpha } } \\partial _ { \\alpha }$ (Fig. 1). ",
+        "bbox": [
+            171,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/66a444b0af7483094d85e1019feae8039ad1ee494275c1042f0548072df9fede.jpg",
+        "image_caption": [
+            "Figure 2: Sketch of the procedure for approximating G-conv using L-conv. First, the kernel is written as the sum of a number of localized kernels $\\kappa _ { k }$ with support around $u _ { k }$ (left). Each of the $\\kappa _ { k }$ is then moved toward identity by composing multiple L-conv layers $Q _ { \\epsilon ^ { \\prime } } \\circ Q _ { \\epsilon } \\dots \\kappa _ { k }$ (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            92,
+            751,
+            194
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lie algebra convolutional (L-conv) layer. Equation 8 states that for a kernel localized near the identity, the effect of the kernel can be summarized in $W ^ { 0 }$ and $\\overline { { \\epsilon } } ^ { i } \\hat { L } _ { i }$ . Note that we do not need to perform the integral over $G$ explicitly anymore. Instead of working with a kernel $\\kappa _ { 0 }$ , we only need to specify $W ^ { 0 }$ and $\\overline { { \\epsilon } } ^ { i }$ . Hence, in general, we define the Lie algebra convolution (L-conv) as ",
+        "bbox": [
+            173,
+            275,
+            825,
+            334
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc6261404ce50c4de41d23e64abf060e0e8a697115ab3b1e1ea6a9ab648e7e47.jpg",
+        "text": "$$\n\\begin{array} { l } { { Q [ f ] ( { \\pmb x } ) = W ^ { 0 } \\left[ I + \\overline { { { \\epsilon } } } ^ { i } \\hat { L } _ { i } \\right] f ( { \\pmb x } ) } } \\\\ { { \\ = W ^ { 0 } \\left[ I + \\overline { { { \\epsilon } } } ^ { i } [ g L _ { i } { \\pmb x } _ { 0 } ] ^ { \\alpha } \\partial _ { \\alpha } \\right] f ( { \\pmb x } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            340,
+            638,
+            390
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Being an expansion of G-conv, L-conv inherits the equivariance of $\\mathbf { G }$ -conv, as we show next. ",
+        "bbox": [
+            173,
+            398,
+            779,
+            414
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 (Equivariance of L-conv). With assumptions above, $L$ -conv is equivariant under $G$ ",
+        "bbox": [
+            173,
+            416,
+            812,
+            433
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof: First, note that the components of $\\hat { L } _ { i }$ transform as $[ \\hat { L } _ { i } ( v \\pmb { x } ) ] ^ { \\alpha } = [ v g L _ { i } \\pmb { x } _ { 0 } ] ^ { \\alpha } = v _ { \\beta } ^ { \\alpha } \\hat { L } _ { i } ( \\pmb { x } ) ^ { \\beta }$ , while the partial transforms as $\\partial / \\partial [ v \\pmb { x } ] ^ { \\alpha } = [ v ^ { - 1 } ] _ { \\alpha } ^ { \\gamma } \\partial _ { \\gamma }$ . As a result in $\\hat { L } _ { i } = [ g L _ { i } { \\pmb x } _ { 0 } ] ^ { \\alpha } \\partial _ { \\alpha }$ all factors of $v$ cancel, meaning for $v \\in G$ , $\\hat { L } _ { i } ( v \\pmb { x } ) = \\hat { L } _ { i } ( \\pmb { x } )$ . This is because of the fact that $\\hat { L } _ { i } \\in T S$ is a vector field (i.e. 1-tensor) and, thus, invariant under change of basis. Plugging into equation 10, for $w \\in G$ ",
+        "bbox": [
+            174,
+            443,
+            825,
+            510
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6fb1cdfaa7491afccd1d33d2bcd9e434bb8c49f0f3133f980591f5ae04c7e7d2.jpg",
+        "text": "$$\n\\begin{array} { l } { w \\cdot Q [ f ] ( { \\pmb x } ) = Q [ f ] ( w ^ { - 1 } { \\pmb x } ) = W ^ { 0 } \\left[ I + \\bar { \\epsilon } ^ { i } \\hat { L } _ { i } ( w ^ { - 1 } { \\pmb x } ) \\right] f ( w ^ { - 1 } { \\pmb x } ) } \\\\ { = W ^ { 0 } \\left[ I + \\bar { \\epsilon } ^ { i } \\hat { L } _ { i } ( g ) \\right] f ( w ^ { - 1 } { \\pmb x } ) = W ^ { 0 } \\left[ I + \\bar { \\epsilon } ^ { i } \\hat { L } _ { i } ( g ) \\right] w \\cdot f ( { \\pmb x } ) = Q [ w \\cdot f ] ( { \\pmb x } ) ( \\bar { \\epsilon } ^ { i } - { \\pmb x } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            513,
+            800,
+            570
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which proves L-conv is equivariant. ",
+        "bbox": [
+            174,
+            574,
+            406,
+            589
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Examples. Using equation 9 we can calculate $\\mathrm { L }$ -conv for specific groups (details in SI A.2). For translations $G = T _ { n } = ( \\mathbb { R } ^ { n } , + )$ , we find the generators become simple partial derivatives $\\hat { L } _ { i } = \\partial _ { i }$ (SI A.2.2), yielding $f ( \\pmb { x } ) + \\epsilon ^ { \\alpha } \\partial _ { \\alpha } f ( \\pmb { x } )$ . For 2D rotations (SI A.2.1) the generator $\\hat { L } \\equiv ( x \\partial _ { y } - y \\partial _ { x } ) = \\partial _ { \\theta }$ , which is the angular momentum operator about the $\\mathbf { Z }$ -axis in quantum mechanics and field theories. For rotations with scaling, $G = S O ( 2 ) \\times \\mathbb { R } ^ { + }$ , we have two $L _ { i }$ , one $\\hat { L } _ { \\theta } = \\partial _ { \\theta }$ from $s o ( 2 )$ and a scaling with $L _ { r } = I$ , yielding $\\hat { L } _ { r } = x \\partial _ { x } + y \\partial _ { y } = r \\partial _ { r }$ . Next, we discuss the form of L-conv on discrete data. ",
+        "bbox": [
+            173,
+            594,
+            826,
+            689
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Approximating G-conv using L-conv ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            703,
+            467,
+            719
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "L-conv can be used as a basic building block to construct G-conv with more general kernels. Figure 2 sketches the argument described here (see also SI A.1). ",
+        "bbox": [
+            173,
+            728,
+            823,
+            757
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (G-conv from L-convs). $G$ -conv equation 3 can be approximated using $L$ -conv layers. ",
+        "bbox": [
+            171,
+            761,
+            813,
+            776
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof: The procedure involves two steps, as illustrated in Fig. 2: 1) approximate the kernel using localized kernels as the $\\delta _ { \\eta }$ in L-conv; 2) move the kernels towards identity using multiple L-conv layers. The following lemma outline the details. \u0003 ",
+        "bbox": [
+            173,
+            786,
+            825,
+            829
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 (Approximating the kernel). Let the kernel $\\kappa : G \\to { \\mathcal { F } } ^ { \\prime } \\otimes { \\mathcal { F } }$ with $\\textstyle \\int _ { G } \\| \\kappa ( g ) \\| ^ { 2 } d g < \\infty$ be continuously differentiable with $\\| d \\kappa ( g ) / d g \\| ^ { 2 } < \\xi ^ { 2 }$ , and with compact support over $G _ { 0 } \\subset G$ . Let The $\\kappa _ { k } ( g ) = c _ { k } \\delta _ { \\eta } ( u _ { k } ^ { - 1 } g )$ set oand $N$ els with ssuch that $\\eta$ neighborhoodapproximates f , $u _ { k } \\in G$ $c _ { k } \\in \\mathcal { F } ^ { \\prime } \\otimes \\mathcal { F }$ $u _ { k } \\in G$ $\\begin{array} { r } { \\tilde { \\kappa } = \\sum _ { k = 1 } ^ { N } \\kappa _ { k } } \\end{array}$ $\\kappa _ { \\ast }$ $\\begin{array} { r } { \\int _ { G } \\| \\kappa ( g ) - \\tilde { \\kappa } ( g ) \\| ^ { 2 } d g < \\zeta ^ { 2 } } \\end{array}$ for arbitrary small $\\zeta \\in \\mathbb { R } _ { + }$ . ",
+        "bbox": [
+            173,
+            832,
+            825,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof: See SI A.1 for details. The intuition is similar to the universal approximation theorem for neural networks (Hornik et al., 1989; Cybenko, 1989), only generalized to a group manifold instead of $\\mathbb { R }$ . Let $B _ { 0 }$ be the set of $v _ { \\epsilon } = I + \\epsilon ^ { i } \\bar { L _ { i } } \\in \\mathfrak { g }$ , with $\\| \\dot { \\epsilon } \\| ^ { 2 } < \\dot { \\eta } ^ { 2 }$ . Choose a set of $u _ { k } \\in G$ such that the k  k 0   that on small enough neighborhoods neighborhoods $B _ { k } = u _ { k } B _ { 0 } \\subset G$ cover the support $B _ { k } \\subset G$ 0 , for any two $G _ { 0 }$ of $u , v \\in B _ { k }$ $\\kappa$ . The bound we have $\\| d \\kappa ( g ) / d g \\| ^ { 2 } < \\xi ^ { 2 }$ $\\| \\kappa ( \\dot { u } ) - \\ddot { \\kappa } ( v ) \\| ^ { 2 } \\leq \\eta ^ { 2 } \\xi ^ { 2 }$ means where $| G _ { 0 } |$ is the volume of the support of $\\kappa$ . Hence, for $g \\in B _ { k }$ , $\\kappa ( g )$ can be approximated with $\\kappa _ { k } ( g ) = \\kappa ( u _ { k } ) \\delta _ { \\eta } ( u _ { k } ^ { - 1 } g )$ , with normalized localized kernels $\\delta _ { \\eta } ( g )$ , and any element $u _ { k } ~ \\in$ $B _ { k }$ . We show that the approximation error of using $\\tilde { \\kappa } = \\sum _ { k } \\kappa _ { k }$ to approximate $\\kappa$ is bounded by $\\begin{array} { r } { \\int _ { G } d g \\| \\kappa ( g ) - \\tilde { \\kappa } ( g ) \\| ^ { 2 } < | G _ { 0 } | \\eta ^ { 2 } \\xi ^ { 2 } } \\end{array}$ . Any desired error bound $\\zeta$ can then be attained by choosing small enough $\\eta$ for neighborhood sizes. \u0003 ",
+        "bbox": [
+            173,
+            90,
+            825,
+            233
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus, we can approximate a large class of kernels as $\\begin{array} { r } { \\kappa ( g ) \\approx \\sum _ { k } \\kappa _ { k } ( g ) } \\end{array}$ where the local kernels $\\kappa _ { k } ( g ) = c _ { k } \\delta _ { \\eta } ( u _ { k } ^ { - 1 } g )$ have support only on an $\\eta$ neighborhood of $u _ { k } \\in G$ . Here $c _ { k } \\in \\mathbb { R } ^ { m ^ { \\prime } } \\otimes \\mathbb { R } ^ { m }$ are constants and $\\delta _ { \\eta } ( u )$ is as in equation 8. Using this, $\\mathbf { G }$ -conv equation 3 becomes ",
+        "bbox": [
+            174,
+            238,
+            825,
+            285
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b971475c7794b3693ec677d869e50cf82c18ac3c7eaaf356ece1e0f8511ede9d.jpg",
+        "text": "$$\n[ \\kappa \\star f ] ( g ) = \\sum _ { k } c _ { k } \\int d v \\delta _ { \\eta } ( u _ { k } ^ { - 1 } v ) f ( g v ) = \\sum _ { k } c _ { k } [ \\delta _ { \\eta } \\star f ] ( g u _ { k } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            287,
+            714,
+            325
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The kernels $\\kappa _ { k }$ are localized around $u _ { k }$ , whereas in L-conv the kernel is around identity. We can compose L-conv layers to move $\\kappa _ { k }$ from $u _ { k }$ to identity. ",
+        "bbox": [
+            174,
+            327,
+            820,
+            356
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2 (Moving kernels to identity). $\\kappa _ { k }$ can be moved near identity using a multilayer $L$ -conv. ",
+        "bbox": [
+            173,
+            358,
+            812,
+            373
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof: In equation 12, write $u _ { k } = v _ { \\epsilon } u _ { k } ^ { \\prime }$ , with $v _ { \\epsilon } = I + \\epsilon ^ { i } L _ { i } \\in \\mathfrak { g }$ . Using the definition equation 10 an L-conv layer $Q _ { \\epsilon } = I - \\epsilon ^ { i } \\hat { L } _ { i }$ performs a first order Taylor expansion (SI A.1) and so $Q _ { \\epsilon } [ \\delta _ { \\eta } ] ( u _ { \\ \\boldsymbol { k } } ^ { \\prime - 1 } v ) =$ $\\delta _ { \\eta } ( u _ { k } ^ { - 1 } v ) + O ( \\epsilon ^ { 2 } )$ . Thus, applying one $\\mathrm { L }$ -conv layer moves the localized kernel along $v _ { \\epsilon }$ on $G$ . Writing $u _ { k }$ as the product of a set of small group elements $\\textstyle u _ { k } = \\prod _ { a = 1 } ^ { p } v _ { a }$ , with $v _ { a } = I + \\epsilon _ { a } ^ { i } L _ { i } \\in \\mathfrak { g }$ Defining L-conv layers $Q _ { a } = I - \\epsilon _ { a } ^ { i } \\hat { L } _ { i }$ , we can write ",
+        "bbox": [
+            173,
+            380,
+            826,
+            462
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10c193589ef6135211da9a6dfb6d3d35b68ae51baac3e1b85875044372424793.jpg",
+        "text": "$$\n\\kappa _ { k } ( g ) \\approx c _ { k } Q _ { p } \\circ \\cdot \\cdot \\cdot \\circ Q _ { 1 } \\circ \\delta _ { \\eta } ( g )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            464,
+            609,
+            482
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "meaning $\\kappa _ { k }$ localized around $u _ { k }$ can be written as a $p$ layer L-conv acting on a kernel $\\delta _ { \\eta } ( g )$ , localized around the identity of the group. With $\\| \\epsilon _ { a } \\| < \\eta$ , the error in $u _ { k }$ is $O ( \\eta ^ { p + 1 } )$ . \u0003 ",
+        "bbox": [
+            174,
+            484,
+            820,
+            515
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus, we conclude that any G-conv equation 3 can be approximated by multilayer L-conv. Furthermore, for compact $G$ , using the theorem in Kondor & Trivedi (2018), we can show that any equivariant feedforward neural network can be approximated using multilayer L-conv with nonlinearities. ",
+        "bbox": [
+            174,
+            518,
+            826,
+            563
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Equivariance of nonlinearity. Pointwise nonlinearities give equivariant maps between scalar feature maps. To see this, let $\\sigma : \\mathbb { R }  \\mathbb { R }$ . We extend $\\sigma : { \\mathcal { F } }  { \\mathcal { F } }$ by applying $\\sigma$ component-wise. Let $f : S  { \\mathcal { F } }$ be a scalar feature map (i.e., $g \\cdot f ( { \\pmb x } ) = f ( g ^ { - 1 } { \\pmb x } ) )$ . Then ",
+        "bbox": [
+            173,
+            568,
+            825,
+            611
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9148922662b2bb94d4152cc4252aa6c8616aa4d8fb4850c51b240e9a81a0de00.jpg",
+        "text": "$$\ng \\cdot ( \\sigma \\circ ( f ) ) ( { \\pmb x } ) = \\sigma \\circ ( f ) ( g ^ { - 1 } { \\pmb x } ) = \\sigma \\circ ( g \\cdot f ) ( { \\pmb x } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            613,
+            668,
+            632
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since the composition of equivariant maps is equivariant, given equivariant linear mapping $Q : { \\mathcal { F } } ^ { S } $ $\\mathcal { F } ^ { \\prime } \\mathcal { S }$ (i.e. $g \\cdot { \\bar { Q } } [ f ] = Q [ g \\cdot { \\bar { f } } ] )$ , the layer $f \\mapsto \\bar { \\sigma } \\circ Q [ f ]$ is equivariant. Hence we have the corollary: ",
+        "bbox": [
+            174,
+            633,
+            825,
+            662
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1. Assume $G$ is compact and acts on $s$ transitively. Then any equivariant feedforward neural network (FNN) can be approximated using multilayer $L$ -conv with point-wise nonlinearities. ",
+        "bbox": [
+            173,
+            665,
+            823,
+            694
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof: A FNN is defined as $\\sigma _ { p } \\circ F _ { p } [ \\cdot \\cdot \\cdot [ \\sigma _ { 1 } \\circ F _ { 1 } [ f ] ] ( \\pmb { x } )$ where $F _ { k }$ are linear and $\\sigma _ { k }$ are point-wise nonlinearities. By Theorem 1 of Kondor & Trivedi (2018), any linear layer in the equivariant FNN is a G-conv, which by Theorem 1 can be approximated by multilayer L-conv. Therefore, multilayer L-conv with nonlinearity can approximate any equivariant FNN. \u0003 ",
+        "bbox": [
+            173,
+            700,
+            825,
+            758
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, to our knowledge it is not known whether every equivariant function can be approximated by equivariant FNN for a Lie group $G$ . Hence, the corollary above is not a universal approximation theorem for equivariant scalar functions in terms of L-conv. However, it does show that multilayer $L$ -conv is equally expressive as other equivariant networks. Next, we discuss implementation details. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            821
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Discretized space and implementation: the tensor notation ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            838,
+            692,
+            856
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In many datasets, such as images, $f ( { \\pmb x } )$ is not given as continuous function, but rather as a discrete array, with ${ \\cal S } = \\{ { \\pmb x } _ { 0 } , \\ldots x _ { d - 1 } \\}$ containing $d$ points. Each ${ \\pmb x } _ { \\mu }$ represents a coordinate in higher dimensional space, e.g. on a $1 0 \\times 1 0$ image, $\\scriptstyle { \\mathbf { { \\vec { x } } } } _ { 0 }$ is $( x , y ) = ( 0 , \\dot { 0 } )$ point and ${ \\pmb x } _ { 9 9 }$ is $( x , y ) = ( 9 , { \\bar { 9 } } )$ . ",
+        "bbox": [
+            176,
+            868,
+            825,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Feature maps and group action In the tensor notation, we encode $\\pmb { x } _ { \\pmb { \\mu } } \\in \\mathcal { S }$ as the canonical basis (one-hot) vectors in ${ \\pmb x } _ { \\pmb { \\mu } } \\in \\mathbb { R } ^ { d }$ with $[ { \\pmb x } _ { \\mu } ] _ { \\nu } = \\delta _ { \\mu \\nu }$ (Kronecker delta), e.g. $\\pmb { x } _ { 0 } = ( 1 , 0 , \\dots , 0 )$ . The features become $\\pmb { f } \\in \\mathcal { F } = \\mathbb { R } ^ { d } \\otimes \\mathbb { R } ^ { m }$ , meaning $d \\times m$ tensors, with $f ( \\pmb { x } _ { \\mu } ) = \\pmb { x } _ { \\mu } ^ { T } \\pmb { f } = \\pmb { f } _ { \\mu }$ . Although $s$ is discrete, the group acting on $\\mathcal { F }$ can be continuous (e.g. image rotations). Any $G \\subseteq { \\mathrm { G L } } _ { d } ( \\mathbb { R } )$ of the general linear group (invertible $d \\times d$ matrices) acts on $\\pmb { x } _ { \\pmb { \\mu } } \\in \\mathbb { R } ^ { d }$ and $f \\in { \\mathcal { F } }$ . We define $f ( g \\cdot { \\pmb x } _ { \\mu } ) = { \\pmb x } _ { \\mu } ^ { T } g ^ { T } { \\pmb f } , \\forall g \\in G$ , so that for $w \\in G$ we have ",
+        "bbox": [
+            173,
+            90,
+            825,
+            184
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae45bc94f7cfa2834e2ff3c29dcfb3c1832a778276087f8812064fccf82c7e7c.jpg",
+        "text": "$$\nw \\cdot f ( \\pmb { x } _ { \\mu } ) = f ( \\pmb { w } ^ { - 1 } \\cdot \\pmb { x } _ { \\mu } ) = \\pmb { x } _ { \\mu } ^ { T } \\pmb { w } ^ { - 1 T } \\pmb { f } = [ \\pmb { w } ^ { - 1 } \\pmb { x } _ { \\mu } ] ^ { T } \\pmb { f }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            193,
+            684,
+            214
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Dropping the position $\\scriptstyle { \\pmb { x } } _ { \\mu }$ , the transformed features are matrix product $w \\cdot f = w ^ { - 1 T } f$ . We can write $\\mathbf { G }$ -conv in this notation (SI B). Similarly, we can rewrite $\\mathrm { L }$ -conv equation 8 in the tensor notation. Defining $v _ { \\epsilon } = I + \\overline { { { \\epsilon } } } ^ { i } L _ { i }$ ",
+        "bbox": [
+            174,
+            223,
+            826,
+            266
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d7581013b75cd82cfcef85af1881b1ec70177a3c9630ad900e314d2623e880e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\qquad Q [ \\pmb { f } ] ( g ) = W ^ { 0 } f \\left( g \\left( I + \\overline { { \\epsilon } } ^ { i } L _ { i } \\right) \\right) = \\pmb { x } _ { 0 } ^ { T } \\left( I + \\overline { { \\epsilon } } ^ { i } L _ { i } \\right) ^ { T } g ^ { T } f W ^ { 0 T } } \\\\ & { \\qquad = \\left( \\pmb { x } + \\overline { { \\epsilon } } ^ { i } [ g L _ { i } \\pmb { x } _ { 0 } ] \\right) ^ { T } \\pmb { f } W ^ { 0 T } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            273,
+            702,
+            323
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\hat { L } _ { i } = g L _ { i } { \\pmb x } _ { 0 }$ is exactly the matrix analogue of pushforward vector field $\\hat { L } _ { i }$ in equation 9. The equivariance of L-conv in tensor notation is again evident from the $g ^ { T } f$ , resulting in ",
+        "bbox": [
+            173,
+            330,
+            825,
+            361
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bced48f10cfffcac661b8c58ee0b54983c1f2df4af7ff9a1e6bf8cf3358d23f3.jpg",
+        "text": "$$\nQ [ w \\cdot f ] ( g ) = \\pmb { x } _ { 0 } ^ { T } \\pmb { v } _ { \\epsilon } ^ { T } g ^ { T } w ^ { - 1 T } \\pmb { f } W ^ { 0 T } = Q [ \\pmb { f } ] ( w ^ { - 1 } g ) = w \\cdot Q [ \\pmb { f } ] ( g )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            368,
+            725,
+            388
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Tensor L-conv layer implementation The discrete space $\\mathrm { L }$ -conv equation 15 can be rewritten using the global Lie algebra basis $\\hat { L } _ { i }$ ",
+        "bbox": [
+            169,
+            402,
+            825,
+            434
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd5ab55eeb80891784dc67dbad6cf67263cc328f191a1df5964ed480fa8da7d9.jpg",
+        "text": "$$\nQ [ f ] = \\left( f + \\hat { L } _ { i } f \\bar { \\epsilon } ^ { i } \\right) W ^ { 0 T } , Q [ f ] _ { \\mu } ^ { a } = f _ { \\mu } ^ { b } [ W ^ { 0 T } ] _ { b } ^ { a } + [ \\hat { L } _ { i } ] _ { \\mu } ^ { \\nu } f _ { \\nu } ^ { c } \\left[ W ^ { i } \\right] _ { c } ^ { a }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            441,
+            756,
+            469
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Where $W ^ { i } = W ^ { 0 } \\overline { { { \\epsilon } } } ^ { i }$ , $W ^ { 0 } \\in \\mathbb { R } ^ { m _ { i n } } \\otimes \\mathbb { R } ^ { m _ { o u t } }$ and $\\overline { { \\epsilon } } ^ { i } \\in \\mathbb { R } ^ { m _ { i n } } \\otimes \\mathbb { R } ^ { m _ { i n } }$ are trainable weights. The $\\hat { L } _ { i }$ can be either inserted as inductive bias or they can be learned to discover symmetries. ",
+        "bbox": [
+            171,
+            479,
+            825,
+            508
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To implement L-conv, note that the formula of equation 17 is quite similar to a Graph Convolutional Network (GCN) (Kipf & Welling, 2016). For each $i$ , the shared convolutional weights are $\\overline { { \\epsilon } } ^ { i } W ^ { 0 T }$ and the aggregation function of the GCN, a function of the graph adjacency matrix, is $\\hat { L } _ { i }$ in L-conv. Thus, Lconv can be implemented as GCN modules for each $\\hat { L } _ { i }$ , plus a residual connection for the $f W ^ { 0 T }$ term. ",
+        "bbox": [
+            173,
+            515,
+            519,
+            631
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the schematic of the L-conv layer. In a naive implementation, $\\hat { L } _ { i }$ can be general $d \\times d$ matrices. However, being vector fields generated by the Lie algebra, $\\hat { L } _ { i }$ has a more constrained structure which allows them to be encoded and learned using much fewer parameters than a $d \\times d$ matrix. Specifically, encoding the topology of $s$ as a graph (see SI B.1), the incidence matrix replaces partial derivatives (Schaub et al., 2020) in equation 9 and the $L _ { i }$ become weighting of the edges. This weighting is similar to ",
+        "bbox": [
+            173,
+            636,
+            517,
+            779
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/7da7ef66be0a2de4ec31dd46c4a64379db96e59b7bf42566f6da3a91ec8a4ec1.jpg",
+        "image_caption": [
+            "Figure 3: L-conv layer architecture. $L _ { i }$ only act on the $d$ flattened spatial dimensions, and $W ^ { i }$ only act on the $m _ { i n }$ input features and returns $m _ { o u t }$ output features. For each $i$ , Lconv is analogous to a GCN with $d$ nodes and $m _ { i n }$ features. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            544,
+            516,
+            816,
+            662
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Gauge Equivariant Mesh (GEM) CNN (Cohen et al., 2019a). Indeed, in L-conv the lift ${ \\pmb x } _ { \\mu } = g _ { \\mu } { \\pmb x } _ { 0 }$ fixes the gauge by mapping neighbors of $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ to neighbors of ${ \\pmb x } _ { \\mu }$ . Changing how the discrete $s$ samples an underlying continuous space will change $g _ { \\mu }$ and hence the gauge. ",
+        "bbox": [
+            176,
+            779,
+            823,
+            820
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Choosing the number of $L _ { i }$ . Beside the width of $W ^ { 0 }$ and $\\bar { \\epsilon } ^ { i }$ , the number $n _ { L }$ of $L _ { i }$ is a hyperparameter in L-conv. For instance, if $s$ is a discretization of $n$ dimensional space the symmetry group is likely $G \\subset { \\mathrm { G L } } _ { n } ( \\mathbb { R } ) \\ltimes T _ { n }$ , with $n _ { L } \\sim O ( n ^ { 2 } )$ . Note that $n _ { L }$ is independent of the size $d$ of the discretized space (e.g. number of pixels) and generally ${ n ^ { 2 } \\ll d }$ . Choosing $n _ { L }$ larger than the true number of $L _ { i }$ only results in an over-complete basis and shouldn’t be a problem. We conducted small controlled experiments to verify how multilayer L-conv approximates $\\mathbf { G }$ -conv (SI C). ",
+        "bbox": [
+            174,
+            827,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/c8c7405c79a9c40c28b49ba9a2f69384016cff3c9158d9a55a31f7886f000548.jpg",
+        "image_caption": [
+            "Figure 4: Learning the infinitesimal generator of $S O ( 2 )$ Left shows the architecture for learning rotation angles between pairs of images (SI C.3). Next to it is the $L$ learned using recursive L-conv in this experiment. Middle $L$ is learned using a fixed small rotation angle $\\theta = \\pi / 1 \\bar { 0 }$ , and right shows $L$ found using the numeric solution from the data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            92,
+            815,
+            209
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Learning symmetries using L-conv. Rao & Ruderman (1999) introduced a basic version of Lconv and showed that it can learn 1D translation and 2D rotation. We conducted experiments to learn large rotation angle between two images (SI C), shown in Fig. 4. Left shows the architecture for learning the rotation angles between a pair of $7 \\times 7$ random images $f$ and $R ( \\theta ) f$ with $\\theta \\in [ 0 , \\pi / 3 )$ . Second left is the learned $L \\in S O ( { \\bar { 2 } } )$ using 3 recursive layer L-conv. Middle is the $L$ learned using L-conv with fixed small rotation angle $\\theta = \\pi / 1 0$ (SI C.2) and right is the exact solution $R = \\overline { { ( Y X ^ { T } ) ( X ^ { T } X ) ^ { - 1 } } }$ . While the middle $L$ is less noisy, it does not capture weights beyond first neighbors of each pixel. (also see SI C for a discussion on symmetry discovery literature.) ",
+        "bbox": [
+            173,
+            303,
+            826,
+            415
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "L-conv can potentially replace other equivariant layers in a neural network. We conducted limited experiments for this on small image datasets (SI D). L-conv allows one to look for potential symmetries in data which may have been scrambled or harbors hidden symmetries. ",
+        "bbox": [
+            174,
+            421,
+            826,
+            463
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Relation to other architectures ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            483,
+            460,
+            501
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "CNN. This is a special case of expressing G-conv as $\\mathrm { L }$ -conv when the group is continuous 1D translations. The arguments here generalize trivially to higher dimensions. Rao & Ruderman (1999, sec. 4) used the Shanperiodic 1D arrays as $f _ { \\rho } ^ { \\prime } = g ( z ) _ { \\rho } ^ { \\nu } f _ { \\nu }$ Inter. He $\\begin{array} { r } { g ( z ) _ { \\rho } ^ { \\nu } = \\frac { 1 } { d } \\sum _ { p = - d / 2 } ^ { d / 2 } \\cos \\left( \\frac { 2 \\pi p } { d } ( z + \\rho - \\nu ) \\right) } \\end{array}$ $z$ $g ( z )$ $G$ $g ( w ) g ( z ) = g ( w + z )$ with $g ( 0 ) _ { \\rho } ^ { \\nu } = \\delta _ { \\rho } ^ { \\nu }$ . For any $z = \\mu \\in \\mathbb { Z }$ , $g _ { \\mu } = g ( z = \\mu )$ are circulant matrices that shift by $\\mu$ as $[ g _ { \\mu } ] _ { \\nu } ^ { \\rho } = \\delta _ { \\nu - \\mu } ^ { \\dot { \\rho } }$ . Thus, a 1D CNN with kernel size $k$ can be written suing $g _ { \\mu }$ as ",
+        "bbox": [
+            173,
+            513,
+            826,
+            623
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edfb761c6fdb297f147b83aef76f82d8223a880e309e0c9a645718147633f895.jpg",
+        "text": "$$\nF ( { \\pmb f } ) _ { \\nu } ^ { a } = \\sigma \\left( \\sum _ { \\mu = 0 } ^ { k } { \\pmb f } _ { \\nu - \\mu } ^ { c } [ W ^ { \\mu } ] _ { c } ^ { a } + b ^ { a } \\right) = \\sigma \\left( \\sum _ { \\mu = 0 } ^ { k } [ g _ { \\mu } { \\pmb f } ] _ { \\nu } ^ { c } [ W ^ { \\mu } ] _ { c } ^ { a } + b ^ { a } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            631,
+            727,
+            676
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $W , b$ are the filter weights and biases. $g _ { \\mu }$ can be approximated using the Lie algebra and written as multi-layer $\\mathrm { L }$ -conv as in sec. 3.1. Using $g ( 0 ) _ { \\rho } ^ { - } \\approx \\delta ( \\rho - \\nu )$ , the single Lie algebra basis $[ \\hat { L } ] _ { 0 } = \\partial _ { z } g ( z ) | _ { z  0 }$ , acts as $\\hat { L } f ( z ) \\approx - \\partial _ { z } f ( z )$ (because $\\begin{array} { r } { \\int \\partial _ { z } \\delta ( z - \\nu ) f ( z ) = - \\partial _ { \\nu } f ( \\nu ) ) } \\end{array}$ . Its components are $\\begin{array} { r } { \\hat { L } _ { \\rho } ^ { \\nu } = L ( \\rho - \\nu ) = \\sum _ { p } \\frac { 2 \\pi p } { d ^ { 2 } } \\sin \\left( \\frac { 2 \\pi p } { d } ( \\rho - \\nu ) \\right) } \\end{array}$ , which are also circulant due to the $( \\rho - \\nu )$ dependence. Hence, $\\begin{array} { r } { [ \\hat { L } \\pmb { f } ] _ { \\rho } = \\sum _ { \\nu } L ( \\rho - \\nu ) \\pmb { f } _ { \\nu } = [ L \\star \\pmb { f } ] _ { \\nu } } \\end{array}$ is a convolution. Rao & Ruderman (1999) already showed that this $\\hat { L }$ can reproduce finite discrete shifts $g _ { \\mu }$ used in CNN. They used a primitive version of L-conv with $g _ { \\mu } = ( I + \\epsilon \\hat { L } ) ^ { N }$ . Thus, $\\mathrm { L }$ -conv can approximate 1D CNN. This result generalizes easily to higher dimensions. ",
+        "bbox": [
+            173,
+            683,
+            825,
+            814
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Graph Convolutional Network (GCN). Let $\\pmb { A }$ be the adjacency matrix of a graph. In equation 17 if ${ \\hat { L } } _ { i } = h ( A )$ , such as $\\hat { L } _ { i } = D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ , we obtain a GCN (Kipf & Welling, 2016) $( D _ { \\mu \\nu } =$ $\\delta _ { \\mu \\nu } \\sum _ { \\rho } A _ { \\mu \\rho }$ being the degree matrix). So in the special case where all neighbors of each node $< \\mu >$ have the same edge weight, meaning $[ \\hat { L } _ { i } ] _ { \\mu } ^ { \\nu } = [ \\hat { L } _ { i } ] _ { \\mu } ^ { \\rho } , \\forall \\nu , \\rho \\in < \\mu >$ , equation 8 is uniformly aggregating over neighbors and $\\mathrm { L }$ -conv reduces to a GCN. Note that this similarity is not just superficial. In GCN $\\mathbf { \\nabla } \\cdot h ( \\mathbf { A } ) = { \\hat { L } }$ is in fact a Lie algebra basis. When $\\hat { L } = h ( A )$ , the vector field is the flow of isotropic diffusion $d f / d t = h ( A ) f$ from each node to its neighbors. This vector field defines one parameter Lie group with elements $\\begin{array} { r } { \\dot { g } ( t ) = \\exp [ h ( A ) t ] } \\end{array}$ . Hence, L-conv for flow groups with a single generator are GCN. These flow groups include Hamiltonian flows and other linear dynamical systems. The main difference between L-conv and GCN is that L-conv can assign a different weight to each neighbor of the same node, similar to GEM-CNN (Cohen et al., 2019a) with a fixed gauge set by $g _ { \\mu }$ . Next, we discuss the mathematical properties of the loss functions for L-conv. ",
+        "bbox": [
+            173,
+            818,
+            825,
+            912
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            175
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Group invariant loss ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            194,
+            377,
+            212
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Loss functions of equivariant networks are rarely discussed. Yet, recent work by Kunin et al. (2020) showed the existence of symmetry directions in the loss landscape. To understand how the symmetry generators in L-conv manifest themselves in the loss landscape, we work out the explicit example of a mean square error (MSE) loss. Because $G$ is the symmetry group, $f$ and $g \\cdot f$ should result in the same optimal parameters. Hence, the minima of the loss function need to be group invariant. One way to satisfy this is for the loss itself to be group invariant, which can be constructed by integrating over $G$ (global pooling (Bronstein et al., 2021)). A function $\\begin{array} { r } { I = \\int _ { G } d g F ( g ) } \\end{array}$ is $G$ -invariant (SI A.3). We can also change the integration to $\\textstyle \\int _ { S } d ^ { n } x$ by change of variable $d g / d x$ (see SI A.3 for discussion on stabilizers). ",
+        "bbox": [
+            173,
+            224,
+            826,
+            353
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "MSE loss and Field Theory. The MSE is given by $\\begin{array} { r } { I = \\sum _ { n } \\int _ { G } d g \\| Q [ f _ { n } ] ( g ) \\| ^ { 2 } } \\end{array}$ , where $f _ { n }$ are data samples and $Q [ f ]$ is L-conv or another $G$ -equivariant function. In supervised learning the input is a pair $f _ { n } , y _ { n }$ . $G$ can also act on the labels $y _ { n }$ . We assme that $y _ { n }$ are either also scalar features $y _ { n } : S  \\mathbb { R } ^ { m _ { y } }$ with a group action $g \\cdot y _ { n } ( x ) \\stackrel { \\cdot } { = } y _ { n } ( g ^ { - 1 } x )$ (e.g. $f _ { n }$ and $y _ { n }$ are both images), or that $y _ { n }$ are categorical. In the latter case $g \\cdot y _ { n } = y _ { n }$ because the only representations of a continuous $G$ on a discrete set are constant. We can concatenate the inputs to $\\bar { \\phi _ { n } } \\equiv [ f _ { n } | y _ { n } ]$ with a well-defined $G$ action $g \\cdot \\phi _ { n } = [ g \\cdot f _ { n } | g \\cdot y _ { n } ]$ . The collection of combined inputs $\\Phi = ( \\phi _ { 1 } , \\ldots , \\phi _ { N } ) ^ { T }$ is an $( m + m _ { y } ) \\times N$ matrix. Using equations 8 and 9, the MSE loss with parameters $W = \\{ W ^ { 0 } , { \\overline { { \\epsilon } } } \\}$ becomes (SI A.3.1) ",
