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+# TransGAN: Two Pure Transformers Can Make One Strong GAN, and That Can Scale Up
+
+Yifan Jiang1, Shiyu Chang2,3, Zhangyang Wang1 1University of Texas at Austin 2UC Santa Barbara 3MIT-IBM Watson AI Lab {yifanjiang97,atlaswang}@utexas.edu, chang87@ucsb.edu
+
+# Abstract
+
+The recent explosive interest on transformers has suggested their potential to become powerful “universal" models for computer vision tasks, such as classification, detection, and segmentation. While those attempts mainly study the discriminative models, we explore transformers on some more notoriously difficult vision tasks, e.g., generative adversarial networks (GANs). Our goal is to conduct the first pilot study in building a GAN completely free of convolutions, using only pure transformer-based architectures. Our vanilla GAN architecture, dubbed TransGAN, consists of a memory-friendly transformer-based generator that progressively increases feature resolution, and correspondingly a multi-scale discriminator to capture simultaneously semantic contexts and low-level textures. On top of them, we introduce the new module of grid self-attention for alleviating the memory bottleneck further, in order to scale up TransGAN to high-resolution generation. We also develop a unique training recipe including a series of techniques that can mitigate the training instability issues of TransGAN, such as data augmentation, modified normalization, and relative position encoding. Our best architecture achieves highly competitive performance compared to current stateof-the-art GANs using convolutional backbones. Specifically, TransGAN sets the new state-of-the-art inception score of 10.43 and FID of 18.28 on STL-10. It also reaches the inception score of 9.02 and FID of 9.26 on CIFAR-10, and 5.28 FID on CelebA $\mathbf { 1 2 8 \times 1 2 8 }$ , respectively: both on par with the current best results. When it comes to higher-resolution (e.g. $\mathbf { 2 5 6 \times 2 5 6 } )$ ) generation tasks, such as on CelebAHQ and LSUN-Church, TransGAN continues to produce diverse visual examples with high fidelity and reasonable texture details. In addition, we dive deep into the transformer-based generation models to understand how their behaviors differ from convolutional ones, by visualizing training dynamics. The code is available at: https://github.com/VITA-Group/TransGAN.
+
+# 1 Introduction
+
+Generative adversarial networks (GANs) have gained considerable success on numerous tasks [1, 2, 3, 4, 5, 6, 7]. Unfortunately, GANs suffer from the notorious training instability, and numerous efforts have been devoted to stabilizing GAN training, introducing various regularization terms [8, 9, 10, 11], better losses [1, 12, 13, 14], and training recipes [15, 16]. Among them, one important route to improving GANs examines their neural architectures. [17, 8] reported a large-scale study of GANs and observed that when serving as (generator) backbones, popular neural architectures perform comparably well across the considered datasets. Their ablation study suggested that most of the variations applied in the ResNet family resulted in very marginal improvements. Nevertheless, neural architecture search (NAS) was later introduced to GANs and suggests enhanced backbone designs are also important for improving GANs, just like for other computer vision tasks. Those works are consistently able to discover stronger GAN architectures beyond the standard ResNet topology [18, 19, 20]. Other efforts include customized modules such as self-attention [21], style-based generator [22], and autoregressive transformer-based part composition [23].
+
+![](images/adcb25935a6e25e1b2ab70151735aa65bcbfa9d2b385186c0dc4a0005b6915f2.jpg)  
+Figure 1: Representative visual examples synthesized by TransGAN, without using convolutional layers. (a) The synthesized visual examples on CelebA-HQ $2 5 6 \times 2 5 6 )$ dataset. (b) The linear interpolation results between two latent vectors, on CelebA-HQ ( $2 5 6 \times 2 5 6 )$ ) dataset.
+
+However, one last “commonsense" seems to have seldomly been challenged: using convolutional neural networks (CNNs) as GAN backbones. The original GAN [24, 25] used fully-connected networks and can only generate small images. DCGAN [26] was the first to scale up GANs using CNN architectures, which allowed for stable training for higher resolution and deeper generative models. Since then, in the computer vision domain, every successful GAN relies on CNN-based generators and discriminators. Convolutions, with the strong inductive bias for natural images, crucially contribute to the appealing visual results and rich diversity achieved by modern GANs.
+
+Can we build a strong GAN completely free of convolutions? This is a question not only arising from intellectual curiosity, but also of practical relevance. Fundamentally, a convolution operator has a local receptive field, and hence CNNs cannot process long-range dependencies unless passing through a sufficient number of layers. However, that is inefficient, and could cause the loss of feature resolution and fine details, in addition to the difficulty of optimization. Vanilla CNN-based models are therefore inherently not well suited for capturing an input image’s “global" statistics, as demonstrated by the benefits from adopting self-attention [21] and non-local [27] operations in computer vision. Moreover, the spatial invariance possessed by convolution poses a bottleneck on its ability of adapting to spatially varying/heterogeneous visual patterns, which also motivates the success of relational network [28], dynamic filters [29, 30] and kernel prediction [31] methods.
+
+# 1.1 Our Contributions
+
+This paper aims to be the first pilot study to build a GAN completely free of convolutions, using only pure transformer-based architectures. We are inspired by the recent success of transformer architectures in computer vision [32, 33, 34]. Compared to parallel generative modeling works [21, 23, 35] that applied self-attention or transformer encoder in conjunction with CNN-based backbones, our goal is more ambitious and faces several daunting gaps ahead. First and foremost, although a pure transformer architecture applied directly to sequences of image patches can perform very well on image classification tasks [34], it is unclear whether the same way remains effective in generating images, which crucially demands the spatial coherency in structure, color, and texture, as well as the richness of fine details. The handful of existing transformers that output images have unanimously leveraged convolutional part encoders [23] or feature extractors [36, 37]. Moreover, even given well-designed CNN-based architectures, training GANs is notoriously unstable and prone to mode collapse [15]. Training vision transformers are also known to be tedious, heavy, and data-hungry [34]. Combining the two will undoubtedly amplify the challenges of training.
+
+In view of those challenges, this paper presents a coherent set of efforts and innovations towards building the pure transformer-based GAN architectures, dubbed TransGAN. A naive option may directly stack multiple transformer blocks from raw pixel inputs, but that would scale poorly due to memory explosion. Instead, we start with a memory-friendly transformer-based generator by gradually increasing the feature map resolution in each stage. Correspondingly, we also improve the discriminator with a multi-scale structure that takes patches of varied size as inputs, which balances between capturing global contexts and local details, in addition to enhancing memory efficiency more. Based on the above generator-discriminator design, we introduce a new module called grid self-attention, that alleviates the memory bottleneck further when scaling up TransGAN to high-resolution generation (e.g. $2 5 6 \times 2 5 6$ ).
+
+To address the aforementioned instability issue brought by both GAN and Transformer, we also develop a unique training recipe in association with our innovative TransGAN architecture, that effectively stabilizes its optimization and generalization. That includes showings the necessity of data augmentation, modifying layer normalization, and replacing absolute token locations with relative position encoding. Our contributions are outlined below:
+
+• Novel Architecture Design: We build the first GAN using purely transformers and no convolution. TransGAN has customized a memory-friendly generator and a multi-scale discriminator, and is further equipped with a new grid self-attention mechanism. Those architectural components are thoughtfully designed to balance memory efficiency, global feature statistics, and local fine details with spatial variances.   
+• New Training Recipe: We study a number of techniques to train TransGAN better, including leveraging data augmentation, modifying layer normalization, and adopting relative position encoding, for both generator and discriminator. Extensive ablation studies, discussions, and insights are presented.   
+Performance and Scalability: TransGAN achieves highly competitive performance compared to current state-of-the-art GANs. Specifically, it sets the new state-of-the-art inception score of 10.43 and FID score of 18.28 on STL-10. It also reaches competitive 9.02 inception score and 9.26 FID on CIFAR-10, and 5.28 FID score on CelebA $1 2 8 \times 1 2 8$ , respectively. Meanwhile, we also evaluate TransGAN on higher-resolution (e.g., $2 5 6 \times 2 5 6 )$ generation tasks, where TransGAN continues to yield diverse and impressive visual examples.
+
+# 2 Related Works
+
+Generative Adversarial Networks. After its origin, GANs quickly embraced fully convolutional backbones [26], and inherited most successful designs from CNNs such as batch normalization, pooling, (Leaky) ReLU and more [38, 39, 40, 18]. GANs are widely adopted in image translation [3, 4, 41], image enhancement [7, 42, 43], and image editing [44, 45]. To alleviate its unstable training, a number of techniques have been studied, including the Wasserstein loss [46], the stylebased generator [22], progressive training [16], lottery ticket [47], and spectral normalization [48].
+
+Transformers in Computer Vision. The original transformer was built for NLP [49], where the multi-head self-attention and feed-forward MLP layer are stacked to capture the long-term correlation between words. A recent work [34] implements highly competitive ImageNet classification using pure transformers, by treating an image as a sequence of $1 6 \times 1 6$ visual words. It has strong representation capability and is free of human-defined inductive bias. In comparison, CNNs exhibit a strong bias towards feature locality, as well as spatial invariance due to sharing filter weights across all locations. However, the success of original vision transformer relies on pretraining on large-scale external data. [50, 51] improve the data efficiency and address the difficulty of optimizing deeper models. Other works introduce the pyramid/hierarchical structure to transformer [52, 53, 54] or combine it with convolutional layers [55, 56]. Besides image classification task, transformer and its variants are also explored on image processing [37], point cloud [57], semantic segmentation [58], object detection [32, 59] and so on. A comprehensive review is referred to [60].
+
+Transformer Modules for Image Generation. There exist several related works combining the transformer modules into image generation models, by replacing certain components of CNNs. [61] firstly formulated image generation as autoregressive sequence generation, for which they adopted a transformer architecture. [62] propose sparse factorization of the attention matrix to reduce its complexity. While those two works did not tackle the GANs, one recent (concurrent) work [23] used a convolutional GAN to learn a codebook of context-rich visual parts, whose composition is subsequently modeled with an autoregressive transformer architecture.The authors demonstrated success in synthesizing high-resolution images. However, the overall CNN architecture remains in place (including CNN encoder/decoder for the generators, and a fully CNN-based discriminator), and the customized designs (e.g, codebook and quantization) also limit their model’s versatility. Another concurrent work [35] employs a bipartite self-attention on StyleGAN and thus it can propagate latent variables to the evolving visual features, yet its main structure is still convolutional, including both the generator and discriminator. To our best knowledge, no other existing work has tried to completely remove convolutions from their generative modeling frameworks.
+
+![](images/e89dc9eb445115b78109b53440aa8db8c120d23b4e23956ef5aedc971cd11810.jpg)  
+Figure 2: The pipeline of the pure transform-based generator and discriminator of TransGAN. We take $2 5 6 \times 2 5 6$ resolution image generation task as a typical example to illustrate the main procedure. Here patch size $p$ is set to 32 as an example for the convenience of illustration, while practically the patch size is normally set to be no more than $8 \times 8$ , depending on the specific dataset. Grid Transformer Blocks refers to the transformer blocks with the proposed grid self-attention. Detailed architecture configurations are included in Appendix B.
+
+# 3 Technical Approach: A Journey Towards GAN with Pure Transformers
+
+In this section, we start by introducing the memory-friendly generator and multi-scale discriminator, equipped with a novel grid self-attention. We then introduce a series of training techniques to stabilize its training procedure, including data augmentation, the modified normalization, and injecting relative position encoding to self-attention.
+
+To start with, we choose the transformer encoder [49] as our basic block and try to make minimal changes. An encoder is a composition of two parts. The first part is constructed by a multi-head self-attention module and the second part is a feed-forward MLP with GELU non-linearity. The normalization layer is applied before both of the two parts. Both parts employ residual connection.
+
+# 3.1 Memory-friendly Generator
+
+The task of generation poses a high standard for spatial coherency in structure, color, and texture, both globally and locally. The transformer encoders take embedding token words as inputs and calculate the interaction between each token recursively. [63, 34]. The main dilemma here is: what is the right “word" for image generation tasks? If we similarly generate an image in a pixel-by-pixel manner through stacking transformer encoders, even a low-resolution image (e.g. $3 2 \times 3 2$ ) can result in an excessively long sequence (1024), causing the explosive cost of self-attention (quadratic w.r.t. the sequence length) and prohibiting the scalability to higher resolutions. To avoid this daunting cost, we are inspired by a common design philosophy in CNN-based GANs, to iteratively upscale the resolution at multiple stages [25, 16]. Our strategy is hence to increase the input sequence and reduce the embedding dimension gradually .
+
+Figure 2 (left) illustrates a memory-friendly transformer-based generator that consists of multiple stages. Each stage stacks several transformer blocks. By stages, we gradually increase the feature map resolution until it meets the target resolution $H \times W$ . Specifically, the generator takes the random noise as its input, and passes it through a multiple-layer perceptron (MLP) to a vector of length $H _ { 0 } \times W _ { 0 } \times C$ . The vector is reshaped into a $H _ { 0 } \times W _ { 0 }$ resolution feature map (by default we use $H _ { 0 } = W _ { 0 } = 8$ ), each point a $C$ -dimensional embedding. This “feature map" is next treated as a length-64 sequence of $C$ -dimensional tokens, combined with the learnable positional encoding.
+
+To scale up to higher-resolution images, we insert an upsampling module after each stage, consisting of a reshaping and resolution-upscaling layer. For lower-resolution stages (resolution lower than $6 4 \times 6 4 )$ ), the upsampling module firstly reshapes the 1D sequence of token embedding back to a 2D feature map $X _ { i } ~ \overset { \cdot } { \in } ~ \mathbb { R } ^ { \cup _ { i } \times W _ { i } \times C }$ and then adopts the bicubic layer to upsample its resolution while the embedded dimension is kept unchanged, resulting in the output $\boldsymbol { X } _ { i } ^ { ' } \in \mathbb { R } ^ { 2 H _ { i } \times 2 W _ { i } \times C }$ . After that, the 2D feature map $X _ { i } ^ { ' }$ is again reshaped into the 1D sequence of embedding tokens. For higher-resolution stages, we replace the bicubic upscaling layer with the pixelshuffle module, which upsamples the resolution of feature map by $2 \times$ ratio and also reduces the embedding dimension to a quarter of the input. This pyramid-structure with modified upscaling layers mitigates the memory and computation explosion. We repeat multiple stages until it reaches the target resolution $( H , W )$ , and then we will project the embedding dimension to 3 and obtain the RGB image $Y \in \mathbb { R } ^ { H \times W \times 3 }$ .
+
+# 3.2 Multi-scale Discriminator
+
+Unlike the generator which synthesizes precise pixels, the discriminator is tasked to distinguish between real/fake images. This allows us to treat it as a typical classifier by simply tokenizing the input image in a coarser patch-level [34], where each patch can be regarded as a “word". However, compared to image recognition tasks where classifiers focus on the semantic differences, the discriminator executes a simpler and more detail-oriented task to distinguish between synthesized and real. Therefore, the local visual cues and artifacts will have an important effect on the discriminator. Practically, we observe that the patch splitting rule plays a crucial role, where large patch size sacrifices low-level texture details, and smaller patch size results in a longer sequence that costs more memory. The above dilemma motivates our design of multi-scale discriminator below.
+
+As shown in Figure 2 (right), a multi-scale discriminator is designed to take varying size of patches as inputs, at its different stages. We firstly split the input images $Y \in R ^ { H \times W \times 3 ^ { \bullet } }$ into three different sequences by choosing different patch sizes $( P , 2 P , 4 P )$ . The longest sequence $\textstyle \left( { \frac { H } { P } } \times { \frac { W } { P } } \right) \times 3$ is linearly transformed to $\begin{array} { r } { \left( \frac { H } { P } \times \frac { W } { P } \right) \times \frac { C } { 4 } } \end{array}$ and then combined with the learnable position encoding to serve as the input of the first stage, where $\frac { C } { 4 }$ is the embedded dimension size. Similarly, the second 4and third sequences are linearly transformed to $\begin{array} { r } { \big ( \frac { H } { 2 P } \times \frac { W } { 2 P } \big ) \times \frac { C } { 4 } } \end{array}$ and $\bigl ( { \frac { H } { 4 P } } \times { \frac { W } { 4 P } } \bigr ) \times { \frac { C } { 2 } }$ , and then separately concatenated into the second and third stages. Thus these three different sequences are able to extract both the semantic structure and texture details. Similar to the generator, we reshape the 1D-sentence to 2D feature map and adopt Average Pooling layer to downsample the feature map resolution, between each stage. By recursively forming the transformer blocks in each stage, we obtain a pyramid architecture where multi-scale representation is extracted. At the end of these blocks, a [cls] token is appended at the beginning of the 1D sequence and then taken by the classification head to output the real/fake prediction.
+
+# 3.3 Grid Self-Attention: A Scalable Variant of Self-Attention for Image Generation
+
+Self-attention allows the generator to capture the global correspondence, yet also impedes the efficiency when modeling long sequences/higher resolutions. That motivates many efficient selfattention designs in both language [64, 65] and vision tasks [66, 67]. To adapt self-attention for higher-resolution generative tasks, we propose a simple yet effective strategy, named Grid SelfAttention, tailored for high-resolution image generation.
+
+As shown in Figure 3, instead of calculating the correspondence between a given token and all other tokens, the grid self-attention partitions the full-size feature map into several non-overlapped grids, and the token interactions are calculated inside each local grid. We add the grid self-attention on high-resolution stages (resolution higher than $3 2 \times 3 2$ ) while still keeping standard self-attention in low-resolution stages, shown as Figure 2, again so as to strategically balance local details and global awareness. The grid self-attention shows surprising effectiveness over other efficient self-attention forms [64, 67] in generative tasks, as compared later in Section 4.1.
+
+One potential concern might arise with the boundary artifact between each grid. We observe that while the artifact indeed occurs at early training stages, it gradually vanishes given enough training iterations and training data, while producing nicely coherent final results. We think this is owing to the larger, multi-scale receptive field of the discriminator that requires generated image fidelity in different scales. For other cases where the large-scale training data is hard to obtain, we discuss several solutions on Sec. 4.6.
+
+![](images/5aa30c4dc12e735df83a2803ff77d51d6c674cced188b389797c086e25b8de54.jpg)  
+Figure 3: Grid Self-Attention across different transformer stages. We replace Standard Self-Attention with Grid Self-Attention when the resolution is higher than $3 2 \times 3 2$ and the grid size is set to be $1 6 \times 1 6$ by default.
+
+# 3.4 Exploring the Training Recipe
+
+Data Augmentation. The transformer-based architectures are known to be highly data-hungry due to removing human-designed bias. Particularly in image recognition task [34], they were inferior to CNNs until much larger external data [68] was used for pre-training. To remove this roadblock, data augmentation was revealed as a blessing in [50], which showed that different types of strong data augmentation could lead us to data-efficient training for vision transformers.
+
+We follow a similar mindset. Traditionally, training CNN-based GANs hardly refers to data augmentation. Recently, there is an interest surge in the few-shot GAN training, aiming to match state-of-the-art GAN results with orders of magnitude fewer real images [69, 70]. Contrary to this “commonsense" in CNNs, data augmentation is found to be crucial in transformer-based architectures, even with $100 \%$ real images being utilized. We show that simply using differential augmentation [69] with three basic operators {T ranslation, Cutout, Color} leads to surprising performance improvement for TransGAN, while CNN-based GANs hardly benefit from it. We conduct a concrete study on the effectiveness of augmentation for both transformer and CNNs: see details in Section 4.2
+
+Relative Position Encoding. While classical transformers [49, 34] used deterministic position encoding or learnable position encoding, the relative position encoding [71] gains increasing popularity [72, 28, 52, 73], by exploiting lags instead of absolute positions. Considering a single head of self-attention layer,
+
+$$
+A t t e n t i o n ( Q , K , V ) = s o f t m a x ( ( \frac { Q K ^ { T } } { \sqrt { d _ { k } } } V )
+$$
+
+where $Q , K , V \in \mathbb { R } ^ { ( H \times W ) \times C }$ represent query, key, value matrices, $H , W , C$ denotes the height, width, embedded dimension of the input feature map. The difference in coordinate between each query and key on $H$ axis lies in the range of $[ - ( H - \bar { 1 } ) , H - 1 ]$ , and similar for $W$ axis. By simultaneously considering both $H$ and $W$ axis, the relative position can be represented by a parameterized matrix $M \in \mathbb { R } ^ { ( 2 \hat { H _ { - 1 } } ) \times ( 2 W - 1 ) }$ . Per coordinate, the relative position encoding $E$ is taken from matrix $M$ and added to the attention map $Q K ^ { T }$ as a bias term, shown as following,
+
+$$
+A t t e n t i o n ( Q , K , V ) = s o f t m a x ( ( ( \frac { Q K ^ { T } } { \sqrt { d _ { k } } } + E ) V )
+$$
+
+Compared to its absolute counterpart, relative position encoding learns a stronger “relationship" between local contents, bringing important performance gains in large-scale cases and enjoying widespread use ever since. We also observe it to consistently improve TransGAN, especially on higher-resolution datasets. We hence apply it on top of the learnable absolute positional encoding for both the generator and discriminator.
+
+Modified Normalization. Normalization layers are known to help stabilize the deep learning training of deep neural networks, sometimes remarkably. While both the original transformer [49] and its variants [52, 54] by default use the layer normalization, we follow previous works [75, 16] and replace it with a token-wise scaling layer to prevent the magnitudes in transformer blocks from being too high, describe as $\begin{array} { r } { Y = X / \sqrt { \frac { 1 } { C } \sum _ { i = 0 } ^ { C - 1 } ( X ^ { i } ) ^ { 2 } + \epsilon } } \end{array}$ , where $\epsilon = 1 e - 8$ by default, $X$ and $Y$ denote the token before and after scaling layer, $C$ represents the embedded dimension. Note that our modified normalization resembles local response normalization that was once used in AlexNet [75]. Unlike other “modern" normalization layers [76, 77, 78] that need affine parameters for both mean and variances, we find that a simple re-scaling without learnable parameters suffices to stabilize TransGAN training – in fact, it makes TransGAN train better and improves the FID.
+
+Table 1: Unconditional image generation results on CIFAR-10, STl-10, and CelebA $( 1 2 8 \times 1 2 8 )$ dataset. We train the models with their official code if the results are unavailable, denoted as “\*”, others are all reported from references.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">CIFAR-10</td><td colspan="2">STL-10</td><td>CelebA</td></tr><tr><td>IS↑</td><td>FID↓</td><td>IS↑</td><td>FID↓</td><td>FID↓</td></tr><tr><td>WGAN-GP [1]</td><td>6.49 ± 0.09</td><td>39.68</td><td>1</td><td>=</td><td></td></tr><tr><td>SN-GAN [48]</td><td>8.22 ± 0.05</td><td></td><td>9.16 ± 0.12</td><td>40.1</td><td></td></tr><tr><td>AutoGAN[18]</td><td>8.55 ±0.10</td><td>12.42</td><td>9.16 ± 0.12</td><td>31.01</td><td></td></tr><tr><td>AdversarialNAS-GAN[18]</td><td>8.74 ±0.07</td><td>10.87</td><td>9.63 ± 0.19</td><td>26.98</td><td>=</td></tr><tr><td>Progressive-GAN[16]</td><td>8.80 ± 0.05</td><td>15.52</td><td>=</td><td></td><td>7.30</td></tr><tr><td>COCO-GAN [74]</td><td></td><td>-</td><td>=</td><td>-</td><td>5.74</td></tr><tr><td>StyleGAN-V2 [69]</td><td>9.18</td><td>11.07</td><td>10.21* ± 0.14</td><td>20.84*</td><td>5.59*</td></tr><tr><td>StyleGAN-V2 + DiffAug. [69]</td><td>9.40</td><td>9.89</td><td>10.31*± 0.12</td><td>19.15*</td><td>5.40*</td></tr><tr><td>TransGAN</td><td>9.02 ± 0.12</td><td>9.26</td><td>10.43 ± 0.16</td><td>18.28</td><td>5.28</td></tr></table>
+
+# 4 Experiments
+
+Datasets We start by evaluating our methods on three common testbeds: CIFAR-10 [79], STL10 [80], and CelebA [81] dataset. The CIFAR-10 dataset consists of $6 0 \mathbf { k } \ 3 2 \times 3 2$ images, with $5 0 \mathrm { k }$ training and $1 0 \mathrm { k }$ testing images, respectively. We follow the standard setting to use the $5 0 \mathrm { k }$ training images without labels. For the STL-10 dataset, we use both the 5k training images and $1 0 0 \mathrm { k }$ unlabeled images, and all are resized to $4 8 \times 4 8$ resolution. For the CelebA dataset, we use $2 0 0 \mathrm { k }$ unlabeled face images (aligned and cropped version), with each image at $1 2 8 \times 1 2 8$ resolution. We further consider the CelebA-HQ and LSUN Church datasets to scale up TransGAN to higher resolution image generation tasks. We use $3 0 \mathrm { k }$ images for CelebA-HQ [16] dataset and $1 2 5 \mathrm { k }$ images for LSUN Church dataset [82], all at $2 5 6 \times 2 5 6$ resolution.
+
+Implementation We follow the setting of WGAN [46], and use the WGAN-GP loss [1]. We adopt a learning rate of $1 e - 4$ for both generator and discriminator, an Adam optimizer with $\beta _ { 1 } = 0$ and $\beta _ { 2 } = 0 . 9 9$ , exponential moving average weights for generator, and a batch size of 128 for generator and 64 for discriminator, for all experiments. We choose DiffAug. [69] as basic augmentation strategy during the training process if not specially mentioned, and apply it to our competitors for a fair comparison. Other popular augmentation strategies ([70, 10]) are not discussed here since it is beyond the scope of this work. We use common evaluation metrics Inception Score (IS) [15] and Frechet Inception Distance (FID) [83], both are measured by 50K samples with their official Tensorflow implementations 12. All experiments are set with 16 V100 GPUs, using PyTorch 1.7.0. We include detailed training cost for each dataset in Appendix D. We focus on the unconditional image generation setting for simplicity.
+
+# 4.1 Comparison with State-of-the-art GANs
+
+CIFAR-10. We compare TransGAN with recently published results by unconditional CNN-based GANs on the CIFAR-10 dataset, shown in Table 1. Note that some promising conditional GANs [21, 8] are not included, due to the different settings. As shown in Table 1, TransGAN surpasses the strong model of Progressive GAN [16], and many other latest competitors such as SN-GAN [48], AutoGAN [18], and AdversarialNAS-GAN [19], in terms of inception score (IS). It is only next to the huge and heavily engineered StyleGAN-v2 [40]. Once we look at the FID results, TransGAN is even found to outperform StyleGAN-v2 [40] with both applied the same data augmentation [69].
+
+STL-10. We then apply TransGAN on another popular benchmark STL-10, which is larger in scale (105k) and higher in resolution (48x48). We compare TransGAN with both the automatic searched and hand-crafted CNN-based GANs, shown in Table 1. Different from the results on CIFAR-10, we find that TransGAN outperforms all current CNN-based GAN models, and sets new state-of-the-art results in terms of both IS and FID score. This is thanks to the fact that the STL-10 dataset size is $2 \times$ larger than CIFAR-10, suggesting that transformer-based architectures benefit much more notably from larger-scale data than CNNs.
+
+![](images/6e96e44a3e68b78d7018919b2cfbce5508a09fd739e9d26cf4824f8d0c72caee.jpg)  
+Figure 4: Representative visual results produced by TransGAN on different datasets, as resolution grows from $3 2 \times 3 2$ to $2 5 6 \times 2 5 6$ . More visual examples are included in Appendix F.