+        "bbox": [
+            171,
+            364,
+            826,
+            492
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06a569b1c36c02d23dec87a73f3f0f5ccf782e6c08d79aee149c27dc78353733.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle I [ \\Phi ; W ] = \\int _ { G } d g { \\mathcal L } [ \\Phi ; W ] = \\int _ { G } d g \\left\\| W ^ { 0 } \\left[ I + \\overline { { { \\epsilon } } } ^ { i } [ \\hat { L } _ { i } ] ^ { \\alpha } \\partial _ { \\alpha } \\right] \\Phi ( g ) \\right\\| ^ { 2 } } } \\\\ { { \\displaystyle ~ = \\int _ { S } \\frac { d ^ { n } x } { \\left| \\frac { \\partial x } { \\partial g } \\right| } \\left[ \\Phi ^ { T } { \\bf m } _ { 2 } \\Phi + \\partial _ { \\alpha } \\Phi ^ { T } { \\bf h } ^ { \\alpha \\beta } \\partial _ { \\beta } \\Phi + [ \\hat { L } _ { i } ] ^ { \\alpha } \\partial _ { \\alpha } \\left( \\Phi ^ { T } { \\bf v } ^ { i } \\Phi \\right) \\right] } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            497,
+            733,
+            575
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Equation 19 generalizes the free field theories in physics (Polyakov, 2018). Here $\\left| { \\frac { \\partial x } { \\partial g } } \\right|$ is the determinant of the Jacobian, $W ^ { i } = W ^ { 0 } \\overline { { { \\epsilon } } } ^ { i }$ and ",
+        "bbox": [
+            174,
+            583,
+            825,
+            619
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5909b87714e06938d8933766e26b6adcaa961769f0fbd33bc0c9202c7076f12d.jpg",
+        "text": "$$\n\\begin{array} { r } { { \\bf m } _ { 2 } = W ^ { 0 T } W ^ { 0 } , \\qquad { \\bf h } ^ { \\alpha \\beta } ( { \\bf x } ) = \\bar { \\epsilon } ^ { i T } { \\bf m } _ { 2 } \\bar { \\epsilon } ^ { j } [ \\hat { L } _ { i } ] ^ { \\alpha } [ \\hat { L } _ { j } ] ^ { \\beta } , \\qquad { \\bf v } ^ { i } = { \\bf m } _ { 2 } \\bar { \\epsilon } ^ { i } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            622,
+            761,
+            642
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that $\\mathbf { h }$ has feature space indices via $[ \\overline { { \\epsilon } } ^ { i T } \\mathbf { m } _ { 2 } \\overline { { \\epsilon } } ^ { j } ] _ { a b }$ , with index symmetry $\\mathbf { h } _ { a b } ^ { \\alpha \\beta } = \\mathbf { h } _ { b a } ^ { \\beta \\alpha }$ When (i.e. is a 1D scalar), becomes a a Riemannian metric for . In general h combines a 2-tensor $\\mathbf { h } _ { a b } = \\mathbf { h } _ { a b } ^ { \\alpha \\beta } \\partial _ { \\alpha } \\partial _ { \\beta } \\in T S \\otimes T S$ with an inner product $h ^ { T } \\mathbf { h } ^ { \\alpha \\beta } f$ on the feature space $\\mathcal { F }$ . ",
+        "bbox": [
+            173,
+            650,
+            825,
+            699
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In field theory, the motivation is to preserve spatial symmetries for the metric $\\mathbf { h }$ . In equation 19, h transforms equivariantly as a 2-tensor $v \\cdot \\mathbf { h } ^ { \\alpha \\beta ^ { \\bullet } } = [ v ^ { - \\bar { 1 } } ] _ { \\rho } ^ { \\alpha } [ v ^ { - 1 } ] _ { \\gamma } ^ { \\beta } \\mathbf { h } ^ { \\rho \\gamma } ( \\pmb { x } )$ for $v \\in G$ (SI A.3). The last term in equation 19 vanishes for many groups (SI A.3) and it is also absent in physics. ",
+        "bbox": [
+            176,
+            704,
+            823,
+            747
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Robustness and Euler-Lagrange Equation. Equivariant neural networks are more robust. To check this, we can quantify how the network would perform for an input $\\phi ^ { \\prime } = \\phi + \\delta \\phi$ which adds a small random perturbation $\\delta \\phi$ to a data point $\\phi$ . Robustness to such perturbation would mean that, for optimal parameters $W ^ { * }$ , the loss function would not change, i.e. $I [ \\phi ^ { \\prime } ; W ^ { * } ] = I [ \\phi ; W ^ { * } ]$ , requiring $I$ to be minimized around real data points $\\phi$ . ",
+        "bbox": [
+            173,
+            752,
+            825,
+            823
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This can be cast as a variational equation $\\delta I [ \\phi ; W ^ { * } ] = 0$ , which yield the familiar Euler-Lagrange (EL) equation (SI A.4). Therefore, for an equivariant network to be robust, i.e. $\\delta I [ \\phi ; W ^ { * } ] / \\bar { \\delta { \\phi } } = \\mathbf { \\bar { 0 } }$ we would require the data points $\\phi$ to satisfy the EL equations for optimal parameters $W ^ { * }$ : ",
+        "bbox": [
+            173,
+            828,
+            825,
+            871
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/774ca8e9371d4803fe5c8f190d4ba1382b8edbbd53747d0e708fe199261f082b.jpg",
+        "text": "$$\n{ \\mathrm { R o b u s t n e s s ~ t o ~ r a n d o m ~ n o i s e } } \\longleftrightarrow \\mathrm { E L : } \\quad { \\frac { \\partial { \\mathcal { L } } } { \\partial \\phi ^ { b } } } - \\partial _ { \\alpha } { \\frac { \\partial { \\mathcal { L } } } { \\partial ( \\partial _ { \\alpha } \\phi ^ { b } ) } } = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            876,
+            714,
+            910
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where the partial derivative terms appear because of the L-conv layer. ",
+        "bbox": [
+            174,
+            90,
+            627,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Equivariance and Conservation laws. Conserved currents, via Noether’s theorem provide a way to find hidden symmetries (see also Kunin et al. (2020)). The idea is that the equivariance condition equation 2 can be written for the integrand of the loss, $\\mathcal { L } [ \\phi , W ]$ . If we write the equivariance equation for infinitesimal $v _ { \\epsilon }$ , we obtain a vector field which is divergence free. Since $G$ is the symmetry of the system, transforming an input $\\phi  w \\cdot \\phi$ by $w \\in G$ the integrand should change equivariantly, meaning $\\mathcal { L } [ w \\cdot \\phi ] = w \\cdot \\mathcal { L } [ \\phi ]$ . When robustness error is minimized as in equation 21, an infinitesimal $w \\approx I + \\eta ^ { i } L _ { i }$ , with $\\delta \\phi = \\epsilon ^ { i } \\hat { L } _ { i } \\phi$ , results in a conserved current (SI A.4) ",
+        "bbox": [
+            173,
+            111,
+            825,
+            212
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17c5a80299f8b2f40ee994c01580f07256a4a40c7d88369044fac46ec22840f2.jpg",
+        "text": "$$\nJ ^ { \\alpha } = \\frac { \\partial \\mathcal { L } } { \\partial ( \\partial _ { \\alpha } \\phi ^ { b } ) } \\delta \\phi ^ { b } - \\frac { \\partial \\mathcal { L } } { \\partial x ^ { \\alpha } } \\delta x ^ { \\alpha } , \\qquad \\delta I [ \\phi ; W ^ { * } ] = 0 \\quad \\Rightarrow \\partial _ { \\alpha } J ^ { \\alpha } = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            214,
+            767,
+            248
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The above equation shows that for equivariant networks with a given symmetry, the deviation in data along the symmetry direction $( \\hat { L } _ { i } )$ yields a divergence free current $J ^ { \\alpha }$ , known as Noether current. It also provides an alternative means to discover symmetry generators $L _ { i }$ by minimizing $\\| \\partial _ { \\alpha } J ^ { \\alpha } \\|$ . Note that this Noether current is the ��stress-energy” tensor, associated with space (or space-time) variations $\\delta \\mathbfit { x }$ (Landau, 2013) (SI A.5). We can potentially design more general equivariant networks leading to other Noether currents. ",
+        "bbox": [
+            174,
+            251,
+            825,
+            337
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion and Discussions ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            354,
+            436,
+            372
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We propose the Lie algebra convolutional neural network (L-conv), an infinitesimal version of Gconv. L-conv layers do not require encoding irreps or discretizing the group, and can be combined to approximate any feedforward equivariant networks on compact groups. Additionally, L-conv’s universal and simple structure allows us to discover symmetries from data. It is easy to implement, with a formula similar to GCN. We validated that L-conv can learn the correct Lie algebra basis in a synthetic experiment. ",
+        "bbox": [
+            174,
+            386,
+            826,
+            469
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We discover several intriguing connections between L-conv and physics. Our derivation shows that equivariant neural networks based on L-conv lead to Noether’s theorem and conservation laws. Conversely, we can also optimize Noether current to discover symmetries. Furthermore, the current equivariance formulation only pertains to “spatial symmetries” (i.e. $G$ acts on $s$ ). In physics, more general “internal symmetries” are quite common (e.g. particle physics). We can potentially design more general equivariant networks with L-conv encoding such symmetries. ",
+        "bbox": [
+            173,
+            476,
+            825,
+            559
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our method also shed lights on scientific machine learning, especially for physical sciences. Physicists generally use simple polynomial forms for the Lagrangian, or the loss function. These “perturbative” Lagrangian lead to divergences in quantum field theory. However, it is believed the true Lagrangian is more complicated. Hence, more expressive L-conv based models can potentially provide more advanced ansatze for solving scientific problems. ",
+        "bbox": [
+            174,
+            565,
+            826,
+            635
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            654,
+            553,
+            670
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R. Walters is supported by a Postdoctoral Fellowship from the Roux Institute and NSF grants #2107256 and #2134178. This work was supported in part by the U. S. Army Research Office under Grant W911NF-20-1-0334, DOE ASCR 2493 and NSF Grant #2134274. N. Dehmamy and D. Wang were supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0354. ",
+        "bbox": [
+            174,
+            684,
+            825,
+            753
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            773,
+            266,
+            789
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Anselmi, F., Evangelopoulos, G., Rosasco, L., and Poggio, T. Symmetry-adapted representation learning. Pattern Recognition, 86:201–208, 2019. ",
+        "bbox": [
+            173,
+            796,
+            825,
+            825
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Benton, G., Finzi, M., Izmailov, P., and Wilson, A. G. Learning invariances in neural networks. arXiv preprint arXiv:2010.11882, 2020. ",
+        "bbox": [
+            173,
+            832,
+            823,
+            861
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bogatskiy, A., Anderson, B., Offermann, J., Roussi, M., Miller, D., and Kondor, R. Lorentz group equivariant neural network for particle physics. In International Conference on Machine Learning, pp. 992–1002. PMLR, 2020. ",
+        "bbox": [
+            173,
+            868,
+            823,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bronstein, M. M., Bruna, J., Cohen, T., and Velickovi ˇ c, P. Geometric deep learning: Grids, groups, ´ graphs, geodesics, and gauges. arXiv preprint arXiv:2104.13478, 2021.   \nCohen, T. and Welling, M. Learning the irreducible representations of commutative lie groups. In International Conference on Machine Learning, pp. 1755–1763, 2014.   \nCohen, T. and Welling, M. Group equivariant convolutional networks. In International conference on machine learning, pp. 2990–2999, 2016a.   \nCohen, T., Weiler, M., Kicanaoglu, B., and Welling, M. Gauge equivariant convolutional networks and the icosahedral cnn. In International Conference on Machine Learning, pp. 1321–1330. PMLR, 2019a.   \nCohen, T. S. and Welling, M. Steerable cnns. 2016b.   \nCohen, T. S., Geiger, M., Köhler, J., and Welling, M. Spherical cnns. In International Conference on Learning Representations, 2018.   \nCohen, T. S., Geiger, M., and Weiler, M. A general theory of equivariant cnns on homogeneous spaces. In Advances in Neural Information Processing Systems, pp. 9142–9153, 2019b.   \nCybenko, G. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems, 2(4):303–314, 1989.   \nFinzi, M., Stanton, S., Izmailov, P., and Wilson, A. G. Generalizing convolutional neural networks for equivariance to lie groups on arbitrary continuous data. arXiv preprint arXiv:2002.12880, 2020.   \nFinzi, M., Welling, M., and Wilson, A. G. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. arXiv preprint arXiv:2104.09459, 2021.   \nGens, R. and Domingos, P. M. Deep symmetry networks. Advances in neural information processing systems, 27:2537–2545, 2014.   \nGilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chemistry. In International Conference on Machine Learning, pp. 1263–1272. PMLR, 2017.   \nHall, B. Lie groups, Lie algebras, and representations: an elementary introduction, volume 222. Springer, 2015.   \nHartford, J., Graham, D., Leyton-Brown, K., and Ravanbakhsh, S. Deep models of interactions across sets. In International Conference on Machine Learning, pp. 1909–1918. PMLR, 2018.   \nHornik, K., Stinchcombe, M., and White, H. Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366, 1989.   \nKipf, T. N. and Welling, M. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.   \nKondor, R. and Trivedi, S. On the generalization of equivariance and convolution in neural networks to the action of compact groups. arXiv preprint arXiv:1802.03690, 2018.   \nKondor, R., Lin, Z., and Trivedi, S. Clebsch–gordan nets: a fully fourier space spherical convolutional neural network. In Advances in Neural Information Processing Systems, pp. 10117–10126, 2018.   \nKunin, D., Sagastuy-Brena, J., Ganguli, S., Yamins, D. L., and Tanaka, H. Neural mechanics: Symmetry and broken conservation laws in deep learning dynamics. arXiv preprint arXiv:2012.04728, 2020.   \nLandau, L. D. The classical theory of fields, volume 2. Elsevier, 2013.   \nLeCun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W., and Jackel, L. D. Backpropagation applied to handwritten zip code recognition. Neural computation, 1(4):541–551, 1989. ",
+        "bbox": [
+            169,
+            42,
+            828,
+            917
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. ",
+        "bbox": [
+            169,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lee, J. M., Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., et al. Manifolds and differential geometry. Topology, 643:658, 2009. ",
+        "bbox": [
+            173,
+            135,
+            821,
+            165
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Pfau, D., Higgins, I., Botev, A., and Racanière, S. Disentangling by subspace diffusion. arXiv preprint arXiv:2006.12982, 2020. ",
+        "bbox": [
+            169,
+            179,
+            823,
+            209
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Polyakov, A. M. Gauge fields and strings. Routledge, 2018. ",
+        "bbox": [
+            173,
+            223,
+            570,
+            239
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Rao, R. P. and Ruderman, D. L. Learning lie groups for invariant visual perception. In Advances in neural information processing systems, pp. 810–816, 1999. ",
+        "bbox": [
+            169,
+            255,
+            825,
+            285
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ravanbakhsh, S. Universal equivariant multilayer perceptrons. arXiv preprint arXiv:2002.02912, 2020. ",
+        "bbox": [
+            171,
+            299,
+            825,
+            328
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ravanbakhsh, S., Schneider, J., and Poczos, B. Equivariance through parameter-sharing. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2892–2901. JMLR. org, 2017. ",
+        "bbox": [
+            173,
+            343,
+            826,
+            387
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Schaub, M. T., Benson, A. R., Horn, P., Lippner, G., and Jadbabaie, A. Random walks on simplicial complexes and the normalized hodge 1-laplacian. SIAM Review, 62(2):353–391, 2020. ",
+        "bbox": [
+            169,
+            401,
+            825,
+            431
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sohl-Dickstein, J., Wang, C. M., and Olshausen, B. A. An unsupervised algorithm for learning lie group transformations. arXiv preprint arXiv:1001.1027, 2010. ",
+        "bbox": [
+            173,
+            445,
+            823,
+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Wang, R., Walters, R., and Yu, R. Incorporating symmetry into deep dynamics models for improved generalization. In International Conference on Learning Representations, 2020. ",
+        "bbox": [
+            169,
+            491,
+            825,
+            520
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Weiler, M. and Cesa, G. General e (2)-equivariant steerable cnns. In Advances in Neural Information Processing Systems, pp. 14334–14345, 2019. ",
+        "bbox": [
+            174,
+            535,
+            825,
+            565
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Weiler, M., Geiger, M., Welling, M., Boomsma, W., and Cohen, T. S. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. In Advances in Neural Information Processing Systems, pp. 10381–10392, 2018a. ",
+        "bbox": [
+            176,
+            579,
+            825,
+            623
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Weiler, M., Hamprecht, F. A., and Storath, M. Learning steerable filters for rotation equivariant cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 849–858, 2018b. ",
+        "bbox": [
+            174,
+            637,
+            828,
+            680
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Weinberg, S. The quantum theory of fields, volume 3. Cambridge university press, 1995. ",
+        "bbox": [
+            168,
+            695,
+            754,
+            712
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Whitaker, E. On the functions which are represented by the expansion of interpolating theory. In Proc. Roy. Soc. Edinburgh, volume 35, pp. 181–194, 1915. ",
+        "bbox": [
+            171,
+            727,
+            825,
+            756
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Worrall, D. E., Garbin, S. J., Turmukhambetov, D., and Brostow, G. J. Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5028–5037, 2017. ",
+        "bbox": [
+            173,
+            770,
+            826,
+            814
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. Deep sets. In Advances in neural information processing systems, pp. 3391–3401, 2017. ",
+        "bbox": [
+            169,
+            829,
+            825,
+            859
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zhou, A., Knowles, T., and Finn, C. Meta-learning symmetries by reparameterization. arXiv preprint arXiv:2007.02933, 2020. ",
+        "bbox": [
+            173,
+            873,
+            823,
+            904
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            254,
+            106
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ",
+        "bbox": [
+            176,
+            121,
+            825,
+            176
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Did you include the license to the code and datasets? [Yes] See Section   \n• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   \n• Did you include the license to the code and datasets? [N/A] ",
+        "bbox": [
+            215,
+            186,
+            825,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below. ",
+        "bbox": [
+            174,
+            263,
+            823,
+            308
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            319,
+            339,
+            333
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [No]   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            338,
+            825,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            433,
+            493,
+            448
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(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] ",
+        "bbox": [
+            236,
+            452,
+            733,
+            483
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            212,
+            488,
+            393,
+            501
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(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]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(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] ",
+        "bbox": [
+            238,
+            506,
+            825,
+            625
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            628,
+            823,
+            643
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [N/A]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            647,
+            823,
+            756
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            758,
+            705,
+            773
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            777,
+            825,
+            867
+        ],
+        "page_idx": 12
+    }
+]

parse/train/m4PC1eUknQG/m4PC1eUknQG.md

@@ -0,0 +1,418 @@
+# L2E: LEARNING TO EXPLOIT YOUR OPPONENT
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Opponent modeling is essential to exploit sub-optimal opponents in strategic interactions. One key challenge facing opponent modeling is how to fast adapt to opponents with diverse styles of strategies. Most previous works focus on building explicit models to predict the opponents’ styles or strategies directly. However, these methods require a large amount of data to train the model and lack the adaptability to new opponents of unknown styles. In this work, we propose a novel Learning to Exploit (L2E) framework for implicit opponent modeling. L2E acquires the ability to exploit opponents by a few interactions with different opponents during training so that it can adapt to new opponents with unknown styles during testing quickly. We propose a novel Opponent Strategy Generation (OSG) algorithm that produces effective opponents for training automatically. By learning to exploit the challenging opponents generated by OSG through adversarial training, L2E gradually eliminates its own strategy’s weaknesses. Moreover, the generalization ability of L2E is significantly improved by training with diverse opponents, which are produced by OSG through diversity-regularized policy optimization. We evaluate the L2E framework on two poker games and one grid soccer game, which are the commonly used benchmark for opponent modeling. Comprehensive experimental results indicate that L2E quickly adapts to diverse styles of unknown opponents.
+
+# 1 INTRODUCTION
+
+One core research topic in modern artificial intelligence is creating agents that can interact effectively with their opponents in different scenarios. To achieve this goal, the agents should have the ability to reason about their opponents’ behaviors, goals, and beliefs. Opponent modeling, which constructs the opponents’ models to reason about them, has been extensively studied in past decades (Albrecht & Stone, 2018). In general, an opponent model is a function that takes some interaction history as its input and predicts some property of interest of the opponent. Specifically, the interaction history may contain the past actions that the opponent took in various situations, and the properties of interest could be the actions that the opponent may take in the future, the style of the opponent (e.g., “defensive”, “aggressive”), or its current goals. The resulting opponent model can inform the agent’s decision-making by incorporating the model’s predictions in its planning procedure to optimize its interactions with the opponent. Opponent modeling has already been used in many practical applications, such as dialogue systems (Grosz & Sidner, 1986), intelligent tutor systems (McCalla et al., 2000), and security systems (Jarvis et al., 2005).
+
+Many opponent modeling algorithms vary greatly in their underlying assumptions and methodology. For example, policy reconstruction based methods (Powers & Shoham, 2005; Banerjee & Sen, 2007) explicitly fit an opponent model to reflect the opponent’s observed behaviors. Type reasoning based methods (Dekel et al., 2004; Nachbar, 2005) reuse pre-learned models of several known opponents by finding the one which most resembles the behavior of the current opponent. Classification based methods (Huynh et al., 2006; Sukthankar & Sycara, 2007) build models that predict the play style of the opponent, and employ the counter-strategy, which is effective against that particular style. Some recent works combine opponent modeling with deep learning methods or reinforcement learning methods and propose many related algorithms (He et al., 2016; Foerster et al., 2018; Wen et al., 2018). Although these algorithms have achieved some success, they also have some obvious disadvantages. First, constructing accurate opponent models requires a lot of data, which is problematic since the agent does not have the time or opportunity to collect enough data about its opponent in most applications. Second, most of these algorithms perform well only when the opponents during testing are similar to the ones used for training, and it is difficult for them to adapt to opponents with new styles quickly. More related works on opponent modeling are in Appendix A.1.
+
+To overcome these shortcomings, we propose a novel Learning to Exploit (L2E) framework in this work for implicit opponent modeling, which has two desirable advantages. First, L2E does not build an explicit model for the opponent, so it does not require a large amount of interactive data and eliminates the modeling errors simultaneously. Second, L2E can quickly adapt to new opponents with unknown styles, with only a few interactions with them. The key idea underlying L2E to train a base policy against various styles of opponents by using only a few interactions between them during training, such that it acquires the ability to exploit different opponents quickly. After training, the base policy can quickly adapt to new opponents using only a few interactions during testing. In effect, our L2E framework optimizes for a base policy that is easy and fast to adapt. It can be seen as a particular case of learning to learn, i.e., meta-learning (Finn et al., 2017). The meta-learning algorithm $( c . f . ,$ Appendix A.2 for details), such as MAML (Finn et al., 2017), is initially designed for single-agent environments. It requires manual design of training tasks, and the final performance largely depends on the user-specified training task distribution. The L2E framework is designed explicitly for the multi-agent competitive environments, which generates effective training tasks (opponents) automatically $( c . f . ,$ Appendix A.3 for details). Some recent works have also initially used meta-learning for opponent modeling. Unlike these works, which either use meta-learning to predict the opponent’s behaviors (Rabinowitz et al., 2018) or to handle the non-stationarity problem in multi-agent reinforcement learning (Al-Shedivat et al., 2018), we focus on how to improve the agent’s ability to adapt to unknown opponents quickly.
+
+In our L2E framework, the base policy is explicitly trained such that a few interactions with a new opponent will produce an opponent-specific policy to effectively exploit this opponent, i.e., the base policy has strong adaptability that is broadly adaptive to many opponents. In specific, if a deep neural network models the base policy, then the opponent-specific policy can be obtained by fine-tuning the parameters of the base policy’s network using the new interactive data with the opponent. A critical step in L2E is how to generate effective opponents to train the base policy. The ideal training opponents should satisfy the following two desiderata. 1) The opponents need to be challenging enough (i.e., hard to exploit). By learning to exploit these challenging opponents, the base policy eliminates its weakness and learns a more robust strategy. 2) The opponents need to have enough diversity. The more diverse the opponents during training, the stronger the base policy’s generalization ability is, and the more adaptable the base policy to the new opponents.
+
+To this end, we propose a novel opponent strategy generation (OSG) algorithm, which can produce challenging and diverse opponents automatically. We use the idea of adversarial training to generate challenging opponents. Some previous works have also been proposed to obtain more robust policies through adversarial training and showed that it improves the generalization (Pinto et al., 2017; Pattanaik et al., 2018). From the perspective of the base policy, giving an opponent, the base policy first adjusts itself to obtain an adapted policy, the base policy is then optimized to maximize the rewards that the adapted policy gets when facing the opponent. The challenging opponents are then adversarially generated by minimizing the base policy’s adaptability by automatically generating difficult to exploit opponents. These hard-to-exploit opponents are trained such that even if the base policy adapts to them, the adapted base policy cannot take advantage of them. Besides, our OSG algorithm can further produce diverse training opponents with a novel diversity-regularized policy optimization procedure. In specific, we use the Maximum Mean Discrepancy (MMD) metric (Gretton et al., 2007) to evaluate the differences between policies. The MMD metric is then incorporated as a regularization term into the policy optimization process to obtain a diverse set of opponent policies. By training with these challenging and diverse training opponents, the robustness and generalization ability of our L2E framework can be significantly improved. To summarize, the main contributions of this work are listed bellow in four-fold:
+
+• We propose a novel learning to exploit (L2E) framework to exploit sub-optimal opponents without building explicit models for it. L2E can quickly adapt to a new opponent with unknown style using only a few interactions.   
+We propose to use an adversarial training procedure to generate challenging opponents automatically. These hard to exploit opponents help L2E eliminate its weakness and improve its robustness effectively.
+
+![](images/b707d933bb844758c8d3fff63a1cfa318bae8884d89228a071e70814ad52be3a.jpg)  
+Figure 1: The overview of our proposed L2E framework. The entire training process is based on the idea of adversarial learning (Alg. 1). The base policy training part maximizes the base policy’s adaptability by continually interacting with opponents of different strengths and styles (Section 2.1). The opponent strategy generation part first generates hard-to-exploit opponents for the current base policy (Hard-OSG, see Section 2.2.1), then generates diverse opponent policies to improve the generalization ability of the base policy (Diverse-OSG, see Section 2.2.2). The resulting base policy can fast adapt to completely new opponents with a few interactions.
+
+• We further propose a diversity-regularized policy optimization procedure to generate diverse opponents automatically. The generalization ability of L2E is improved significantly by training with these diverse opponents. We conduct detailed experiments to evaluate the L2E framework in three different environments. The experimental results demonstrate that the base policy trained with L2E quickly exploits a wide range of opponents compared to other algorithms.
+
+# 2 METHOD
+
+In this paper, we propose a novel L2E framework to endow the agents to adapt to diverse opponents quickly. As shown in Fig. 1, L2E mainly consists of two modules, i.e., the base policy training part, and the opponent strategy generation part. In the base policy training part, our goal is to find a base policy that, given the unknown opponent, can fast adapt to it by using only a few interactions. To this end, the base policy is trained to be able to adapt to many opponents. In specific, giving an opponent $O$ , the base policy $B$ first adjusts itself to obtain an adapted policy $B ^ { \prime }$ by using a little interaction data between $O$ and $B$ , the base policy is then optimized to maximize the rewards that $B ^ { \prime }$ gets when facing $O$ . In other words, the base policy has learned how to adapt to its opponents and exploit them quickly.
+
+The opponent strategy generation provides the base policy training part with challenging and diverse training opponents automatically. First, our proposed opponent strategy generation (OSG) algorithm can produce difficult to exploit opponents. In specific, the base policy $B$ first adjusts itself to obtain an adapted policy $B ^ { \prime }$ by using a little interaction data between $O$ and $B$ , the opponent $O$ is then optimized to minimize the rewards that $B ^ { \prime }$ gets when facing $O$ . The resulting opponent $O$ is hard to exploit since even if the base policy $B$ adapts to $O$ , the adapted policy $B ^ { \prime }$ can not take advantage of $O$ . By training with these hard to exploit opponents, the base policy can eliminate its weakness and improve its robustness effectively. Second, our OSG algorithm can further produce diverse training opponents with a novel diversity-regularized policy optimization procedure. More specifically, we first formalize the difference between opponent policies as the difference between the distribution of trajectories induced by each policy. The difference between distributions can be evaluated by the Maximum Mean Discrepancy (MMD) metric (Gretton et al., 2007). Then, MMD is integrated as a regularization term in the policy optimization process to identify various opponent policies. By training with these diverse opponents, the base policy can improve its generalization ability significantly. Next, we introduce these two modules in detail.
+
+# 2.1 BASE POLICY TRAINING
+
+Our goal is to find a base policy $B$ that can fast adapt to an unknown opponent $O$ by updating the parameters of $B$ using only a few interactions between $B$ and $O$ . The key idea is to train the base policy $B$ against many opponents to maximize its payoffs by using only a small amount of interactive data during training, such that it acquires the ability to exploit different opponents quickly. In effect, our L2E framework treats each opponent as a training example. After training, the resulting base policy $B$ can quickly adapt to new and unknown opponents using only a few interactions. Without loss of generality, the base policy $B$ is modeled by a deep neural network in this work, i.e., a parameterized function $\pi _ { \theta }$ with parameters $\theta$ . Similarly, the opponent $O$ for training is also a deep neural network $\pi _ { \phi }$ with parameters $\phi$ . We model the base policy as playing against an opponent in a two-player Markov game (Shapley, 1953). This Markov game $M \ =$ $( \bar { S } , ( A _ { B } , A _ { O } ) , \bar { T } , ( R _ { B } , R _ { O } ) )$ consists of the state space $S$ , the action space $A _ { B }$ and $A _ { O }$ , and a state transition function $T : S \times A _ { B } \times A _ { O } \to \Delta ( S )$ where $\Delta \left( S \right)$ is a probability distribution on $S$ . The reward function $R _ { i } : S \times A _ { B } \times A _ { O } \times S \to \mathbb { R }$ for each player $i \in \{ B , { \dot { O } } \}$ depends on the current state, the next state and both players’ actions. Given a training opponent $O$ whose policy is known and fixed, this two-player Markov game $M$ reduces to a single-player Markov Decision Process (MDP), i.e., $M _ { B } ^ { O } = ( \mathbf { \bar { { S } } } , \mathbf { \bar { { A } } } _ { B } , T _ { B } ^ { O } , R _ { B } ^ { O } ) $ . The state and action space of $\dot { M } _ { B } ^ { O }$ are the same as in $M$ . The transition and reward functions have the opponent policy embedded:
+
+$$
+T _ { B } ^ { O } ( s , a _ { B } ) = T ( s , a _ { B } , a _ { O } ) , \quad R _ { B } ^ { O } ( s , a _ { B } , s ^ { \prime } ) = R _ { B } ( s , a _ { B } , a _ { O } , s ^ { \prime } ) ,
+$$
+
+where the opponent’s action is sampled from its policy $a _ { O } \sim \pi _ { \phi } ( \cdot \mid s )$ . Throughout the paper, $M _ { X } ^ { Y }$ represents a single-player MDP, which is reduced from a two-player Markov game (i.e., player $X$ and player $Y$ ). In this MDP, the player $Y$ is fixed and can be regarded as part of the environment.
+
+Suppose a set of training opponents can be constructed as described ab $\{ O _ { i } \} _ { i = 1 } ^ { N }$ is given. Fore base policy ch tra, i.e., ng opponent is allowed $O _ { i }$ , an MDP query a li $M _ { B } ^ { O _ { i } }$ $B$ $\pi _ { \theta }$   
+number of sample trajectories $\tau$ to adapt to $O _ { i }$ . In our method, the adapted parameters $\theta ^ { O _ { i } }$ of the   
+base policy are computed using one or more gradient descent updates with the sample trajectories   
+$\tau$ . For example, when using one gradient update:
+
+$$
+\theta ^ { O _ { i } } = \theta - \alpha \nabla _ { \theta } \mathcal { L } _ { B } ^ { O _ { i } } ( \pi _ { \theta } ) ,
+$$
+
+$$
+\mathcal { L } _ { B } ^ { O _ { i } } ( \pi _ { \theta } ) = - \mathbb { E } _ { \tau \sim M _ { B } ^ { O _ { i } } } [ \sum _ { t } \gamma ^ { t } R _ { B } ^ { O _ { i } } ( s ^ { ( t ) } , a _ { B } ^ { ( t ) } , s ^ { ( t + 1 ) } ) ] .
+$$
+
+$\tau \sim M _ { B } ^ { O _ { i } }$ represents that the trajectory $\tau = \{ s ^ { ( 1 ) } , a _ { B } ^ { ( 1 ) } , s ^ { ( 2 ) } , \ldots , s ^ { ( t ) } , a _ { B } ^ { ( t ) } , s ^ { ( t + 1 ) } , \ldots \}$ is sampled from the MDP $M _ { B } ^ { O _ { i } }$ , where $s ^ { ( t + 1 ) } \sim T _ { B } ^ { O _ { i } } ( s ^ { ( t ) } , a _ { B } ^ { ( t ) } )$ and $a _ { B } ^ { ( t ) } \sim \pi _ { \theta } ( \cdot \mid s ^ { ( t ) } )$ .
+
+We use $B ^ { O _ { i } }$ to denote the updated base policy, i.e., $\pi _ { \theta } o _ { i }$ . $B ^ { O _ { i } }$ can be seen as an opponent-specific policy, which is updated from the base policy through fast adaptation. Our goal is to find a generalizable base policy whose opponent-specific policy $B ^ { O _ { i } }$ can exploit its opponent $O _ { i }$ as much as possible. To this end, we optimize the parameters $\theta$ of the base policy to maximize the rewards that $\mathbf { \bar { \it B } } ^ { O _ { i } }$ gets when interacting with $O _ { i }$ . More concretely, the learning to exploit objective function is defined as follows:
+
+$$
+\operatorname* { m i n } _ { \theta } \sum _ { i = 1 } ^ { N } \mathcal { L } _ { B ^ { O _ { i } } } ^ { O _ { i } } ( \pi _ { \theta ^ { O _ { i } } } ) = \operatorname* { m i n } _ { \theta } \sum _ { i = 1 } ^ { N } \mathcal { L } _ { B ^ { O _ { i } } } ^ { O _ { i } } ( \pi _ { \theta - \alpha \nabla _ { \theta } \mathcal { L } _ { B } ^ { O _ { i } } ( \pi _ { \theta } ) } ) .
+$$
+
+It is worth noting that the optimization is performed over the base policy’s parameters $\theta$ , whereas the objective is computed using the adapted based policy’s parameters $\theta ^ { \boldsymbol { O } _ { i } }$ . The parameters $\theta$ of the base policy are updated as follows:
+
+$$
+\theta = \theta - \beta \nabla _ { \theta } \sum _ { i = 1 } ^ { N } \mathcal { L } _ { B ^ { O _ { i } } } ^ { O _ { i } } \big ( \pi _ { \theta ^ { O _ { i } } } \big ) .
+$$
+
+In effect, our L2E framework aims to find a base policy that can significantly exploit the opponent with only a few interactions with it (i.e., with a few gradient steps). The resulting base policy has learned how to adapt to different opponents and exploit them quickly. An overall description of the base policy training procedure is shown in Alg. 1. The algorithm consists of three main steps. First, generating hard to exploit opponents through the Hard-OSG module. Second, generating diverse opponent policies through the Diverse-OSG module. Third, training the base policy with these opponents to obtain fast adaptability.
+
+# 2.2 AUTOMATIC OPPONENT GENERATION
+
+Previously, we assumed that the set of opponents had been given. How to automatically generate effective opponents for training is the key to the success of our L2E framework. The training opponents should be challenging enough (i.e., hard to exploit). By learning to exploit these hard-toexploit opponents, the base policy $B$ can eliminate its weakness and become more robust. Besides, they should be sufficiently diverse. The more diverse they are, the stronger the generalization ability of the resulting base policy. We propose a novel opponent strategy generation (OSG) algorithm to achieve these goals.
+
+# 2.2.1 HARD-TO-EXPLOIT OPPONENTS GENERATION
+
+We use the idea of adversarial learning to generate challenging training opponents for the base policy $B$ . From the perspective of the base policy $B$ , giving an opponent $O$ , $B$ first adjusts itself to obtain an adapted policy, i.e., the opponent-specific policy $B ^ { O }$ , the base policy is then optimized to maximize the rewards that $B ^ { O }$ gets when interacting with $O$ . Contrary to the base policy’s goal, we want to find a hard-to-exploit opponent $\widehat { O }$ for the current base policy $B$ , such that even if $B$ adapts to $\widehat { O }$ , the adapted policy $B ^ { \widehat { O } }$ cannot take advantage of $\widehat { O }$ . In other words, the hard-to-exploit opponent $\widehat { O }$ is trained to minimize the rewards that $B ^ { \bar { \hat { O } } }$ gets when interacting with $\widehat { O }$ . The base policy attempts to increase its adaptability by learning to exploit different opponents, while the hardto-exploit opponent adversarially tries to minimize the base policy’s adaptability, i.e., maximize its counter-adaptability.
+
+More concretely, the hard-to-exploit opponent $\widehat { O }$ is also a deep neural network $\pi _ { \widehat { \phi } }$ with randomly initialized parameters $\widehat { \phi }$ . At each training iteration, an MDP $M _ { B } ^ { \widehat { O } }$ can be constructed. The base policy $B$ first query a limited number of trajectories to adapt to $\widehat { O }$ . The parameters $\theta ^ { \hat { O } }$ of the adapted policy $B ^ { \widehat { O } }$ are computed using one gradient descent update,
+
+$$
+\theta ^ { \widehat { O } } = \theta - \alpha \nabla _ { \theta } \mathcal { L } _ { B } ^ { \widehat { O } } ( \pi _ { \theta } ) .
+$$
+
+$$
+\mathcal { L } _ { B } ^ { \widehat { O } } ( \pi _ { \theta } ) = - \mathbb { E } _ { \tau \sim M _ { B } ^ { \widehat { O } } } [ \sum _ { t } \gamma ^ { t } R _ { B } ^ { \widehat { O } } ( s ^ { ( t ) } , a _ { B } ^ { ( t ) } , s ^ { ( t + 1 ) } ) ] .
+$$
+
+The parameters $\widehat { \phi }$ of $\widehat { O }$ is optimized to minimize the rewards that $B ^ { \widehat { O } }$ gets when interacting with $\widehat { O }$ . This is equivalent to maximize the rewards that $\widehat { O }$ gets since we only consider the competitive setting in this work. More concretely, the parameters $\widehat { \phi }$ are updated as follows:
+
+$$
+\widehat { \phi } = \widehat { \phi } - \alpha \nabla _ { \widehat { \phi } } \mathcal { L } _ { \widehat { O } } ^ { B ^ { \widehat { O } } } ( \pi _ { \widehat { \phi } } )
+$$
+
+$$
+{ \mathcal L } _ { \hat { O } } ^ { B ^ { \hat { O } } } ( \pi _ { \widehat { \phi } } ) = - \mathbb { E } _ { \tau ^ { \prime } \sim M _ { \hat { O } } ^ { B ^ { \hat { O } } } } [ \sum _ { t } \gamma ^ { t } R _ { \hat { O } } ^ { B ^ { \hat { O } } } ( s ^ { ( t ) } , a _ { \hat { O } } ^ { ( t ) } , s ^ { ( t + 1 ) } ) ] .
+$$
+
+After several rounds of iteration, we can obtain a hard-to-exploit opponent $\pi _ { \widehat { \phi } }$ for the current base policy $B$ . An overall description of this procedure is shown in Alg. 2.
+
+# 2.2.2 DIVERSE OPPONENTS GENERATION
+
+Training an effective base policy requires not only the hard-to-exploit opponents, but also diverse opponents of different styles. The more diverse the opponents used for training, the stronger the generalization ability of the resulting base policy. From a human player’s perspective, the opponent style is usually defined as different types, such as aggressive, defensive, elusive, etc. The most significant difference between opponents with different styles lies in the actions taken in the same state.
+
+Take poker as an example; different opponents’ styles tend to take different actions when holding the same hand. Based on the above analysis, we formalize the difference between opponent policies as the difference between the distribution of trajectories induced by each policy when interacting with the base policy. We argue that differences in trajectories better capture the differences between different opponent policies.
+
+Formally, given a base policy $B$ , i.e., $\pi _ { \theta }$ and an opponent policy $O _ { i }$ , i.e., $\pi _ { \phi _ { i } }$ , our diversityregularized policy optimization algorithm is to generate a new opponent $O _ { j }$ , i.e., $\pi _ { \phi _ { j } }$ whose style is different from $O _ { i }$ . We first construct two MDPs, i.e., $M _ { O _ { i } } ^ { B }$ and $M _ { O _ { j } } ^ { B }$ , and then sample two sets of trajectories, i.e., $\mathrm { T } _ { i } = \{ \tau \sim M _ { O _ { i } } ^ { B } \}$ and $\mathrm { T } _ { j } = \{ \tau \sim M _ { O _ { j } } ^ { B } \}$ from this two MDPs. The stochasticity in the MDP and the policy will induce a distribution over trajectories. We use the Maximum Mean Discrepancy (MMD) (Gretton et al., 2007) metric $( c . f$ . Appendix $\textrm { C }$ for details) to measure the differences between $\mathrm { T } _ { i }$ and $\mathrm { T } _ { j }$ :
+
+$$
+\begin{array} { r } { \mathbf { M M D } ^ { 2 } ( \mathrm { T } _ { i } , \mathrm { T } _ { j } ) = \mathbb { E } _ { \tau , \tau ^ { \prime } \sim M _ { \mathcal { O } _ { i } } ^ { B } } k \left( \tau , \tau ^ { \prime } \right) - 2 \mathbb { E } _ { \tau \sim M _ { \mathcal { O } _ { i } } ^ { B } , \tau ^ { \prime } \sim M _ { \mathcal { O } _ { j } } ^ { B } } k \left( \tau , \tau ^ { \prime } \right) + \mathbb { E } _ { \tau , \tau ^ { \prime } \sim M _ { \mathcal { O } _ { j } } ^ { B } } k \left( \tau , \tau ^ { \prime } \right) . } \end{array}
+$$
+
+$k$ is the Gaussian radial basis function kernel defined over a pair of trajectories:
+
+$$
+k ( \tau , \tau ^ { \prime } ) = \exp ( - \frac { \| g ( \tau ) - g ( \tau ^ { \prime } ) \| ^ { 2 } } { 2 } ) ,
+$$
+
+where $g$ stacks the states and actions of a trajectory into a vector. For trajectories with different length, we clip the long trajectory to the same length as the short one. There overall objective function of our proposed diversity-regularized policy optimization algorithm is as follows:
+
+$$
+\mathcal { L } ^ { \phi _ { i } } ( \phi _ { j } ) = - \mathbb { E } _ { \tau \sim M _ { \partial _ { j } } ^ { B } } [ \sum _ { t } \gamma ^ { t } R _ { O _ { j } } ^ { B } ( s ^ { ( t ) } , a _ { O _ { j } } ^ { ( t ) } , s ^ { ( t + 1 ) } ) ] - \alpha _ { m m d } \mathrm { M M D } ^ { 2 } ( \mathrm { T } _ { i } , \mathrm { T } _ { j } ) .
+$$
+
+The first term is to maximize the rewards that $O _ { j }$ gets when interacting with the base policy $B$ . The second term measures the difference between $O _ { j }$ and the existing opponent $O _ { i }$ . By this diversityregularized policy optimization, the resulting opponent $O _ { j }$ is not only useful in performance but also diverse relative to the existing policy.
+
+We can iteratively apply the above algorithm to find a set of $N$ distinct and diverse opponents. In specific, subsequent opponents are learned by encouraging diversity with respect to previously generated opponent set $S$ . The distance between an opponent $O _ { m }$ and an opponent set $S$ is defined by the distance between have obtained a set of ocan be obtained by opti $O _ { m }$ and nentsg: $O _ { n }$ $O _ { n } \in S$ most s. The olicy to -th opp $O _ { m }$ . Suppt, i.e., $S = \{ O _ { m } \} _ { m = 1 } ^ { M } , M < N$ $M + 1$ $\pi _ { \phi _ { M + 1 } }$
+
+$$
+\mathcal { L } ^ { S } ( \phi _ { M + 1 } ) = - \mathbb { E } _ { \tau \sim M _ { \mathcal { O } _ { M + 1 } } ^ { B } } [ \sum _ { t } \gamma ^ { t } R _ { \mathcal { O } _ { M + 1 } } ^ { B } ( s ^ { ( t ) } , a _ { \mathcal { O } _ { M + 1 } } ^ { ( t ) } , s ^ { ( t + 1 ) } ) ] - \operatorname* { m i n } _ { \mathcal { O } _ { i } \in S } \mathrm { M M D } ^ { 2 } ( \mathrm { T } _ { i } , \mathrm { T } _ { M + 1 } ) .
+$$
+
+By doing so, the resulting $M + 1$ -th opponent remains diverse relative to the opponent set $S$ . An overall description of this procedure is shown in Alg. 3.
+
+# 3 EXPERIMENTS
+
+In this section, we conduct extensive experiments to evaluate the proposed L2E framework. We evaluate algorithm performance on the Leduc poker, the BigLeduc poker and a Grid Soccer environment, which are the commonly used benchmark for opponent modeling (Lanctot et al., 2017; Steinberger, 2019; He et al., 2016). We first verify the trained base policy using our L2E framework can fast exploit a wide range of opponents with only a few gradient updates. Then, we compare with other baseline methods to show the superiority of our L2E framework. Finally, we conduct a series of ablation experiments to demonstrate each part of our L2E framework’s effectiveness.
+
+# 3.1 RAPID ADAPTABILITY
+
+In this section, we verify the trained base policy’s ability to quickly adapt to different opponents in the Leduc poker environment $_ { . c . f }$ . Appendix $\mathrm { D }$ for details). We provide four opponents with different styles and strengths. 1) The random opponent randomly takes actions whose strategy is relatively weak but hard to exploit since it does not have an evident decision-making style. 2) The call opponent always takes call actions and has a fixed decision-making style that is easy to exploit. 3) The rocks opponent takes actions based on its hand-strength whose strategy is relatively strong. 4) The oracle opponent is a cheating, and the strongest player who can see the other players’ hands and make decisions based on this perfect information. As shown in Fig. 2, the base policy achieves a
+
+![](images/02cf3677b5c06c9ea5edc630829fc0af50cedbe0c7fd4b2e330cb28afec856ac.jpg)
+
+Figure 2: The trained base policy using our L2E framework can quickly adapt to different opponents of different styles and strengths in the Leduc poker environment.
+
+rapid increase in its average returns with a few gradient updates against all four opponent strategies. For the call opponent, which has a clear and monotonous style, the base policy can significantly exploit it. Against the random opponent with no clear style, the base policy can also exploit it quickly. When facing the strong rocks opponent or even the strongest oracle opponent, the base policy can quickly improve its average returns. One significant advantage of the proposed L2E framework is that the same base policy can exploit a wide range of opponents with different styles, demonstrating its strong generalization ability.
+
+3.2 COMPARISONS WITH OTHER BASELINE METHODS  
+
+<table><tr><td></td><td>Random</td><td>Call</td><td>Rocks</td><td>Nash</td><td>Oracle</td></tr><tr><td>L2E</td><td>0.42±0.32</td><td>1.34±0.14</td><td>0.38±0.17</td><td>-0.03±0.14</td><td>-1.15±0.27</td></tr><tr><td>MAML</td><td>1.27±0.17</td><td>-0.23±0.22</td><td>-1.42±0.07</td><td>-0.77±0.23</td><td>-2.93±0.17</td></tr><tr><td>Random</td><td>-0.02±3.77</td><td>-0.02±3.31</td><td>-0.68±3.75</td><td>-0.74±4.26</td><td>-1.90±4.78</td></tr><tr><td>TRPO</td><td>0.07±0.08</td><td>-0.22±0.09</td><td>-0.77±0.12</td><td>-0.42±0.07</td><td>-1.96±0.46</td></tr><tr><td>TRPO (pretrained)</td><td>0.15±0.17</td><td>-0.05±0.14</td><td>-0.70±0.27</td><td>-0.61±0.32</td><td>-1.32±0.27</td></tr><tr><td>EOM (Explicit Opponent Modeling)</td><td>0.30±0.15</td><td>-0.01±0.05</td><td>-0.13±0.20</td><td>-0.36±0.11</td><td>-1.82±0.28</td></tr></table>
+
+Table 1: The average return of each method when performing rapid adaptation against different opponents in the Leduc Poker environment. The adaptation process is restricted to a three-step gradient update.
+
+As discussed in Section 1, most previous opponent modeling methods require constructing explicit opponent models from a large amount of data before learning to adapt to new opponents. To the best of our knowledge, our L2E framework is the first attempt to use meta-learning to learn to exploit opponents without building explicit opponent models. To demonstrate the effectiveness of the L2E framework, we design several competitive baseline methods. As with the previous experiments, we also use three gradient updates when adapting to a new opponent. 1) MAML. The seminal meta-learning algorithm MAML (Finn et al., 2017) is designed for single-agent environments. We have redesigned and reimplemented the MAML algorithm for the two-player competitive environments. The MAML baseline trains a base policy by continually sampling the opponent’s strategies, either manually specified or randomly generated. 2) TRPO. The TRPO baseline does not perform pre-training and uses the TRPO algorithm (Schulman et al., 2015) to updated its parameters via three-step gradient updates to adapt to different opponents. 3) Random. The Random baseline is neither pre-trained nor updated online. To evaluate different algorithms more comprehensively, we additionally add a new Nash opponent. This opponent’s policy is a part of an approximate Nash Equilibrium generated iteratively by the CFR (Zinkevich et al., 2008) algorithm. Playing a strategy from a Nash Equilibrium in a two-player zero-sum game is guaranteed not to lose in expectation even if the opponent is the best response strategy when the value of the game is zero. We show the performance of the various algorithms in Table 1. It is clear that L2E maintains the highest profitability against all four types of opponents other than the random type. L2E can exploit the opponent with evident style significantly, such as the the Call opponent. Compared to other baseline methods, L2E achieved the highest average return against opponents with unclear styles, such as the Rocks opponent, the Nash opponent, and the cheating Oracle opponent.
+
+# 3.3 ABLATION STUDIES
+
+![](images/b1d09b3308ce6796c8f703dbd5ac51a7a687e1e467bec10dd91e8a98bcd1ab4a.jpg)  
+Figure 3: Visualization of the styles of the strategies generated with or without the MMD regularization term in the Leduc poker environment.
+
+In this section, we verify whether our proposed diversity-regularized policy optimization algorithm can effectively generate policies with different styles. In Leduc poker, hand-action pairs represent different combinations of hands and actions. In the pre-flop phase, each player’s hand has three possibilities, i.e., J, Q, and K. Meanwhile, each player also has three optional actions, i.e., Call (c), Rise (r), and Fold (f). For example, ‘Jc’ means to call when getting the jack. Action probability is the probability that a player will take a corresponding action with a particular hand. Fig. 3 demonstrates that without the MMD regularization term, the two sets of strategies generated in both the pre-flop and flop phases have similar styles. By optimizing with the MMD regularization term, the generated strategies are diverse enough which cover a wide range of different states and actions.
+
+# 3.3.2 EFFECTS OF THE HARD-OSG AND THE DIVERSE-OSG
+
+As discussed previously, a crucial step in L2E is the automatic generation of training opponents. The HardOSG and Diverse-OSG modules are used to generate opponents that are difficult to exploit and diverse in styles. Fig. 4 shows the impact of each module on the performance of L2E. ‘L2E w/o counter’ is L2E without the Hard-OSG module. Similarly, ‘L2E w/o diverse’ is L2E without the Diverse-OSG module. ‘L2E w/o diverse&counter’ removes both modules altogether. The results show that both Hard-OSG and Diverse-OSG have a crucial influence on L2E’s performance. It is clear that the Hard-OSG module helps to enhance the stability of the base policy, and the Diverse-OSG module can improve the base policy’s performance significantly. To further demonstrate the generalization ability of L2E, we conducted a series of additional experiments on the BigLeduc poker and a
+
+![](images/af475094bf5da7f87e35d8896c718dea65774c05e7ada4c61cc8398b927d199f.jpg)  
+Figure 4: Each curve shows the average normalized returns of the base policy trained with different variants of L2E in the grid soccer environment.
+
+Grid Soccer game environment $( c . f .$ . Appendix E and Appendix F for details).
+
+# 4 CONCLUSION
+
+We propose a learning to exploit (L2E) framework to exploit sub-optimal opponents without building explicit opponent models. L2E acquires the ability to exploit opponents by a few interactions with different opponents during training, so that it adapts to new opponents during testing quickly. We propose a novel opponent strategy generation algorithm that produces effective training opponents for L2E automatically. We first design an adversarial training procedure to generate challenging opponents to improve L2E’s robustness effectively. We further exploit a diversity-regularized policy optimization procedure to generate diverse opponents to improve L2E’s generalization ability significantly. Detailed experimental results in three challenging environments demonstrate the effectiveness of the proposed L2E framework.
+
+# REFERENCES
+
+Mohammed Amin Abdullah, Hang Ren, Haitham Bou Ammar, Vladimir Milenkovic, Rui Luo, Mingtian Zhang, and Jun Wang. Wasserstein robust reinforcement learning. arXiv preprint arXiv:1907.13196, 2019.
+
+Alexandros Agapitos, Julian Togelius, Simon M Lucas, Jurgen Schmidhuber, and Andreas Konstantinidis. Generating diverse opponents with multiobjective evolution. In IEEE Symposium On Computational Intelligence and Games, pp. 135–142, 2008.
+
+Maruan Al-Shedivat, Trapit Bansal, Yura Burda, Ilya Sutskever, Igor Mordatch, and Pieter Abbeel. Continuous adaptation via meta-learning in nonstationary and competitive environments. In International Conference on Learning Representations, 2018.
+
+Stefano V Albrecht and Peter Stone. Reasoning about hypothetical agent behaviours and their parameters. In International Conference on Autonomous Agents and Multiagent Systems, pp. 547– 555, 2017.
+
+Stefano V Albrecht and Peter Stone. Autonomous agents modelling other agents: A comprehensive survey and open problems. Artificial Intelligence, 258:66–95, 2018.
+
+Dipyaman Banerjee and Sandip Sen. Reaching pareto-optimality in prisoner’s dilemma using conditional joint action learning. Autonomous Agents and Multi-Agent Systems, 15(1):91–108, 2007.
+
+George W Brown. Iterative solution of games by fictitious play. Activity analysis of production and allocation, 13(1):374–376, 1951.
+
+Harmen de Weerd, Rineke Verbrugge, and Bart Verheij. Negotiating with other minds: the role of recursive theory of mind in negotiation with incomplete information. Autonomous Agents and Multi-Agent Systems, 31(2):250–287, 2017.
+
+Eddie Dekel, Drew Fudenberg, and David K Levine. Learning to play bayesian games. Games and Economic Behavior, 46(2):282–303, 2004.
+
+Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. Diversity is all you need: Learning skills without a reward function. In International Conference on Learning Representations, 2018.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In International Conference on Machine Learning, pp. 1126–1135, 2017.
+
+Jakob Foerster, Richard Y Chen, Maruan Al-Shedivat, Shimon Whiteson, Pieter Abbeel, and Igor Mordatch. Learning with opponent-learning awareness. In International Conference on Autonomous Agents and MultiAgent Systems, pp. 122–130, 2018.
+
+Meire Fortunato, Mohammad Gheshlaghi Azar, Bilal Piot, Jacob Menick, Matteo Hessel, Ian Osband, Alex Graves, Volodymyr Mnih, Remi Munos, Demis Hassabis, et al. Noisy networks for exploration. In International Conference on Learning Representations, 2018.
+
+Arthur Gretton, Karsten Borgwardt, Malte Rasch, Bernhard Scholkopf, and Alex J Smola. A kernel ¨ method for the two-sample-problem. In Advances in neural information processing systems, pp. 513–520, 2007.
+
+Barbara Grosz and Candace L Sidner. Attention, intentions, and the structure of discourse. Computational linguistics, 1986.
+
+Abhishek Gupta, Benjamin Eysenbach, Chelsea Finn, and Sergey Levine. Unsupervised metalearning for reinforcement learning. arXiv preprint arXiv:1806.04640, 2018.
+
+He He, Jordan Boyd-Graber, Kevin Kwok, and Hal Daume III. Opponent modeling in deep rein- ´ forcement learning. In International conference on machine learning, pp. 1804–1813, 2016.
+
+Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey. Meta-learning in neural networks: A survey. arXiv preprint arXiv:2004.05439, 2020.
+
+Trung Dong Huynh, Nicholas R Jennings, and Nigel R Shadbolt. An integrated trust and reputation model for open multi-agent systems. Autonomous Agents and Multi-Agent Systems, 13(2):119– 154, 2006.
+
+Max Jaderberg, Valentin Dalibard, Simon Osindero, Wojciech M Czarnecki, Jeff Donahue, Ali Razavi, Oriol Vinyals, Tim Green, Iain Dunning, Karen Simonyan, et al. Population based training of neural networks. arXiv preprint arXiv:1711.09846, 2017.
+
+Max Jaderberg, Wojciech M Czarnecki, Iain Dunning, Luke Marris, Guy Lever, Antonio Garcia Castaneda, Charles Beattie, Neil C Rabinowitz, Ari S Morcos, Avraham Ruderman, et al. Humanlevel performance in 3d multiplayer games with population-based reinforcement learning. Science, 364(6443):859–865, 2019.
+
+Peter A Jarvis, Teresa F Lunt, and Karen L Myers. Identifying terrorist activity with ai plan recognition technology. AI magazine, 26(3):73–73, 2005.
+
+Marc Lanctot, Vinicius Zambaldi, Audrunas Gruslys, Angeliki Lazaridou, Karl Tuyls, Julien Perolat, David Silver, and Thore Graepel. A unified game-theoretic approach to multiagent re- ´ inforcement learning. In Advances in neural information processing systems, pp. 4190–4203, 2017.
+
+Gordon McCalla, Julita Vassileva, Jim Greer, and Susan Bull. Active learner modelling. In International Conference on Intelligent Tutoring Systems, pp. 53–62, 2000.
+
+Richard Mealing and Jonathan L Shapiro. Opponent modeling by expectation–maximization and sequence prediction in simplified poker. IEEE Transactions on Computational Intelligence and AI in Games, 9(1):11–24, 2015.
+
+Christian Muise, Vaishak Belle, Paolo Felli, Sheila McIlraith, Tim Miller, Adrian R Pearce, and Liz Sonenberg. Planning over multi-agent epistemic states: A classical planning approach. In AAAI Conference on Artificial Intelligence, 2015.
+
+John H Nachbar. Beliefs in repeated games. Econometrica, 73(2):459–480, 2005.
+
+Alex Nichol, Joshua Achiam, and John Schulman. On first-order meta-learning algorithms. arXiv preprint arXiv:1803.02999, 2018.
+
+Anay Pattanaik, Zhenyi Tang, Shuijing Liu, Gautham Bommannan, and Girish Chowdhary. Robust deep reinforcement learning with adversarial attacks. In International Conference on Autonomous Agents and MultiAgent Systems, pp. 2040–2042, 2018.
+
+Lerrel Pinto, James Davidson, Rahul Sukthankar, and Abhinav Gupta. Robust adversarial reinforcement learning. In International Conference on Machine Learning, pp. 2817–2826, 2017.
+
+Matthias Plappert, Rein Houthooft, Prafulla Dhariwal, Szymon Sidor, Richard Y Chen, Xi Chen, Tamim Asfour, Pieter Abbeel, and Marcin Andrychowicz. Parameter space noise for exploration. In International Conference on Learning Representations, 2018.
+
+Rob Powers and Yoav Shoham. Learning against opponents with bounded memory. In International joint conference on Artificial intelligence, pp. 817–822, 2005.
+
+Neil Rabinowitz, Frank Perbet, Francis Song, Chiyuan Zhang, SM Ali Eslami, and Matthew Botvinick. Machine theory of mind. In International Conference on Machine Learning, pp. 4218–4227, 2018.
+
+John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In International conference on machine learning, pp. 1889–1897, 2015.
+
+Lloyd S Shapley. Stochastic games. Proceedings of the national academy of sciences, 39(10): 1095–1100, 1953.
+
+Eric Steinberger. Single deep counterfactual regret minimization. arXiv preprint arXiv:1901.07621, 2019.
+
+Gita Sukthankar and Katia Sycara. Policy recognition for multi-player tactical scenarios. In International joint conference on Autonomous agents and multiagent systems, pp. 1–8, 2007.
+
+Gabriel Synnaeve and Pierre Bessiere. A bayesian model for opening prediction in rts games with application to starcraft. In IEEE Conference on Computational Intelligence and Games, pp. 281– 288, 2011.
+
+Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michael Mathieu, Andrew Dudzik, Juny- ¨ oung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. Nature, 575(7782):350–354, 2019.
+
+Jane X Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z Leibo, Remi Munos, Charles Blundell, Dharshan Kumaran, and Matt Botvinick. Learning to reinforcement learn. arXiv preprint arXiv:1611.05763, 2016.
+
+Rui Wang, Joel Lehman, Jeff Clune, and Kenneth O Stanley. Paired open-ended trailblazer (poet): Endlessly generating increasingly complex and diverse learning environments and their solutions. arXiv preprint arXiv:1901.01753, 2019.
+
+Ben G Weber and Michael Mateas. A data mining approach to strategy prediction. In IEEE Symposium on Computational Intelligence and Games, pp. 140–147, 2009.
+
+Ying Wen, Yaodong Yang, Rui Luo, Jun Wang, and Wei Pan. Probabilistic recursive reasoning for multi-agent reinforcement learning. In International Conference on Learning Representations, 2018.
+
+Zhongwen Xu, Hado P van Hasselt, and David Silver. Meta-gradient reinforcement learning. In Advances in neural information processing systems, pp. 2396–2407, 2018.
+
+Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In Advances in neural information processing systems, pp. 1729–1736, 2008.