+
+Table 2: The effectiveness of Data Augmentation on both CNN-based GANs and TransGAN. We use the full CIFAR-10 training set and DiffAug [69].   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">WGAN-GP</td><td colspan="2">AutoGAN</td><td colspan="2">StyleGAN-V2</td><td colspan="2">TransGAN</td></tr><tr><td>IS个</td><td>FID←</td><td>IS个</td><td>FID←</td><td>IS个</td><td>FID←</td><td>IS个</td><td>FID↓</td></tr><tr><td>Original</td><td>6.49</td><td>39.68</td><td>8.55</td><td>12.42</td><td>9.18</td><td>11.07</td><td>8.36</td><td>22.53</td></tr><tr><td>+ DiffAug [69]</td><td>6.29</td><td>37.14</td><td>8.60</td><td>12.72</td><td>9.40</td><td>9.89</td><td>9.02</td><td>9.26</td></tr></table>
+
+Table 3: The ablation study of proposed techniques in three common dataset CelebA $( 6 4 \times 6 4 )$ , CelebA $( 1 2 8 \times 1 2 8$ , and LSUN Church $( 2 5 6 \times 2 5 6 )$ ). “OOM” represents out-of-momery issue.   
+
+<table><tr><td>Training Configuration</td><td>CelebA (64x64)</td><td>CelebA (128x128)</td><td>LSUN Church (256x256)</td></tr><tr><td>(A). Standard Self-Attention</td><td>8.92</td><td>00M</td><td>0OM</td></tr><tr><td>(B). Nystrom Self-Attention [64]</td><td>13.47</td><td>17.42</td><td>39.92</td></tr><tr><td>(C). Axis Self-Attention [67]</td><td>12.39</td><td>13.95</td><td>29.30</td></tr><tr><td>(D). Grid Self-Attention</td><td>9.89</td><td>10.58</td><td>20.39</td></tr><tr><td>+Multi-scale Discriminator</td><td>9.28</td><td>8.03</td><td>15.29</td></tr><tr><td>+Modified Normalization</td><td>7.05</td><td>7.13</td><td>13.27</td></tr><tr><td>+Relative Position Encoding</td><td>6.14</td><td>6.32</td><td>11.93</td></tr><tr><td>(E). Converge</td><td>5.01</td><td>5.28</td><td>8.94</td></tr></table>
+
+CelebA (128x128). We continue to examine another common benchmark: CelebA dataset $( 1 2 8 \times 1 2 8$ resolution). As shown in Table 1, TransGAN largely outperforms Progressive-GAN [16] and COCOGAN [74], and is slightly better than the strongest competitor StyleGAN-v2 [40], by reaching a FID score of 5.28. Visual examples generated on CIFAR-10, STL-10, and CelebA $( 1 2 8 \times 1 2 8 )$ are shown in Figure 4, from which we observe pleasing visual details and diversity.
+
+# 4.2 Scaling Up to Higher-Resolution
+
+We further scale up TransGAN to higher-resolution $( 2 5 6 \times 2 5 6 )$ generation, including on CelebAHQ [16] and LSUN Church [82]. These high-resolution datasets are significantly more challenging due to their much richer and detailed low-level texture as well as the global composition. Thanks to the proposed multi-scale discriminator, TransGAN produces pleasing visual results, reaching competitive quantitative results with 10.28 FID on CelebA-HQ $2 5 6 \times 2 5 6$ and $8 . 9 4 \mathrm { F I D }$ on LSUN Church dataset, respectively. As shown in Figure 4, diverse examples with rich textures details are produced. We discuss the memory cost reduction brought by the Grid Self-Attention in Appendix E.
+
+# 4.3 Data Augmentation is Crucial for TransGAN
+
+We study the effectiveness of data augmentation for both CNN-based GANs and Our TransGAN. We apply the differentiable augmentation [69] to all these methods. As shown in Table 2, for three CNN-based GANs, the performance gains of data augmentation seems to diminish in the full-data regime. Only the largest model, StyleGAN-V2, is improved on both IS and FID. In sharp contrast, TransGAN sees a shockingly large margin of improvement: IS improving from 8.36 to 9.02 and FID improving from 22.53 to 9.26. This phenomenon suggests that CIFAR-10 is still “small-scale " when fitting transformers; it re-confirms our assumption that transformer-based architectures are much more data-hungry than CNNs, and that can be helped by stronger data augmentation.
+
+![](images/77da0030a49f68b1242125a5ca2cb5670b6ba0577419be181bcc412c753e9bdb.jpg)  
+Figure 5: Left: training dynamic with training epochs for both TransGAN and MSG-GAN on CelebA-HQ $( 2 5 6 \times 2 5 6 )$ . Right: Interpolation on latent space produced by TransGAN.
+
+# 4.4 Ablation Study
+
+To further evaluate the proposed grid self-attention, multi-scale discriminator, and unique training recipe, we conduct the ablation study by separately adding these techniques to the baseline method and report their FID score on different datasets. Due to the fact that most of our contributions are tailored for the challenges brought by higher-resolution tasks, we choose CelebA and LSUN Church as the main testbeds, with details shown in Table 3. We start by constructing our memory-friendly with vanilla discriminator as our baseline method (A), both applied with standard self-attention. The baseline method achieves relatively good results with 8.92 FID on CelebA $( 6 4 \times 6 4 )$ dataset, however, it fail on higher-resolution tasks due to the memory explosion issue brought by self-attention. This motivates us to evaluate two efficient form of self-attention, (B) Nyström Self-Attention [64] and (C) Axis Self-Attention [67]
+
+By replacing all self-attention layers in high-resolution stages (feature map resolution higher than $3 2 \times 3 2 )$ with these efficient variants, both two methods (B)(C) are able to produce reasonable results. However, they still show to be inferior to standard self-attention, even on the $6 4 \times 6 4$ resolution dataset. By adopting our proposed Grid Self-Attention (D), we observe a significant improvement on both three datasets, reaching 9.89, 10.58, 20.39 FID on CelebA $6 4 \times 6 4$ , $1 2 8 \times 1 2 8$ and LSUN Church $2 5 6 \times 2 5 6$ , respectively. Based on the configuration (D), we continue to add the proposed techniques, including the multi-scale discriminator, modified normalization, and relative position encoding. All these three techniques significantly improve the performance of TransGAN on three datasets. At the end, we train our final configuration (E) until it converges, resulting in the best FID on CelebA $6 4 \times 6 4$ (5.01), CelebA $1 2 8 \times 1 2 8$ (5.28), and LSUN Church $2 5 6 \times 2 5 6$ (8.94).
+
+# 4.5 Understanding Transformer-based Generative Model
+
+We dive deep into our transformer-based GAN by conducting interpolation on latent space and comparing its behavior with CNN-based GAN, through visualizing their training dynamics. We choose MSG-GAN [84] for comparison since it extracts multi-scale representation as well. As shown in Figure 5, the CNN-based GAN quickly extracts face representation in the early stage of training process while transformer only produces rough pixels with no meaningful global shape due to missing any inductive bias. However, given enough training iterations, TransGAN gradually learns informative position representation and is able to produce impressive visual examples at convergence. Meanwhile, the boundary artifact also vanishes at the end. For the latent space interpolation, TransGAN continues to show encouraging results where smooth interpolation are maintained on both local and global levels. More high-resolution visual examples will be presented in Appendix F.
+
+# 4.6 Analyzing the Failure Cases and Improving High-resolution Tasks
+
+While TransGAN shows competitive or even better results on common low-resolution benchmarks, we still see large improvement room of its performance on high-resolution synthesis tasks, by analyzing the failure cases shown in appendix C. Here we discuss several alternatives tailored for high-resolution synthesis tasks, as potential remedies to address these failure cases. Specifically, we apply the self-modulation [85, 22, 35] to our generator and use cross-attention [53, 86] to map the latent space to the global region. Besides, we replace the current $2 \times$ upsampling layer, and instead firstly upsample it to $4 \times$ lager resolution using bicubic interpolation, and then downsample it back to $2 \times$ larger one. This simple modification not only helps the cross-boundary information interaction, but also help enhances the high-frequency details [87]. Moreover, an overlapped patch splitting strategy for discriminator can slightly improve the FID score. Additionally, we follow the previous work [22, 40] to conduct noise injection before the self-attention layer, which is found to further improve the generation fidelity and diversity of TransGAN. By applying these techniques to our high-resolution GAN frameworks, we observe additional improvement on both qualitative and quantitative results, e.g., the FID score on CelebA $2 5 6 \times 2 5 6$ dataset is further improved from 10.26 to 8.93.
+
+# 5 Conclusions, Limitation, and Discussions of Broad Impact
+
+In this work, we provide the first pilot study of building GAN with pure transformers. We have carefully crafted the architectures and thoughtfully designed training techniques. As a result, the proposed TransGAN has achieved state-of-the-art performance across multiple popular datasets, and easily scales up to higher-resolution generative tasks. Although TransGAN provides an encouraging starting point, there is still a large room to explore further, such as achieving state-of-the-art results on $2 5 6 \times 2 5 6$ generation tasks or going towards extremely high resolution generation tasks (e.g., $1 0 2 4 \times 1 0 2 4 ,$ ), which would be our future directions.
+
+Broader Impact. The proposed generative model can serve as a data engine to alleviate the challenge of data collection. More importantly, using synthesized image examples helps avoid privacy concerns. However, the abuse of advanced generative models may create fake media materials, which demands caution in the future.
+
+# Acknowledgements
+
+We would like to express our deepest gratitude to the MIT-IBM Watson AI Lab, in particular John Cohn for generously providing us with the computing resources necessary to conduct this research. Z Wang’s work is in part supported by an IBM Faculty Research Award, and the NSF AI Institute for Foundations of Machine Learning (IFML).
+
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+        "text": "The recent explosive interest on transformers has suggested their potential to become powerful “universal\" models for computer vision tasks, such as classification, detection, and segmentation. While those attempts mainly study the discriminative models, we explore transformers on some more notoriously difficult vision tasks, e.g., generative adversarial networks (GANs). Our goal is to conduct the first pilot study in building a GAN completely free of convolutions, using only pure transformer-based architectures. Our vanilla GAN architecture, dubbed TransGAN, consists of a memory-friendly transformer-based generator that progressively increases feature resolution, and correspondingly a multi-scale discriminator to capture simultaneously semantic contexts and low-level textures. On top of them, we introduce the new module of grid self-attention for alleviating the memory bottleneck further, in order to scale up TransGAN to high-resolution generation. We also develop a unique training recipe including a series of techniques that can mitigate the training instability issues of TransGAN, such as data augmentation, modified normalization, and relative position encoding. Our best architecture achieves highly competitive performance compared to current stateof-the-art GANs using convolutional backbones. Specifically, TransGAN sets the new state-of-the-art inception score of 10.43 and FID of 18.28 on STL-10. It also reaches the inception score of 9.02 and FID of 9.26 on CIFAR-10, and 5.28 FID on CelebA $\\mathbf { 1 2 8 \\times 1 2 8 }$ , respectively: both on par with the current best results. When it comes to higher-resolution (e.g. $\\mathbf { 2 5 6 \\times 2 5 6 } )$ ) generation tasks, such as on CelebAHQ and LSUN-Church, TransGAN continues to produce diverse visual examples with high fidelity and reasonable texture details. In addition, we dive deep into the transformer-based generation models to understand how their behaviors differ from convolutional ones, by visualizing training dynamics. The code is available at: https://github.com/VITA-Group/TransGAN. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Generative adversarial networks (GANs) have gained considerable success on numerous tasks [1, 2, 3, 4, 5, 6, 7]. Unfortunately, GANs suffer from the notorious training instability, and numerous efforts have been devoted to stabilizing GAN training, introducing various regularization terms [8, 9, 10, 11], better losses [1, 12, 13, 14], and training recipes [15, 16]. Among them, one important route to improving GANs examines their neural architectures. [17, 8] reported a large-scale study of GANs and observed that when serving as (generator) backbones, popular neural architectures perform comparably well across the considered datasets. Their ablation study suggested that most of the variations applied in the ResNet family resulted in very marginal improvements. Nevertheless, neural architecture search (NAS) was later introduced to GANs and suggests enhanced backbone designs are also important for improving GANs, just like for other computer vision tasks. Those works are consistently able to discover stronger GAN architectures beyond the standard ResNet topology [18, 19, 20]. Other efforts include customized modules such as self-attention [21], style-based generator [22], and autoregressive transformer-based part composition [23]. ",
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+        "img_path": "images/adcb25935a6e25e1b2ab70151735aa65bcbfa9d2b385186c0dc4a0005b6915f2.jpg",
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+            "Figure 1: Representative visual examples synthesized by TransGAN, without using convolutional layers. (a) The synthesized visual examples on CelebA-HQ $2 5 6 \\times 2 5 6 )$ dataset. (b) The linear interpolation results between two latent vectors, on CelebA-HQ ( $2 5 6 \\times 2 5 6 )$ ) dataset. "
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+        "text": "However, one last “commonsense\" seems to have seldomly been challenged: using convolutional neural networks (CNNs) as GAN backbones. The original GAN [24, 25] used fully-connected networks and can only generate small images. DCGAN [26] was the first to scale up GANs using CNN architectures, which allowed for stable training for higher resolution and deeper generative models. Since then, in the computer vision domain, every successful GAN relies on CNN-based generators and discriminators. Convolutions, with the strong inductive bias for natural images, crucially contribute to the appealing visual results and rich diversity achieved by modern GANs. ",
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+        "text": "Can we build a strong GAN completely free of convolutions? This is a question not only arising from intellectual curiosity, but also of practical relevance. Fundamentally, a convolution operator has a local receptive field, and hence CNNs cannot process long-range dependencies unless passing through a sufficient number of layers. However, that is inefficient, and could cause the loss of feature resolution and fine details, in addition to the difficulty of optimization. Vanilla CNN-based models are therefore inherently not well suited for capturing an input image’s “global\" statistics, as demonstrated by the benefits from adopting self-attention [21] and non-local [27] operations in computer vision. Moreover, the spatial invariance possessed by convolution poses a bottleneck on its ability of adapting to spatially varying/heterogeneous visual patterns, which also motivates the success of relational network [28], dynamic filters [29, 30] and kernel prediction [31] methods. ",
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+        "text": "1.1 Our Contributions ",