+
+# A RELATED WORK
+
+# A.1 OPPONENT MODELING
+
+Opponent modeling is a long-standing research topic in artificial intelligence, and some of the earliest works go back to the early days of game theory research (Brown, 1951). The main goal of opponent modeling is to interact more effectively with other agents by building models to reason about their intentions, predicting their next moves or other properties (Albrecht & Stone, 2018). The commonly used opponent modeling methods can be roughly divided into four categories: policy reconstruction, classification, type-based reasoning and recursive reasoning. Policy reconstruction methods (Mealing & Shapiro, 2015) reconstruct the opponents’ decision making process by building models which make explicit predictions about their actions. Classification methods (Weber & Mateas, 2009; Synnaeve & Bessiere, 2011) produce models which assign class labels (e.g., “aggressive” or “defensive”) to the opponent and employ a precomputed strategy which is effective against that particular class of opponent. Type-based reasoning methods (He et al., 2016; Albrecht & Stone, 2017) assume that the opponent has one of several known types and update the belief using the new observations obtained during the real-time interactions. Recursive reasoning based methods (Muise et al., 2015; de Weerd et al., 2017) model the nested beliefs (e.g., “I believe that you believe that I believe...”) and simulate the reasoning processes of the opponents to predict their actions. Different from these existing methods which usually require a large amount of interactive data to generate useful opponent models, our L2E framework does not explicitly model the opponent and acquires the ability to exploit different opponents by training with limited interactions with different styles of opponents.
+
+# A.2 META-LEARNING
+
+Meta-learning is a new trend of research in the machine learning community which tackles the problem of learning to learn (Hospedales et al., 2020). It leverages past experiences in the training phase to learn how to learn, acquiring the ability to generalize to new environments or new tasks. Recent progress in meta-learning has achieved impressive results ranging from classification and regression in supervised learning (Finn et al., 2017; Nichol et al., 2018) to new task adaption in reinforcement learning (Wang et al., 2016; Xu et al., 2018). Some recent works have also initially explored the application of meta-learning in opponent modeling. For example, the theory of mind network (ToMnet) (Rabinowitz et al., 2018) uses meta-learning to improve the predictions about the opponents’ future behaviors. Another related work (Al-Shedivat et al., 2018) uses meta-learning to handle the non-stationarity problem in the multi-agent interactions. Different from these methods, we focus on how to improve the agents’ ability to quickly adapt to different and unknown opponents.
+
+# A.3 STRATEGY GENERATION
+
+The automatic generation of effective opponent strategies for training with is a critical step in our approach, and how to generate diverse strategies has been preliminarily studied in the reinforcement learning community. In specific, diverse strategies can be obtained in a variety of ways, including adding some diversity regularization to the optimization objective (Abdullah et al., 2019), randomly searching in some diverse parameter space (Plappert et al., 2018; Fortunato et al., 2018), using information-based strategy proposal (Eysenbach et al., 2018; Gupta et al., 2018) and searching diverse strategies with evolutionary algorithms (Agapitos et al., 2008; Wang et al., 2019; Jaderberg et al., 2017; 2019). More recently, the researchers from DeepMind propose a league training paradigm to obtain a Grandmaster level StarCraft II AI (i.e., AlphaStar) by training a diverse league of clorntinually adapting strategies and counter-strategies (Vinyals et al., 2019). Different from AlphaStar, our opponent strategy generation algorithm exploits adversarial training and diversityregularized policy optimization to produce challenging and diverse opponents respectively.
+
+# B ALGORITHM
+
+Algorithm 1: The base policy training procedure of our L2E framework.   
+Input: Step size hyper parameters $\alpha , \beta$ ; base policy $B$ with parameters $\theta$ ; opponent policy $O$ with parameters $\phi$ .   
+Output: An adaptive base policy $\mathbf { B }$ with parameters $\theta$   
+randomly initialize $\theta , \phi$ ;   
+initialize policy pool $\mathcal { M } = \{ O \}$ ;   
+for $1 < e \leq e p o c h s$ do $O = \mathbf { H } \mathbf { a r d - } \mathbf { O S } \mathbf { G } ( B )$ .(see Alg. 2 ) ; $\mathcal { P } =$ Diverse- $\mathbf { 0 s G } ( B , O , N )$ .(see Alg. 3) ; Update opponent policy pool ${ \mathcal { M } } = { \mathcal { M } } \cup { \mathcal { P } }$ ; Sample batch of opponents $O _ { i } \sim { \mathcal { M } }$ ; for Each opponent $O _ { i }$ do Construct a single-player MDP $M _ { B } ^ { O _ { i } }$ ; Sample trajectories $\tau$ using $B$ against fixed opponent $O _ { i }$ ; Use Eqn. (1) to update the parameters of $B$ to obtain an adapted policy $B ^ { O _ { i } }$ ; Resample trajectories $\tau ^ { \prime }$ using $B ^ { O _ { i } }$ against $O _ { i }$ ; Update the parameters $\theta$ of $B$ according to Eqn. (4);
+
+# C MAXIMUM MEAN DISCREPANCY
+
+We use the Maximum Mean Difference (MMD) (Gretton et al., 2007) metric to measure the differences between the distributions of trajectories induced by different opponent strategies.
+
+Algorithm 2: Hard-OSG, the hard-to-exploit training opponent generation algorithm.
+
+Input: The latest base policy $B$ with parameters $\theta$
+
+Output: A hard-to-exploit opponent $\hat { \boldsymbol { O } }$ for $B$
+
+Randomly initialize $\widehat { O }$ ’s parameters $\widehat { \phi }$ for $1 \leq i \leq$ epochs do
+
+Construct a single-player MDP $M _ { B } ^ { \widehat { O } }$ ;   
+Sample a small number of trajectories $\tau \sim M _ { B } ^ { \widehat { O } }$ using $B$ against $\widehat { O }$ ;   
+Use Eqn. (5) to update the parameters of $B$ to obtain an adapted policy $B ^ { \widehat { O } }$ ;   
+Sample trajectories τ 0 ∼ M BObO u sing $\widehat { O }$ against $B ^ { \widehat { O } }$ ;   
+Update the parameters $\widehat { \phi }$ of $\widehat { O }$ according to Eqn. (7) ;
+
+Algorithm 3: Diverse-OSG, the proposed diversity-regularized policy optimization algorithm to generate diverse training opponents.
+
+Input: The latest base policy $B$ , an exsiting opponent $O _ { 1 }$ , the total number of opponents that to be generated $N$ .
+
+$S = \{ O _ { m } \} _ { m = 1 } ^ { N }$
+
+Initialize the opponent set $S = \{ O _ { 1 } \}$ ;
+
+for $i = 2$ to $N$ do
+
+Update the opponent set $S$ , i.e., $S = S \cup O _ { i }$
+
+Definition 1 Let $\mathcal { F }$ be a function space $f : \mathcal { X }  \mathbb { R }$ . Suppose we have two distributions $p$ and $q$ , $X : = \{ x _ { 1 } , . . . , x _ { m } \} \sim p$ , $Y : = \{ y _ { 1 } , . . . , y _ { n } \} \sim q$ . The MMD between $p$ and $q$ using test functions from the function space $\mathcal { F }$ is defined as follows:
+
+$$
+\mathrm { M M D } [ { \mathcal F } , p , q ] : = \operatorname* { s u p } _ { f \in { \mathcal F } } \left( { \mathbf E } _ { x \sim p } [ f ( x ) ] - { \mathbf E } _ { y \sim q } [ f ( y ) ] \right) .
+$$
+
+If we can pick a suitable function space $\mathcal { F }$ , we get the following important theorem (Gretton et al., 2007).
+
+Theorem 1 Let $\mathcal { F } = \{ f \ | \ \| f \| _ { \mathcal { H } } \leq 1 \}$ be a unit ball in a Reproducing Kernel Hilbert Space (RKHS).   
+Then $\mathrm { M M D } [ \mathcal { F } , p , q ] = 0$ if and only if $p = q$ .
+
+So the MMD distance between two strategies is 0 when the distributions of trajectories induced by them are identical. To obtain a set of strategies with diverse styles, we should increase the MMD distances between different strategies. $\varphi$ is a feature space mapping from $x$ to RKHS, we can easily
+
+# Algorithm 4: The testing procedure of our L2E framework.
+
+Input: Step size hyper parameters $\alpha$ ; The trained base policy $B$ with parameters $\theta$ ; an unknown opponent $O$ .   
+Output: The updated base policy $B ^ { O }$ that has been adapted to the opponent $O$ .   
+Construct a single-player MDP $\dot { M } _ { B } ^ { O }$ ;   
+for $0 < s t e p \leq s t e p s$ do Sample trajectories $\tau$ from the MDP $M _ { B } ^ { O }$ ; Update the parameters $\theta$ of $B$ according to Eqn. (1);
+
+calculate the MMD distance using the kernel method $k ( x , x ^ { \prime } ) : = \langle \varphi ( x ) , \varphi ( x ^ { \prime } ) \rangle _ { \mathcal { H } }$ :
+
+$$
+\begin{array} { r l } {  { \mathrm { M M D } ^ { 2 } ( \mathcal { F } , p , q ) } } \\ & { = \bigl \| \mathbb { E } _ { X \sim p } \varphi ( X ) - \mathbb { E } _ { Y \sim q } \varphi ( Y ) \bigr \| _ { \mathcal { H } } ^ { 2 } } \\ & { = \langle \mathbb { E } _ { X \sim p } \varphi ( X ) - \mathbb { E } _ { Y \sim q } \varphi ( Y ) , \mathbb { E } _ { X \sim p } \varphi ( X ) - \mathbb { E } _ { Y \sim q } \varphi ( Y ) \rangle } \\ & { = \mathbb { E } _ { X , X ^ { \prime } \sim p } k ( X , X ^ { \prime } ) - 2 \mathbb { E } _ { X \sim p , Y \sim q } k ( X , Y ) + \mathbb { E } _ { Y , Y ^ { \prime } \sim q } k ( Y , Y ^ { \prime } ) . } \end{array}
+$$
+
+The expectation terms in Eqn. (14) can be approximated using samples:
+
+$$
+\begin{array} { r l r } & { } & { \mathrm { M M D } ^ { 2 } [ { \mathcal F } , X , Y ] = \frac { 1 } { m ( m - 1 ) } \sum _ { i \neq j } ^ { m } k \left( x _ { i } , x _ { j } \right) } \\ & { } & { \qquad + \frac { 1 } { n ( n - 1 ) } \sum _ { i \neq j } ^ { n } k \left( y _ { i } , y _ { j } \right) - \frac { 2 } { m n } \sum _ { i , j = 1 } ^ { m , n } k \left( x _ { i } , y _ { j } \right) . } \end{array}
+$$
+
+The gradient of the MMD term with respect to the policy’s parameter $\phi _ { j }$ in our L2E framework can be calculated as follows:
+
+$$
+\begin{array} { r l } & { \nabla _ { \phi _ { j } } \mathrm { M M D } ^ { 2 } ( \mathrm { T } _ { i } , \mathrm { T } _ { j } ) = \nabla _ { \phi _ { j } } \mathrm { M M D } ^ { 2 } ( \{ \tau \sim M _ { O _ { i } } ^ { B } \} , \{ \tau \sim M _ { O _ { j } } ^ { B } \} ) } \\ & { \qquad = \mathbb { E } _ { \tau , \tau ^ { \prime } \sim M _ { O _ { i } } ^ { B } } [ k \left( \tau , \tau ^ { \prime } \right) \nabla _ { \phi _ { j } } \log ( p ( \tau ) p ( \tau ^ { \prime } ) ) ] } \\ & { \qquad - 2 \mathbb { E } _ { \tau \sim M _ { O _ { i } } ^ { B } , \tau ^ { \prime } \sim M _ { O _ { j } } ^ { B } } [ k \left( \tau , \tau ^ { \prime } \right) \nabla _ { \phi _ { j } } \log ( p ( \tau ) p ( \tau ^ { \prime } ) ) ] } \\ & { \qquad + \mathbb { E } _ { \tau , \tau ^ { \prime } \sim M _ { O _ { j } } ^ { B } } [ k \left( \tau , \tau ^ { \prime } \right) \nabla _ { \phi _ { j } } \log ( p ( \tau ) p ( \tau ^ { \prime } ) ) ] , } \end{array}
+$$
+
+where $p ( \tau )$ is the probability of the trajectory. Since $\mathrm { T } _ { i } = \{ \tau \sim M _ { O _ { i } } ^ { B } \}$ , $O _ { i }$ is the known opponent policy that has no dependence on $\phi _ { j }$ . The gradient with respect to the parameters $\phi _ { j }$ in first term is 0. The gradient of the second and third terms can be easily calculated as follows:
+
+$$
+\nabla _ { \phi _ { j } } \log ( p ( \tau ) ) = \sum _ { t = 0 } ^ { T } \nabla _ { \phi _ { j } } \log \pi _ { \phi _ { j } } ( a _ { t } | s _ { t } ) .
+$$
+
+# D LEDUC POKER
+
+The Leduc poker generally uses a deck of six cards that includes two suites, each with three ranks (Jack, Queen, and King of Spades, Jack, Queen, and King of Hearts). The game has a total of two rounds. Each player is dealt with a private card in the first round, with the opponent’s deck information hidden. In the second round, another card is dealt with as a community card, and the information about this card is open to both players. If a player’s private card is paired with the community card, that player wins the game; otherwise, the player with the highest private card wins the game. Both players bet one chip into the pot before the cards are dealt. Moreover, a betting round follows at the end of each dealing round. The betting wheel alternates between two players, where each player can choose between the following actions: call, check, raise, or fold. If a player chooses to call, that player will need to increase his bet until both players have the same number of chips. If one player raises, that player must first make up the chip difference and then place an additional bet. Check means that a player does not choose any action on the round, but can only check if both players have the same chips. If a player chooses to fold, the hand ends, and the other player wins the game. When all players have equal chips for the round, the game moves on to the next round. The final winner wins all the chips in the game.
+
+Next, we introduce how to define the state vector. Position in a poker game is a critical piece of information that determines the order of action. We define the button (the pre-flop first-hand position), the action position (whose turn it is to take action), and the current game round as one dimension of the state, respectively. In Poker, the combination of a player’s hole cards and board cards determines the game’s outcome. We encode the hole cards and the board cards separately. The amount of chips is an essential consideration in a player’s decision-making process. We encode this information into two dimensions of the state. The number of chips in the pot can reflect the action history of both players. The difference in bets between players in this round affects the choice of action (the game goes to the next round until both players have the same number of chips). In summary, the state vector has seven dimensions, i.e., button, to act, round, hole cards, board cards, chips to call, and pot.
+
+# E BIGLEDUC POKER
+
+We use a larger and more challenging BigLeduc poker environment to further verify the effectiveness of our L2E framework. The BigLeduc poker has the same rules as Leduc but uses a deck of 24 cards with 12 ranks. In addition to the larger state space, BigLeduc allows a maximum of 6 instead of 2 raises per round. As shown in Fig. 5, L2E still achieves fast adaptation to different opponents. In comparison with other baseline methods, L2E achieves the highest average return in Table 2.
+
+Table 2: The average return of each method when performing rapid adaptation against different opponents in the BigLeduc poker environment.   
+
+<table><tr><td></td><td>Random</td><td>Call</td><td>Raise</td><td>Oracle</td></tr><tr><td>L2E</td><td>0.82±0.28</td><td>0.74±0.22</td><td>0.68±0.09</td><td>-1.02±0.24</td></tr><tr><td>MAML</td><td>0.77±0.30</td><td>0.17±0.08</td><td>-2.02±0.99</td><td>-1.21±0.30</td></tr><tr><td>Random</td><td>-0.00±3.08</td><td>-0.00±2.78</td><td>-2.83±5.25</td><td>-1.88±4.28</td></tr><tr><td>TRPO</td><td>0.19±0.09</td><td>0.10±0.16</td><td>-2.22±0.71</td><td>-1.42±0.47</td></tr><tr><td>TRPO (pretrained)</td><td>0.36±0.42</td><td>0.23±0.21</td><td>-2.03±1.61</td><td>-1.69±0.69</td></tr><tr><td>EOM</td><td>0.56±0.43</td><td>0.15±0.13</td><td>-1.15±0.12</td><td>-1.63±0.62</td></tr></table>
+
+![](images/f6d477f4d11aa1e84d11d6d7f38f20b6984973eea4df047f10e6c1463e92c88b.jpg)  
+Figure 5: The trained base policy using our L2E framework can quickly adapt to different opponents of different styles and strengths in the BigLeduc poker environment.
+
+# F GRID SOCCER
+
+This game contains a board with a $6 \times 9$ grid, two players, and their respective target areas. The position of the target area is fixed, and the two players appear randomly in their respective areas at the start of the game. One of the two players randomly has the ball. The goal of all players is to move the ball to the other player’s target position. When the two players move to the same grid, the player with the ball loses the ball. Players gain one point for moving the ball to the opponent’s area. The player can move in all four directions within the grid, and action is invalid when it moves to the boundary.
+
+We train the L2E algorithm in this soccer environment in which both players are modeled by a neural network. Inputs to the network include information about the position of both players, the position of the ball, and the boundary. We provide two types of opponents to test the effectiveness of the resulting base policy. 1) A defensive opponent who adopts a strategy of not leaving the target area and preventing opposing players from attacking. 2) An aggressive opponent who adopts a strategy of continually stealing the ball and approaching the target area with the ball. Facing a defensive opponent won’t lose points, but the agent must learn to carry the ball and avoid the opponent moving to the target area to score points. Against an aggressive opponent, the agent must learn to defend at the target area to avoid losing points. Fig. 6 shows the comparisons between L2E, MAML, and TRPO. L2E adapts quickly to both types of opponents; TRPO works well against defensive opponents but loses many points against aggressive opponents; MAML is unstable due to its reliance on task specification during the training process.
+
+![](images/a5942e5de626a37bb169b7469505e9133863977d2f517cc76ea5ed8fe5f93911.jpg)  
+Figure 6: The trained base policy using our L2E framework can quickly adapt to opponents with different styles in a Grid Soccer environment.
+
+# G CONVERGENCE
+
+Since convergence in game theory is difficult to analyze theoretically, we have designed a series of small-scale experiments to empirically verify the convergence of L2E with the help of Rock-PaperScissors(RPS) game. There are several reasons why RPS game is chosen: 1. RPS game is easy to analyze due to the small state and action space. 2. RPS game is easy to visualize and analyze due to the small state and action space. 3. RPS game is often used in game theory for theoretical analysis. The experiments we designed contains the following parts: 1. Testing the adaptability of Policy Gradient (PG), Self Play (SF), and L2E by visualizing the adaptation process. 2. Analyzing the relationship between L2E strategy and Nash strategy. 3. Analyzing the convergence of L2E.
+
+![](images/ed71134ace0492d76a4a43e931ec1a04f3b1860ec90ac99979fb483e2fb31395.jpg)  
+Figure 7: A. Policy Gradient $:$ Optimize iteratively for the initial opponent (the orange dot), eventually converging to the best response of the initial opponent’s strategy. B. Self Play $:$ Each iteration seeks the best response to the previous round of strategy, which does not converge in an intransitive game like RPS. C. L2E’s adaptation process when facing a new opponent (the orange dot). D. Nash policy’s adaptation process when facing a new opponent (the orange dot).
+
+As shown in Fig. 7, we can draw the following conclusions: 1. PG eventually converged to the best response, but it took dozens of gradient descent steps in our experiments (Each blue dot represents a ten-step gradient descent). SP failed to converge in the RPS game due to the intransitive nature of the RPS game (Rock>Scissors $>$ Paper $\textgreater$ Rock). In contrast, our L2E quickly converged to the best response strategy (Each blue dot represents a one-step gradient descent). 2. The strategy visualization in the Fig. 7.C shows that the base policy of L2E does not converge to the Nash equilibrium strategy after training but converges to the vicinity of the Nash equilibrium strategy. 3. If we fix the base policy to the Nash strategy by imitation learning and then adapting it, we do not get good results either. This further illustrates the difference between the L2E strategy and the Nash equilibrium strategy. And the Fig. 8 further shows the performance of L2E and Nash strategy in RPS game when facing new opponents.
+
+![](images/6879e74de3c4fb30ca264321f42c43f5b3727a65e23ec57434b2117ca295c813.jpg)  
+Figure 8: The performance of L2E and Nash strategy in RPS game when facing new opponents.
+
+Although it is theoretically difficult to analyze the convergence properties of L2E, from the experimental results in Fig. 9, it can be seen that as the training progresses, L2E’s adaptability becomes stronger and stronger. After reaching a certain number of iterations, the improvement eventually reaches a plateau, which provides some empirical evidence for the convergence of L2E.
+
+![](images/dfd855071cd7625c082997c44e325ec3e6150976c1ca3ad88b044c6b93f1333b.jpg)  
+Figure 9: The convergence properties of L2E.
+
+# H HYPER-PARAMETERS
+
+The hyper-parameters of all experiments are shown in the table below.
+
+Table 3: Hyper-parameters of L2E.   
+
+<table><tr><td>Hyper-parameter</td><td>Value</td></tr><tr><td>Training step size hyper parameters (α,β)</td><td>(0.1,0.01)</td></tr><tr><td>Testing step size hyper parameter y</td><td>0.1</td></tr><tr><td>Number of opponents sampled per batch</td><td>40</td></tr><tr><td>Number of trajectories to sample for each opponent</td><td>20</td></tr><tr><td>Number of gradient steps in the training loop</td><td>1</td></tr><tr><td>Number of gradient steps in the testing loop</td><td>3</td></tr><tr><td>Policy network size(Leduc,BigLeduc,Grid Soccer)</td><td>[64,64,(4,4,5)]</td></tr><tr><td>Number of training steps required for convergence in Leduc</td><td>300</td></tr><tr><td>Number of training steps required for convergence in BigLeduc</td><td>400</td></tr><tr><td>Number of training steps required for convergence in GridSoccer</td><td>300</td></tr><tr><td>Hard-to-exploit opponent training epochs</td><td>20</td></tr><tr><td>Diverse opponent training epochs</td><td>50</td></tr><tr><td>Weight of the MMD term αMMD</td><td>0.8</td></tr><tr><td>Bandwidth of RBF kernel</td><td>1</td></tr><tr><td>Minimum trajectory length N to calculate MMD term</td><td>20</td></tr><tr><td>Number of opponent strategies generated by OSG per round of iteration</td><td>N≤5</td></tr><tr><td>Number of trajectories sampled to compute MMD term</td><td>8</td></tr></table>

parse/train/rJqBEPcxe/rJqBEPcxe.md

@@ -0,0 +1,247 @@
+# ZONEOUT: REGULARIZING RNNS BY RANDOMLY PRESERVING HIDDEN ACTIVATIONS
+
+David Krueger1,?, Tegan Maharaj2,?, János Kramár2   
+Mohammad Pezeshki1 Nicolas Ballas1, Nan Rosemary ${ \bf K } \mathbf { e } ^ { 2 }$ , Anirudh Goyal1   
+Yoshua Bengio1†, Aaron Courville1‡, Christopher Pal2
+
+1 MILA, Université de Montréal, firstname.lastname@umontreal.ca.   
+2 École Polytechnique de Montréal, firstname.lastname@polymtl.ca.   
+? Equal contributions. †CIFAR Senior Fellow. ‡CIFAR Fellow.
+
+# ABSTRACT
+
+We propose zoneout, a novel method for regularizing RNNs. At each timestep, zoneout stochastically forces some hidden units to maintain their previous values. Like dropout, zoneout uses random noise to train a pseudo-ensemble, improving generalization. But by preserving instead of dropping hidden units, gradient information and state information are more readily propagated through time, as in feedforward stochastic depth networks. We perform an empirical investigation of various RNN regularizers, and find that zoneout gives significant performance improvements across tasks. We achieve competitive results with relatively simple models in character- and word-level language modelling on the Penn Treebank and Text8 datasets, and combining with recurrent batch normalization (Cooijmans et al., 2016) yields state-of-the-art results on permuted sequential MNIST.
+
+# 1 INTRODUCTION
+
+Regularizing neural nets can significantly improve performance, as indicated by the widespread use of early stopping, and success of regularization methods such as dropout and its recurrent variants (Hinton et al., 2012; Srivastava et al., 2014; Zaremba et al., 2014; Gal, 2015). In this paper, we address the issue of regularization in recurrent neural networks (RNNs) with a novel method called zoneout.
+
+RNNs sequentially construct fixed-length representations of arbitrary-length sequences by folding new observations into their hidden state using an input-dependent transition operator. The repeated application of the same transition operator at the different time steps of the sequence, however, can make the dynamics of an RNN sensitive to minor perturbations in the hidden state; the transition dynamics can magnify components of these perturbations exponentially. Zoneout aims to improve RNNs’ robustness to perturbations in the hidden state in order to regularize transition dynamics.
+
+Like dropout, zoneout injects noise during training. But instead of setting some units’ activations to 0 as in dropout, zoneout randomly replaces some units’ activations with their activations from the previous timestep. As in dropout, we use the expectation of the random noise at test time. This results in a simple regularization approach which can be applied through time for any RNN architecture, and can be conceptually extended to any model whose state varies over time.
+
+Compared with dropout, zoneout is appealing because it preserves information flow forwards and backwards through the network. This helps combat the vanishing gradient problem (Hochreiter, 1991; Bengio et al., 1994), as we observe experimentally.
+
+We also empirically evaluate zoneout on classification using the permuted sequential MNIST dataset, and on language modelling using the Penn Treebank and Text8 datasets, demonstrating competitive or state of the art performance across tasks. In particular, we show that zoneout performs competitively with other proposed regularization methods for RNNs, including recently-proposed dropout variants. Code for replicating all experiments can be found at: http://github.com/teganmaharaj/zoneout
+
+# 2 RELATED WORK
+
+# 2.1 RELATIONSHIP TO DROPOUT
+
+Zoneout can be seen as a selective application of dropout to some of the nodes in a modified computational graph, as shown in Figure 1. In zoneout, instead of dropping out (being set to 0), units zone out and are set to their previous value $( h _ { t } = h _ { t - 1 }$ ). Zoneout, like dropout, can be viewed as a way to train a pseudo-ensemble (Bachman et al., 2014), injecting noise using a stochastic “identity-mask” rather than a zero-mask. We conjecture that identity-masking is more appropriate for RNNs, since it makes it easier for the network to preserve information from previous timesteps going forward, and facilitates, rather than hinders, the flow of gradient information going backward, as we demonstrate experimentally.
+
+![](images/fe3d0d0d8949fb13b47017abb5a00b1b00be4e3299f8baf423a1a7fc3e0fae82.jpg)  
+Figure 1: Zoneout as a special case of dropout; $\tilde { h } _ { t }$ is the unit $h$ ’s hidden activation for the next time step (if not zoned out). Zoneout can be seen as applying dropout on the hidden state delta, $\tilde { h } _ { t } - h _ { t - 1 }$ . When this update is dropped out (represented by the dashed line), $h _ { t }$ becomes $h _ { t - 1 }$ .
+
+# 2.2 DROPOUT IN RNNS
+
+Initially successful applications of dropout in RNNs (Pham et al., 2013; Zaremba et al., 2014) only applied dropout to feed-forward connections (“up the stack”), and not recurrent connections (“forward through time”), but several recent works (Semeniuta et al., 2016; Moon et al., 2015; Gal, 2015) propose methods that are not limited in this way. Bayer et al. (2013) successfully apply fast dropout (Wang & Manning, 2013), a deterministic approximation of dropout, to RNNs.
+
+Semeniuta et al. (2016) apply recurrent dropout to the updates to LSTM memory cells (or GRU states), i.e. they drop out the input/update gate in LSTM/GRU. Like zoneout, their approach prevents the loss of long-term memories built up in the states/cells of GRUs/LSTMS, but zoneout does this by preserving units’ activations exactly. This difference is most salient when zoning out the hidden states (not the memory cells) of an LSTM, for which there is no analogue in recurrent dropout. Whereas saturated output gates or output nonlinearities would cause recurrent dropout to suffer from vanishing gradients (Bengio et al., 1994), zoned-out units still propagate gradients effectively in this situation. Furthermore, while the recurrent dropout method is specific to LSTMs and GRUs, zoneout generalizes to any model that sequentially builds distributed representations of its input, including vanilla RNNs.
+
+Also motivated by preventing memory loss, Moon et al. (2015) propose rnnDrop. This technique amounts to using the same dropout mask at every timestep, which the authors show results in improved performance on speech recognition in their experiments. Semeniuta et al. (2016) show, however, that past states’ influence vanishes exponentially as a function of dropout probability when taking the expectation at test time in rnnDrop; this is problematic for tasks involving longer-term dependencies.
+
+Gal (2015) propose another technique which uses the same mask at each timestep. Motivated by variational inference, they drop out the rows of weight matrices in the input and output embeddings and LSTM gates, instead of dropping units’ activations. The proposed variational RNN technique achieves single-model state-of-the-art test perplexity of 73.4 on word-level language modelling of Penn Treebank.
+
+# 2.3 RELATIONSHIP TO STOCHASTIC DEPTH
+
+Zoneout can also be viewed as a per-unit version of stochastic depth (Huang et al., 2016), which randomly drops entire layers of feed-forward residual networks (ResNets (He et al., 2015)). This is equivalent to zoning out all of the units of a layer at the same time. In a typical RNN, there is a new input at each timestep, causing issues for a naive implementation of stochastic depth. Zoning out an entire layer in an RNN means the input at the corresponding timestep is completely ignored, whereas zoning out individual units allows the RNN to take each element of its input sequence into account. We also found that using residual connections in recurrent nets led to instability, presumably due to the parameter sharing in RNNs. Concurrent with our work, Singh et al. (2016) propose zoneout for ResNets, calling it SkipForward. In their experiments, zoneout is outperformed by stochastic depth, dropout, and their proposed Swapout technique, which randomly drops either or both of the identity or residual connections. Unlike Singh et al. (2016), we apply zoneout to RNNs, and find it outperforms stochastic depth and recurrent dropout.
+
+# 2.4 SELECTIVELY UPDATING HIDDEN UNITS
+
+Like zoneout, clockwork RNNs (Koutnik et al., 2014) and hierarchical RNNs (Hihi & Bengio, 1996) update only some units’ activations at every timestep, but their updates are periodic, whereas zoneout’s are stochastic. Inspired by clockwork RNNs, we experimented with zoneout variants that target different update rates or schedules for different units, but did not find any performance benefit. Hierarchical multiscale LSTMs (Chung et al., 2016) learn update probabilities for different units using the straight-through estimator (Bengio et al., 2013; Courbariaux et al., 2015), and combined with recently-proposed Layer Normalization (Ba et al., 2016), achieve competitive results on a variety of tasks. As the authors note, their method can be interpreted as an input-dependent form of adaptive zoneout.
+
+In recent work, Ha et al. (2016) use a hypernetwork to dynamically rescale the row-weights of a primary LSTM network, achieving state-of-the-art 1.21 BPC on character-level Penn Treebank when combined with layer normalization (Ba et al., 2016) in a two-layer network. This scaling can be viewed as an adaptive, differentiable version of the variational LSTM (Gal, 2015), and could similarly be used to create an adaptive, differentiable version of zoneout. Very recent work conditions zoneout probabilities on suprisal (a measure of the discrepancy between the predicted and actual state), and sets a new state of the art on enwik8 (Rocki et al., 2016).
+
+# 3 ZONEOUT AND PRELIMINARIES
+
+We now explain zoneout in full detail, and compare with other forms of dropout in RNNs. We start by reviewing recurrent neural networks (RNNs).
+
+# 3.1 RECURRENT NEURAL NETWORKS
+
+Recurrent neural networks process data $x _ { 1 } , x _ { 2 } , \ldots , x _ { T }$ sequentially, constructing a corresponding sequence of representations, $h _ { 1 } , h _ { 2 } , \ldots , h _ { T }$ . Each hidden state is trained (implicitly) to remember and emphasize all task-relevant aspects of the preceding inputs, and to incorporate new inputs via a transition operator, $\tau$ , which converts the present hidden state and input into a new hidden state: $h _ { t } = \mathcal { T } ( h _ { t - 1 } , x _ { t } )$ . Zoneout modifies these dynamics by mixing the original transition operator $\tilde { \tau }$ with the identity operator (as opposed to the null operator used in dropout), according to a vector of Bernoulli masks, $d _ { t }$ :
+
+$$
+{ \mathcal { T } } = d _ { t } \odot { \tilde { \mathcal { T } } } + ( 1 - d _ { t } ) \odot 1 \qquad { \mathrm { D r o p o u t : } } \qquad { \mathcal { T } } = d _ { t } \odot { \tilde { \mathcal { T } } } + ( 1 - d _ { t } ) \odot 0
+$$
+
+# 3.2 LONG SHORT-TERM MEMORY
+
+In long short-term memory RNNs (LSTMs) (Hochreiter & Schmidhuber, 1997), the hidden state is divided into memory cell $c _ { t }$ , intended for internal long-term storage, and hidden state $h _ { t }$ , used as a transient representation of state at timestep $t$ . In the most widely used formulation of an LSTM (Gers et al., 2000), $c _ { t }$ and $h _ { t }$ are computed via a set of four “gates”, including the forget gate, $f _ { t }$ , which directly connects $c _ { t }$ to the memories of the previous timestep $c _ { t - 1 }$ , via an element-wise multiplication. Large values of the forget gate cause the cell to remember most (not all) of its previous value. The other gates control the flow of information in $( i _ { t } , g _ { t } )$ and out $\left( o _ { t } \right)$ of the cell. Each gate has a weight matrix and bias vector; for example the forget gate has $W _ { x f }$ , $W _ { h f }$ , and $b _ { f }$ . For brevity, we will write these as $W _ { x } , W _ { h } , b$ .