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+        "text": "This paper aims to be the first pilot study to build a GAN completely free of convolutions, using only pure transformer-based architectures. We are inspired by the recent success of transformer architectures in computer vision [32, 33, 34]. Compared to parallel generative modeling works [21, 23, 35] that applied self-attention or transformer encoder in conjunction with CNN-based backbones, our goal is more ambitious and faces several daunting gaps ahead. First and foremost, although a pure transformer architecture applied directly to sequences of image patches can perform very well on image classification tasks [34], it is unclear whether the same way remains effective in generating images, which crucially demands the spatial coherency in structure, color, and texture, as well as the richness of fine details. The handful of existing transformers that output images have unanimously leveraged convolutional part encoders [23] or feature extractors [36, 37]. Moreover, even given well-designed CNN-based architectures, training GANs is notoriously unstable and prone to mode collapse [15]. Training vision transformers are also known to be tedious, heavy, and data-hungry [34]. Combining the two will undoubtedly amplify the challenges of training. ",
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+        "text": "In view of those challenges, this paper presents a coherent set of efforts and innovations towards building the pure transformer-based GAN architectures, dubbed TransGAN. A naive option may directly stack multiple transformer blocks from raw pixel inputs, but that would scale poorly due to memory explosion. Instead, we start with a memory-friendly transformer-based generator by gradually increasing the feature map resolution in each stage. Correspondingly, we also improve the discriminator with a multi-scale structure that takes patches of varied size as inputs, which balances between capturing global contexts and local details, in addition to enhancing memory efficiency more. Based on the above generator-discriminator design, we introduce a new module called grid self-attention, that alleviates the memory bottleneck further when scaling up TransGAN to high-resolution generation (e.g. $2 5 6 \\times 2 5 6$ ). ",
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+        "text": "To address the aforementioned instability issue brought by both GAN and Transformer, we also develop a unique training recipe in association with our innovative TransGAN architecture, that effectively stabilizes its optimization and generalization. That includes showings the necessity of data augmentation, modifying layer normalization, and replacing absolute token locations with relative position encoding. Our contributions are outlined below: ",
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+        "text": "• Novel Architecture Design: We build the first GAN using purely transformers and no convolution. TransGAN has customized a memory-friendly generator and a multi-scale discriminator, and is further equipped with a new grid self-attention mechanism. Those architectural components are thoughtfully designed to balance memory efficiency, global feature statistics, and local fine details with spatial variances.   \n• New Training Recipe: We study a number of techniques to train TransGAN better, including leveraging data augmentation, modifying layer normalization, and adopting relative position encoding, for both generator and discriminator. Extensive ablation studies, discussions, and insights are presented.   \nPerformance and Scalability: TransGAN achieves highly competitive performance compared to current state-of-the-art GANs. Specifically, it sets the new state-of-the-art inception score of 10.43 and FID score of 18.28 on STL-10. It also reaches competitive 9.02 inception score and 9.26 FID on CIFAR-10, and 5.28 FID score on CelebA $1 2 8 \\times 1 2 8$ , respectively. Meanwhile, we also evaluate TransGAN on higher-resolution (e.g., $2 5 6 \\times 2 5 6 )$ generation tasks, where TransGAN continues to yield diverse and impressive visual examples. ",
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+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Generative Adversarial Networks. After its origin, GANs quickly embraced fully convolutional backbones [26], and inherited most successful designs from CNNs such as batch normalization, pooling, (Leaky) ReLU and more [38, 39, 40, 18]. GANs are widely adopted in image translation [3, 4, 41], image enhancement [7, 42, 43], and image editing [44, 45]. To alleviate its unstable training, a number of techniques have been studied, including the Wasserstein loss [46], the stylebased generator [22], progressive training [16], lottery ticket [47], and spectral normalization [48]. ",
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+        "text": "Transformers in Computer Vision. The original transformer was built for NLP [49], where the multi-head self-attention and feed-forward MLP layer are stacked to capture the long-term correlation between words. A recent work [34] implements highly competitive ImageNet classification using pure transformers, by treating an image as a sequence of $1 6 \\times 1 6$ visual words. It has strong representation capability and is free of human-defined inductive bias. In comparison, CNNs exhibit a strong bias towards feature locality, as well as spatial invariance due to sharing filter weights across all locations. However, the success of original vision transformer relies on pretraining on large-scale external data. [50, 51] improve the data efficiency and address the difficulty of optimizing deeper models. Other works introduce the pyramid/hierarchical structure to transformer [52, 53, 54] or combine it with convolutional layers [55, 56]. Besides image classification task, transformer and its variants are also explored on image processing [37], point cloud [57], semantic segmentation [58], object detection [32, 59] and so on. A comprehensive review is referred to [60]. ",
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+    {
+        "type": "text",
+        "text": "Transformer Modules for Image Generation. There exist several related works combining the transformer modules into image generation models, by replacing certain components of CNNs. [61] firstly formulated image generation as autoregressive sequence generation, for which they adopted a transformer architecture. [62] propose sparse factorization of the attention matrix to reduce its complexity. While those two works did not tackle the GANs, one recent (concurrent) work [23] used a convolutional GAN to learn a codebook of context-rich visual parts, whose composition is subsequently modeled with an autoregressive transformer architecture.The authors demonstrated success in synthesizing high-resolution images. However, the overall CNN architecture remains in place (including CNN encoder/decoder for the generators, and a fully CNN-based discriminator), and the customized designs (e.g, codebook and quantization) also limit their model’s versatility. Another concurrent work [35] employs a bipartite self-attention on StyleGAN and thus it can propagate latent variables to the evolving visual features, yet its main structure is still convolutional, including both the generator and discriminator. To our best knowledge, no other existing work has tried to completely remove convolutions from their generative modeling frameworks. ",
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+        "img_path": "images/e89dc9eb445115b78109b53440aa8db8c120d23b4e23956ef5aedc971cd11810.jpg",
+        "image_caption": [
+            "Figure 2: The pipeline of the pure transform-based generator and discriminator of TransGAN. We take $2 5 6 \\times 2 5 6$ resolution image generation task as a typical example to illustrate the main procedure. Here patch size $p$ is set to 32 as an example for the convenience of illustration, while practically the patch size is normally set to be no more than $8 \\times 8$ , depending on the specific dataset. Grid Transformer Blocks refers to the transformer blocks with the proposed grid self-attention. Detailed architecture configurations are included in Appendix B. "
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+        "text": "3 Technical Approach: A Journey Towards GAN with Pure Transformers ",
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+        "text": "In this section, we start by introducing the memory-friendly generator and multi-scale discriminator, equipped with a novel grid self-attention. We then introduce a series of training techniques to stabilize its training procedure, including data augmentation, the modified normalization, and injecting relative position encoding to self-attention. ",
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+        "type": "text",
+        "text": "To start with, we choose the transformer encoder [49] as our basic block and try to make minimal changes. An encoder is a composition of two parts. The first part is constructed by a multi-head self-attention module and the second part is a feed-forward MLP with GELU non-linearity. The normalization layer is applied before both of the two parts. Both parts employ residual connection. ",
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+        "text": "3.1 Memory-friendly Generator ",
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+        "text": "The task of generation poses a high standard for spatial coherency in structure, color, and texture, both globally and locally. The transformer encoders take embedding token words as inputs and calculate the interaction between each token recursively. [63, 34]. The main dilemma here is: what is the right “word\" for image generation tasks? If we similarly generate an image in a pixel-by-pixel manner through stacking transformer encoders, even a low-resolution image (e.g. $3 2 \\times 3 2$ ) can result in an excessively long sequence (1024), causing the explosive cost of self-attention (quadratic w.r.t. the sequence length) and prohibiting the scalability to higher resolutions. To avoid this daunting cost, we are inspired by a common design philosophy in CNN-based GANs, to iteratively upscale the resolution at multiple stages [25, 16]. Our strategy is hence to increase the input sequence and reduce the embedding dimension gradually . ",
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+        "text": "Figure 2 (left) illustrates a memory-friendly transformer-based generator that consists of multiple stages. Each stage stacks several transformer blocks. By stages, we gradually increase the feature map resolution until it meets the target resolution $H \\times W$ . Specifically, the generator takes the random noise as its input, and passes it through a multiple-layer perceptron (MLP) to a vector of length $H _ { 0 } \\times W _ { 0 } \\times C$ . The vector is reshaped into a $H _ { 0 } \\times W _ { 0 }$ resolution feature map (by default we use $H _ { 0 } = W _ { 0 } = 8$ ), each point a $C$ -dimensional embedding. This “feature map\" is next treated as a length-64 sequence of $C$ -dimensional tokens, combined with the learnable positional encoding. ",
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+        "type": "text",
+        "text": "To scale up to higher-resolution images, we insert an upsampling module after each stage, consisting of a reshaping and resolution-upscaling layer. For lower-resolution stages (resolution lower than $6 4 \\times 6 4 )$ ), the upsampling module firstly reshapes the 1D sequence of token embedding back to a 2D feature map $X _ { i } ~ \\overset { \\cdot } { \\in } ~ \\mathbb { R } ^ { \\cup _ { i } \\times W _ { i } \\times C }$ and then adopts the bicubic layer to upsample its resolution while the embedded dimension is kept unchanged, resulting in the output $\\boldsymbol { X } _ { i } ^ { ' } \\in \\mathbb { R } ^ { 2 H _ { i } \\times 2 W _ { i } \\times C }$ . After that, the 2D feature map $X _ { i } ^ { ' }$ is again reshaped into the 1D sequence of embedding tokens. For higher-resolution stages, we replace the bicubic upscaling layer with the pixelshuffle module, which upsamples the resolution of feature map by $2 \\times$ ratio and also reduces the embedding dimension to a quarter of the input. This pyramid-structure with modified upscaling layers mitigates the memory and computation explosion. We repeat multiple stages until it reaches the target resolution $( H , W )$ , and then we will project the embedding dimension to 3 and obtain the RGB image $Y \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ . ",
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+        "text": "3.2 Multi-scale Discriminator ",
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+        "text": "Unlike the generator which synthesizes precise pixels, the discriminator is tasked to distinguish between real/fake images. This allows us to treat it as a typical classifier by simply tokenizing the input image in a coarser patch-level [34], where each patch can be regarded as a “word\". However, compared to image recognition tasks where classifiers focus on the semantic differences, the discriminator executes a simpler and more detail-oriented task to distinguish between synthesized and real. Therefore, the local visual cues and artifacts will have an important effect on the discriminator. Practically, we observe that the patch splitting rule plays a crucial role, where large patch size sacrifices low-level texture details, and smaller patch size results in a longer sequence that costs more memory. The above dilemma motivates our design of multi-scale discriminator below. ",
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+        "text": "As shown in Figure 2 (right), a multi-scale discriminator is designed to take varying size of patches as inputs, at its different stages. We firstly split the input images $Y \\in R ^ { H \\times W \\times 3 ^ { \\bullet } }$ into three different sequences by choosing different patch sizes $( P , 2 P , 4 P )$ . The longest sequence $\\textstyle \\left( { \\frac { H } { P } } \\times { \\frac { W } { P } } \\right) \\times 3$ is linearly transformed to $\\begin{array} { r } { \\left( \\frac { H } { P } \\times \\frac { W } { P } \\right) \\times \\frac { C } { 4 } } \\end{array}$ and then combined with the learnable position encoding to serve as the input of the first stage, where $\\frac { C } { 4 }$ is the embedded dimension size. Similarly, the second 4and third sequences are linearly transformed to $\\begin{array} { r } { \\big ( \\frac { H } { 2 P } \\times \\frac { W } { 2 P } \\big ) \\times \\frac { C } { 4 } } \\end{array}$ and $\\bigl ( { \\frac { H } { 4 P } } \\times { \\frac { W } { 4 P } } \\bigr ) \\times { \\frac { C } { 2 } }$ , and then separately concatenated into the second and third stages. Thus these three different sequences are able to extract both the semantic structure and texture details. Similar to the generator, we reshape the 1D-sentence to 2D feature map and adopt Average Pooling layer to downsample the feature map resolution, between each stage. By recursively forming the transformer blocks in each stage, we obtain a pyramid architecture where multi-scale representation is extracted. At the end of these blocks, a [cls] token is appended at the beginning of the 1D sequence and then taken by the classification head to output the real/fake prediction. ",
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+        "type": "text",
+        "text": "3.3 Grid Self-Attention: A Scalable Variant of Self-Attention for Image Generation ",
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+        "text": "Self-attention allows the generator to capture the global correspondence, yet also impedes the efficiency when modeling long sequences/higher resolutions. That motivates many efficient selfattention designs in both language [64, 65] and vision tasks [66, 67]. To adapt self-attention for higher-resolution generative tasks, we propose a simple yet effective strategy, named Grid SelfAttention, tailored for high-resolution image generation. ",
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+        "type": "text",
+        "text": "As shown in Figure 3, instead of calculating the correspondence between a given token and all other tokens, the grid self-attention partitions the full-size feature map into several non-overlapped grids, and the token interactions are calculated inside each local grid. We add the grid self-attention on high-resolution stages (resolution higher than $3 2 \\times 3 2$ ) while still keeping standard self-attention in low-resolution stages, shown as Figure 2, again so as to strategically balance local details and global awareness. The grid self-attention shows surprising effectiveness over other efficient self-attention forms [64, 67] in generative tasks, as compared later in Section 4.1. ",