+
+An LSTM is defined as follows:
+
+$$
+\begin{array} { r l } & { i _ { t } , f _ { t } , o _ { t } = \sigma ( W _ { x } x _ { t } + W _ { h } h _ { t - 1 } + b ) } \\ & { \qquad g _ { t } = \operatorname { t a n h } ( W _ { x g } x _ { t } + W _ { h g } h _ { t - 1 } + b _ { g } ) } \\ & { \qquad c _ { t } = f _ { t } \odot c _ { t - 1 } + i _ { t } \odot g _ { t } } \\ & { \qquad h _ { t } = o _ { t } \odot \operatorname { t a n h } ( c _ { t } ) } \end{array}
+$$
+
+A naive application of dropout in LSTMs would zero-mask either or both of the memory cells and hidden states, without changing the computation of the gates $( i , f , o , g )$ . Dropping memory cells, for example, changes the computation of $c _ { t }$ as follows:
+
+$$
+c _ { t } = d _ { t } \odot ( f _ { t } \odot c _ { t - 1 } + i _ { t } \odot g _ { t } )
+$$
+
+Alternatives abound, however; masks can be applied to any subset of the gates, cells, and states. Semeniuta et al. (2016), for instance, zero-mask the input gate:
+
+$$
+c _ { t } = \left( f _ { t } \odot c _ { t - 1 } + d _ { t } \odot i _ { t } \odot g _ { t } \right)
+$$
+
+When the input gate is masked like this, there is no additive contribution from the input or hidden state, and the value of the memory cell simply decays according to the forget gate.
+
+![](images/fca1f5e28d087dfc871c894a9c9f28a5b350c663ef282e077425a04520303fb9.jpg)  
+Figure 2: (a) Zoneout, vs (b) the recurrent dropout strategy of (Semeniuta et al., 2016) in an LSTM. Dashed lines are zero-masked; in zoneout, the corresponding dotted lines are masked with the corresponding opposite zero-mask. Rectangular nodes are embedding layers.
+
+In zoneout, the values of the hidden state and memory cell randomly either maintain their previous value or are updated as usual. This introduces stochastic identity connections between subsequent time steps:
+
+$$
+\begin{array} { r l } & { c _ { t } = d _ { t } ^ { c } \odot c _ { t - 1 } + ( 1 - d _ { t } ^ { c } ) \odot \left( f _ { t } \odot c _ { t - 1 } + i _ { t } \odot g _ { t } \right) } \\ & { h _ { t } = d _ { t } ^ { h } \odot h _ { t - 1 } + ( 1 - d _ { t } ^ { h } ) \odot \left( o _ { t } \odot \operatorname { t a n h } \left( f _ { t } \odot c _ { t - 1 } + i _ { t } \odot g _ { t } \right) \right) } \end{array}
+$$
+
+We usually use different zoneout masks for cells and hiddens. We also experiment with a variant of recurrent dropout that reuses the input dropout mask to zoneout the corresponding output gates:
+
+$$
+\begin{array} { r l } & { c _ { t } = \left( f _ { t } \odot c _ { t - 1 } + d _ { t } \odot i _ { t } \odot g _ { t } \right) } \\ & { h _ { t } = \left( \left( 1 - d _ { t } \right) \odot o _ { t } + d _ { t } \odot o _ { t - 1 } \right) \odot \operatorname { t a n h } ( c _ { t } ) } \end{array}
+$$
+
+The motivation for this variant is to prevent the network from being forced (by the output gate) to expose a memory cell which has not been updated, and hence may contain misleading information.
+
+# 4 EXPERIMENTS AND DISCUSSION
+
+We evaluate zoneout’s performance on the following tasks: (1) Character-level language modelling on the Penn Treebank corpus (Marcus et al., 1993); (2) Word-level language modelling on the Penn Treebank corpus (Marcus et al., 1993); (3) Character-level language modelling on the Text8 corpus (Mahoney, 2011); (4) Classification of hand-written digits on permuted sequential MNIST (pMNIST) (Le et al., 2015). We also investigate the gradient flow to past hidden states, using pMNIST.
+
+# 4.1 PENN TREEBANK LANGUAGE MODELLING DATASET
+
+The Penn Treebank language model corpus contains 1 million words. The model is trained to predict the next word (evaluated on perplexity) or character (evaluated on BPC: bits per character) in a sequence. 1
+
+# 4.1.1 CHARACTER-LEVEL
+
+For the character-level task, we train networks with one layer of 1000 hidden units. We train LSTMs with a learning rate of 0.002 on overlapping sequences of 100 in batches of 32, optimize using Adam, and clip gradients with threshold 1. These settings match those used in Cooijmans et al. (2016). We also train GRUs and tanh-RNNs with the same parameters as above, except sequences are nonoverlapping and we use learning rates of 0.001, and 0.0003 for GRUs and tanh-RNNs respectively. Small values (0.1, 0.05) of zoneout significantly improve generalization performance for all three models. Intriguingly, we find zoneout increases training time for GRU and tanh-RNN, but decreases training time for LSTMs.
+
+We focus our investigation on LSTM units, where the dynamics of zoning out states, cells, or both provide interesting insight into zoneout’s behaviour. Figure 3 shows our exploration of zoneout in LSTMs, for various zoneout probabilities of cells and/or hiddens. Zoneout on cells with probability 0.5 or zoneout on states with probability 0.05 both outperform the best-performing recurrent dropout $( p = 0 . 2 5 )$ . Combining $z _ { c } = 0 . 5$ and $z _ { h } = 0 . 0 5$ leads to our best-performing model, which achieves 1.27 BPC, competitive with recent state-of-the-art set by (Ha et al., 2016). We compare zoneout to recurrent dropout (for $p \in \{ 0 . 0 5 , 0 . 2 , 0 . 2 5 , 0 . 5 , 0 . 7 \} $ ), weight noise $\langle \sigma = 0 . 0 7 5$ ), norm stabilizer $( \beta = 5 0 )$ (Krueger & Memisevic, 2015), and explore stochastic depth (Huang et al., 2016) in a recurrent setting (analagous to zoning out an entire timestep). We also tried a shared-mask variant of zoneout as used in $p \mathrm { M N I S T }$ experiments, where the same mask is used for both cells and hiddens. Neither stochastic depth or shared-mask zoneout performed as well as separate masks, sampled per unit. Figure 3 shows the best performance achieved with each regularizer, as well as an unregularized LSTM baseline. Results are reported in Table 1, and learning curves shown in Figure 4.
+
+Low zoneout probabilities (0.05-0.25) also improve over baseline in GRUs and tanh-RNNs, reducing BPC from 1.53 to 1.41 for GRU and 1.67 to 1.52 for tanh-RNN. Similarly, low zoneout probabilities work best on the hidden states of LSTMs. For memory cells in LSTMs, however, higher probabilities (around 0.5) work well, perhaps because large forget-gate values approximate the effect of cells zoning out. We conjecture that best performance is achieved with zoneout LSTMs because of the stability of having both state and cell. The probability that both will be zoned out is very low, but having one or the other zoned out carries information from the previous timestep forward, while having the other react ’normally’ to new information.
+
+# 4.1.2 WORD-LEVEL
+
+For the word-level task, we replicate settings from Zaremba et al. (2014)’s best single-model performance. This network has 2 layers of 1500 units, with weights initialized uniformly [-0.04, $+ 0 . 0 4 ]$ . The model is trained for 14 epochs with learning rate 1, after which the learning rate is reduced by a factor of 1.15 after each epoch. Gradient norms are clipped at 10.
+
+With no dropout on the non-recurrent connections (i.e. zoneout as the only regularization), we do not achieve competitive results. We did not perform any search over models, and conjecture that the large model size requires regularization of the feed-forward connections. Adding zoneout ( $z _ { c } = 0 . 2 5$ and $z _ { h } = 0 . 0 2 5$ ) on the recurrent connections to the model optimized for dropout on the non-recurrent connections however, we are able to improve test perplexity from 78.4 to 77.4. We report the best performance achieved with a given technique in Table 1.
+
+![](images/9f7fa52a777f922d2e80196f24b93c61af9e5225c5a045e17e810a7dd2b487f8.jpg)  
+Figure 3: Validation BPC (bits per character) on Character-level Penn Treebank, for different probabilities of zoneout on cells $z _ { c }$ and hidden states $z _ { h }$ (left), and comparison of an unregularized LSTM, zoneout $z _ { c } = 0 . 5 , z _ { h } = 0 . 0 5$ , stochastic depth zoneout $z = 0 . 0 5$ , recurrent dropout $p = 0 . 2 5$ , norm stabilizer $\beta = 5 0$ , and weight noise $\sigma = 0 . 0 7 5$ (right).
+
+![](images/d87991b3e61dcb303c30c939ac2c248a52e2f3e56599db39e367897f863308b5.jpg)  
+Figure 4: Training and validation bits-per-character (BPC) comparing LSTM regularization methods on character-level Penn Treebank (left) and Text8. (right)
+
+# 4.2 TEXT8
+
+Enwik8 is a corpus made from the first $1 0 ^ { 9 }$ bytes of Wikipedia dumped on Mar. 3, 2006. Text8 is a "clean text" version of this corpus; with html tags removed, numbers spelled out, symbols converted to spaces, all lower-cased. Both datasets were created and are hosted by Mahoney (2011).
+
+We use a single-layer network of 2000 units, initialized orthogonally, with batch size 128, learning rate 0.001, and sequence length 180. We optimize with Adam (Kingma & Ba, 2014), clip gradients to a maximum norm of 1 (Pascanu et al., 2012), and use early stopping, again matching the settings of Cooijmans et al. (2016). Results are reported in Table 1, and Figure 4 shows training and validation learning curves for zoneout $( z _ { c } = 0 . 5 , z _ { h } = 0 . 0 5 )$ ) compared to an unregularized LSTM and to recurrent dropout.
+
+# 4.3 PERMUTED SEQUENTIAL MNIST
+
+In sequential MNIST, pixels of an image representing a number [0-9] are presented one at a time, left to right, top to bottom. The task is to classify the number shown in the image. In $p \mathrm { M N I S T }$ , the pixels are presented in a (fixed) random order.
+
+We compare recurrent dropout and zoneout to an unregularized LSTM baseline. All models have a single layer of 100 units, and are trained for 150 epochs using RMSProp (Tieleman & Hinton, 2012) with a decay rate of 0.5 for the moving average of gradient norms. The learning rate is set to 0.001 and the gradients are clipped to a maximum norm of 1 (Pascanu et al., 2012).
+
+As shown in Figure 5 and Table 2, zoneout gives a significant performance boost compared to the LSTM baseline and outperforms recurrent dropout (Semeniuta et al., 2016), although recurrent batch normalization (Cooijmans et al., 2016) outperforms all three. However, by adding zoneout to the recurrent batch normalized LSTM, we achieve state of the art performance. For this setting, the zoneout mask is shared between cells and states, and the recurrent dropout probability and zoneout probabilities are both set to 0.15.
+
+Table 1: Validation and test results of different models on the three language modelling tasks. Results are reported for the best-performing settings. Performance on Char-PTB and Text8 is measured in bitsper-character (BPC); Word-PTB is measured in perplexity. For Char-PTB and Text8 all models are 1-layer unless otherwise noted; for Word-PTB all models are 2-layer. Results above the line are from our own implementation and experiments. Models below the line are: NR-dropout (non-recurrent dropout), V-Dropout (variational dropout), RBN (recurrent batchnorm), H-LSTM+LN (HyperLSTM $^ +$ LayerNorm), 3-HM-LSTM+LN (3-layer Hierarchical Multiscale LSTM $^ +$ LayerNorm).   
+
+<table><tr><td></td><td colspan="2">Char-PTB</td><td colspan="2">Word-PTB</td><td colspan="2">Text8</td></tr><tr><td>Model</td><td>Valid</td><td>Test</td><td>Valid</td><td>Test</td><td>Valid</td><td>Test</td></tr><tr><td>Unregularized LSTM</td><td>1.466</td><td>1.356</td><td>120.7</td><td>114.5</td><td>1.396</td><td>1.408</td></tr><tr><td>Weight noise</td><td>1.507</td><td>1.344</td><td>1</td><td>1</td><td>1.356</td><td>1.367</td></tr><tr><td>Norm stabilizer</td><td>1.459</td><td>1.352</td><td>1</td><td>1</td><td>1.382</td><td>1.398</td></tr><tr><td>Stochastic depth</td><td>1.432</td><td>1.343</td><td>1</td><td>一</td><td>1.337</td><td>1.343</td></tr><tr><td>Recurrent dropout</td><td>1.396</td><td>1.286</td><td>91.6</td><td>87.0</td><td>1.386</td><td>1.401</td></tr><tr><td>Zoneout</td><td>1.362</td><td>1.252</td><td>81.4</td><td>77.4</td><td>1.331</td><td>1.336</td></tr><tr><td>NR-dropout (Zaremba et al., 2014)</td><td>、</td><td>1</td><td>82.2</td><td>78.4</td><td>1</td><td>1</td></tr><tr><td>V-dropout (Gal, 2015)</td><td>一</td><td>1</td><td>1</td><td>73.4</td><td>1</td><td></td></tr><tr><td>RBN (Cooijmans et al., 2016)</td><td>1</td><td>1.32</td><td>1</td><td>1</td><td>1</td><td>1.36</td></tr><tr><td>H-LSTM+ LN (Ha et al., 2016)</td><td>1.281</td><td>1.250</td><td></td><td></td><td></td><td>一</td></tr><tr><td>3-HM-LSTM + LN (Chung et al., 2016)</td><td>1</td><td>1.24</td><td>一</td><td>1</td><td>1</td><td>1.29</td></tr></table>
+
+Table 2: Error rates on the pMNIST digit classification task. Zoneout outperforms recurrent dropout, and sets state of the art when combined with recurrent batch normalization.   
+
+<table><tr><td>Model</td><td>Valid</td><td>Test</td></tr><tr><td>Unregularized LSTM</td><td>0.092</td><td>0.102</td></tr><tr><td>Recurrent dropout p = 0.5</td><td>0.083</td><td>0.075</td></tr><tr><td>Zoneout zc = zh = 0.15</td><td>0.063</td><td>0.069</td></tr><tr><td>Recurrent batchnorm</td><td>1</td><td>0.046</td></tr><tr><td>Recurrent batchnorm &amp; Zoneout zc = zh = 0.15</td><td>0.045</td><td>0.041</td></tr></table>
+
+![](images/04d068b1519c44044614c13f0e36091977576dbcf61d04ab2adf3529545f36cc.jpg)  
+Figure 5: Training and validation error rates for an unregularized LSTM, recurrent dropout, and zoneout on the task of permuted sequential MNIST digit classification.
+
+# 4.4 GRADIENT FLOW
+
+We investigate the hypothesis that identity connections introduced by zoneout facilitate gradient flow to earlier timesteps. Vanishing gradients are a perennial issue in RNNs. As effective as many techniques are for mitigating vanishing gradients (notably the LSTM architecture Hochreiter & Schmidhuber (1997)), we can always imagine a longer sequence to train on, or a longer-term dependence we want to capture.
+
+We compare gradient flow in an unregularized LSTM to zoning out (stochastic identity-mapping) and dropping out (stochastic zero-mapping) the recurrent connections after one epoch of training on $p \mathrm { M N I S T }$ . We compute the average gradient norms $\| \frac { \partial L } { \partial c _ { t } } \|$ of loss $L$ with respect to cell activations $c _ { t }$ at each timestep $t$ , and for each method, normalize the average gradient norms by the sum of average gradient norms for all timesteps.
+
+Figure 6 shows that zoneout propagates gradient information to early timesteps much more effectively than dropout on the recurrent connections, and even more effectively than an unregularized LSTM. The same effect was observed for hidden states $h _ { t }$ .
+
+![](images/d526f126a76865692c5df5e0c7478955ede7d58821c1eb1cc722700e79c86161.jpg)  
+Figure 6: Normalized $\sum { \| \frac { \partial L } { \partial c _ { t } } \| }$ of loss $L$ with respect to cell activations $c _ { t }$ at each timestep $t$ for zoneout $( z _ { c } = 0 . 5 )$ , dropout $( \dot { z } _ { c } = 0 . 5 )$ , and an unregularized LSTM on one epoch of $p \mathrm { M N I S T }$
+
+# 5 CONCLUSION
+
+We have introduced zoneout, a novel and simple regularizer for RNNs, which stochastically preserves hidden units’ activations. Zoneout improves performance across tasks, outperforming many alternative regularizers to achieve results competitive with state of the art on the Penn Treebank and Text8 datasets, and state of the art results on $p \mathrm { M N I S T }$ . While searching over zoneout probabilites allows us to tune zoneout to each task, low zoneout probabilities $( 0 . 0 5 - 0 . 2 )$ on states reliably improve performance of existing models.
+
+We perform no hyperparameter search to achieve these results, simply using settings from the previous state of the art. Results on $p \mathbf { M N I S T }$ and word-level Penn Treebank suggest that Zoneout works well in combination with other regularizers, such as recurrent batch normalization, and dropout on feedforward/embedding layers. We conjecture that the benefits of zoneout arise from two main factors: (1) Introducing stochasticity makes the network more robust to changes in the hidden state; (2) The identity connections improve the flow of information forward and backward through the network.
+
+# ACKNOWLEDGMENTS
+
+We are grateful to Hugo Larochelle, Jan Chorowski, and students at MILA, especially Çaglar ˘ Gülçehre, Marcin Moczulski, Chiheb Trabelsi, and Christopher Beckham, for helpful feedback and discussions. We thank the developers of Theano (Theano Development Team, 2016), Fuel, and Blocks (van Merriënboer et al., 2015). We acknowledge the computing resources provided by ComputeCanada and CalculQuebec. We also thank IBM and Samsung for their support. We would also like to acknowledge the work of Pranav Shyam on learning RNN hierarchies. This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA) and the Air
+
+Force Research Laboratory (AFRL). The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.
+
+REFERENCES   
+Lei Jimmy Ba, Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. CoRR, abs/1607.06450, 2016. URL http://arxiv.org/abs/1607.06450.   
+Philip Bachman, Ouais Alsharif, and Doina Precup. Learning with pseudo-ensembles. In Advances in Neural Information Processing Systems, pp. 3365–3373, 2014.   
+J. Bayer, C. Osendorfer, D. Korhammer, N. Chen, S. Urban, and P. van der Smagt. On Fast Dropout and its Applicability to Recurrent Networks. ArXiv e-prints, November 2013.   
+Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difficult. Neural Networks, IEEE Transactions on, 5(2):157–166, 1994.   
+Yoshua Bengio, Nicholas Léonard, and Aaron C. Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. CoRR, abs/1308.3432, 2013. URL http://arxiv.org/abs/1308.3432.   
+Junyoung Chung, Sungjin Ahn, and Yoshua Bengio. Hierarchical multiscale recurrent neural networks. CoRR, abs/1609.01704, 2016. URL http://arxiv.org/abs/1609.01704.   
+Tim Cooijmans, Nicolas Ballas, César Laurent, Caglar Gulcehre, and Aaron Courville. Recurrent batch normalization. arXiv preprint arXiv:1603.09025, 2016.   
+Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. In NIPS, pp. 3123–3131, 2015.   
+Yarin Gal. A Theoretically Grounded Application of Dropout in Recurrent Neural Networks. ArXiv e-prints, December 2015.   
+Felix A. Gers, Jürgen Schmidhuber, and Fred A. Cummins. Learning to forget: Continual prediction with LSTM. Neural Computation, 12(10):2451–2471, 2000.   
+David Ha, Andrew M. Dai, and Quoc V. Le. Hypernetworks. CoRR, abs/1609.09106, 2016. URL http://arxiv.org/abs/1609.09106.   
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385, 2015.   
+Salah El Hihi and Yoshua Bengio. Hierarchical recurrent neural networks for long-term dependencies. In Advances in Neural Information Processing Systems. 1996.   
+Geoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan R Salakhutdinov. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint arXiv:1207.0580, 2012.   
+Sepp Hochreiter. Untersuchungen zu dynamischen neuronalen netzen. Master’s thesis, Institut fur Informatik, Technische Universitat, Munchen, 1991.   
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735–1780, 1997.   
+Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Weinberger. Deep networks with stochastic depth. arXiv preprint arXiv:1603.09382, 2016.   
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   
+Jan Koutnik, Klaus Greff, Faustino Gomez, and Juergen Schmidhuber. A clockwork rnn. arXiv preprint arXiv:1402.3511, 2014.
+
+David Krueger and Roland Memisevic. Regularizing rnns by stabilizing activations. arXiv preprint arXiv:1511.08400, 2015.
+
+Quoc V Le, Navdeep Jaitly, and Geoffrey E Hinton. A simple way to initialize recurrent networks of rectified linear units. arXiv preprint arXiv:1504.00941, 2015.
+
+Matt Mahoney. About the test data, 2011. URL http://mattmahoney.net/dc/textdata.
+
+Mitchell P Marcus, Mary Ann Marcinkiewicz, and Beatrice Santorini. Building a large annotated corpus of english: The penn treebank. Computational linguistics, 19(2):313–330, 1993.
+
+Taesup Moon, Heeyoul Choi, Hoshik Lee, and Inchul Song. Rnndrop: A novel dropout for rnns in asr. Automatic Speech Recognition and Understanding (ASRU), 2015.
+
+Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. Understanding the exploding gradient problem. CoRR, abs/1211.5063, 2012. URL http://arxiv.org/abs/1211.5063.
+
+V. Pham, T. Bluche, C. Kermorvant, and J. Louradour. Dropout improves Recurrent Neural Networks for Handwriting Recognition. ArXiv e-prints, November 2013.
+
+Kamil Rocki, Tomasz Kornuta, and Tegan Maharaj. Surprisal-driven zoneout. CoRR, abs/1610.07675, 2016. URL http://arxiv.org/abs/1610.07675.
+
+Stanislau Semeniuta, Aliaksei Severyn, and Erhardt Barth. Recurrent dropout without memory loss. arXiv preprint arXiv:1603.05118, 2016.
+
+S. Singh, D. Hoiem, and D. Forsyth. Swapout: Learning an ensemble of deep architectures. ArXiv e-prints, May 2016.
+
+Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929–1958, 2014.
+
+Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016.
+
+Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 4:2, 2012.
+
+Bart van Merriënboer, Dzmitry Bahdanau, Vincent Dumoulin, Dmitriy Serdyuk, David Warde-Farley, Jan Chorowski, and Yoshua Bengio. Blocks and fuel: Frameworks for deep learning. CoRR, abs/1506.00619, 2015.
+
+Sida Wang and Christopher Manning. Fast dropout training. In Proceedings of the 30th International Conference on Machine Learning, pp. 118–126, 2013.
+
+Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014.
+
+# 6 APPENDIX
+
+# 6.1 STATIC IDENTITY CONNECTIONS EXPERIMENT
+
+This experiment was suggested by AnonReviewer2 during the ICLR review process with the goal of disentangling the effects zoneout has (1) through noise injection in the training process and (2) through identity connections. Based on these results, we observe that noise injection is essential for obtaining the regularization benefits of zoneout.
+
+In this experiment, one zoneout mask is sampled at the beginning of training, and used for all examples. This means the identity connections introduced are static across training examples (but still different for each timestep). Using static identity connections resulted in slightly lower training (but not validation) error than zoneout, but worse performance than an unregularized LSTM on both train and validation sets, as shown in Figure 7.
+
+![](images/a985466faa11d6ed4abef00e1246963841c9478e7fe1f1610b9930e083a2b59a.jpg)  
+Figure 7: Training and validation curves for an LSTM with static identity connections compared to zoneout (both $Z _ { c } = 0 . 5$ and $Z _ { h } = 0 . 0 5 )$ ) and compared to a vanilla LSTM, showing that static identity connections fail to capture the benefits of zoneout.

parse/train/rJqBEPcxe/rJqBEPcxe_content_list.json

@@ -0,0 +1,1196 @@
+[
+    {
+        "type": "text",
+        "text": "ZONEOUT: REGULARIZING RNNS BY RANDOMLY PRESERVING HIDDEN ACTIVATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "David Krueger1,?, Tegan Maharaj2,?, János Kramár2   \nMohammad Pezeshki1 Nicolas Ballas1, Nan Rosemary ${ \\bf K } \\mathbf { e } ^ { 2 }$ , Anirudh Goyal1   \nYoshua Bengio1†, Aaron Courville1‡, Christopher Pal2 ",
+        "bbox": [
+            181,
+            169,
+            714,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 MILA, Université de Montréal, firstname.lastname@umontreal.ca.   \n2 École Polytechnique de Montréal, firstname.lastname@polymtl.ca.   \n? Equal contributions. †CIFAR Senior Fellow. ‡CIFAR Fellow. ",
+        "bbox": [
+            184,
+            214,
+            710,
+            258
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            295,
+            544,
+            310
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose zoneout, a novel method for regularizing RNNs. At each timestep, zoneout stochastically forces some hidden units to maintain their previous values. Like dropout, zoneout uses random noise to train a pseudo-ensemble, improving generalization. But by preserving instead of dropping hidden units, gradient information and state information are more readily propagated through time, as in feedforward stochastic depth networks. We perform an empirical investigation of various RNN regularizers, and find that zoneout gives significant performance improvements across tasks. We achieve competitive results with relatively simple models in character- and word-level language modelling on the Penn Treebank and Text8 datasets, and combining with recurrent batch normalization (Cooijmans et al., 2016) yields state-of-the-art results on permuted sequential MNIST. ",
+        "bbox": [
+            233,
+            329,
+            766,
+            481
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            513,
+            336,
+            529
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Regularizing neural nets can significantly improve performance, as indicated by the widespread use of early stopping, and success of regularization methods such as dropout and its recurrent variants (Hinton et al., 2012; Srivastava et al., 2014; Zaremba et al., 2014; Gal, 2015). In this paper, we address the issue of regularization in recurrent neural networks (RNNs) with a novel method called zoneout. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            616
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "RNNs sequentially construct fixed-length representations of arbitrary-length sequences by folding new observations into their hidden state using an input-dependent transition operator. The repeated application of the same transition operator at the different time steps of the sequence, however, can make the dynamics of an RNN sensitive to minor perturbations in the hidden state; the transition dynamics can magnify components of these perturbations exponentially. Zoneout aims to improve RNNs’ robustness to perturbations in the hidden state in order to regularize transition dynamics. ",
+        "bbox": [
+            174,
+            623,
+            823,
+            707
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Like dropout, zoneout injects noise during training. But instead of setting some units’ activations to 0 as in dropout, zoneout randomly replaces some units’ activations with their activations from the previous timestep. As in dropout, we use the expectation of the random noise at test time. This results in a simple regularization approach which can be applied through time for any RNN architecture, and can be conceptually extended to any model whose state varies over time. ",
+        "bbox": [
+            174,
+            714,
+            823,
+            784
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Compared with dropout, zoneout is appealing because it preserves information flow forwards and backwards through the network. This helps combat the vanishing gradient problem (Hochreiter, 1991; Bengio et al., 1994), as we observe experimentally. ",
+        "bbox": [
+            176,
+            791,
+            823,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We also empirically evaluate zoneout on classification using the permuted sequential MNIST dataset, and on language modelling using the Penn Treebank and Text8 datasets, demonstrating competitive or state of the art performance across tasks. In particular, we show that zoneout performs competitively with other proposed regularization methods for RNNs, including recently-proposed dropout variants. Code for replicating all experiments can be found at: http://github.com/teganmaharaj/zoneout ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            339,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 RELATIONSHIP TO DROPOUT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            411,
+            148
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Zoneout can be seen as a selective application of dropout to some of the nodes in a modified computational graph, as shown in Figure 1. In zoneout, instead of dropping out (being set to 0), units zone out and are set to their previous value $( h _ { t } = h _ { t - 1 }$ ). Zoneout, like dropout, can be viewed as a way to train a pseudo-ensemble (Bachman et al., 2014), injecting noise using a stochastic “identity-mask” rather than a zero-mask. We conjecture that identity-masking is more appropriate for RNNs, since it makes it easier for the network to preserve information from previous timesteps going forward, and facilitates, rather than hinders, the flow of gradient information going backward, as we demonstrate experimentally. ",
+        "bbox": [
+            173,
+            160,
+            826,
+            272
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/fe3d0d0d8949fb13b47017abb5a00b1b00be4e3299f8baf423a1a7fc3e0fae82.jpg",
+        "image_caption": [
+            "Figure 1: Zoneout as a special case of dropout; $\\tilde { h } _ { t }$ is the unit $h$ ’s hidden activation for the next time step (if not zoned out). Zoneout can be seen as applying dropout on the hidden state delta, $\\tilde { h } _ { t } - h _ { t - 1 }$ . When this update is dropped out (represented by the dashed line), $h _ { t }$ becomes $h _ { t - 1 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            446,
+            285,
+            549,
+            372
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 DROPOUT IN RNNS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            455,
+            352,
+            469
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Initially successful applications of dropout in RNNs (Pham et al., 2013; Zaremba et al., 2014) only applied dropout to feed-forward connections (“up the stack”), and not recurrent connections (“forward through time”), but several recent works (Semeniuta et al., 2016; Moon et al., 2015; Gal, 2015) propose methods that are not limited in this way. Bayer et al. (2013) successfully apply fast dropout (Wang & Manning, 2013), a deterministic approximation of dropout, to RNNs. ",
+        "bbox": [
+            174,
+            482,
+            825,
+            553
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Semeniuta et al. (2016) apply recurrent dropout to the updates to LSTM memory cells (or GRU states), i.e. they drop out the input/update gate in LSTM/GRU. Like zoneout, their approach prevents the loss of long-term memories built up in the states/cells of GRUs/LSTMS, but zoneout does this by preserving units’ activations exactly. This difference is most salient when zoning out the hidden states (not the memory cells) of an LSTM, for which there is no analogue in recurrent dropout. Whereas saturated output gates or output nonlinearities would cause recurrent dropout to suffer from vanishing gradients (Bengio et al., 1994), zoned-out units still propagate gradients effectively in this situation. Furthermore, while the recurrent dropout method is specific to LSTMs and GRUs, zoneout generalizes to any model that sequentially builds distributed representations of its input, including vanilla RNNs. ",
+        "bbox": [
+            174,
+            559,
+            825,
+            698
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Also motivated by preventing memory loss, Moon et al. (2015) propose rnnDrop. This technique amounts to using the same dropout mask at every timestep, which the authors show results in improved performance on speech recognition in their experiments. Semeniuta et al. (2016) show, however, that past states’ influence vanishes exponentially as a function of dropout probability when taking the expectation at test time in rnnDrop; this is problematic for tasks involving longer-term dependencies. ",
+        "bbox": [
+            174,
+            704,
+            825,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Gal (2015) propose another technique which uses the same mask at each timestep. Motivated by variational inference, they drop out the rows of weight matrices in the input and output embeddings and LSTM gates, instead of dropping units’ activations. The proposed variational RNN technique achieves single-model state-of-the-art test perplexity of 73.4 on word-level language modelling of Penn Treebank. ",
+        "bbox": [
+            174,
+            781,
+            825,
+            852
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.3 RELATIONSHIP TO STOCHASTIC DEPTH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            868,
+            488,
+            883
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Zoneout can also be viewed as a per-unit version of stochastic depth (Huang et al., 2016), which randomly drops entire layers of feed-forward residual networks (ResNets (He et al., 2015)). This is equivalent to zoning out all of the units of a layer at the same time. In a typical RNN, there is a new input at each timestep, causing issues for a naive implementation of stochastic depth. Zoning out an entire layer in an RNN means the input at the corresponding timestep is completely ignored, whereas zoning out individual units allows the RNN to take each element of its input sequence into account. We also found that using residual connections in recurrent nets led to instability, presumably due to the parameter sharing in RNNs. Concurrent with our work, Singh et al. (2016) propose zoneout for ResNets, calling it SkipForward. In their experiments, zoneout is outperformed by stochastic depth, dropout, and their proposed Swapout technique, which randomly drops either or both of the identity or residual connections. Unlike Singh et al. (2016), we apply zoneout to RNNs, and find it outperforms stochastic depth and recurrent dropout. ",
+        "bbox": [
+            176,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            242
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.4 SELECTIVELY UPDATING HIDDEN UNITS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            260,
+            490,
+            273
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Like zoneout, clockwork RNNs (Koutnik et al., 2014) and hierarchical RNNs (Hihi & Bengio, 1996) update only some units’ activations at every timestep, but their updates are periodic, whereas zoneout’s are stochastic. Inspired by clockwork RNNs, we experimented with zoneout variants that target different update rates or schedules for different units, but did not find any performance benefit. Hierarchical multiscale LSTMs (Chung et al., 2016) learn update probabilities for different units using the straight-through estimator (Bengio et al., 2013; Courbariaux et al., 2015), and combined with recently-proposed Layer Normalization (Ba et al., 2016), achieve competitive results on a variety of tasks. As the authors note, their method can be interpreted as an input-dependent form of adaptive zoneout. ",
+        "bbox": [
+            173,
+            285,
+            826,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In recent work, Ha et al. (2016) use a hypernetwork to dynamically rescale the row-weights of a primary LSTM network, achieving state-of-the-art 1.21 BPC on character-level Penn Treebank when combined with layer normalization (Ba et al., 2016) in a two-layer network. This scaling can be viewed as an adaptive, differentiable version of the variational LSTM (Gal, 2015), and could similarly be used to create an adaptive, differentiable version of zoneout. Very recent work conditions zoneout probabilities on suprisal (a measure of the discrepancy between the predicted and actual state), and sets a new state of the art on enwik8 (Rocki et al., 2016). ",
+        "bbox": [
+            174,
+            416,
+            825,
+            515
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 ZONEOUT AND PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            535,
+            465,
+            550
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now explain zoneout in full detail, and compare with other forms of dropout in RNNs. We start by reviewing recurrent neural networks (RNNs). ",
+        "bbox": [
+            173,
+            565,
+            823,
+            594
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 RECURRENT NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            611,
+            450,
+            625
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks process data $x _ { 1 } , x _ { 2 } , \\ldots , x _ { T }$ sequentially, constructing a corresponding sequence of representations, $h _ { 1 } , h _ { 2 } , \\ldots , h _ { T }$ . Each hidden state is trained (implicitly) to remember and emphasize all task-relevant aspects of the preceding inputs, and to incorporate new inputs via a transition operator, $\\tau$ , which converts the present hidden state and input into a new hidden state: $h _ { t } = \\mathcal { T } ( h _ { t - 1 } , x _ { t } )$ . Zoneout modifies these dynamics by mixing the original transition operator $\\tilde { \\tau }$ with the identity operator (as opposed to the null operator used in dropout), according to a vector of Bernoulli masks, $d _ { t }$ : ",
+        "bbox": [
+            173,
+            637,
+            826,
+            736
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d89d1f9f1daf641ea2cb1ac09d31e18d6be1a14cb809d6fffd0ee37d6780c77d.jpg",
+        "text": "$$\n{ \\mathcal { T } } = d _ { t } \\odot { \\tilde { \\mathcal { T } } } + ( 1 - d _ { t } ) \\odot 1 \\qquad { \\mathrm { D r o p o u t : } } \\qquad { \\mathcal { T } } = d _ { t } \\odot { \\tilde { \\mathcal { T } } } + ( 1 - d _ { t } ) \\odot 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            737,