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+        "type": "text",
+        "text": "One potential concern might arise with the boundary artifact between each grid. We observe that while the artifact indeed occurs at early training stages, it gradually vanishes given enough training iterations and training data, while producing nicely coherent final results. We think this is owing to the larger, multi-scale receptive field of the discriminator that requires generated image fidelity in different scales. For other cases where the large-scale training data is hard to obtain, we discuss several solutions on Sec. 4.6. ",
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+        "img_path": "images/5aa30c4dc12e735df83a2803ff77d51d6c674cced188b389797c086e25b8de54.jpg",
+        "image_caption": [
+            "Figure 3: Grid Self-Attention across different transformer stages. We replace Standard Self-Attention with Grid Self-Attention when the resolution is higher than $3 2 \\times 3 2$ and the grid size is set to be $1 6 \\times 1 6$ by default. "
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+        "type": "text",
+        "text": "3.4 Exploring the Training Recipe ",
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+        "type": "text",
+        "text": "Data Augmentation. The transformer-based architectures are known to be highly data-hungry due to removing human-designed bias. Particularly in image recognition task [34], they were inferior to CNNs until much larger external data [68] was used for pre-training. To remove this roadblock, data augmentation was revealed as a blessing in [50], which showed that different types of strong data augmentation could lead us to data-efficient training for vision transformers. ",
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+        "type": "text",
+        "text": "We follow a similar mindset. Traditionally, training CNN-based GANs hardly refers to data augmentation. Recently, there is an interest surge in the few-shot GAN training, aiming to match state-of-the-art GAN results with orders of magnitude fewer real images [69, 70]. Contrary to this “commonsense\" in CNNs, data augmentation is found to be crucial in transformer-based architectures, even with $100 \\%$ real images being utilized. We show that simply using differential augmentation [69] with three basic operators {T ranslation, Cutout, Color} leads to surprising performance improvement for TransGAN, while CNN-based GANs hardly benefit from it. We conduct a concrete study on the effectiveness of augmentation for both transformer and CNNs: see details in Section 4.2 ",
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+        "type": "text",
+        "text": "Relative Position Encoding. While classical transformers [49, 34] used deterministic position encoding or learnable position encoding, the relative position encoding [71] gains increasing popularity [72, 28, 52, 73], by exploiting lags instead of absolute positions. Considering a single head of self-attention layer, ",
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+        "type": "equation",
+        "img_path": "images/dbb01f6ced9089b3c2847770df83453bea906785efbac2a8e617ec65bab96ef6.jpg",
+        "text": "$$\nA t t e n t i o n ( Q , K , V ) = s o f t m a x ( ( \\frac { Q K ^ { T } } { \\sqrt { d _ { k } } } V )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $Q , K , V \\in \\mathbb { R } ^ { ( H \\times W ) \\times C }$ represent query, key, value matrices, $H , W , C$ denotes the height, width, embedded dimension of the input feature map. The difference in coordinate between each query and key on $H$ axis lies in the range of $[ - ( H - \\bar { 1 } ) , H - 1 ]$ , and similar for $W$ axis. By simultaneously considering both $H$ and $W$ axis, the relative position can be represented by a parameterized matrix $M \\in \\mathbb { R } ^ { ( 2 \\hat { H _ { - 1 } } ) \\times ( 2 W - 1 ) }$ . Per coordinate, the relative position encoding $E$ is taken from matrix $M$ and added to the attention map $Q K ^ { T }$ as a bias term, shown as following, ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d642168023319f11cbc80f8d9df34a9ce6f6ffd07f0e3008d315f28bd6603d49.jpg",
+        "text": "$$\nA t t e n t i o n ( Q , K , V ) = s o f t m a x ( ( ( \\frac { Q K ^ { T } } { \\sqrt { d _ { k } } } + E ) V )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Compared to its absolute counterpart, relative position encoding learns a stronger “relationship\" between local contents, bringing important performance gains in large-scale cases and enjoying widespread use ever since. We also observe it to consistently improve TransGAN, especially on higher-resolution datasets. We hence apply it on top of the learnable absolute positional encoding for both the generator and discriminator. ",
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+    {
+        "type": "text",
+        "text": "Modified Normalization. Normalization layers are known to help stabilize the deep learning training of deep neural networks, sometimes remarkably. While both the original transformer [49] and its variants [52, 54] by default use the layer normalization, we follow previous works [75, 16] and replace it with a token-wise scaling layer to prevent the magnitudes in transformer blocks from being too high, describe as $\\begin{array} { r } { Y = X / \\sqrt { \\frac { 1 } { C } \\sum _ { i = 0 } ^ { C - 1 } ( X ^ { i } ) ^ { 2 } + \\epsilon } } \\end{array}$ , where $\\epsilon = 1 e - 8$ by default, $X$ and $Y$ denote the token before and after scaling layer, $C$ represents the embedded dimension. Note that our modified normalization resembles local response normalization that was once used in AlexNet [75]. Unlike other “modern\" normalization layers [76, 77, 78] that need affine parameters for both mean and variances, we find that a simple re-scaling without learnable parameters suffices to stabilize TransGAN training – in fact, it makes TransGAN train better and improves the FID. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3736709c036ee2e90a0a961b83e5eadc3a8c00fa0505f62df932fa0420a35521.jpg",
+        "table_caption": [
+            "Table 1: Unconditional image generation results on CIFAR-10, STl-10, and CelebA $( 1 2 8 \\times 1 2 8 )$ dataset. We train the models with their official code if the results are unavailable, denoted as “\\*”, others are all reported from references. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td colspan=\"2\">CIFAR-10</td><td colspan=\"2\">STL-10</td><td>CelebA</td></tr><tr><td>IS↑</td><td>FID↓</td><td>IS↑</td><td>FID↓</td><td>FID↓</td></tr><tr><td>WGAN-GP [1]</td><td>6.49 ± 0.09</td><td>39.68</td><td>1</td><td>=</td><td></td></tr><tr><td>SN-GAN [48]</td><td>8.22 ± 0.05</td><td></td><td>9.16 ± 0.12</td><td>40.1</td><td></td></tr><tr><td>AutoGAN[18]</td><td>8.55 ±0.10</td><td>12.42</td><td>9.16 ± 0.12</td><td>31.01</td><td></td></tr><tr><td>AdversarialNAS-GAN[18]</td><td>8.74 ±0.07</td><td>10.87</td><td>9.63 ± 0.19</td><td>26.98</td><td>=</td></tr><tr><td>Progressive-GAN[16]</td><td>8.80 ± 0.05</td><td>15.52</td><td>=</td><td></td><td>7.30</td></tr><tr><td>COCO-GAN [74]</td><td></td><td>-</td><td>=</td><td>-</td><td>5.74</td></tr><tr><td>StyleGAN-V2 [69]</td><td>9.18</td><td>11.07</td><td>10.21* ± 0.14</td><td>20.84*</td><td>5.59*</td></tr><tr><td>StyleGAN-V2 + DiffAug. [69]</td><td>9.40</td><td>9.89</td><td>10.31*± 0.12</td><td>19.15*</td><td>5.40*</td></tr><tr><td>TransGAN</td><td>9.02 ± 0.12</td><td>9.26</td><td>10.43 ± 0.16</td><td>18.28</td><td>5.28</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Datasets We start by evaluating our methods on three common testbeds: CIFAR-10 [79], STL10 [80], and CelebA [81] dataset. The CIFAR-10 dataset consists of $6 0 \\mathbf { k } \\ 3 2 \\times 3 2$ images, with $5 0 \\mathrm { k }$ training and $1 0 \\mathrm { k }$ testing images, respectively. We follow the standard setting to use the $5 0 \\mathrm { k }$ training images without labels. For the STL-10 dataset, we use both the 5k training images and $1 0 0 \\mathrm { k }$ unlabeled images, and all are resized to $4 8 \\times 4 8$ resolution. For the CelebA dataset, we use $2 0 0 \\mathrm { k }$ unlabeled face images (aligned and cropped version), with each image at $1 2 8 \\times 1 2 8$ resolution. We further consider the CelebA-HQ and LSUN Church datasets to scale up TransGAN to higher resolution image generation tasks. We use $3 0 \\mathrm { k }$ images for CelebA-HQ [16] dataset and $1 2 5 \\mathrm { k }$ images for LSUN Church dataset [82], all at $2 5 6 \\times 2 5 6$ resolution. ",
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+        "type": "text",
+        "text": "Implementation We follow the setting of WGAN [46], and use the WGAN-GP loss [1]. We adopt a learning rate of $1 e - 4$ for both generator and discriminator, an Adam optimizer with $\\beta _ { 1 } = 0$ and $\\beta _ { 2 } = 0 . 9 9$ , exponential moving average weights for generator, and a batch size of 128 for generator and 64 for discriminator, for all experiments. We choose DiffAug. [69] as basic augmentation strategy during the training process if not specially mentioned, and apply it to our competitors for a fair comparison. Other popular augmentation strategies ([70, 10]) are not discussed here since it is beyond the scope of this work. We use common evaluation metrics Inception Score (IS) [15] and Frechet Inception Distance (FID) [83], both are measured by 50K samples with their official Tensorflow implementations 12. All experiments are set with 16 V100 GPUs, using PyTorch 1.7.0. We include detailed training cost for each dataset in Appendix D. We focus on the unconditional image generation setting for simplicity. ",
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+    {
+        "type": "text",
+        "text": "4.1 Comparison with State-of-the-art GANs ",
+        "text_level": 1,
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+        "text": "CIFAR-10. We compare TransGAN with recently published results by unconditional CNN-based GANs on the CIFAR-10 dataset, shown in Table 1. Note that some promising conditional GANs [21, 8] are not included, due to the different settings. As shown in Table 1, TransGAN surpasses the strong model of Progressive GAN [16], and many other latest competitors such as SN-GAN [48], AutoGAN [18], and AdversarialNAS-GAN [19], in terms of inception score (IS). It is only next to the huge and heavily engineered StyleGAN-v2 [40]. Once we look at the FID results, TransGAN is even found to outperform StyleGAN-v2 [40] with both applied the same data augmentation [69]. ",
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+    {
+        "type": "text",
+        "text": "STL-10. We then apply TransGAN on another popular benchmark STL-10, which is larger in scale (105k) and higher in resolution (48x48). We compare TransGAN with both the automatic searched and hand-crafted CNN-based GANs, shown in Table 1. Different from the results on CIFAR-10, we find that TransGAN outperforms all current CNN-based GAN models, and sets new state-of-the-art results in terms of both IS and FID score. This is thanks to the fact that the STL-10 dataset size is $2 \\times$ larger than CIFAR-10, suggesting that transformer-based architectures benefit much more notably from larger-scale data than CNNs. ",
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+    {
+        "type": "image",
+        "img_path": "images/6e96e44a3e68b78d7018919b2cfbce5508a09fd739e9d26cf4824f8d0c72caee.jpg",
+        "image_caption": [
+            "Figure 4: Representative visual results produced by TransGAN on different datasets, as resolution grows from $3 2 \\times 3 2$ to $2 5 6 \\times 2 5 6$ . More visual examples are included in Appendix F. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f1c4e2b09d5e2f8e5edf0927060448a8a0cc45c22cecb15cb148ce6fe11b716d.jpg",
+        "table_caption": [
+            "Table 2: The effectiveness of Data Augmentation on both CNN-based GANs and TransGAN. We use the full CIFAR-10 training set and DiffAug [69]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td colspan=\"2\">WGAN-GP</td><td colspan=\"2\">AutoGAN</td><td colspan=\"2\">StyleGAN-V2</td><td colspan=\"2\">TransGAN</td></tr><tr><td>IS个</td><td>FID←</td><td>IS个</td><td>FID←</td><td>IS个</td><td>FID←</td><td>IS个</td><td>FID↓</td></tr><tr><td>Original</td><td>6.49</td><td>39.68</td><td>8.55</td><td>12.42</td><td>9.18</td><td>11.07</td><td>8.36</td><td>22.53</td></tr><tr><td>+ DiffAug [69]</td><td>6.29</td><td>37.14</td><td>8.60</td><td>12.72</td><td>9.40</td><td>9.89</td><td>9.02</td><td>9.26</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/25c0032de9117db6ffe66c1bcb8ba86cc9bfa5f079bdbf7a78a3daa9c1d02d03.jpg",
+        "table_caption": [
+            "Table 3: The ablation study of proposed techniques in three common dataset CelebA $( 6 4 \\times 6 4 )$ , CelebA $( 1 2 8 \\times 1 2 8$ , and LSUN Church $( 2 5 6 \\times 2 5 6 )$ ). “OOM” represents out-of-momery issue. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Training Configuration</td><td>CelebA (64x64)</td><td>CelebA (128x128)</td><td>LSUN Church (256x256)</td></tr><tr><td>(A). Standard Self-Attention</td><td>8.92</td><td>00M</td><td>0OM</td></tr><tr><td>(B). Nystrom Self-Attention [64]</td><td>13.47</td><td>17.42</td><td>39.92</td></tr><tr><td>(C). Axis Self-Attention [67]</td><td>12.39</td><td>13.95</td><td>29.30</td></tr><tr><td>(D). Grid Self-Attention</td><td>9.89</td><td>10.58</td><td>20.39</td></tr><tr><td>+Multi-scale Discriminator</td><td>9.28</td><td>8.03</td><td>15.29</td></tr><tr><td>+Modified Normalization</td><td>7.05</td><td>7.13</td><td>13.27</td></tr><tr><td>+Relative Position Encoding</td><td>6.14</td><td>6.32</td><td>11.93</td></tr><tr><td>(E). Converge</td><td>5.01</td><td>5.28</td><td>8.94</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "CelebA (128x128). We continue to examine another common benchmark: CelebA dataset $( 1 2 8 \\times 1 2 8$ resolution). As shown in Table 1, TransGAN largely outperforms Progressive-GAN [16] and COCOGAN [74], and is slightly better than the strongest competitor StyleGAN-v2 [40], by reaching a FID score of 5.28. Visual examples generated on CIFAR-10, STL-10, and CelebA $( 1 2 8 \\times 1 2 8 )$ are shown in Figure 4, from which we observe pleasing visual details and diversity. ",
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+    {
+        "type": "text",
+        "text": "4.2 Scaling Up to Higher-Resolution ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We further scale up TransGAN to higher-resolution $( 2 5 6 \\times 2 5 6 )$ generation, including on CelebAHQ [16] and LSUN Church [82]. These high-resolution datasets are significantly more challenging due to their much richer and detailed low-level texture as well as the global composition. Thanks to the proposed multi-scale discriminator, TransGAN produces pleasing visual results, reaching competitive quantitative results with 10.28 FID on CelebA-HQ $2 5 6 \\times 2 5 6$ and $8 . 9 4 \\mathrm { F I D }$ on LSUN Church dataset, respectively. As shown in Figure 4, diverse examples with rich textures details are produced. We discuss the memory cost reduction brought by the Grid Self-Attention in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "4.3 Data Augmentation is Crucial for TransGAN ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We study the effectiveness of data augmentation for both CNN-based GANs and Our TransGAN. We apply the differentiable augmentation [69] to all these methods. As shown in Table 2, for three CNN-based GANs, the performance gains of data augmentation seems to diminish in the full-data regime. Only the largest model, StyleGAN-V2, is improved on both IS and FID. In sharp contrast, TransGAN sees a shockingly large margin of improvement: IS improving from 8.36 to 9.02 and FID improving from 22.53 to 9.26. This phenomenon suggests that CIFAR-10 is still “small-scale \" when fitting transformers; it re-confirms our assumption that transformer-based architectures are much more data-hungry than CNNs, and that can be helped by stronger data augmentation. ",