+            794,
+            757
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 LONG SHORT-TERM MEMORY",
+        "text_level": 1,
+        "bbox": [
+            176,
+            772,
+            416,
+            786
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In long short-term memory RNNs (LSTMs) (Hochreiter & Schmidhuber, 1997), the hidden state is divided into memory cell $c _ { t }$ , intended for internal long-term storage, and hidden state $h _ { t }$ , used as a transient representation of state at timestep $t$ . In the most widely used formulation of an LSTM (Gers et al., 2000), $c _ { t }$ and $h _ { t }$ are computed via a set of four “gates”, including the forget gate, $f _ { t }$ , which directly connects $c _ { t }$ to the memories of the previous timestep $c _ { t - 1 }$ , via an element-wise multiplication. Large values of the forget gate cause the cell to remember most (not all) of its previous value. The other gates control the flow of information in $( i _ { t } , g _ { t } )$ and out $\\left( o _ { t } \\right)$ of the cell. Each gate has a weight matrix and bias vector; for example the forget gate has $W _ { x f }$ , $W _ { h f }$ , and $b _ { f }$ . For brevity, we will write these as $W _ { x } , W _ { h } , b$ . ",
+        "bbox": [
+            173,
+            797,
+            826,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An LSTM is defined as follows: ",
+        "bbox": [
+            174,
+            103,
+            385,
+            117
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f2a30860b961c2f3ff9fda39538701584e210084f6af9395ec397e17036fb87.jpg",
+        "text": "$$\n\\begin{array} { r l } & { i _ { t } , f _ { t } , o _ { t } = \\sigma ( W _ { x } x _ { t } + W _ { h } h _ { t - 1 } + b ) } \\\\ & { \\qquad g _ { t } = \\operatorname { t a n h } ( W _ { x g } x _ { t } + W _ { h g } h _ { t - 1 } + b _ { g } ) } \\\\ & { \\qquad c _ { t } = f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot g _ { t } } \\\\ & { \\qquad h _ { t } = o _ { t } \\odot \\operatorname { t a n h } ( c _ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            123,
+            642,
+            196
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A naive application of dropout in LSTMs would zero-mask either or both of the memory cells and hidden states, without changing the computation of the gates $( i , f , o , g )$ . Dropping memory cells, for example, changes the computation of $c _ { t }$ as follows: ",
+        "bbox": [
+            173,
+            208,
+            826,
+            251
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5eec57ab58d33ed1db50a70abab1a95fc05dd39a57d0352444da3f288ade8d91.jpg",
+        "text": "$$\nc _ { t } = d _ { t } \\odot ( f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot g _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            257,
+            606,
+            275
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Alternatives abound, however; masks can be applied to any subset of the gates, cells, and states. Semeniuta et al. (2016), for instance, zero-mask the input gate: ",
+        "bbox": [
+            173,
+            289,
+            825,
+            318
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/210f284578b66e72858122738bbce9ca433179e8ad0c72ee96bfdda2ba1a3421.jpg",
+        "text": "$$\nc _ { t } = \\left( f _ { t } \\odot c _ { t - 1 } + d _ { t } \\odot i _ { t } \\odot g _ { t } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            323,
+            606,
+            340
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When the input gate is masked like this, there is no additive contribution from the input or hidden state, and the value of the memory cell simply decays according to the forget gate. ",
+        "bbox": [
+            173,
+            354,
+            825,
+            383
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/fca1f5e28d087dfc871c894a9c9f28a5b350c663ef282e077425a04520303fb9.jpg",
+        "image_caption": [
+            "Figure 2: (a) Zoneout, vs (b) the recurrent dropout strategy of (Semeniuta et al., 2016) in an LSTM. Dashed lines are zero-masked; in zoneout, the corresponding dotted lines are masked with the corresponding opposite zero-mask. Rectangular nodes are embedding layers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            393,
+            818,
+            515
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In zoneout, the values of the hidden state and memory cell randomly either maintain their previous value or are updated as usual. This introduces stochastic identity connections between subsequent time steps: ",
+        "bbox": [
+            174,
+            583,
+            825,
+            625
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0a569a429aea399bf019c8ad1b72e9a9b263e665e1512da14371aa252c35545b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { c _ { t } = d _ { t } ^ { c } \\odot c _ { t - 1 } + ( 1 - d _ { t } ^ { c } ) \\odot \\left( f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot g _ { t } \\right) } \\\\ & { h _ { t } = d _ { t } ^ { h } \\odot h _ { t - 1 } + ( 1 - d _ { t } ^ { h } ) \\odot \\left( o _ { t } \\odot \\operatorname { t a n h } \\left( f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot g _ { t } \\right) \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            630,
+            715,
+            674
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We usually use different zoneout masks for cells and hiddens. We also experiment with a variant of recurrent dropout that reuses the input dropout mask to zoneout the corresponding output gates: ",
+        "bbox": [
+            173,
+            684,
+            821,
+            713
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e41f88196992878fd10249ab39e9323b7324acd7770764fb94d4f50507793cce.jpg",
+        "text": "$$\n\\begin{array} { r l } & { c _ { t } = \\left( f _ { t } \\odot c _ { t - 1 } + d _ { t } \\odot i _ { t } \\odot g _ { t } \\right) } \\\\ & { h _ { t } = \\left( \\left( 1 - d _ { t } \\right) \\odot o _ { t } + d _ { t } \\odot o _ { t - 1 } \\right) \\odot \\operatorname { t a n h } ( c _ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            717,
+            651,
+            756
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The motivation for this variant is to prevent the network from being forced (by the output gate) to expose a memory cell which has not been updated, and hence may contain misleading information. ",
+        "bbox": [
+            174,
+            760,
+            826,
+            789
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            808,
+            478,
+            825
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We evaluate zoneout’s performance on the following tasks: (1) Character-level language modelling on the Penn Treebank corpus (Marcus et al., 1993); (2) Word-level language modelling on the Penn Treebank corpus (Marcus et al., 1993); (3) Character-level language modelling on the Text8 corpus (Mahoney, 2011); (4) Classification of hand-written digits on permuted sequential MNIST (pMNIST) (Le et al., 2015). We also investigate the gradient flow to past hidden states, using pMNIST. ",
+        "bbox": [
+            173,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 PENN TREEBANK LANGUAGE MODELLING DATASET ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            578,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The Penn Treebank language model corpus contains 1 million words. The model is trained to predict the next word (evaluated on perplexity) or character (evaluated on BPC: bits per character) in a sequence. 1 ",
+        "bbox": [
+            174,
+            135,
+            825,
+            178
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 CHARACTER-LEVEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            208,
+            367,
+            222
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the character-level task, we train networks with one layer of 1000 hidden units. We train LSTMs with a learning rate of 0.002 on overlapping sequences of 100 in batches of 32, optimize using Adam, and clip gradients with threshold 1. These settings match those used in Cooijmans et al. (2016). We also train GRUs and tanh-RNNs with the same parameters as above, except sequences are nonoverlapping and we use learning rates of 0.001, and 0.0003 for GRUs and tanh-RNNs respectively. Small values (0.1, 0.05) of zoneout significantly improve generalization performance for all three models. Intriguingly, we find zoneout increases training time for GRU and tanh-RNN, but decreases training time for LSTMs. ",
+        "bbox": [
+            174,
+            237,
+            826,
+            348
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We focus our investigation on LSTM units, where the dynamics of zoning out states, cells, or both provide interesting insight into zoneout’s behaviour. Figure 3 shows our exploration of zoneout in LSTMs, for various zoneout probabilities of cells and/or hiddens. Zoneout on cells with probability 0.5 or zoneout on states with probability 0.05 both outperform the best-performing recurrent dropout $( p = 0 . 2 5 )$ . Combining $z _ { c } = 0 . 5$ and $z _ { h } = 0 . 0 5$ leads to our best-performing model, which achieves 1.27 BPC, competitive with recent state-of-the-art set by (Ha et al., 2016). We compare zoneout to recurrent dropout (for $p \\in \\{ 0 . 0 5 , 0 . 2 , 0 . 2 5 , 0 . 5 , 0 . 7 \\} $ ), weight noise $\\langle \\sigma = 0 . 0 7 5$ ), norm stabilizer $( \\beta = 5 0 )$ (Krueger & Memisevic, 2015), and explore stochastic depth (Huang et al., 2016) in a recurrent setting (analagous to zoning out an entire timestep). We also tried a shared-mask variant of zoneout as used in $p \\mathrm { M N I S T }$ experiments, where the same mask is used for both cells and hiddens. Neither stochastic depth or shared-mask zoneout performed as well as separate masks, sampled per unit. Figure 3 shows the best performance achieved with each regularizer, as well as an unregularized LSTM baseline. Results are reported in Table 1, and learning curves shown in Figure 4. ",
+        "bbox": [
+            173,
+            356,
+            825,
+            536
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Low zoneout probabilities (0.05-0.25) also improve over baseline in GRUs and tanh-RNNs, reducing BPC from 1.53 to 1.41 for GRU and 1.67 to 1.52 for tanh-RNN. Similarly, low zoneout probabilities work best on the hidden states of LSTMs. For memory cells in LSTMs, however, higher probabilities (around 0.5) work well, perhaps because large forget-gate values approximate the effect of cells zoning out. We conjecture that best performance is achieved with zoneout LSTMs because of the stability of having both state and cell. The probability that both will be zoned out is very low, but having one or the other zoned out carries information from the previous timestep forward, while having the other react ’normally’ to new information. ",
+        "bbox": [
+            174,
+            542,
+            825,
+            654
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 WORD-LEVEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            684,
+            325,
+            698
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the word-level task, we replicate settings from Zaremba et al. (2014)’s best single-model performance. This network has 2 layers of 1500 units, with weights initialized uniformly [-0.04, $+ 0 . 0 4 ]$ . The model is trained for 14 epochs with learning rate 1, after which the learning rate is reduced by a factor of 1.15 after each epoch. Gradient norms are clipped at 10. ",
+        "bbox": [
+            174,
+            714,
+            825,
+            770
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With no dropout on the non-recurrent connections (i.e. zoneout as the only regularization), we do not achieve competitive results. We did not perform any search over models, and conjecture that the large model size requires regularization of the feed-forward connections. Adding zoneout ( $z _ { c } = 0 . 2 5$ and $z _ { h } = 0 . 0 2 5$ ) on the recurrent connections to the model optimized for dropout on the non-recurrent connections however, we are able to improve test perplexity from 78.4 to 77.4. We report the best performance achieved with a given technique in Table 1. ",
+        "bbox": [
+            174,
+            777,
+            825,
+            861
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9f7fa52a777f922d2e80196f24b93c61af9e5225c5a045e17e810a7dd2b487f8.jpg",
+        "image_caption": [
+            "Figure 3: Validation BPC (bits per character) on Character-level Penn Treebank, for different probabilities of zoneout on cells $z _ { c }$ and hidden states $z _ { h }$ (left), and comparison of an unregularized LSTM, zoneout $z _ { c } = 0 . 5 , z _ { h } = 0 . 0 5$ , stochastic depth zoneout $z = 0 . 0 5$ , recurrent dropout $p = 0 . 2 5$ , norm stabilizer $\\beta = 5 0$ , and weight noise $\\sigma = 0 . 0 7 5$ (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            117,
+            792,
+            285
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/d87991b3e61dcb303c30c939ac2c248a52e2f3e56599db39e367897f863308b5.jpg",
+        "image_caption": [
+            "Figure 4: Training and validation bits-per-character (BPC) comparing LSTM regularization methods on character-level Penn Treebank (left) and Text8. (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            382,
+            797,
+            541
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 TEXT8 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
+            261,
+            625
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Enwik8 is a corpus made from the first $1 0 ^ { 9 }$ bytes of Wikipedia dumped on Mar. 3, 2006. Text8 is a \"clean text\" version of this corpus; with html tags removed, numbers spelled out, symbols converted to spaces, all lower-cased. Both datasets were created and are hosted by Mahoney (2011). ",
+        "bbox": [
+            174,
+            637,
+            825,
+            679
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use a single-layer network of 2000 units, initialized orthogonally, with batch size 128, learning rate 0.001, and sequence length 180. We optimize with Adam (Kingma & Ba, 2014), clip gradients to a maximum norm of 1 (Pascanu et al., 2012), and use early stopping, again matching the settings of Cooijmans et al. (2016). Results are reported in Table 1, and Figure 4 shows training and validation learning curves for zoneout $( z _ { c } = 0 . 5 , z _ { h } = 0 . 0 5 )$ ) compared to an unregularized LSTM and to recurrent dropout. ",
+        "bbox": [
+            173,
+            686,
+            825,
+            770
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 PERMUTED SEQUENTIAL MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            791,
+            444,
+            805
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In sequential MNIST, pixels of an image representing a number [0-9] are presented one at a time, left to right, top to bottom. The task is to classify the number shown in the image. In $p \\mathrm { M N I S T }$ , the pixels are presented in a (fixed) random order. ",
+        "bbox": [
+            174,
+            819,
+            825,
+            861
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare recurrent dropout and zoneout to an unregularized LSTM baseline. All models have a single layer of 100 units, and are trained for 150 epochs using RMSProp (Tieleman & Hinton, 2012) with a decay rate of 0.5 for the moving average of gradient norms. The learning rate is set to 0.001 and the gradients are clipped to a maximum norm of 1 (Pascanu et al., 2012). ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 5 and Table 2, zoneout gives a significant performance boost compared to the LSTM baseline and outperforms recurrent dropout (Semeniuta et al., 2016), although recurrent batch normalization (Cooijmans et al., 2016) outperforms all three. However, by adding zoneout to the recurrent batch normalized LSTM, we achieve state of the art performance. For this setting, the zoneout mask is shared between cells and states, and the recurrent dropout probability and zoneout probabilities are both set to 0.15. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/cde91cd0ad07123b92e7945a468af7925e655ff8f4d495f50e3c14aae2691a38.jpg",
+        "table_caption": [
+            "Table 1: Validation and test results of different models on the three language modelling tasks. Results are reported for the best-performing settings. Performance on Char-PTB and Text8 is measured in bitsper-character (BPC); Word-PTB is measured in perplexity. For Char-PTB and Text8 all models are 1-layer unless otherwise noted; for Word-PTB all models are 2-layer. Results above the line are from our own implementation and experiments. Models below the line are: NR-dropout (non-recurrent dropout), V-Dropout (variational dropout), RBN (recurrent batchnorm), H-LSTM+LN (HyperLSTM $^ +$ LayerNorm), 3-HM-LSTM+LN (3-layer Hierarchical Multiscale LSTM $^ +$ LayerNorm). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Char-PTB</td><td colspan=\"2\">Word-PTB</td><td colspan=\"2\">Text8</td></tr><tr><td>Model</td><td>Valid</td><td>Test</td><td>Valid</td><td>Test</td><td>Valid</td><td>Test</td></tr><tr><td>Unregularized LSTM</td><td>1.466</td><td>1.356</td><td>120.7</td><td>114.5</td><td>1.396</td><td>1.408</td></tr><tr><td>Weight noise</td><td>1.507</td><td>1.344</td><td>1</td><td>1</td><td>1.356</td><td>1.367</td></tr><tr><td>Norm stabilizer</td><td>1.459</td><td>1.352</td><td>1</td><td>1</td><td>1.382</td><td>1.398</td></tr><tr><td>Stochastic depth</td><td>1.432</td><td>1.343</td><td>1</td><td>一</td><td>1.337</td><td>1.343</td></tr><tr><td>Recurrent dropout</td><td>1.396</td><td>1.286</td><td>91.6</td><td>87.0</td><td>1.386</td><td>1.401</td></tr><tr><td>Zoneout</td><td>1.362</td><td>1.252</td><td>81.4</td><td>77.4</td><td>1.331</td><td>1.336</td></tr><tr><td>NR-dropout (Zaremba et al., 2014)</td><td>、</td><td>1</td><td>82.2</td><td>78.4</td><td>1</td><td>1</td></tr><tr><td>V-dropout (Gal, 2015)</td><td>一</td><td>1</td><td>1</td><td>73.4</td><td>1</td><td></td></tr><tr><td>RBN (Cooijmans et al., 2016)</td><td>1</td><td>1.32</td><td>1</td><td>1</td><td>1</td><td>1.36</td></tr><tr><td>H-LSTM+ LN (Ha et al., 2016)</td><td>1.281</td><td>1.250</td><td></td><td></td><td></td><td>一</td></tr><tr><td>3-HM-LSTM + LN (Chung et al., 2016)</td><td>1</td><td>1.24</td><td>一</td><td>1</td><td>1</td><td>1.29</td></tr></table>",
+        "bbox": [
+            178,
+            308,
+            816,
+            517
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/7ef20d89d54a19f9e5e35c549760147141c1842edf65e1b00e30a6ed94054f2e.jpg",
+        "table_caption": [
+            "Table 2: Error rates on the pMNIST digit classification task. Zoneout outperforms recurrent dropout, and sets state of the art when combined with recurrent batch normalization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Valid</td><td>Test</td></tr><tr><td>Unregularized LSTM</td><td>0.092</td><td>0.102</td></tr><tr><td>Recurrent dropout p = 0.5</td><td>0.083</td><td>0.075</td></tr><tr><td>Zoneout zc = zh = 0.15</td><td>0.063</td><td>0.069</td></tr><tr><td>Recurrent batchnorm</td><td>1</td><td>0.046</td></tr><tr><td>Recurrent batchnorm &amp; Zoneout zc = zh = 0.15</td><td>0.045</td><td>0.041</td></tr></table>",
+        "bbox": [
+            256,
+            575,
+            736,
+            675
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/04d068b1519c44044614c13f0e36091977576dbcf61d04ab2adf3529545f36cc.jpg",
+        "image_caption": [
+            "Figure 5: Training and validation error rates for an unregularized LSTM, recurrent dropout, and zoneout on the task of permuted sequential MNIST digit classification. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            352,
+            699,
+            642,
+            873
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 GRADIENT FLOW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            333,
+            117
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We investigate the hypothesis that identity connections introduced by zoneout facilitate gradient flow to earlier timesteps. Vanishing gradients are a perennial issue in RNNs. As effective as many techniques are for mitigating vanishing gradients (notably the LSTM architecture Hochreiter & Schmidhuber (1997)), we can always imagine a longer sequence to train on, or a longer-term dependence we want to capture. ",
+        "bbox": [
+            174,
+            128,
+            825,
+            199
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare gradient flow in an unregularized LSTM to zoning out (stochastic identity-mapping) and dropping out (stochastic zero-mapping) the recurrent connections after one epoch of training on $p \\mathrm { M N I S T }$ . We compute the average gradient norms $\\| \\frac { \\partial L } { \\partial c _ { t } } \\|$ of loss $L$ with respect to cell activations $c _ { t }$ at each timestep $t$ , and for each method, normalize the average gradient norms by the sum of average gradient norms for all timesteps. ",
+        "bbox": [
+            174,
+            207,
+            825,
+            277
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows that zoneout propagates gradient information to early timesteps much more effectively than dropout on the recurrent connections, and even more effectively than an unregularized LSTM. The same effect was observed for hidden states $h _ { t }$ . ",
+        "bbox": [
+            174,
+            285,
+            826,
+            327
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d526f126a76865692c5df5e0c7478955ede7d58821c1eb1cc722700e79c86161.jpg",
+        "image_caption": [
+            "Figure 6: Normalized $\\sum { \\| \\frac { \\partial L } { \\partial c _ { t } } \\| }$ of loss $L$ with respect to cell activations $c _ { t }$ at each timestep $t$ for zoneout $( z _ { c } = 0 . 5 )$ , dropout $( \\dot { z } _ { c } = 0 . 5 )$ , and an unregularized LSTM on one epoch of $p \\mathrm { M N I S T }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            359,
+            352,
+            619,
+            484
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            566,
+            318,
+            583
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We have introduced zoneout, a novel and simple regularizer for RNNs, which stochastically preserves hidden units’ activations. Zoneout improves performance across tasks, outperforming many alternative regularizers to achieve results competitive with state of the art on the Penn Treebank and Text8 datasets, and state of the art results on $p \\mathrm { M N I S T }$ . While searching over zoneout probabilites allows us to tune zoneout to each task, low zoneout probabilities $( 0 . 0 5 - 0 . 2 )$ on states reliably improve performance of existing models. ",
+        "bbox": [
+            174,
+            598,
+            825,
+            681
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We perform no hyperparameter search to achieve these results, simply using settings from the previous state of the art. Results on $p \\mathbf { M N I S T }$ and word-level Penn Treebank suggest that Zoneout works well in combination with other regularizers, such as recurrent batch normalization, and dropout on feedforward/embedding layers. We conjecture that the benefits of zoneout arise from two main factors: (1) Introducing stochasticity makes the network more robust to changes in the hidden state; (2) The identity connections improve the flow of information forward and backward through the network. ",
+        "bbox": [
+            174,
+            689,
+            825,
+            786
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            803,
+            326,
+            815
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We are grateful to Hugo Larochelle, Jan Chorowski, and students at MILA, especially Çaglar ˘ Gülçehre, Marcin Moczulski, Chiheb Trabelsi, and Christopher Beckham, for helpful feedback and discussions. We thank the developers of Theano (Theano Development Team, 2016), Fuel, and Blocks (van Merriënboer et al., 2015). We acknowledge the computing resources provided by ComputeCanada and CalculQuebec. We also thank IBM and Samsung for their support. We would also like to acknowledge the work of Pranav Shyam on learning RNN hierarchies. This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA) and the Air ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Force Research Laboratory (AFRL). The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nLei Jimmy Ba, Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. CoRR, abs/1607.06450, 2016. URL http://arxiv.org/abs/1607.06450.   \nPhilip Bachman, Ouais Alsharif, and Doina Precup. Learning with pseudo-ensembles. In Advances in Neural Information Processing Systems, pp. 3365–3373, 2014.   \nJ. Bayer, C. Osendorfer, D. Korhammer, N. Chen, S. Urban, and P. van der Smagt. On Fast Dropout and its Applicability to Recurrent Networks. ArXiv e-prints, November 2013.   \nYoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difficult. Neural Networks, IEEE Transactions on, 5(2):157–166, 1994.   \nYoshua Bengio, Nicholas Léonard, and Aaron C. Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. CoRR, abs/1308.3432, 2013. URL http://arxiv.org/abs/1308.3432.   \nJunyoung Chung, Sungjin Ahn, and Yoshua Bengio. Hierarchical multiscale recurrent neural networks. CoRR, abs/1609.01704, 2016. URL http://arxiv.org/abs/1609.01704.   \nTim Cooijmans, Nicolas Ballas, César Laurent, Caglar Gulcehre, and Aaron Courville. Recurrent batch normalization. arXiv preprint arXiv:1603.09025, 2016.   \nMatthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. In NIPS, pp. 3123–3131, 2015.   \nYarin Gal. A Theoretically Grounded Application of Dropout in Recurrent Neural Networks. ArXiv e-prints, December 2015.   \nFelix A. Gers, Jürgen Schmidhuber, and Fred A. Cummins. Learning to forget: Continual prediction with LSTM. Neural Computation, 12(10):2451–2471, 2000.   \nDavid Ha, Andrew M. Dai, and Quoc V. Le. Hypernetworks. CoRR, abs/1609.09106, 2016. URL http://arxiv.org/abs/1609.09106.   \nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385, 2015.   \nSalah El Hihi and Yoshua Bengio. Hierarchical recurrent neural networks for long-term dependencies. In Advances in Neural Information Processing Systems. 1996.   \nGeoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan R Salakhutdinov. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint arXiv:1207.0580, 2012.   \nSepp Hochreiter. Untersuchungen zu dynamischen neuronalen netzen. Master’s thesis, Institut fur Informatik, Technische Universitat, Munchen, 1991.   \nSepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735–1780, 1997.   \nGao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Weinberger. Deep networks with stochastic depth. arXiv preprint arXiv:1603.09382, 2016.   \nDiederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \nJan Koutnik, Klaus Greff, Faustino Gomez, and Juergen Schmidhuber. A clockwork rnn. arXiv preprint arXiv:1402.3511, 2014. ",
+        "bbox": [
+            171,
+            121,
+            828,
+            931
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "David Krueger and Roland Memisevic. Regularizing rnns by stabilizing activations. arXiv preprint arXiv:1511.08400, 2015. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Quoc V Le, Navdeep Jaitly, and Geoffrey E Hinton. A simple way to initialize recurrent networks of rectified linear units. arXiv preprint arXiv:1504.00941, 2015. ",
+        "bbox": [
+            174,
+            140,
+            823,
+            170
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Matt Mahoney. About the test data, 2011. URL http://mattmahoney.net/dc/textdata. ",
+        "bbox": [
+            173,
+            178,
+            825,
+            194
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mitchell P Marcus, Mary Ann Marcinkiewicz, and Beatrice Santorini. Building a large annotated corpus of english: The penn treebank. Computational linguistics, 19(2):313–330, 1993. ",
+        "bbox": [
+            173,
+            202,
+            823,
+            232
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Taesup Moon, Heeyoul Choi, Hoshik Lee, and Inchul Song. Rnndrop: A novel dropout for rnns in asr. Automatic Speech Recognition and Understanding (ASRU), 2015. ",
+        "bbox": [
+            173,
+            239,
+            823,
+            270
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. Understanding the exploding gradient problem. CoRR, abs/1211.5063, 2012. URL http://arxiv.org/abs/1211.5063. ",
+        "bbox": [
+            174,
+            277,
+            823,
+            308
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "V. Pham, T. Bluche, C. Kermorvant, and J. Louradour. Dropout improves Recurrent Neural Networks for Handwriting Recognition. ArXiv e-prints, November 2013. ",
+        "bbox": [
+            173,
+            315,
+            823,
+            344
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kamil Rocki, Tomasz Kornuta, and Tegan Maharaj. Surprisal-driven zoneout. CoRR, abs/1610.07675, 2016. URL http://arxiv.org/abs/1610.07675. ",
+        "bbox": [
+            171,
+            353,
+            825,
+            383
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stanislau Semeniuta, Aliaksei Severyn, and Erhardt Barth. Recurrent dropout without memory loss. arXiv preprint arXiv:1603.05118, 2016. ",
+        "bbox": [
+            173,
+            391,
+            825,
+            420
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "S. Singh, D. Hoiem, and D. Forsyth. Swapout: Learning an ensemble of deep architectures. ArXiv e-prints, May 2016. ",
+        "bbox": [
+            171,
+            429,
+            825,
+            458
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929–1958, 2014. ",
+        "bbox": [
+            176,
+            467,
+            823,
+            510
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016. ",
+        "bbox": [
+            169,
+            517,
+            825,
+            547
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 4:2, 2012. ",
+        "bbox": [
+            174,
+            555,
+            823,
+            585
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bart van Merriënboer, Dzmitry Bahdanau, Vincent Dumoulin, Dmitriy Serdyuk, David Warde-Farley, Jan Chorowski, and Yoshua Bengio. Blocks and fuel: Frameworks for deep learning. CoRR, abs/1506.00619, 2015. ",
+        "bbox": [
+            173,
+            593,
+            825,
+            637
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sida Wang and Christopher Manning. Fast dropout training. In Proceedings of the 30th International Conference on Machine Learning, pp. 118–126, 2013. ",
+        "bbox": [
+            173,
+            645,
+            823,
+            675
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014. ",
+        "bbox": [
+            174,
+            683,
+            821,
+            712
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.1 STATIC IDENTITY CONNECTIONS EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            535,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This experiment was suggested by AnonReviewer2 during the ICLR review process with the goal of disentangling the effects zoneout has (1) through noise injection in the training process and (2) through identity connections. Based on these results, we observe that noise injection is essential for obtaining the regularization benefits of zoneout. ",
+        "bbox": [
+            174,
+            160,
+            825,
+            215
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, one zoneout mask is sampled at the beginning of training, and used for all examples. This means the identity connections introduced are static across training examples (but still different for each timestep). Using static identity connections resulted in slightly lower training (but not validation) error than zoneout, but worse performance than an unregularized LSTM on both train and validation sets, as shown in Figure 7. ",
+        "bbox": [
+            173,
+            222,
+            825,
+            292
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/a985466faa11d6ed4abef00e1246963841c9478e7fe1f1610b9930e083a2b59a.jpg",
+        "image_caption": [
+            "Figure 7: Training and validation curves for an LSTM with static identity connections compared to zoneout (both $Z _ { c } = 0 . 5$ and $Z _ { h } = 0 . 0 5 )$ ) and compared to a vanilla LSTM, showing that static identity connections fail to capture the benefits of zoneout. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            310,
+            329,
+            673,
+            545
+        ],
+        "page_idx": 10
+    }
+]

parse/train/rkevMnRqYQ/rkevMnRqYQ.md

@@ -0,0 +1,397 @@
+# PREFERENCES IMPLICIT IN THE STATE OF THE WORLD
+
+Rohin Shah ∗ † UC Berkeley
+
+Dmitrii Krasheninnikov ∗ † ‡ University of Amsterdam
+
+Jordan Alexander † ‡ Stanford University
+
+Pieter Abbeel UC Berkeley
+
+Anca D. Dragan UC Berkeley
+
+# ABSTRACT
+
+Reinforcement learning (RL) agents optimize only the features specified in a reward function and are indifferent to anything left out inadvertently. This means that we must not only specify what to do, but also the much larger space of what not to do. It is easy to forget these preferences, since these preferences are already satisfied in our environment. This motivates our key insight: when a robot is deployed in an environment that humans act in, the state of the environment is already optimized for what humans want. We can therefore use this implicit preference information from the state to fill in the blanks. We develop an algorithm based on Maximum Causal Entropy IRL and use it to evaluate the idea in a suite of proof-of-concept environments designed to show its properties. We find that information from the initial state can be used to infer both side effects that should be avoided as well as preferences for how the environment should be organized. Our code can be found at https://github.com/HumanCompatibleAI/rlsp.
+
+# 1 INTRODUCTION
+
+Deep reinforcement learning (deep RL) has been shown to succeed at a wide variety of complex tasks given a correctly specified reward function. Unfortunately, for many real-world tasks it can be challenging to specify a reward function that captures human preferences, particularly the preference for avoiding unnecessary side effects while still accomplishing the goal (Amodei et al., 2016). As a result, there has been much recent work (Christiano et al., 2017; Fu et al., 2017; Sadigh et al., 2017) that aims to learn specifications for tasks a robot should perform.
+
+Typically when learning about what people want and don’t want, we look to human action as evidence: what reward they specify (Hadfield-Menell et al., 2017), how they perform a task (Ziebart et al., 2010; Fu et al., 2017), what choices they make (Christiano et al., 2017; Sadigh et al., 2017), or how they rate certain options (Daniel et al., 2014). Here, we argue that there is an additional source of information that is potentially rather helpful, but that we have been ignoring thus far:
+
+The key insight of this paper is that when a robot is deployed in an environment that humans have been acting in, the state of the environment is already optimized for what humans want.
+
+For example, consider an environment in which a household robot must navigate to a goal location without breaking any vases in its path, illustrated in Figure 1. The human operator, Alice, asks the robot to go to the purple door, forgetting to specify that it should also avoid breaking vases along the way. However, since the robot has been deployed in a state that only contains unbroken vases, it can infer that while acting in the environment (prior to robot’s deployment), Alice was using one of the relatively few policies that do not break vases, and so must have cared about keeping vases intact.
+
+![](images/f9aa5b013d6e89cceeedb74ce4122761126fe5629ca0ebfb51a700904a443afc.jpg)  
+Figure 1: An illustration of learning preferences from an initial state. Alice attempts to accomplish a goal in an environment with an easily breakable vase in the center. The robot observes the state of the environment, $s _ { 0 }$ , after Alice has acted for some time from an even earlier state $s _ { - T }$ . It considers multiple possible human reward functions, and infers that states where vases are intact usually occur when Alice’s reward penalizes breaking vases. In contrast, it doesn’t matter much what the reward function says about carpets, as we would observe the same final state either way. Note that while we consider a specific $s _ { - T }$ for clarity here, the robot could also reason using a distribution over $s _ { - T }$ .
+
+The initial state $s _ { 0 }$ can contain information about arbitrary preferences, including tasks that the robot should actively perform. For example, if the robot observes a basket full of apples near an apple tree, it can reasonably infer that Alice wants to harvest apples. However, $s _ { 0 }$ is particularly useful for inferring which side effects humans care about. Recent approaches avoid unnecessary side effects by penalizing changes from an inaction baseline (Krakovna et al., 2018; Turner, 2018). However, this penalizes all side effects. The inaction baseline is appealing precisely because the initial state has already been optimized for human preferences, and action is more likely to ruin $s _ { 0 }$ than inaction. If our robot infers preferences from $s _ { 0 }$ , it can avoid negative side effects while allowing positive ones.
+
+This work is about highlighting the potential of this observation, and as such makes unrealistic assumptions, such as known dynamics and hand-coded features. Given just $s _ { 0 }$ , these assumptions are necessary: without dynamics, it is hard to tell whether some feature of $s _ { 0 }$ was created by humans or not. Nonetheless, we are optimistic that these assumptions can be relaxed, so that this insight can be used to improve deep RL systems. We suggest some approaches in our discussion.
+
+Our contributions are threefold. First, we identify the state of the world at initialization as a source of information about human preferences. Second, we leverage this insight to derive an algorithm, Reward Learning by Simulating the Past (RLSP), which infers reward from initial state based on a Maximum Causal Entropy (Ziebart et al., 2010) model of human behavior. Third, we demonstrate the properties and limitations of RLSP on a suite of proof-of-concept environments: we use it to avoid side effects, as well as to learn implicit preferences that require active action. In Figure 1 the robot moves to the purple door without breaking the vase, despite the lack of a penalty for breaking vases.
+
+# 2 RELATED WORK
+
+Preference learning. Much recent work has learned preferences from different sources of data, such as demonstrations (Ziebart et al., 2010; Ramachandran and Amir, 2007; Ho and Ermon, 2016; Fu et al., 2017; Finn et al., 2016), comparisons (Christiano et al., 2017; Sadigh et al., 2017; Wirth et al., 2017), ratings (Daniel et al., 2014), human reinforcement signals (Knox and Stone, 2009; Warnell et al., 2017; MacGlashan et al., 2017), proxy rewards (Hadfield-Menell et al., 2017), etc. We suggest preference learning with a new source of data: the state of the environment when the robot is first deployed. It can also be seen as a variant of Maximum Causal Entropy Inverse Reinforcement Learning (Ziebart et al., 2010): while inverse reinforcement learning (IRL) requires demonstrations, or at least state sequences without actions (Edwards et al., 2018; Yu et al., 2018), we learn a reward function from a single state, albeit with the simplifying assumption of known dynamics. This can also be seen as an instance of IRL from summary data (Kangasra¨asi ¨ o and Kaski, 2018). ¨
+
+Frame properties. The frame problem in AI (McCarthy and Hayes, 1981) refers to the issue that we must specify what stays the same in addition to what changes. In formal verification, this manifests as a requirement to explicitly specify the many quantities that the program does not change (Andreescu, 2017). Analogously, rewards are likely to specify what to do (the task), but may forget to say what not to do (the frame properties). One of our goals is to infer frame properties automatically.
+
+Side effects. An impact penalty can mitigate reward specification problems, since it penalizes unnecessary “large” changes (Armstrong and Levinstein, 2017). We could penalize a reduction in the number of reachable states (Krakovna et al., 2018) or attainable utility (Turner, 2018). However, such approaches will penalize all irreversible effects, including ones that humans want. In contrast, by taking a preference inference approach, we can infer which effects humans care about.
+
+Goal states as specifications. Desired behavior in RL can be specified with an explicitly chosen goal state (Kaelbling, 1993; Schaul et al., 2015; Nair et al., 2018; Bahdanau et al., 2018; Andrychowicz et al., 2017). In our setting, the robot observes the initial state $s _ { 0 }$ where it starts acting, which is not explicitly chosen by the designer, but nonetheless contains preference information.
+
+# 3 PRELIMINARIES
+
+A finite-horizon Markov decision process (MDP) is a tuple $\mathcal { M } = \langle \mathcal { S } , \mathcal { A } , \mathcal { T } , r , T \rangle$ , where $s$ is the set of states, $\mathcal { A }$ is the set of actions, $\mathcal { T } : \mathcal { S } \times \mathcal { A } \times \mathcal { S } \mapsto [ 0 , 1 ]$ is the transition probability function, $r : S \mapsto \mathbb { R }$ is the reward function, and $T \in \mathbb { Z } _ { + }$ is the finite planning horizon. We consider MDPs where the reward is linear in features, and does not depend on action: ${ \bf \nabla } _ { r ( s ; \theta ) } = \theta ^ { T } f ( s )$ , where $\theta$ are the parameters defining the reward function and $f$ computes features of a given state.
+
+Inverse Reinforcement Learning (IRL). In IRL, the aim is to infer the reward function $r$ given an MDP without reward $\mathcal { M } \backslash r$ and expert demonstrations $\mathcal { D } = \{ \tau _ { 1 } , . . . , \tau _ { n } \}$ , where each $\tau _ { i } =$ $( s _ { 0 } , a _ { 0 } , . . . , s _ { T } , a _ { T } )$ is a trajectory sampled from the expert policy acting in the MDP. It is assumed that each $\tau _ { i }$ is feasible, so that $\mathcal { T } ( s _ { j + 1 } \mid s _ { j } , a _ { j } ) > 0$ for every $j$ .
+
+Maximum Causal Entropy IRL (MCEIRL). As human demonstrations are rarely optimal, Ziebart et al. (2010) models the expert as a Boltzmann-rational agent that maximizes total reward and causal entropy of the policy. This leads to the policy $\pi _ { t } ( a \mathbin | \mathbin { \bar { \ s } } , \theta ) = \exp ( Q _ { t } ( s , a ; \theta ) - V _ { t } ( s ; \theta ) )$ , where $\begin{array} { r } { V _ { t } ( s ; \bar { \theta } ) = \ln \bar { \sum _ { a } } e x p ( Q _ { t } ( s , a ; \theta ) ) } \end{array}$ plays the role of a normalizing constant. Intuitively, the expert is assumed to act close to randomly when the difference in expected total reward across the actions is small, but nearly always chooses the best action when it leads to a substantially higher expected return. The soft Bellman backup for the state-action value function $Q$ is the same as usual, and is given by $\begin{array} { r } { Q _ { t } ( s , a ; \theta ) = \theta ^ { T } f ( s ) \dot { + } \sum _ { s ^ { \prime } } \mathcal { T } ( s ^ { \prime } \mid s , a ) V _ { t + 1 } ( s ^ { \prime } ; \theta ) } \end{array}$ .
+
+The likelihood of a trajectory $\tau$ given the reward parameters $\theta$ is:
+
+$$
+p ( \tau \mid \theta ) = p ( s _ { 0 } ) \bigg ( \prod _ { t = 0 } ^ { T - 1 } \mathcal { T } ( s _ { t + 1 } \mid s _ { t } , a _ { t } ) \pi _ { t } ( a _ { t } \mid s _ { t } , \theta ) \bigg ) \pi _ { T } ( a _ { T } \mid s _ { T } , \theta ) .
+$$
+
+MCEIRL finds the reward parameters $\theta ^ { * }$ that maximize the log-likelihood of the demonstrations:
+
+$$
+\theta ^ { * } = \operatorname * { a r g m a x } _ { \theta } \ln { p ( \mathcal { D } \mid \theta ) } = \operatorname * { a r g m a x } _ { \theta } \sum _ { i } \sum _ { t } \ln { \pi } _ { t } ( a _ { i , t } \mid s _ { i , t } , \theta ) .
+$$
+
+$\theta ^ { * }$ gives rise to a policy whose feature expectations match those of the expert demonstrations.
+
+# 4 REWARD LEARNING BY SIMULATING THE PAST
+
+We solve the problem of learning the reward function of an expert Alice given a single final state of her trajectory; we refer to this problem as IRL from a single state. Formally, we aim to infer Alice’s reward $\theta$ given an environment $\mathcal { M } \backslash r$ and the last state of the expert’s trajectory $s _ { 0 }$ .
+
+Formulation. To adapt MCEIRL to the one state setting we modify the observation model from Equation 1. Since we only have a single end state $s _ { 0 }$ of the trajectory $\tau _ { 0 } = ( s _ { - T } , a _ { - T } , . . . , s _ { 0 } , a _ { 0 } )$ , we marginalize over all of the other variables in the trajectory:
+
+$$
+p ( s _ { 0 } \mid \theta ) = \sum _ { s _ { - T } , a _ { - T } , \ldots s _ { - 1 } , a _ { - 1 } , a _ { 0 } } p ( \tau _ { 0 } \mid \theta ) ,
+$$
+
+where $p ( \tau _ { 0 } \mid \theta )$ is given in Equation 1. We could invert this and sample from $p ( \theta \mid s _ { 0 } )$ ; the resulting algorithm is presented in Appendix C, but is relatively noisy and slow. We instead find the MLE:
+
+$$
+\theta ^ { * } = \operatorname * { a r g m a x } _ { \theta } \ln p ( s _ { 0 } \mid \theta ) .
+$$
+
+Solution. Similarly to MCEIRL, we use a gradient ascent algorithm to solve the IRL from one state problem. We explain the key steps here and give the full derivation in Appendix B. First, we express the gradient in terms of the gradients of trajectories:
+
+$$
+\nabla _ { \theta } \ln p ( s _ { 0 } \mid \theta ) = \sum _ { \tau _ { - T ; - 1 } } p ( \tau _ { - T ; - 1 } \mid s _ { 0 } , \theta ) \nabla _ { \theta } \ln p ( \tau _ { - T ; 0 } \mid \theta ) .
+$$
+
+This has a nice interpretation – compute the Maximum Causal Entropy gradients for each trajectory, and then take their weighted sum, where each weight is the probability of the trajectory given the evidence $s _ { 0 }$ and current reward $\theta$ . We derive the exact gradient for a trajectory instead of the approximate one in Ziebart et al. (2010) in Appendix A and substitute it in to get:
+
+$$
+\nabla _ { \theta } \ln p ( s _ { 0 } ) = \frac { 1 } { p ( s _ { 0 } ) } \sum _ { \tau _ { - } , \tau _ { : - 1 } } \left[ p ( \tau _ { - T : - 1 } , s _ { 0 } ) \sum _ { t = - T } ^ { - 1 } \left( f ( s _ { t } ) + \mathbb { E } _ { s _ { t + 1 } ^ { \prime } } \left[ \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ^ { \prime } ) \right] - \mathcal { F } _ { t } ( s _ { t } ) \right) \right] ,
+$$
+
+where we have suppressed the dependence on $\theta$ for readability. $\mathcal { F } _ { t } ( s _ { t } )$ denotes the expected features when starting at $s _ { t }$ at time $t$ and acting until time 0 under the policy implied by $\theta$ .
+
+Since we combine gradients from simulated past trajectories, we name our algorithm Reward Learning by Simulating the Past (RLSP). The algorithm computes the gradient using dynamic programming, detailed in Appendix B. We can easily incorporate a prior on $\theta$ by adding the gradient of the log prior to the gradient in Equation 5.
+
+# 5 EVALUATION
+
+Evaluation of RLSP is non-trivial. The inferred reward is very likely to assign state $s _ { 0 }$ maximal reward, since it was inferred under the assumption that when Alice optimized the reward she ended up at $s _ { 0 }$ . If the robot then starts in state $s _ { 0 }$ , if a no-op action is available (as it often is), the RLSP reward is likely to incentivize no-ops, which is not very interesting.
+
+Ultimately, we hope to use RLSP to correct badly specified instructions or reward functions. So, we created a suite of environments with a true reward $R _ { \mathrm { t r u e } }$ , a specified reward $R _ { \mathrm { s p e c } }$ , Alice’s first state $s _ { - T }$ , and the robot’s initial state $s _ { 0 }$ , where $R _ { \mathrm { s p e c } }$ ignores some aspect(s) of $R _ { \mathrm { t r u e } }$ . RLSP is used to infer a reward $\theta _ { \mathrm { A l i c e } }$ from $s _ { 0 }$ , which is then combined with the specified reward to get a final reward $\theta _ { \mathrm { f i n a l } } = \theta _ { \mathrm { A l i c e } } + \lambda \theta _ { \mathrm { s p e c } }$ . (We considered another method for combining rewards; see Appendix D for details.) We inspect the inferred reward qualitatively and measure the expected amount of true reward obtained when planning with $\theta _ { \mathrm { f i n a l } }$ , as a fraction of the expected true reward from the optimal policy. We tune the hyperparameter $\lambda$ controlling the tradeoff between $R _ { \mathrm { s p e c } }$ and the human reward for all algorithms, including baselines. We use a Gaussian prior over the reward parameters.
+
+# 5.1 BASELINES
+
+Specified reward policy $\pi _ { \mathbf { s p e c } }$ . We act as if the true reward is exactly the specified reward.
+
+Policy that penalizes deviations πdeviation. This baseline minimizes change by penalizing deviations from the observed features $f ( s _ { 0 } )$ , giving $R _ { \mathrm { f i n a l } } ( s ) = \theta _ { \mathrm { s p e c } } ^ { T } f ( s ) + \lambda \vert \vert f ( s ) ^ { } - \bar { f ( s _ { 0 } ) } \vert \vert$ .
+
+Relative reachability policy πreachability. Relative reachability (Krakovna et al., 2018) considers a change to be negative when it decreases coverage, relative to what would have happened had the agent done nothing. Here, coverage is a measure of how easily states can be reached from the current state. We compare against the variant of relative reachability that uses undiscounted coverage and a baseline policy where the agent takes no-op actions, as in the original paper. Relative reachability requires known dynamics but not a handcoded featurization. A version of relative reachability that operates in feature space instead of state space would behave similarly.
+
+# 5.2 COMPARISON TO BASELINES
+
+We compare RLSP to our baselines with the assumption of known $s _ { - T }$ , because it makes it easier to analyze RLSP’s properties. We consider the case of unknown $s _ { - T }$ in Section 5.3. We summarize the results in Table 1, and show the environments and trajectories in Figure 2.
+
+Table 1: Performance of algorithms on environments designed to test particular properties.   
+
+<table><tr><td></td><td>Side effects Room</td><td>Env effect Toy train</td><td>Implicitreward Apple collection</td><td colspan="2">Desirable effect Batteries</td><td>Unseen effect Far away vase</td></tr><tr><td>Tspec</td><td>×</td><td></td><td></td><td>Easy √</td><td>Hard</td><td></td></tr><tr><td>Tdeviation Treachability</td><td>√</td><td>xx/</td><td>xxx</td><td>~ ~ 厂</td><td>xxx √</td><td>√</td></tr></table>
+
+![](images/5bc1b06f0cdfb776a4e0850326a57f20ab09faf9e97b0af2a2156e6de40728b7.jpg)  
+Figure 2: Evaluation of RLSP on our environments. Silhouettes indicate the initial position of an object or agent, while filled in version indicate their positions after an agent has acted. The first row depicts the information given to RLSP. The second row shows the trajectory taken by the robot when following the policy $\pi _ { \mathrm { s p e c } }$ that is optimal for $\theta _ { \mathrm { s p e c } }$ . The third row shows the trajectory taken when following the policy $\pi _ { \mathrm { R L S P } }$ that is optimal for $\dot { \theta } _ { \mathrm { f i n a l } } = \theta _ { \mathrm { A l i c e } } + \lambda \theta _ { \mathrm { s p e c } }$ . (a) Side effects: Room with vase (b) Distinguishing environment effects: Toy train (c) Implicit reward: Apple collection (d) Desirable side effect: Batteries (e) “Unseen” side effect: Room with far away vase.
+
+Side effects: Room with vase (Figure 2a). The room tests whether the robot can avoid breaking a vase as a side effect of going to the purple door. There are features for the number of broken vases, standing on a carpet, and each door location. Since Alice didn’t walk over the vase, RLSP infers a negative reward on broken vases, and a small positive reward on carpets (since paths to the top door usually involve carpets). So, $\pi _ { \mathrm { R L S P } }$ successfully avoids breaking the vase. The penalties also achieve the desired behavior: πdeviation avoids breaking the vase since it would change the “number of broken vases” feature, while relative reachability avoids breaking the vase since doing so would result in all states with intact vases becoming unreachable.
+
+Distinguishing environment effects: Toy train (Figure 2b). To test whether algorithms can distinguish between effects caused by the agent and effects caused by the environment, as suggested in Krakovna et al. (2018), we add a toy train that moves along a predefined track. The train breaks if the agent steps on it. We add a new feature indicating whether the train is broken and new features for each possible train location. As before, the specified reward only has a positive weight on the purple door, while the true reward also penalizes broken trains and vases.
+
+RLSP infers a negative reward on broken vases and broken trains, for the same reason as before. It also infers not to put any weight on any particular train location, even though it changes frequently, because it doesn’t help explain $s _ { 0 }$ . As a result, $\pi _ { \mathrm { R L S P } }$ walks over a carpet, but not a vase or a train. πdeviation immediately breaks the train to keep the train location the same. πreachability deduces that breaking the train is irreversible, and so follows the same trajectory as πRLSP.
+
+Implicit reward: Apple collection (Figure 2d). This environment tests whether the algorithms can learn tasks implicit in $s _ { 0 }$ . There are three trees that grow apples, as well as a basket for collecting apples, and the goal is for the robot to harvest apples. However, the specified reward is zero: the robot must infer the task from the observed state. We have features for the number of apples in baskets, the number of apples on trees, whether the robot is carrying an apple, and each location that the agent could be in. $s _ { 0 }$ has two apples in the basket, while $s _ { - T }$ has none.
+
+$\pi _ { \mathrm { s p e c } }$ is arbitrary since every policy is optimal for the zero reward. πdeviation does nothing, achieving zero reward, since its reward can never be positive. πreachability also does not harvest apples. RLSP infers a positive reward on apples in baskets, a negative reward for apples on trees, and a small positive reward for carrying apples. Despite the spurious weights, $\pi _ { \mathrm { R L S P } }$ harvests apples as desired.
+
+Desirable side effect: Batteries (Figure $2 c$ ). This environment tests whether the algorithms can tell when a side effect is allowed. We take the toy train environment, remove vases and carpets, and add batteries. The robot can pick up batteries and put them into the (now unbreakable) toy train, but the batteries are never replenished. If the train runs for 10 timesteps without a new battery, it stops operating. There are features for the number of batteries, whether the train is operational, each train location, and each door location. There are two batteries at $s _ { - T }$ but only one at $s _ { 0 }$ . The true reward incentivizes an operational train and being at the purple door. We consider two variants for the task reward – an “easy” case, where the task reward equals the true reward, and a “hard” case, where the task reward only rewards being at the purple door.
+
+Unsurprisingly, $\pi _ { \mathrm { s p e c } }$ succeeds at the easy case, and fails on the hard case by allowing the train to run out of power. Both πdeviation and πreachability see the action of putting a battery in the train as a side effect to be penalized, and so neither can solve the hard case. They penalize picking up the batteries, and so only solve the easy case if the penalty weight is small. RLSP sees that one battery is gone and that the train is operational, and infers that Alice wants the train to be operational and doesn’t want batteries (since a preference against batteries and a preference for an operational train are nearly indistinguishable). So, it solves both the easy and the hard case, with πRLSP picking up the battery, then staying at the purple door except to deliver the battery to the train.
+
+“Unseen” side effect: Room with far away vase (Figure 2e). This environment demonstrates a limitation of our algorithm: it cannot identify side effects that Alice would never have triggered. In this room, the vase is nowhere close to the shortest path from the Alice’s original position to her goal, but is on the path to the robot’s goal. Since our baselines don’t care about the trajectory the human takes, they all perform as before: $\pi _ { \mathrm { s p e c } }$ walks over the vase, while $\pi _ { \mathrm { d e v i a t i o n } }$ and πreachability both avoid it. Our method infers a near zero weight on the broken vase feature, since it is not present on any reasonable trajectory to the goal, and so breaks it when moving to the goal. Note that this only applies when Alice is known to be at the bottom left corner at $s _ { - T }$ : if we have a uniform prior over $s _ { - T }$ (considered in Section 5.3) then we do consider trajectories where vases are broken.
+
+# 5.3 COMPARISON BETWEEN KNOWING $s _ { - T }$ VS. A DISTRIBUTION OVER $s _ { - T }$
+
+So far, we have considered the setting where the robot knows $s _ { - T }$ , since it is easier to analyze what happens. However, typically we will not know $s _ { - T }$ , and will instead have some prior over $s _ { - T }$ . Here, we compare RLSP in two settings: perfect knowledge of $s _ { - T }$ (as in Section 5.2), and a uniform distribution over all states.
+
+Side effects: Room with vase (Figure 2a) and toy train (Figure $2 b$ ). In both room with vase and toy train, RLSP learns a smaller negative reward on broken vases when using a uniform prior. This is because RLSP considers many more feasible trajectories when using a uniform prior, many of which do not give Alice a chance to break the vase, as in Room with far away vase in Section 5.2. In room with vase, the small positive reward on carpets changes to a near-zero negative reward on carpets. With known $s _ { - T }$ , RLSP overfits to the few consistent trajectories, which usually go over carpets, whereas with a uniform prior it considers many more trajectories that often don’t go over carpets, and so it correctly infers a near-zero weight. In toy train, the negative reward on broken trains becomes slightly more negative, while other features remain approximately the same. This may be because when Alice starts out closer to the toy train, she has more of an opportunity to break it, compared to the known $s _ { - T }$ case.
+
+Implicit preference: Apple collection (Figure 2d). Here, a uniform prior leads to a smaller positive weight on the number of apples in baskets compared to the case with known $s _ { - T }$ . Intuitively, this is because RLSP is considering cases where $s _ { - T }$ already has one or two apples in the basket, which implies that Alice has collected fewer apples and so must have been less interested in them. States where the basket starts with three or more apples are inconsistent with the observed $s _ { 0 }$ and so are not considered. Following the inferred reward still leads to good apple harvesting behavior.
+
+Desirable side effects: Batteries (Figure $2 c$ ). With the uniform prior, we see the same behavior as in Apple collection, where RLSP with a uniform prior learns a slightly smaller negative reward on the batteries, since it considers states $s _ { - T }$ where the battery was already gone. In addition, due to the particular setup the battery must have been given to the train two timesteps prior, which means that in any state where the train started with very little charge, it was allowed to die even though a battery could have been provided before, leading to a near-zero positive weight on the train losing charge. Despite this, RLSP successfully delivers the battery to the train in both easy and hard cases.
+
+“Unseen” side effect: Room with far away vase (Figure 2e). With a uniform prior, we “see” the side effect: if Alice started at the purple door, then the shortest trajectory to the black door would break a vase. As a result, πRLSP successfully avoids the vase (whereas it previously did not). Here, uncertainty over the initial state $s _ { - T }$ can counterintuitively improve the results, because it increases the diversity of trajectories considered, which prevents RLSP from “overfitting” to the few trajectories consistent with a known $s _ { - T }$ and $s _ { 0 }$ .
+
+Overall, RLSP is quite robust to the use of a uniform prior over $s _ { - T }$ , suggesting that we do not need to be particularly careful in the design of that prior.
+
+# 5.4 ROBUSTNESS TO THE CHOICE OF ALICE’S PLANNING HORIZON
+
+We investigate how RLSP performs when assuming the wrong value of Alice’s planning horizon $T$ . We vary the value of $T$ assumed by RLSP, and report the true return achieved by $\pi _ { \mathrm { R L S P } }$ obtained using the inferred reward and a fixed horizon for the robot to act. For this experiment, we used a uniform prior over $s _ { - T }$ , since with known $s _ { - T }$ , RLSP often detects that the given $s _ { - T }$ and $s _ { 0 }$ are incompatible (when $T$ is misspecified). The results are presented in Figure 3.
+
+The performance worsens when RLSP assumes that Alice had a smaller planning horizon than she actually had. Intuitively, if we assume that Alice has only taken one or two actions ever, then even if we knew the actions they could have been in service of many goals, and so we end up quite uncertain about Alice’s reward.
+
+When the assumed $T$ is larger than the true horizon, RLSP correctly infers things the robot should not do. Knowing that the vase was not broken for longer than $T$ timesteps is more evidence to suspect that Alice cared about not breaking the vase. However, overestimated $T$ leads to worse performance at inferring implicit preferences, as in the Apples environment. If we assume Alice has only collected two apples in 100 timesteps, she must not have cared about them much, since she could have collected many more. The batteries environment is unusual – assuming that Alice has been acting for 100 timesteps, the only explanation for the observed $s _ { 0 }$ is that Alice waited until the $9 8 \mathrm { t h }$ timestep to put the battery into the train. This is not particularly consistent with any reward function, and performance degrades.
+
+![](images/6b3a606294f2021c91c945d45ed1bd8c8aa90d0b1c89de72e8a4bf50375e9ceb.jpg)  
+Figure 3: Reward achieved by $\pi _ { \mathrm { R L S P } }$ , as a fraction of the expected reward of the optimal policy, for different values of Alice’s planning horizon $T$ .
+
+Overall, $T$ is an important parameter and needs to be set appropriately. However, even when $T$ is misspecified, performance degrades gracefully to what would have happened if we optimized $\theta _ { \mathrm { s p e c } }$ by itself, so RLSP does not hurt. In addition, if $T$ is larger than it should be, then RLSP still tends to accurately infer parts of the reward that specify what not to do.
+
+# 6 LIMITATIONS AND FUTURE WORK
+
+Summary. Our key insight is that when a robot is deployed, the state that it observes has already been optimized to satisfy human preferences. This explains our preference for a policy that generally avoids side effects. We formalized this by assuming that Alice has been acting in the environment prior to the robot’s deployment. We developed an algorithm, RLSP, that computes a MAP estimate of Alice’s reward function. The robot then acts according to a tradeoff between Alice’s reward function and the specified reward function. Our evaluation showed that information from the initial state can be used to successfully infer side effects to avoid as well as tasks to complete, though there are cases in which we cannot infer the relevant preferences. While we believe this is an important step forward, there is still much work to be done to make this accurate and practical.
+
+Realistic environments. The primary avenue for future work is to scale to realistic environments, where we cannot enumerate states, we don’t know dynamics, and the reward function may be nonlinear. This could be done by adapting existing IRL algorithms (Fu et al., 2017; Ho and Ermon, 2016; Finn et al., 2016). Unknown dynamics is particularly challenging, since we cannot learn dynamics from a single state observation. While acting in the environment, we would have to learn a dynamics model or an inverse dynamics model that can be used to simulate the past, and update the learned preferences as our model improves over time. Alternatively, if we use unsupervised skill learning (Achiam et al., 2018; Eysenbach et al., 2018; Nair et al., 2018) or exploration (Burda et al., 2018), or learn a goal-conditioned policy (Schaul et al., 2015; Andrychowicz et al., 2017), we could compare the explored states with the observed $s _ { 0 }$ .
+
+Hyperparameter choice. While our evaluation showed that RLSP is reasonably robust to the choice of planning horizon $T$ and prior over $s _ { - T }$ , this may be specific to our gridworlds. In the real world, we often make long term hierarchical plans, and if we don’t observe the entire plan (corresponding to a choice of $\mathrm { T }$ that is too small) it seems possible that we infer bad rewards, especially if we have an uninformative prior over $s _ { - T }$ . We do not know whether this will be a problem, and if so how bad it will be, and hope to investigate it in future work with more realistic environments.
+
+Conflicts between $\theta _ { \mathbf { s p e c } }$ and $\theta _ { \mathbf { A l i c e } }$ . RLSP allows us to infer $\theta _ { \mathrm { A l i c e } }$ from $s _ { 0 }$ , which we must somehow combine with $\theta _ { \mathrm { s p e c } }$ to produce a reward $\theta _ { \mathrm { f i n a l } }$ for the robot to optimize. $\theta _ { \mathrm { A l i c e } }$ will usually prefer the status quo of keeping the state similar to $s _ { 0 }$ , while $\theta _ { \mathrm { s p e c } }$ will probably incentivize some change to the state, leading to conflict. We traded off between the two by optimizing their sum, but future work could improve upon this. For example, $\theta _ { \mathrm { A l i c e } }$ could be decomposed into $\theta _ { \mathrm { A l i c e , t a s k } }$ , which says which task Alice is performing (“go to the black door”), and $\theta _ { \mathrm { f r a m e } }$ , which consists of the frame conditions (“don’t break vases”). The robot then optimizes $\theta _ { \mathrm { f r a m e } } + \lambda \theta _ { \mathrm { s p e c } }$ . This requires some way of performing the decomposition. We could model the human as pursuing multiple different subgoals, or the environment as being created by multiple humans with different goals. $\theta _ { \mathrm { f r a m e } }$ would be shared, while $\theta _ { \mathrm { t a s k } }$ would vary, allowing us to distinguish between them. However, combination may not be the answer – instead, perhaps the robot ought to use the inferred reward to inform Alice of any conflicts and actively query her for more information, along the lines of Amin et al. (2017).
+
+Learning tasks to perform. The apples and batteries environments demonstrate that RLSP can learn preferences that require the robot to actively perform a task. It is not clear that this is desirable, since the robot may perform an inferred task instead of the task Alice explicitly sets for it.
+
+Preferences that are not a result of human optimization. While the initial state is optimized for human preferences, this may not be a result of human optimization, as assumed in this paper. For example, we prefer that the atmosphere contain oxygen for us to breathe. The atmosphere meets this preference in spite of human action, and so RLSP would not infer this preference. While this is of limited relevance for household robots, it may become important for more capable AI systems.
+
+# ACKNOWLEDGMENTS
+
+We thank the researchers at the Center for Human Compatible AI for valuable feedback. This work was supported by the Open Philanthropy Project, AFOSR, and National Science Foundation Graduate Research Fellowship Grant No. DGE 1752814.
+
+# REFERENCES
+
+Joshua Achiam, Harrison Edwards, Dario Amodei, and Pieter Abbeel. Variational option discovery algorithms. arXiv preprint arXiv:1807.10299, 2018.
+
+Kareem Amin, Nan Jiang, and Satinder Singh. Repeated inverse reinforcement learning. In Advances in Neural Information Processing Systems, pages 1815–1824, 2017.
+
+Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mane.´ Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016.
+
+Oana Fabiana Andreescu. Static analysis of functional programs with an application to the frame problem in deductive verification. PhD thesis, Rennes 1, 2017.
+
+Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. In Advances in Neural Information Processing Systems, pages 5048–5058, 2017.
+
+Stuart Armstrong and Benjamin Levinstein. Low impact artificial intelligences. arXiv preprint arXiv:1705.10720, 2017.
+
+Dzmitry Bahdanau, Felix Hill, Jan Leike, Edward Hughes, Pushmeet Kohli, and Edward Grefenstette. Learning to understand goal specifications by modelling reward. arXiv preprint arXiv:1806.01946, 2018.
+
+Yuri Burda, Harri Edwards, Deepak Pathak, Amos Storkey, Trevor Darrell, and Alexei A Efros. Large-scale study of curiosity-driven learning. arXiv preprint arXiv:1808.04355, 2018.
+
+Paul F Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei. Deep reinforcement learning from human preferences. In Advances in Neural Information Processing Systems, pages 4299–4307, 2017.
+
+Christian Daniel, Malte Viering, Jan Metz, Oliver Kroemer, and Jan Peters. Active reward learning. In Robotics: Science and Systems, 2014.
+
+Ashley D Edwards, Himanshu Sahni, Yannick Schroeker, and Charles L Isbell. Imitating latent policies from observation. arXiv preprint arXiv:1805.07914, 2018.
+
+Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. Diversity is all you need: Learning skills without a reward function. arXiv preprint arXiv:1802.06070, 2018.
+
+Chelsea Finn, Sergey Levine, and Pieter Abbeel. Guided cost learning: Deep inverse optimal control via policy optimization. In International Conference on Machine Learning, pages 49–58, 2016.
+
+Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adversarial inverse reinforcement learning. arXiv preprint arXiv:1710.11248, 2017.
+
+Dylan Hadfield-Menell, Smitha Milli, Pieter Abbeel, Stuart J Russell, and Anca Dragan. Inverse reward design. In Advances in Neural Information Processing Systems, pages 6765–6774, 2017.
+
+Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems, pages 4565–4573, 2016.
+
+Leslie Pack Kaelbling. Learning to achieve goals. In IJCAI, pages 1094–1099. Citeseer, 1993.
+
+Antti Kangasra¨asi ¨ o and Samuel Kaski. Inverse reinforcement learning from summary data. ¨ Machine Learning, 107(8-10):1517–1535, 2018.
+
+W Bradley Knox and Peter Stone. Interactively shaping agents via human reinforcement: The tamer framework. In Proceedings of the fifth international conference on Knowledge capture, pages 9–16. ACM, 2009.
+
+Victoria Krakovna, Laurent Orseau, Miljan Martic, and Shane Legg. Measuring and avoiding side effects using relative reachability. arXiv preprint arXiv:1806.01186, 2018.
+
+James MacGlashan, Mark K Ho, Robert Loftin, Bei Peng, David Roberts, Matthew E Taylor, and Michael L Littman. Interactive learning from policy-dependent human feedback. arXiv preprint arXiv:1701.06049, 2017.
+
+John McCarthy and Patrick J Hayes. Some philosophical problems from the standpoint of artificial intelligence. In Readings in artificial intelligence, pages 431–450. Elsevier, 1981.
+
+Ashvin Nair, Vitchyr Pong, Murtaza Dalal, Shikhar Bahl, Steven Lin, and Sergey Levine. Visual reinforcement learning with imagined goals. arXiv preprint arXiv:1807.04742, 2018.
+
+Deepak Ramachandran and Eyal Amir. Bayesian inverse reinforcement learning. Urbana, 51(61801): 1–4, 2007.