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+        "img_path": "images/77da0030a49f68b1242125a5ca2cb5670b6ba0577419be181bcc412c753e9bdb.jpg",
+        "image_caption": [
+            "Figure 5: Left: training dynamic with training epochs for both TransGAN and MSG-GAN on CelebA-HQ $( 2 5 6 \\times 2 5 6 )$ . Right: Interpolation on latent space produced by TransGAN. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.4 Ablation Study ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To further evaluate the proposed grid self-attention, multi-scale discriminator, and unique training recipe, we conduct the ablation study by separately adding these techniques to the baseline method and report their FID score on different datasets. Due to the fact that most of our contributions are tailored for the challenges brought by higher-resolution tasks, we choose CelebA and LSUN Church as the main testbeds, with details shown in Table 3. We start by constructing our memory-friendly with vanilla discriminator as our baseline method (A), both applied with standard self-attention. The baseline method achieves relatively good results with 8.92 FID on CelebA $( 6 4 \\times 6 4 )$ dataset, however, it fail on higher-resolution tasks due to the memory explosion issue brought by self-attention. This motivates us to evaluate two efficient form of self-attention, (B) Nyström Self-Attention [64] and (C) Axis Self-Attention [67] ",
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+    {
+        "type": "text",
+        "text": "By replacing all self-attention layers in high-resolution stages (feature map resolution higher than $3 2 \\times 3 2 )$ with these efficient variants, both two methods (B)(C) are able to produce reasonable results. However, they still show to be inferior to standard self-attention, even on the $6 4 \\times 6 4$ resolution dataset. By adopting our proposed Grid Self-Attention (D), we observe a significant improvement on both three datasets, reaching 9.89, 10.58, 20.39 FID on CelebA $6 4 \\times 6 4$ , $1 2 8 \\times 1 2 8$ and LSUN Church $2 5 6 \\times 2 5 6$ , respectively. Based on the configuration (D), we continue to add the proposed techniques, including the multi-scale discriminator, modified normalization, and relative position encoding. All these three techniques significantly improve the performance of TransGAN on three datasets. At the end, we train our final configuration (E) until it converges, resulting in the best FID on CelebA $6 4 \\times 6 4$ (5.01), CelebA $1 2 8 \\times 1 2 8$ (5.28), and LSUN Church $2 5 6 \\times 2 5 6$ (8.94). ",
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+    {
+        "type": "text",
+        "text": "4.5 Understanding Transformer-based Generative Model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We dive deep into our transformer-based GAN by conducting interpolation on latent space and comparing its behavior with CNN-based GAN, through visualizing their training dynamics. We choose MSG-GAN [84] for comparison since it extracts multi-scale representation as well. As shown in Figure 5, the CNN-based GAN quickly extracts face representation in the early stage of training process while transformer only produces rough pixels with no meaningful global shape due to missing any inductive bias. However, given enough training iterations, TransGAN gradually learns informative position representation and is able to produce impressive visual examples at convergence. Meanwhile, the boundary artifact also vanishes at the end. For the latent space interpolation, TransGAN continues to show encouraging results where smooth interpolation are maintained on both local and global levels. More high-resolution visual examples will be presented in Appendix F. ",
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+    {
+        "type": "text",
+        "text": "4.6 Analyzing the Failure Cases and Improving High-resolution Tasks ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While TransGAN shows competitive or even better results on common low-resolution benchmarks, we still see large improvement room of its performance on high-resolution synthesis tasks, by analyzing the failure cases shown in appendix C. Here we discuss several alternatives tailored for high-resolution synthesis tasks, as potential remedies to address these failure cases. Specifically, we apply the self-modulation [85, 22, 35] to our generator and use cross-attention [53, 86] to map the latent space to the global region. Besides, we replace the current $2 \\times$ upsampling layer, and instead firstly upsample it to $4 \\times$ lager resolution using bicubic interpolation, and then downsample it back to $2 \\times$ larger one. This simple modification not only helps the cross-boundary information interaction, but also help enhances the high-frequency details [87]. Moreover, an overlapped patch splitting strategy for discriminator can slightly improve the FID score. Additionally, we follow the previous work [22, 40] to conduct noise injection before the self-attention layer, which is found to further improve the generation fidelity and diversity of TransGAN. By applying these techniques to our high-resolution GAN frameworks, we observe additional improvement on both qualitative and quantitative results, e.g., the FID score on CelebA $2 5 6 \\times 2 5 6$ dataset is further improved from 10.26 to 8.93. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Conclusions, Limitation, and Discussions of Broad Impact ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we provide the first pilot study of building GAN with pure transformers. We have carefully crafted the architectures and thoughtfully designed training techniques. As a result, the proposed TransGAN has achieved state-of-the-art performance across multiple popular datasets, and easily scales up to higher-resolution generative tasks. Although TransGAN provides an encouraging starting point, there is still a large room to explore further, such as achieving state-of-the-art results on $2 5 6 \\times 2 5 6$ generation tasks or going towards extremely high resolution generation tasks (e.g., $1 0 2 4 \\times 1 0 2 4 ,$ ), which would be our future directions. ",
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+    {
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+        "text": "Broader Impact. The proposed generative model can serve as a data engine to alleviate the challenge of data collection. More importantly, using synthesized image examples helps avoid privacy concerns. However, the abuse of advanced generative models may create fake media materials, which demands caution in the future. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We would like to express our deepest gratitude to the MIT-IBM Watson AI Lab, in particular John Cohn for generously providing us with the computing resources necessary to conduct this research. Z Wang’s work is in part supported by an IBM Faculty Research Award, and the NSF AI Institute for Foundations of Machine Learning (IFML). ",
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+        "text": "References ",
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parse/train/3ZeGLibhFo0/3ZeGLibhFo0.md

@@ -0,0 +1,285 @@
+# ENABLING COUNTERFACTUAL SURVIVAL ANALYSIS WITH BALANCED REPRESENTATIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Balanced representation learning methods have been applied successfully to counterfactual inference from observational data. However, approaches that account for survival outcomes are relatively limited. Survival data are frequently encountered across diverse medical applications, i.e., drug development, risk profiling, and clinical trials, and such data are also relevant in fields like manufacturing (for equipment monitoring). When the outcome of interest is time-to-event, special precautions for handling censored events need to be taken, as ignoring censored outcomes may lead to biased estimates. We propose a theoretically grounded unified framework for counterfactual inference applicable to survival outcomes. Further, we formulate a nonparametric hazard ratio metric for evaluating average and individualized treatment effects. Experimental results on real-world and semi-synthetic datasets, the latter which we introduce, demonstrate that the proposed approach significantly outperforms competitive alternatives in both survival-outcome predictions and treatment-effect estimation.
+
+# 1 INTRODUCTION
+
+Survival analysis or time-to-event studies focus on modeling the time of a future event, such as death or failure, and investigate its relationship with covariates or predictors of interest. Specifically, we may be interested in the causal effect of a given intervention or treatment on survival time. A typical question may be: will a given therapy increase the chances of survival of an individual or population? Such causal inquiries on survival outcomes are common in the fields of epidemiology and medicine (Robins, 1986; Hammer et al., 1996; Yusuf et al., 2016). As an important current example, the COVID-19 pandemic is creating a demand for methodological development to address such questions, specifically, when evaluating the effectiveness of a potential vaccine or therapeutic outside randomized controlled trial settings.
+
+Traditional causal survival analysis is typically carried out in the context of a randomized controlled trial (RCT), where the treatment assignment is controlled by researchers. Though they are the gold standard for causal inference, RCTs are usually long-term engagements, expensive and limited in sample size. Alternatively, the availability of observational data with comprehensive information about patients, such as electronic health records (EHRs), constitutes a more accessible but also more challenging source for estimating causal effects (Häyrinen et al., 2008; Jha et al., 2009). Such observational data may be used to augment and verify an RCT, after a particular treatment is approved and in use (Gombar et al., 2019; Frankovich et al., 2011; Longhurst et al., 2014). Moreover, the wealth of information from observational data also allows for the estimation of the individualized treatment effect (ITE), namely, the causal effect of an intervention at the individual level. In this work, we develop a novel framework for counterfactual time-to-event prediction to estimate the ITE for survival or time-to-event outcomes from observational data.
+
+Estimating the causal effect for survival outcomes in observational data manifests two principal challenges. First, the treatment assignment mechanism is not known a priori. Therefore, there may be variables, known as confounders, affecting both the treatment and survival time, which lead to selection bias (Bareinboim & Pearl, 2012), i.e., that the distributions across treatment groups are not the same. In this work, we focus on selection biases due to confounding, but other sources may also be considered. For instance, patients who are severely ill are likely to receive more aggressive therapy, however, their health status may also inevitably influence survival. Traditional survival analysis neglects such bias, leading to incorrect causal estimation. Second, the exact time-to-event is not always observed, i.e., sometimes we only know that an event has not occurred up to a certain point in time. This is known as the censoring problem. Moreover, censoring might be informative depending on the characteristics of the individuals and their treatment assignments, thus proper adjustment is required for accurate causal estimation (Cole & Hernán, 2004; Díaz, 2019).
+
+Traditional causal survival-analysis approaches typically model the effect of the treatment or covariates (not time or survival) in a parametric manner. Two commonly used models are the Cox proportional hazards (CoxPH) model (Cox, 1972) and the accelerated failure time (AFT) model (Wei, 1992), which presume a linear relationship between the covariates and survival probability. Further, proper weighting for each individual has been employed to account for confounding bias from these models (Austin, 2007; 2014; Hernán et al., 2005). For instance, probability weighting schemes that account for both selection bias and covariate dependent censoring have been considered for adjusted survival curves (Cole & Hernán, 2004; Díaz, 2019). Moreover, such probability weighting schemes have been applied to causal survival-analysis under time-varying treatment and confounding (Robins, 1986; Hernán et al., 2000). See van der Laan & Robins (2003); Tsiatis (2007); Van der Laan & Rose (2011); Hernán & Robins (2020) for an overview. Such linear specification makes these models interpretable but compromises their flexibility, and makes it difficult to adapt them for high-dimensional data or to capture complex interactions among covariates. Importantly, these methods lack a counterfactual prediction mechanism, which is key for ITE estimation (see Section 2).
+
+Fortunately, recent advances in machine learning, such as representation learning or generative modeling, have enabled causal inference methods to handle high-dimensional data and to characterize complex interactions effectively. For instance, there has been recent interest in tree-based (Chipman et al., 2010; Wager & Athey, 2018) and neural-network-based (Shalit et al., 2017; Zhang et al., 2020) approaches. For pre-specified time-horizons, the nonparametric Random Survival Forest (RSF) (Ishwaran et al., 2008) and Bayesian Additive regression trees (BART) (Chipman et al., 2010) have been extended to causal survival analysis. RSF has been applied to causal survival forests with weighted bootstrap inference (Shen et al., 2018; Cui et al., 2020) while a BART is extended to account for survival outcomes in Surv-BART (Sparapani et al., 2016), and AFT-BART (Henderson et al., 2020). See Hu et al. (2020) for an extensive investigation of the causal survival tree-based methods. Alternatively, when estimating the ITE, neural-network-based methods propose to regularize the transformed covariates or representations for an individual to have balanced distributions across treatment groups, thus accounting for the confounding bias and improving ITE prediction. However, most approaches employing representation learning techniques for counterfactual inference deal with continuous or binary outcomes, instead of time-to-event outcomes with censoring (informative or non-informative). Hence, a principled generalization to the context of counterfactual survival analysis is needed.
+
+In this work we leverage balanced (latent) representation learning to estimate ITE via counterfactual prediction of survival outcomes in observational studies. We develop a framework to predict event times from a low-dimensional transformation of the original covariate space. To address the specific challenges associated with counterfactual survival analysis, we make the following contributions:
+
+• We develop an optimization objective incorporating adjustments for informative censoring, as well as a balanced regularization term bounding the generalization error for ITE prediction. For the latter, we repurpose a recently proposed bound (Shalit et al., 2017) for our time-to-event scenario.   
+• We propose a generative model for event times to relax restrictive survival linear and parametric assumptions, thus allowing for more flexible modeling. Our approach can also provide nonparametric uncertainty quantification for ITE predictions.   
+• We provide survival-specific evaluation metrics, including a new nonparametric hazard ratio estimator, and discuss how to perform model selection for survival outcomes. The proposed model demonstrates superior performance relative to the commonly used baselines in real-world and semi-synthetic datasets.   
+• We introduce a survival-specific semi-synthetic dataset and demonstrate an approach for leveraging prior randomized experiments in longitudinal studies for model validation.