+
+Dorsa Sadigh, Anca Dragan, Shankar Sastry, and Sanjit A Seshia. Active preference-based learning of reward functions. In Robotics: Science and Systems (RSS), 2017.
+
+Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. Universal value function approximators. In International Conference on Machine Learning, pages 1312–1320, 2015.
+
+Alex Turner. Towards a new impact measure, 2018. https://www.alignmentforum.org/ posts/yEa7kwoMpsBgaBCgb/towards-a-new-impact-measure, Last accessed on 2018-09-26.
+
+Garrett Warnell, Nicholas Waytowich, Vernon Lawhern, and Peter Stone. Deep tamer: Interactive agent shaping in high-dimensional state spaces. arXiv preprint arXiv:1709.10163, 2017.
+
+Christian Wirth, Riad Akrour, Gerhard Neumann, and Johannes Furnkranz. A Survey of Preference- ¨ Based Reinforcement Learning Methods. Journal of Machine Learning Research, 18(136):1–46, 2017. ISSN 15337928. doi: 10.1109/CIS.2008.204. URL http://jmlr.org/papers/ v18/16-634.html.
+
+Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and Sergey Levine. One-shot imitation from observing humans via domain-adaptive meta-learning. arXiv preprint arXiv:1802.01557, 2018.
+
+Brian D Ziebart, J Andrew Bagnell, and Anind K Dey. Modeling interaction via the principle of maximum causal entropy. 2010.
+
+A
+
+Here, we derive an exact gradient for the maximum causal entropy distribution introduced in Ziebart et al. (2010), as the existing approximation is insufficient for our purposes. Given a trajectory $\tau _ { T } = s _ { 0 } a _ { 0 } \ldots s _ { T } a _ { T }$ , we seek the gradient $\nabla _ { \boldsymbol { \theta } } \ln { p ( \tau _ { T } ) }$ . We assume that the expert has been acting according to the maximum causal entropy IRL model given in Section 3 (where we have dropped $\theta$ from the notation for clarity):
+
+$$
+\begin{array} { r l r } & { \displaystyle \pi _ { t } ( a \mid s ) = \exp ( Q _ { t } ( s , a ) - V _ { t } ( s ) ) , } & \\ & { \displaystyle V _ { t } ( s ) = \ln \sum _ { a } \exp ( Q _ { t } ( s , a ) ) } & { \qquad \mathrm { f o r ~ } 1 \leq t \leq T , } \\ & { \displaystyle Q _ { t } ( s , a ) = \theta ^ { T } f ( s ) + \sum _ { s ^ { \prime } } \mathcal { T } ( s ^ { \prime } \mid s , a ) V _ { t + 1 } ( s ^ { \prime } ) } & { \qquad \mathrm { f o r ~ } 1 \leq t \leq T , } \\ & { \displaystyle V _ { T + 1 } ( s ) = 0 . } & \end{array}
+$$
+
+In the following, unless otherwise specified, all expectations over states and actions use the probability distribution over trajectories from the above model, starting from the state and action just prior. For example, Es0T ,a0T [X (s0T , a0T )] = Ps0 ,a0 $\begin{array} { r } { \mathbb { E } _ { s _ { T } ^ { \prime } , a _ { T } ^ { \prime } } \left[ \bar { X } ( s _ { T } ^ { \prime } , a _ { T } ^ { \prime } ) \right] = \sum _ { s _ { T } ^ { \prime } , a _ { T } ^ { \prime } } \mathcal { T } ( s _ { T } ^ { \prime } \mid s _ { T - 1 } , a _ { T - 1 } ) \pi _ { T } ( a _ { T } ^ { \prime } \mid s _ { T } ^ { \prime } ) X ( s _ { T } ^ { \prime } , a _ { T } ^ { \prime } ) } \end{array}$ . In addition, for all probability distributions over states and actions, we drop the dependence on $\theta$ for readability, so the probability of reaching state $s _ { T }$ is written as $p ( { \boldsymbol { s } } _ { T } )$ instead of $p ( s _ { T } \mid \theta )$ .
+
+First, we compute the gradient of $V _ { t } ( s )$ . We have $\nabla _ { \boldsymbol { \theta } } V _ { T + 1 } ( s ) = 0$ , and for $0 \leq t \leq T$
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } V _ { \lfloor \epsilon ( s ) \rfloor } } \\ &  = \nabla _ { \theta } \log \Big [ \operatorname* { m i n } _ { \epsilon ^ { \prime } \in \mathcal { N } _ { \epsilon ^ { \prime } } ( \epsilon _ { s } , a _ { \epsilon } ^ { \prime } ) \big \} } \\ & { \quad \times _ { \epsilon ^ { \prime } } ^ { \epsilon ^ { \prime } } } \\ & { = \frac { 1 } { \exp { [ ( V _ { \epsilon } ( s _ { \epsilon } ) ) ] } } \sum _ { \epsilon ^ { \prime } \in \mathcal { N } ( \mathcal { G } _ { \epsilon } ( s _ { \epsilon } , a _ { \epsilon } ^ { \prime } ) ) \in \mathcal { G } _ { \epsilon } ( s _ { \epsilon } , a _ { \epsilon } ^ { \prime } ) } } \\ & { = \frac { 1 } { \exp { [ ( V _ { \epsilon } ( s _ { \epsilon } ) ) ] } } \sum _ { \epsilon ^ { \prime } \in \mathcal { N } ( \mathcal { G } _ { \epsilon } ( s _ { \epsilon } , a _ { \epsilon } ^ { \prime } ) ) \in \mathcal { V } _ { \epsilon } } \Big [ \theta ^ { T } \int ( s _ { \epsilon } ) + \mathbb { E } _ { s _ { \epsilon } ^ { \prime } ( s _ { \epsilon } ^ { \prime } ) \sim \mathcal { V } ( \epsilon ^ { \prime } \cup s , a _ { \epsilon } ^ { \prime } ) } \left[ V _ { \epsilon + 1 } ( s _ { \epsilon + 1 } ^ { \prime } ) \right] \Big ] } \\ &  = \sum _ { \epsilon ^ { \prime } \in \mathcal { N } ( \mathcal { G } _ { \epsilon } ( s _ { \epsilon } , a _ { \epsilon } ^ { \prime } ) ) \sim \mathcal { V } _ { \epsilon } ( s _ { \epsilon } , a _ { \epsilon } ^ { \prime } ) \in \mathcal { F } _ { \epsilon } ( s _ { \epsilon } ) \sim \mathbb { V } _ { \epsilon } ( s _ { \epsilon + 1 } ^ { \prime } ) \sim \mathcal { V } ( \epsilon | a _ { \epsilon } , a _ { \epsilon } ^ { \prime } ) \left[ V _ { \epsilon } ( s _ { \epsilon + 1 } ^ { \prime } ) - V _ { \epsilon } ( s _ { \epsilon + 1 } ^ { \prime } ) \right] } \\ &  = \sum _  \epsilon ^ { \prime } \in \mathcal { F } ( a _ { \epsilon } ^ { \prime } \mid s _ { \epsilon } ) \in \mathcal { F } _ { \epsilon } ( s _ { \epsilon } ^ { \prime } ) \sim \mathcal { V } _ { \epsilon } ( s _ { \epsilon } ^ { \prime } ) \sim \mathcal { V } _  \epsilon \end{array}
+$$
+
+Unrolling the recursion, we get that the gradient is the expected feature counts under the policy implied by $\theta$ from $s _ { t }$ onwards, which we could prove using induction. Define:
+
+$$
+\mathcal { F } _ { t } ( s _ { t } ) \equiv f ( s _ { t } ) + \mathbb { E } _ { a _ { t : T - 1 } ^ { \prime } , s _ { t + 1 : T } ^ { \prime } } \left[ \sum _ { t ^ { \prime } = t + 1 } ^ { T } f ( s _ { t ^ { \prime } } ^ { \prime } ) \right] .
+$$
+
+Then we have:
+
+$$
+\nabla _ { \boldsymbol { \theta } } V _ { t } ( s _ { t } ) = \mathcal { F } _ { t } ( s _ { t } ) .
+$$
+
+We can now calculate the gradient we actually care about:
+
+$$
+\begin{array} { l } { { \nabla _ { 0 } \ln p ( \hat { \rho } _ { T } ) } } \\ { { { } } } \\ { { \displaystyle = \nabla _ { \theta } \left[ \ln p ( s _ { 0 } ) + \sum _ { k = 0 } ^ { T } \ln \pi ( i \alpha _ { k } \mid s _ { k } ) + \sum _ { t = 0 } ^ { T - 1 } \ln \pi ( s _ { t + 1 } \mid s _ { k } , \alpha _ { k } ) \right] } } \\ { { { } } } \\ { { { } } } \\ { { { } = \displaystyle \sum _ { \ell = 0 } ^ { T } \nabla _ { \theta } \ln \pi _ { \ell } ( \alpha _ { \ell } \mid s _ { \ell } ) } } \\ { { { } } } \\ { { { } = \displaystyle \sum _ { \ell = 0 } ^ { T } \nabla _ { \theta } \left[ Q _ { \ell } ( s _ { \ell } , \alpha _ { \ell } ) - V _ { \ell } ( s _ { \ell } ) \right] } } \\ { { { } } } \\ { { { } = \displaystyle \sum _ { \ell = 0 } ^ { T } \nabla _ { \theta } \left[ \theta ^ { \ell } J ( s _ { \ell } ) + \mathbb { E } _ { s _ { \ell + 1 } } \left[ \mathbb { V } _ { \ell + 1 } ( s _ { \ell + 1 } ^ { \ell } ) \right] - V _ { \ell } ( s _ { \ell } ) \right] } } \\ { { { } } } \\ { { { } = \displaystyle \sum _ { \ell = 0 } ^ { T } \left( f ( s _ { \ell } ) + \mathbb { E } _ { s _ { \ell - 1 } } \left[ \nabla _ { \theta } V _ { \ell + 1 } ( s _ { \ell + 1 } ^ { \ell } ) \right] - \nabla _ { \theta } V _ { \ell } ( s _ { \ell } ) \right) . } } \end{array}
+$$
+
+only $\pi _ { t }$ depends on $\theta$
+
+The last term of the summation is $f ( s _ { T } ) + \mathbb { E } _ { s _ { T + 1 } ^ { \prime } } \left[ \nabla _ { \theta } V _ { T + 1 } ( s _ { T + 1 } ^ { \prime } ) \right] - \nabla _ { \theta } V _ { T } ( s _ { T } )$ , which simplifies to $f ( s _ { T } ) + 0 - \mathcal { F } _ { T } ( s _ { T } ) = f ( s _ { T } ) - f ( s _ { T } ) = { \bar { 0 } }$ , so we can drop it. Thus, our gradient is:
+
+$$
+\nabla _ { \theta } \ln p ( \tau _ { T } ) = \sum _ { t = 0 } ^ { T - 1 } \left( f ( s _ { t } ) + \mathbb { E } _ { s _ { t + 1 } ^ { \prime } } \left[ \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ^ { \prime } ) \right] - \mathcal { F } _ { t } ( s _ { t } ) \right) .
+$$
+
+This is the gradient we will use in Appendix $\mathbf { B }$ , but a little more manipulation allows us to compare with the gradient in Ziebart et al. (2010). We reintroduce the terms that we cancelled above:
+
+$$
+\begin{array} { r l } & { = \left( \displaystyle \sum _ { t = 0 } ^ { T } f ( s _ { t } ) \right) + \left( \displaystyle \sum _ { t = 0 } ^ { T - 1 } \mathbb { E } _ { s _ { t + 1 } ^ { \prime } } \left[ \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ^ { \prime } ) \right] \right) - \left( \mathcal { F } _ { 0 } ( s _ { 0 } ) + \displaystyle \sum _ { t = 0 } ^ { T - 1 } \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ) \right) } \\ & { = \left( \displaystyle \sum _ { t = 0 } ^ { T } f ( s _ { t } ) \right) - \mathcal { F } _ { 0 } ( s _ { 0 } ) + \displaystyle \sum _ { t = 0 } ^ { T - 1 } \left( \mathbb { E } _ { s _ { t + 1 } ^ { \prime } } \left[ \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ^ { \prime } ) \right] - \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ) \right) . } \end{array}
+$$
+
+Ziebart et al. (2010) states that the gradient is given by the expert policy feature expectations minus the learned policy feature expectations, and in practice uses the feature expectations from demonstrations to approximate the expert policy feature expectations. Assuming we have $N$ trajectories $\{ \tau _ { i } \}$ , the gradient would be $\begin{array} { r } { \Big ( \frac { 1 } { N } \sum _ { i } \sum _ { t = 0 } ^ { T } f ( s _ { t , i } ) \Big ) - \mathbb { E } _ { s _ { 0 } } \left[ \mathscr { F } _ { 0 } ( s _ { 0 } ) \right] } \end{array}$ . The first term matches our first term exactly. Our second term matches the second term in the limit of sufficiently many trajectories, so that the starting states $s _ { 0 }$ follow the distribution $p ( s _ { 0 } )$ . Our third term converges to zero with sufficiently many trajectories, since any $s _ { t } , a _ { t }$ pair in a demonstration will be present sufficiently often that the empirical counts of $s _ { t + 1 }$ will match the expected proportions prescribed by $\mathcal { T } ( \cdot \mid s _ { t } , \mathbf { \bar { \alpha } } { a } _ { t } )$ .
+
+In a deterministic environment, we have $\begin{array} { r } { \mathcal { T } ( s _ { t + 1 } ^ { \prime } \mid s _ { t } , a _ { t } ) = 1 [ s _ { t + 1 } ^ { \prime } = s _ { t + 1 } ] } \end{array}$ since only one transition is possible. Thus, the third term is zero and even for one trajectory the gradient reduces to $\begin{array} { r l } {  { ( \sum _ { t = 0 } ^ { T } f ( s _ { t } ) ) - \mathcal { F } _ { 0 } ( s _ { 0 } ) } } & { { } } \end{array}$ . This differs from the gradient in Ziebart et al. (2010) only in that it computes feature expectations from the observed starting state $s _ { 0 }$ instead of the MDP distribution over initial states $p ( s _ { 0 } )$ .
+
+In a stochastic environment, the third term need not be zero, and corrects for the “bias” in the observed states $s _ { t + 1 }$ . Intuitively, when the expert chose action $a _ { t }$ , she did not know which next state $s _ { t + 1 } ^ { \prime }$ would arise, but the first term of our gradient upweights the particular next state $s _ { t + 1 }$ that we observed. The third term downweights the future value of the observed state and upweights the future value of all other states, all in proportion to their prior probability $\mathcal { T } ( s _ { t + 1 } ^ { \prime } \mid s _ { t } , \bar { a _ { t } } )$ .
+
+# B
+
+This section provides a derivation of the gradient $\nabla _ { \theta } \ln p ( s _ { 0 } )$ , which is needed to solve argmax ${ } _ { \theta } \ln p ( s _ { 0 } )$ with gradient ascent. We provide the results first as a quick reference:
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } \ln p ( s _ { 0 } ) = \displaystyle \frac { G _ { 0 } ( s _ { 0 } ) } { p ( s _ { 0 } ) } , } \\ & { \qquad p ( s _ { t + 1 } ) = \displaystyle \sum _ { s _ { t } , a _ { t } } p ( s _ { t } ) \pi _ { t } ( a _ { t } \mid s _ { t } ) { \mathcal { T } } ( s _ { t + 1 } \mid s _ { t } , a _ { t } ) , } \\ & { G _ { t + 1 } ( s _ { t + 1 } ) = \displaystyle \sum _ { s _ { t } , a _ { t } } { \mathcal { T } } ( s _ { t + 1 } \mid s _ { t } , a _ { t } ) \pi _ { t } ( a _ { t } \mid s _ { t } ) \bigg ( p ( s _ { t } ) g ( s _ { t } , a _ { t } ) + G _ { t } ( s _ { t } ) \bigg ) , } \\ & { \qquad g ( s _ { t } , a _ { t } ) = f ( s _ { t } ) + \mathbb { E } _ { s _ { t + 1 } ^ { \prime } } \left[ { \mathcal { F } } _ { t + 1 } ( s _ { t + 1 } ^ { \prime } ) \right] - { \mathcal { F } } _ { t } ( s _ { t } ) , } \\ & { { \mathcal { F } } _ { t - 1 } ( s _ { t - 1 } ) = f ( s _ { t - 1 } ) + \displaystyle \sum _ { a _ { t - 1 } ^ { \prime } , s _ { t } ^ { \prime } } { \pi } _ { t - 1 } ( a _ { t - 1 } ^ { \prime } \mid s _ { t - 1 } ) { \mathcal { T } } ( s _ { t } ^ { \prime } \mid s _ { t - 1 } , a _ { t - 1 } ^ { \prime } ) { \mathcal { F } } _ { t } ( s _ { t } ) . } \end{array}
+$$
+
+Base cases: first, $p ( s _ { - T } )$ is given, second, $G _ { - T } ( s _ { - T } ) = 0$ , and third, $\mathcal { F } _ { 0 } ( s _ { 0 } ) = f ( s _ { 0 } )$ .
+
+For the derivation, we start by expressing the gradient in terms of gradients of trajectories, so that we can use the result from Appendix A. Note that, by inspecting the final form of the gradient in Appendix A, we can see that $\nabla _ { \theta } p \big ( \tau _ { - T : 0 } \big )$ is independent of $a _ { 0 }$ . Then, we have:
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } \ln p ( s _ { 0 } ) = \displaystyle \frac { 1 } { p ( s _ { 0 } ) } \nabla _ { \theta } p ( s _ { 0 } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \end{array}
+$$
+
+This has a nice interpretation – compute the gradient for each trajectory and take the weighted sum, where each weight is the probability of the trajectory given the evidence $s _ { 0 }$ and current reward $\theta$ .
+
+We can rewrite the gradient in Equation 6 as $\begin{array} { r } { \nabla _ { \theta } \ln p ( \tau _ { T } ) = \sum _ { t = 0 } ^ { T - 1 } g ( s _ { t } , a _ { t } ) } \end{array}$ , where
+
+$$
+\begin{array} { r } { g ( s _ { t } , a _ { t } ) \equiv f ( s _ { t } ) + \mathbb { E } _ { s _ { t + 1 } ^ { \prime } } \left[ \mathcal { F } _ { t + 1 } ( s _ { t + 1 } ^ { \prime } ) \right] - \mathcal { F } _ { t } ( s _ { t } ) . } \end{array}
+$$
+
+We can now substitute this to get:
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \theta } \ln p ( s _ { 0 } ) = \sum _ { s - { T } : - 1 , a - { T } : - 1 } p ( \tau _ { - T : - 1 } \mid s _ { 0 } ) \left( \sum _ { t = - { T } } ^ { - 1 } g ( s _ { t } , a _ { t } ) \right) } \\ { = \displaystyle \frac { 1 } { p ( s _ { 0 } ) } \sum _ { s - { T } : - 1 , a - { T } : - 1 } \left[ p ( \tau _ { - T : - 1 } , s _ { 0 } ) \sum _ { t = - { T } } ^ { - 1 } g ( s _ { t } , a _ { t } ) \right] } \\ { = \displaystyle \frac { 1 } { p ( s _ { 0 } ) } \sum _ { s - { T } : - 1 , a - { T } : - 1 } \left[ p ( \tau _ { - T : - 1 } , s _ { 0 } ) \sum _ { t = - { T } } ^ { - 1 } g ( s _ { t } , a _ { t } ) \right] . } \end{array}
+$$
+
+Note that we can compute $p ( s _ { t } )$ since we are given the distribution $p ( s _ { - T } )$ and we can use the recursive rule $\begin{array} { r } { p ( s _ { t + 1 } ) = \sum _ { s _ { t } , a _ { t } } p ( s _ { t } ) \pi _ { t } ( a _ { t } \mid s _ { t } ) \mathcal { T } ( s _ { t + 1 } \mid s _ { t } , a _ { t } ) } \end{array}$ .
+
+In order to compute $g ( s _ { t } , a _ { t } )$ we need to compute $\mathcal { F } _ { t } ( s _ { t } )$ , which has base case $\mathcal { F } _ { 0 } ( s _ { 0 } ) = f ( s _ { 0 } )$ and recursive rule:
+
+$$
+\begin{array} { r l } & { \mathcal { F } _ { t - 1 } \big ( s _ { t - 1 } \big ) } \\ & { = f ( s _ { t - 1 } ) + \mathbb { E } _ { a _ { t - 1 : 1 } ^ { \prime } , s _ { t } ^ { \prime } , 0 } \left[ \underset { t ^ { \prime } = t } { \overset { 0 } { \sum } } f \big ( s _ { t ^ { \prime } } ^ { \prime } \big ) \right] } \\ & { = f ( s _ { t - 1 } ) + \underset { a _ { t - 1 } ^ { \prime } , s _ { t } ^ { \prime } } { \sum } \pi _ { t - 1 } \big ( a _ { t - 1 } ^ { \prime } \mid s _ { t - 1 } \big ) \mathcal { T } \big ( s _ { t } ^ { \prime } \mid s _ { t - 1 } , a _ { t - 1 } ^ { \prime } \big ) \left[ f \big ( s _ { t } ^ { \prime } \big ) + \mathbb { E } _ { a _ { t - 1 } ^ { \prime } , s _ { t + 1 : 0 } ^ { \prime } } \left[ \underset { t ^ { \prime } = t + 1 } { \overset { 0 } { \sum } } f \big ( s _ { t ^ { \prime } } ^ { \prime } \big ) \right] \right] } \\ & { = f ( s _ { t - 1 } ) + \underset { a _ { t - 1 } ^ { \prime } , s _ { t } ^ { \prime } } { \sum } \pi _ { t - 1 } \big ( a _ { t - 1 } ^ { \prime } \mid s _ { t - 1 } \big ) \mathcal { T } \big ( s _ { t } ^ { \prime } \mid s _ { t - 1 } , a _ { t - 1 } ^ { \prime } \big ) \mathcal { F } _ { t } \big ( s _ { t } \big ) . } \end{array}
+$$
+
+For the remaining part of the gradient, define $G _ { t }$ such that $\begin{array} { r } { \nabla _ { \theta } \ln { p ( s _ { 0 } ) } = \frac { G _ { 0 } ( s _ { 0 } ) } { p ( s _ { 0 } ) } } \end{array}$ :
+
+$$
+G _ { t } ( s _ { t } ) \equiv \sum _ { s _ { - T : t - 1 } , a _ { - T : t - 1 } } \left[ p ( \tau _ { - T : t - 1 } , s _ { t } ) \sum _ { t ^ { \prime } = - T } ^ { t - 1 } g ( s _ { t ^ { \prime } } , a _ { t ^ { \prime } } ) \right] .
+$$
+
+We now derive a recursive relation for $G$ :
+
+$$
+\begin{array} { r l } & { \mathcal { G } _ { + + } [ s _ { + + } ] } \\ & { = \displaystyle \sum _ { s = - \infty } \Bigg [ p ( \sigma _ { - 2 : s } , s _ { + + } ) \sum _ { \psi = - \infty } ^ { s } g ( s _ { \psi } , \sigma _ { \psi } ) } \\ & { \quad - \sum _ { s = \infty } ^ { s } \gamma _ { s } \left[ \mathcal { F } _ { - \mathcal { R } _ { + + } } ^ { \prime } , s _ { + + } ) \sum _ { \psi = - \infty } ^ { s } ( s _ { \psi } , \sigma _ { \psi } ) g ( s _ { \psi } , \sigma _ { \psi } ) \right] } \\ & { = \displaystyle \sum _ { s \neq \infty } \sum _ { s \neq \infty } \sum _ { s = - \infty + \infty - 1 } \mathcal { F } _ { \left( s _ { \psi + + + } \right. } [ s _ { s _ { \psi } , \sigma _ { \psi } } ] s _ { \psi } [ \sigma _ { \psi } , | s _ { \psi } | ) ( \sigma _ { \psi } - \mathbb { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal { R } _ { - \mathcal } { R _ - \mathcal { R } _ { - \mathcal } { R _ } } } } } } } } } } } ) } \Bigg ( g ( s _ { \psi } , u _ { s } , u _ { s + + } ) + \displaystyle \sum _ { s = - \infty } ^ { - 1 } g ( s _ { \psi } , \pi _ { \psi } )  \\ & { = \displaystyle \sum _ { s = \infty } \Bigg [ \mathcal { T } _ { \left( s _ { + + } \right. } [ s _ { + } , \sigma _ { s } ] ) \pi _ { \mathfrak { c } _ { \psi } } ( \sigma _ { \psi } ) \left( s _ { \psi } \right) \left( \displaystyle \sum _ { s = - \infty + \infty - \infty + \infty } p ( \sigma _ { - 2 : s _ { - } \{ s _ { - } \} , s _ { \psi } } ) \right) g ( s _ { \psi } , \sigma _ { \psi } ) \Bigg ] } \\ &  \quad + \displaystyle \sum _ { s = \infty } \Bigg [ \mathcal { T } _ { \left( s _ { + } \right. } [ s _ { + } , \sigma _ { s } ] ) \pi _ { \mathfrak { c } _ { \psi } } ( \sigma _ { \psi } ) \underset { s = \pm \infty } { \sum _ { s = - \infty } ^ { s } } \ \end{array}
+$$
+
+For the base case, note that
+
+$$
+\begin{array} { l } { \tilde { \mathfrak { r } } _ { - T + 1 } \big ( \mathfrak { s } _ { - T + 1 } \big ) = \displaystyle \sum _ { \substack { s _ { - T } , a _ { - T } } } \left[ p \big ( \mathfrak { s } _ { - T } , a _ { - T } , \mathfrak { s } _ { - T + 1 } \big ) g \big ( \mathfrak { s } _ { - T } , a _ { - T } , \mathfrak { s } _ { - T + 1 } \big ) \right] } \\ { = \displaystyle \sum _ { \substack { s _ { - T } , a _ { - T } } } \mathcal { T } \big ( \mathfrak { s } _ { - T + 1 } \mid \mathfrak { s } _ { - T } , a _ { - T } \big ) \pi _ { - T } \big ( a _ { - T } \mid \mathfrak { s } _ { - T } \big ) \bigg ( p \big ( \mathfrak { s } _ { - T } \big ) g \big ( \mathfrak { s } _ { - T } , a _ { - T } , \mathfrak { s } _ { - T + 1 } \big ) \bigg ) . } \end{array}
+$$
+
+Comparing this to the recursive rule, for the base case we can set $G _ { - T } ( s _ { - T } ) = 0$ .
+
+# C
+
+Instead of estimating the MLE (or MAP if we have a prior) using RLSP, we could approximate the entire posterior distribution. One standard way to address the computational challenges involved with the continuous and high-dimensional nature of $\theta$ is to use MCMC sampling to sample from $p ( \theta \mid s _ { 0 } ) \propto p ( s _ { 0 } \mid \theta ) p ( \theta )$ . The resulting algorithm resembles Bayesian IRL (Ramachandran and Amir, 2007) and is presented in Algorithm 1.
+
+While this algorithm is less efficient and noisier than RLSP, it gives us an estimate of the full posterior distribution. In our experiments, we collapsed the full distribution into a point estimate by taking the mean. Initial experiments showed that the algorithm was slower and noisier than the gradientbased RLSP, so we did not test it further. However, in future work we could better leverage the full distribution, for example to create risk-averse policies, to identify features that are uncertain, or to identify features that are certain but conflict with the specified reward, after which we could actively query Alice for more information.
+
+# Algorithm 1 MCMC sampling from the one state IRL posterior
+
+Require: MDP $\mathcal { M }$ , prior $p ( \theta )$ , step size $\delta$   
+1: $\theta \gets$ random sample $( p ( \theta ) )$   
+2: $\pi , V = \operatorname { s o f t }$ value iteration $( \mathcal { M } , \theta )$   
+3: $p \gets p ( s _ { 0 } \mid \theta ) p ( \theta )$   
+4: repeat   
+5: $\theta ^ { \prime } \gets$ random sample $\left( \mathcal { N } ( \theta , \delta ) \right)$   
+6: π0, V 0 = soft value iteration $( { \mathcal { M } } , \theta ^ { \prime } )$ . The value function is initialized with $V$ .   
+7: $p ^ { \prime } \gets p ( s _ { 0 } \mid \theta ^ { \prime } ) p ( \theta ^ { \prime } )$   
+8: if random sample $( { \mathrm { U n i f } } ( 0 , 1 ) ) \leq \operatorname* { m i n } ( 1 , \frac { p ^ { \prime } } { p } )$ then   
+9: $\theta  \theta ^ { \prime } ; \ V  V ^ { \prime }$   
+10: end if   
+11: append $\theta$ to the list of samples   
+12: until have generated the desired number of samples
+
+![](images/c7573fdcd26b33dcaa2892a029cc2bd0e7b6ea73a9d5666d72622d443f5d6366.jpg)  
+Figure 4: Comparison of the Additive and Bayesian methods. We show how the percentage of true reward obtained by $\pi _ { \mathrm { R L S P } }$ varies as we change the tradeoff between $\theta _ { \mathrm { A l i c e } }$ and $\theta _ { \mathrm { s p e c } }$ . The zero temperature case corresponds to traditional value iteration; this often leads to identical behavior and so the lines overlap. So, we also show the results when planning with soft value iteration, varying the softmax temperature, to introduce some noise into the policy. Overall, there is not much difference between the two methods. We did not include the Apples environment because $\theta _ { \mathrm { s p e c } }$ is uniformly zero and the Additive and Bayesian methods do exactly the same thing.
+
+# D COMBINING THE SPECIFIED REWARD WITH THE INFERRED REWARD
+
+In Section 5, we evaluated RLSP by combining the reward it infers with a specified reward to get a final reward $\theta _ { \mathrm { f i n a l } } = \theta _ { \mathrm { A l i c e } } + \lambda \theta _ { \mathrm { s p e c } }$ . As discussed in Section 6, the problem of combining $\theta _ { \mathrm { A l i c e } }$ and $\theta _ { \mathrm { s p e c } }$ is difficult, since the two rewards incentivize different behaviors and will conflict. The Additive method above is a simple way of trading off between the two.
+
+Both RLSP and the sampling algorithm of Appendix C can incorporate a prior over $\theta$ . Another way to combine the two rewards is to condition the prior on $\theta _ { \mathrm { s p e c } }$ before running the algorithms. In particular, we could replace our prior $P ( \theta _ { \mathrm { A l i c e } } )$ with a new prior $P ( \theta _ { \mathrm { A l i c e } } \mid \theta _ { \mathrm { s p e c } } )$ , such as a Gaussian distribution centered at $\theta _ { \mathrm { s p e c } }$ . When we use this prior, the reward returned by RLSP can be used as the final reward $\theta _ { \mathrm { f i n a l } }$ .
+
+It might seem like this is a principled Bayesian method that allows us to combine the two rewards. However, the conflict between the two reward functions still exists. In this formulation, it arises in the new prior $P ( \theta _ { \mathrm { A l i c e } } \mid \theta _ { \mathrm { s p e c } } )$ . Modeling this as a Gaussian centered at $\theta _ { \mathrm { s p e c } }$ suggests that before knowing $s _ { 0 }$ , it seems likely that $\theta _ { \mathrm { A l i c e } }$ is very similar to $\theta _ { \mathrm { s p e c } }$ . However, this is not true – Alice is probably providing the reward $\theta _ { \mathrm { s p e c } }$ to the robot so that it causes some change to the state that she has optimized, and so it will be predictably different from $\theta _ { \mathrm { s p e c } }$ . On the other hand, we do need to put high probability on $\theta _ { \mathrm { s p e c } }$ , since otherwise $\theta _ { \mathrm { f i n a l } }$ will not incentivize any of the behaviors that $\theta _ { \mathrm { s p e c } }$ did.
+
+Nonetheless, this is another simple heuristic for how we might combine the two rewards, that manages the tradeoff between $\theta _ { \mathrm { s p e c } }$ and $\theta _ { \mathrm { A l i c e } }$ . We compared the Additive and Bayesian methods by evaluating their robustness. We vary the parameter that controls the tradeoff and report the true reward obtained by $\pi _ { \mathrm { R L S P } }$ , as a fraction of the expected true reward under the optimal policy. For the Bayesian method, we vary the standard deviation $\sigma$ of the Gaussian prior over $\theta _ { \mathrm { A l i c e } }$ that is centered at $\theta _ { \mathrm { s p e c } }$ . For the Additive method, the natural choice would be to vary $\lambda$ ; however, in order to make the results more comparable, we instead set $\lambda = 1$ and vary the standard deviation of the Gaussian prior used while inferring $\theta _ { \mathrm { A l i c e } }$ , which is centered at zero instead of at $\theta _ { \mathrm { s p e c } }$ . A larger standard deviation allows $\theta _ { \mathrm { A l i c e } }$ to become larger in magnitude (since it is penalized less for deviating from the mean of zero reward), which effectively corresponds to a smaller $\lambda$ .
+
+While we typically create $\pi _ { \mathrm { R L S P } }$ using value iteration, this leads to deterministic policies with very sharp changes in behavior that make it hard to see differences between methods, and so we also show results with soft value iteration, which creates stochastic policies that vary more continuously. As demonstrated in Figure 4, our experiments show that overall the two methods perform very similarly, with some evidence that the Additive method is slightly more robust. The Additive method also has the benefit that it can be applied in situations where the inferred reward and specified reward are over different feature spaces, by creating the final reward $R _ { \mathrm { f i n a l } } ( s ) = { \theta _ { \mathrm { A l i c e } } } ^ { T } f _ { \mathrm { A l i c e } } ( s ) + \lambda R _ { \mathrm { s p e c } } ( s )$ .