+
+![](images/482e9df6a7b5d031a444403a4be41d381d10c980808568d8cbf42f012f3882e6.jpg)  
+Figure 1: (a) Illustration of the proposed counterfactual survival analysis (CSA). Covariates $X =$ $x$ are mapped into latent representation $r$ via deterministic mapping $r \ = \ \Phi ( x )$ . The potential outcomes are sampled from $\bar { t } _ { a } \sim p ( T _ { A } | X = x )$ for $A = a$ via stochastic mapping $h _ { A } ( r , \tilde { \epsilon } )$ , where randomness is induced with a flow-based transformation, $\tilde { \epsilon }$ , of a simple distribution $p ( \epsilon )$ , i.e., uniform or Gaussian. (b) and (c) show the proposed causal graphs for non-informative and informative censoring, respectively.
+
+# 2 PROBLEM FORMULATION
+
+We first introduce the basic setup for performing causal survival analysis in observational studies. Suppose we have $N$ units, with $N _ { 1 }$ units being treated and $N _ { 0 }$ in the control group $( N = N _ { 1 } + N _ { 0 } )$ . For each unit (individual), we have covariates $X$ , which can be heterogeneous, e.g., a mixture of categorical and continuous covariates which, in the context of medicine, may include labs, vitals, procedure codes, etc. We also have a treatment indicator $A$ , where $A = 0$ for the controls and $A = 1$ for the treated, as well as the outcome (event) of interest $T$ . Under the potential-outcomes framework (Rubin, 2005), let $T _ { 0 }$ and $T _ { 1 }$ be the potential event times for a given subject under control and treatment, respectively. In practice we only observe one realization of the potential outcomes, i.e., the factual outcome $T = T _ { A }$ , while the counterfactual outcome $T _ { 1 - A }$ is unobserved.
+
+In survival analysis, the problem becomes more difficult because we do not always observe the exact event time for each individual, but rather the time up to which we are certain that the event has not occurred; specifically, we have a (right) censoring problem, most likely due to the loss of follow-up. We denote the censoring time as $C$ and censoring indicator as $\delta \in \{ 0 , 1 \}$ . The actual observed time is $Y = \operatorname* { m i n } ( T _ { A } , C )$ , i.e., the outcome is observed (non-censored) if $T _ { A } < C$ and $\delta = 1$ .
+
+In this work, we are interested in the expected difference between the $T _ { 1 }$ and $T _ { 0 }$ conditioned on $X$ for a given unit (individual), which is commonly known as the individualized treatment effect (ITE). Specifically, we wish to perform inference on the conditional distributions of $T _ { 1 }$ and $T _ { 0 }$ , i.e., $p ( T _ { 1 } | X )$ and $p ( T _ { 0 } | X )$ , respectively, as shown in Figure 1a. In practice, we observe $N$ realizations of $( Y , \delta , X , A )$ for observed time, censoring indicator, covariates and treatment indicator, respectively; hence, from an observational study the dataset takes the form $\mathcal { D } = \{ ( y _ { i } , \delta _ { i } , x _ { i } , a _ { i } ) \} _ { i = 1 } ^ { N }$ . Below, we discuss several common choices of estimands in survival analysis.
+
+Estimands of Interest We begin by considering survival analysis in the absence of an intervening treatment choice, $A$ . Let $F ( t | x ) \triangleq P ( T \leq t | X = x )$ be the cumulative distribution function of the event (failure) time, $t$ , given a realization of the covariates, $x$ . Survival analysis is primarily concerned with characterization of the survival function conditioned on covariates $S ( t | x ) \triangleq 1 - F ( t | x )$ , and the hazard function or risk score, $\lambda ( t | x )$ , defined below. $S ( t | x )$ is a monotonically decreasing function indicating the probability of survival up to time $t$ . The hazard function measures the instantaneous probability of the event occurring between $\{ t , t + \Delta t \}$ given $T > t$ and $\Delta t  0$ . From standard definitions (Kleinbaum & Klein, 2010), the relationship between cumulative and hazard function is formulated as
+
+$$
+\lambda ( t | x ) = \operatorname* { l i m } _ { d t  0 } { \frac { P ( t < T < t + d t | X = x ) } { P ( T > t | X = x ) d t } } = - { \frac { d \log S ( t | x ) } { d t } } = { \frac { f ( t | x ) } { S ( t | x ) } } .
+$$
+
+From (1) we see that $f ( t | x ) \triangleq P ( T = t | X = x ) = \lambda ( t | x ) S ( t | x )$ , is the conditional event time density function (Kleinbaum $\&$ Klein, 2010).
+
+Given the binary treatment $A$ , we are interested in its impact on the survival time. For ITE estimation, we are also interested in the difference between the two potential outcomes $T _ { 1 } , T _ { 0 }$ . Let $S _ { A } ( t | x )$ and $\lambda _ { A } ( t | x )$ denote the survival and hazard functions for the potential outcomes $T _ { A }$ , i.e., $T _ { 1 }$ and $T _ { 0 }$ . Several common estimands of interest include (Zhao et al., 2012; Trinquart et al., 2016): difference in expected lifetime: $\begin{array} { r } { \mathrm { { I T E } } ( t , x ) = \int _ { 0 } ^ { t _ { \operatorname* { m a x } } } \{ S _ { 1 } ( t | x ) - S _ { 0 } ( t | x ) \} \mathrm { d } t = \mathbb { E } \{ T _ { 1 } - T _ { 0 } | X = x \} . } \end{array}$ , difference in survival function: $\mathrm { I T E } ( t , x ) = S _ { 1 } ( t | x ) - S _ { 0 } ( t | x )$ , and hazard ratio: $\mathrm { I T E } ( t , x ) = \lambda _ { 1 } ( t | x ) / \lambda _ { 0 } ( t | x )$ . The inference difficulties associated with the above estimands from observational data are two-fold. First, there are confounders affecting both the treatment assignment and outcomes, which stem from selection bias, i.e., the treatment and control covariate distributions are not necessarily the same. Also, we do not have direct knowledge of the conditional treatment assignment mechanism, i.e., $P ( A = a | X = x )$ , also known as the propensity score. Let ⊥⊥ denote statistical independence. For estimands to be identifiable from observational data, we make two assumptions: (i) $\{ T _ { 1 } , T _ { 0 } \} \perp \perp A | X$ , i.e., no unobserved confounders or ignorability, and $( i i )$ overlap in the covariate support $0 < P ( A =$ $1 | X = x ) < 1$ almost surely if $p ( X = x ) > 0$ . Second, the censoring mechanism is also unknown and may lead to bias without proper adjustment. We consider two censoring mechanisms in our work, $( i )$ conditionally independent or informative censoring: $T \perp \perp C | X , A$ , and $( i i )$ random or non-informative censoring: $T \perp \perp C$ . Note that for informative censoring, we also have to consider potential censoring times $C _ { 1 }$ and $C _ { 0 }$ and their conditionals $p ( C _ { 1 } | X )$ and $p ( C _ { 0 } | X )$ , respectively. Figure 1 shows causal graphs illustrating these modeling assumptions.
+
+# 3 MODELING
+
+To overcome the above challenges and adjust for observational biases, we propose a unified framework for counterfactual survival analysis (CSA). Specifically, we repurpose the counterfactual bound in Shalit et al. (2017) for our time-to-event scenario and introduce a nonparametric approach for stochastic survival outcome predictions. Below we formulate a theoretically grounded and unified approach for estimating $( i )$ the encoder function $r = \Phi ( x )$ , which deterministically maps covariates $x$ to their corresponding latent representation $r \in  { \mathbb { R } } ^ { d }$ , and $( i i )$ two stochastic time-to-event generative functions, $h _ { A } ( \cdot )$ , to implicitly draw samples from both potential outcome conditionals $\bar { t } _ { a } \sim p _ { h , \Phi } ( T _ { A } | X = x )$ , for $A = \{ 1 , 0 \}$ , and where $t _ { a }$ indicates the sample from $p _ { h , \Phi } ( T _ { A } | X = x )$ is for $A = a$ . Further, we formulate a general extension that accounts for informative censoring by introducing two stochastic censoring generative functions, $\nu _ { A } ( \cdot )$ , to draw samples for potential censoring times $c _ { a } \sim p _ { \nu , \Phi } ( C _ { A } | X = x )$ . The model-specifying functions, $\{ h _ { A } ( \cdot ) , \nu _ { A } ( \cdot ) , \Phi ( \cdot ) \}$ , are parameterized via neural networks. See the Supplementary Material (SM) for details. Figure 1a summarizes our modeling approach.
+
+Accounting for selection bias We wish to estimate the potential outcomes, i.e., event times, which are sampled by distributions parameterized by functions $\{ h _ { A } ( \cdot ) , \Phi ( \cdot ) \}$ , i.e.,
+
+$$
+\begin{array} { c } { { t \sim p _ { h , \Phi } ( T | X = x , A = a ) } } \\ { { { \it t } _ { a } \sim p _ { h , \Phi } ( T _ { a } | X = x ) } } \end{array}
+$$
+
+We obtain (3) from (2) via the strong ignorability assumption, i.e., $\{ T _ { 0 } , T _ { 1 } \}$ ⊥⊥ $A | X$ (consistent with the causal graphs in Figure 1b and 1c) and $0 < P ( A = \mathsf { \bar { a } } | X \doteq x ) < 1 .$ , and the consistency assumption, i.e., $T = T _ { A } | A = a$ . A similar argument can be made for informative censoring based on Figure 1c, so we can also write $c _ { a } \sim p _ { \nu , \Phi } ( C _ { A } | X = x )$ . Given (3), model functions $\{ h _ { A } ( \cdot ) , \Phi ( \cdot ) \bar  \}$ and $\nu _ { A } ( \cdot )$ for informative censoring can be learned by leveraging standard statistical optimization approaches, that minimize a loss hypothesis $\mathcal { L }$ given samples from the empirical distribution $( y , \delta , x , a ) \sim p ( Y , \delta , X , A )$ , i.e., from dataset $\mathcal { D }$ . Specifically, we write $\mathcal { L } = \mathbb { E } _ { ( y , \delta , x , a ) \sim p ( Y , \delta , X , A ) } \left[ \ell _ { h , \Phi } ( t _ { a } , y , \delta ) \right]$ , where $\ell _ { h , \Phi } ( t _ { a } , y , \delta )$ is a loss function that measures the agreement of $t _ { a } \sim p _ { h , \Phi } ( T _ { A } | X = x )$ (and $c _ { a } \sim p _ { \nu , \Phi } ( C _ { A } | X = x )$ for informative censoring) with ground truth $\{ y , \delta \}$ , the observed time and censoring indicator, respectively.
+
+For some parametric formulations of event time distribution $p _ { h , \Phi } ( T _ { A } | X = x )$ , e.g., exponential, Weibull, log-Normal, etc., and provided the censoring mechanism is non-informative, $- \ell _ { h , \Phi } ( t _ { a } , y , \delta )$ is the closed form log likelihood. Specifically, $- \ell _ { h , \Phi } ( t _ { a } , y , \delta ) \triangleq \log p _ { h , \Phi } ( T _ { a } | X = x ) = \delta$ · $\log f _ { h , \Phi } ( t _ { a } | x ) + ( 1 - \bar { \delta } ) \cdot \log S _ { h , \Phi } ( t _ { a } | x )$ , which implies that the conditional event time density and survival functions can be calculated in closed form from transformations $\{ h _ { A } ( \cdot ) , \Phi ( \cdot ) \}$ of $x$ . See the SM for parametric examples of $\mathcal { L }$ accounting for informative censoring.
+
+We further define the expected loss for a given realization of covariates $x$ and treatment assignment $a$ over observed times $y$ (censored and non-censored), and the censoring indicator $\delta$ as $\zeta _ { h , \Phi } ( x , a ) \triangleq$ $\mathbb { E } _ { ( y , \delta , x ) \sim p ( Y , \delta | X ) } \ell _ { h , \Phi } ( t _ { a } , y , \delta )$ as in Shalit et al. (2017). For a given subject with covariates $x$ and treatment assignment $a$ , we wish to minimize both the factual and counterfactual losses, $\mathcal { L } _ { \mathrm { F } }$ and $\mathcal { L } _ { \mathrm { C F } }$ respectively, by decomposing $\mathcal { L } = \mathcal { L } _ { \mathrm { F } } + \mathcal { L } _ { \mathrm { C F } }$ as follows
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { F } } = \mathbb { E } _ { ( x , a ) \sim p ( A , X ) } \zeta _ { h , \Phi } ( x , a ) , \quad \mathcal { L } _ { \mathrm { C F } } = \mathbb { E } _ { ( x , a ) \sim p ( 1 - A , X ) } \zeta _ { h , \Phi } ( x , a ) . } \end{array}
+$$
+
+Let $u \triangleq P ( A = 1 )$ denote the marginal probability of treatment assignment. We can readily decompose the losses in (4) according to treatment assignments. The decomposed factual $\mathcal { L } _ { \mathrm { F } } =$ $u \cdot \mathcal { L } _ { \mathrm { F } } ^ { A = 1 } + \left( 1 - u \right) \cdot \mathcal { L } _ { \mathrm { F } } ^ { A = 0 }$ , and similarly, the decomposed counterfactual $\mathcal { L } _ { \mathrm { C F } } = \dot { ( } 1 - u ) \cdot \mathcal { L } _ { \mathrm { C F } } ^ { A = 1 } + u$ $\mathcal { L } _ { \mathrm { C F } } ^ { A = 0 }$ F CF . In practice, only factual outcomes are observed, hence, for a non-randomized non-controlled experiment, we cannot obtain an unbiased estimate of $\mathcal { L } _ { \mathrm { C F } }$ from data due to selection bias (or confounding). Therefore, we bound $\mathcal { L } _ { \mathrm { C F } }$ and $\mathcal { L }$ below following Shalit et al. (2017).
+
+Corollary 1 Assume $\Phi ( \cdot )$ is an invertible map, and $\alpha ^ { - 1 } \zeta _ { h , \Phi } ( x , a ) \in G$ , where $G$ is a family of functions, $p _ { \Phi } ^ { A = a } \triangleq p _ { \Phi } ( R | A = a )$ is the latent distribution for group $A = a$ , and $\alpha > 0$ is a constant. Then, we have:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { C F } } \leq ( 1 - u ) \cdot \mathcal { L } _ { \mathrm { F } } ^ { A = 1 } + u \cdot \mathcal { L } _ { \mathrm { F } } ^ { A = 0 } + \alpha \cdot \mathrm { I P M } _ { G } ( p _ { \Phi } ^ { A = 1 } , p _ { \Phi } ^ { A = 0 } ) } \\ & { \quad \mathcal { L } \leq \mathcal { L } _ { \mathrm { F } } ^ { A = 1 } + \mathcal { L } _ { \mathrm { F } } ^ { A = 0 } + \alpha \cdot \mathrm { I P M } _ { G } ( p _ { \Phi } ^ { A = 1 } , p _ { \Phi } ^ { A = 0 } ) . } \end{array}
+$$
+
+The integral probability metric (IPM) (Müller, 1997; Sriperumbudur et al., 2012) measures the distance between two probability distributions $p$ and $q$ defined over $M$ , i.e., the latent space of $R$ . Formally, $\begin{array} { r } { \mathrm { I P M } _ { G } ( p , q ) \triangleq \operatorname* { s u p } _ { g \in G } \vert \int _ { M } g ( m ) \left( p ( m ) - q ( m ) \right) d m \vert } \end{array}$ , where $g : m  \mathbb { R }$ , represents a class of real-valued bounded measurable functions on $M$ (Shalit et al., 2017). Therefore, model functions $\{ h _ { a } ( \cdot ) , \Phi ( \cdot ) \}$ can be learned by minimizing the upper bound in (5) consisting of $( i )$ only factual losses under both treatment assignments and $( i i )$ an IPM regularizer enforcing latent distributional equivalence between the treatment groups. Note that if the data originates from a RCT it follows (by construction) that $\mathrm { I P M } _ { G } ( p _ { \Phi } ^ { A = 1 } , p _ { \Phi } ^ { \tilde { A = 0 } } ) \stackrel { . } { = } 0$ .
+
+Accounting for censoring bias Below we formulate an approach for estimating functions $h _ { A } ( \cdot )$ and $\nu _ { A } ( \cdot )$ for synthesizing (sampling) non-censored $t _ { a } \sim \stackrel { \sim } { p } _ { h , \Phi } ( T _ { A } | X = x )$ and censored $c _ { a } \sim$ $p _ { \nu , \Phi } ( C _ { A } | X = x )$ times, respectively. While some parametric assumptions for $p _ { h , \Phi } ( T _ { A } | X = x )$ yield easy-to-evaluate closed forms for $S _ { h , \Phi } ( t _ { a } | x )$ that can be used as likelihood for censored observations, they are restrictive, and have been shown to generate unrealistic high variance samples (Chapfuwa et al., 2018). So motivated, we seek a nonparametric likelihood-based approach that can model a flexible family of distributions, with an easy-to-sample approach for event times $t _ { a } \sim$ $p _ { h , \Phi } ( T _ { a } | X = x )$ . We model the event time generation process with a source of randomness, $p ( \epsilon )$ , e.g. Gaussian or uniform, which is obtained from a neural-network-based nonlinear transformation. In the experiments we use a planar flow formulation parameterized by $\{ U _ { h } , W _ { h } , b _ { h } \}$ (Rezende & Mohamed, 2015), however, other specifications can also be used. Note that Miscouridou et al. (2018) has previously leveraged normalizing flows for survival analysis, however, our approach is very different in that it focuses on formulating $\romannumeral 1$ ) a counterfactual survival analysis framework that accounts for informative or non-informative censoring mechanisms and confounding, and $_ { i i }$ ) model event times as a continuous variable instead of discretizing them. Specifically, we transform the source of randomness, $\epsilon$ , using a single layer specification as follows
+
+$$
+{ \widetilde { \epsilon } } _ { h } = \epsilon + U _ { h } \operatorname { t a n h } ( W _ { h } \epsilon + b _ { h } ) , \quad \epsilon \sim \operatorname { U n i f o r m } ( 0 , 1 ) , t _ { a } = h _ { A } ( r , { \widetilde { \epsilon } } _ { h } ) , \quad r = \Phi ( x )
+$$
+
+where $\{ U _ { h } , W _ { h } \} \in \mathbb { R } ^ { d \times d }$ , $\{ b _ { h } , \epsilon \} \in \mathbb { R } ^ { d }$ , $d$ is the dimensionality of the normalizing flow; each component of $\epsilon$ is drawn independently from Uniform $( 0 , 1 )$ , and $\tilde { \epsilon } _ { h }$ may be viewed as a skip connection with stochasticity in $\epsilon$ . Further, $h _ { A } ( r , \tilde { \epsilon } _ { h } )$ and $\Phi ( x )$ are time-to-event generative and encoding functions, respectively, parameterized as neural networks. For simplicity, the dimensions of $r$ and $\epsilon$ are set to $d$ , however, they can be set independently if desired. In practice, we are interested in generating realistic event-time samples; therefore, we account for both censored and non-censored observations by adopting the objective from Chapfuwa et al. (2018), formulated as
+
+$$
+\mathcal { L } _ { \mathrm { F } } ^ { \mathrm { C S A } } \triangleq \mathbb { E } _ { ( y , \delta , x , a ) \sim p ( Y , \delta , X , A ) , \epsilon \sim p ( \epsilon ) } \left[ \delta \cdot \left( \left| y - t _ { a } \right| \right) + \left( 1 - \delta \right) \cdot \left( \operatorname* { m a x } ( 0 , y - t _ { a } ) \right) \right] ,
+$$
+
+where the first term encourages sampled event times $t _ { a }$ to be close to $y$ , the ground truth for observed events, i.e., $\delta = 1$ , while penalizing $t _ { a }$ for being smaller than the censoring time when $\delta = 0$ . Further, the expectation is taken over samples (a minibatch) from empirical distribution $p ( Y , \delta , X , A )$ .
+
+Informative censoring We model informative censoring similar to (7) but mirroring the censoring indicators to encourage accurate censoring time samples $c _ { a }$ for $\delta = 0$ , while penalizing $c _ { a }$ for being smaller than $y$ for $\delta \ : = \ : 1$ (observed events). Specifically, we set an independent source of randomness like in (6) but parameterized by $\{ U _ { \nu } , W _ { \nu } , b _ { \nu } \}$ and censoring generative functions $\nu _ { A } ( r , \tilde { \epsilon } _ { \nu } )$ , parameterized as neural networks, where $c _ { a } \sim p _ { \nu , \Phi } ( C _ { A } | X = x )$ formulated as
+
+$$
+\ell _ { c } ( \nu , \Phi ) = \mathbb { E } _ { ( y , \delta , x , a ) \sim p ( y , \delta , X , A ) , \epsilon \sim p ( \epsilon ) } \left[ ( 1 - \delta ) \cdot ( | y - c _ { a } | ) + \delta \cdot ( \operatorname* { m a x } ( 0 , y - c _ { a } ) ) \right] .
+$$
+
+Further, we introduce an additional time-order-consistency loss that enforces the correct order of the observed time relative to the censoring indicator, i.e., $c _ { a } < t _ { a }$ if $\delta = 0$ and $t \sb a < c \sb a$ if $\delta = 1$ , thus
+
+$$
+\ell _ { \mathrm { T C } } ( h , \nu , \Phi ) = \mathbb { E } _ { ( \delta , x , a ) \sim p ( \delta , X , A ) , \epsilon \sim p ( \epsilon ) } \left[ \delta \cdot ( \operatorname* { m a x } ( 0 , t _ { a } - c _ { a } ) ) + ( 1 - \delta ) \cdot ( \operatorname* { m a x } ( 0 , c _ { a } - t _ { a } ) ) \right] .
+$$
+
+Note that $\ell _ { \mathrm { T C } } ( h , \nu , \Phi )$ does not depend on the observed event times but only on the censoring indicators. Finally, we write the consolidated CSA loss for informative censoring (CSA-INFO) by aggregating (7), (8) and (9) as $\mathcal { L } _ { \mathrm { F } } ^ { \mathrm { C S A - I N F O } } \triangleq \mathcal { L } _ { \mathrm { F } } ^ { \mathrm { C S A } } + \ell _ { \mathrm { c } } + \ell _ { \mathrm { T C } }$ .
+
+Learning Model functions $\{ h _ { A } ( \cdot ) , \Phi ( \cdot ) , \nu _ { A } ( \cdot ) \}$ are learned by minimizing the bound (5), via stochastic gradient descent on minibatches from $\mathcal { D }$ , with $\mathcal { L } _ { \mathrm { F } } ^ { \mathrm { C S A } }$ for non-informative censoring and $\mathcal { L } _ { \mathrm { F } } ^ { \mathrm { C S A - I N F O } }$ for informative censoring. Further, for the IPM regularization loss in (5), we optimize the dual formulation of the Wasserstein distance, via the regularized optimal transport (Villani, 2008; Cuturi, 2013). Consequently, we only require $\alpha ^ { - 1 } \zeta _ { h , \Phi } ( \bar { x , } a )$ to be 1-Lipschitz (Shalit et al., 2017) and $\alpha$ is selected by grid search on the validation set using only factual data (details below).
+
+# 4 METRICS
+
+We propose a comprehensive evaluation approach that accounts for both factual and causal metrics. Factual survival outcome predictions are evaluated according to standard survival metrics that measure diverse performance characteristics, such as concordance index (C-Index) (Harrell Jr et al., 1984), mean coefficient of variation (COV) and calibration slope (C-slope) (Chapfuwa et al., 2020). See the SM for more details on these metrics. For causal metrics, defined below, we introduce a nonparametric hazard ratio (HR) between treatment outcomes, and adopt the conventional precision in estimation of heterogeneous effect (PEHE) and average treatment effect (ATE) performance metrics (Hill, 2011). Note that PEHE and ATE require ground truth counterfactual event times, which is only possible in (semi-)synthetic data. For HR, we compare our findings with those independently reported in the literature from gold-standard RCT data.
+
+Nonparametric Hazard Ratio In a medical setting, the population hazard ratio $\mathrm { H R } ( t )$ between treatment groups is considered informative thus has been widely used in drug development and RCT (Yusuf et al., 2016; Mihaylova et al., 2012). For example, $\mathrm { H R } ( t ) < 1 , > 1$ , or $\approx 1$ indicate population positive, negative and neutral treatment effects at time $t$ , respectively. Moreover, $\mathrm { H R } ( t )$ naturally accounts for both censored and non-censored outcomes. Standard approaches for computing $\mathrm { H R } ( { \dot { t } } )$ rely on the restrictive proportional hazard assumption from CoxPH (Cox, 1972), which is constituted as a semi-parametric linear model $\lambda ( t | a ) = \lambda _ { \mathrm { b } } \bar { ( } t ) \exp ( a \beta )$ . However, the constant covariate (time independent) effect is often violated in practice (see Figure 2b). For CoxPH, the marginal HR between treatment and control can be obtained from regression coefficient $\beta$ learned via maximum likelihood without the need for specifying the baseline hazard $\begin{array} { r } { \mathrm {  ~ \cdot ~ } _ { \mathrm { b } } ( t ) \colon \mathrm { H R } _ { \mathrm { C o x P H } } ( t ) = \frac { \lambda ( t | a = 1 ) } { \lambda ( t | a = 0 ) } = \exp ( \beta ) } \end{array}$ . So motivated, we propose a nonparametric, model-free approach for computing $\mathrm { H R } ( t )$ , in which we do not assume a parametric form for the event time distribution or the proportional hazard assumption from CoxPH. This approach only relies on samples from the conditional event time density functions, $f ( t _ { 1 } | x )$ and $f ( t _ { 0 } | x )$ , via $t _ { a } = h _ { A } ( \cdot )$ from (6).
+
+Definition 1 We define the nonparametric marginal Hazard Ratio and its approximation, $\mathrm { \hat { H R } } ( t )$ , as
+
+$$
+\mathrm { H R } ( t ) = \frac { \lambda _ { 1 } ( t ) } { \lambda _ { 0 } ( t ) } = \frac { S _ { 0 } ( t ) } { S _ { 1 } ( t ) } \cdot \frac { S _ { 1 } ^ { \prime } ( t ) } { S _ { 0 } ^ { \prime } ( t ) } , \quad \mathrm { H R } ( t ) = \frac { \hat { S } _ { 0 } ^ { \mathrm { P K M } } ( t ) } { \hat { S } _ { 1 } ^ { \mathrm { P K M } } ( t ) } \cdot \frac { m _ { 1 } ( t ) } { m _ { 0 } ( t ) } ,
+$$
+
+where for $\mathrm { H R } ( t )$ we leveraged (1) to obtain (10) and $S ^ { \prime } ( t ) \triangleq d S ( t ) / d t$ . The nonparametric assumption for $S ( t )$ makes the computation of $S ^ { \prime } ( t )$ challenging. Provided that $S ( t )$ is a monotonically decreasing function, for simplicity, we fit a linear function $S ( t ) = m \cdot t + c .$ , and set $S ^ { \prime } ( t ) \approx m$ . Note that the linear model is only used for estimating $S ^ { \prime } ( t )$ from the nonparametric estimation of $S ( t )$ .
+
+Table 1: Performance comparisons on ACTG-SYNTHETIC data, with $9 5 \%$ $\mathrm { H R } ( t )$ confidence interval. The ground truth, test set, hazard ratio is $\mathrm { H R ( t ) } = 0 . 5 2 _ { ( 0 . 3 9 , 0 . 7 1 ) }$ .   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Causal</td><td colspan="2">Factual</td><td rowspan="2">C-Slope (A=0,A=1)</td></tr><tr><td>EPEHE</td><td>EATE</td><td>HR(t)</td><td>C-Index (A=0,A=1)</td><td>Mean COV</td></tr><tr><td>CoxPH-Uniform</td><td>NA</td><td>NA</td><td>0.97(0.86,1.09)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-IPW</td><td>NA</td><td>NA</td><td>0.48(0.03,7.21)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-OW</td><td>NA</td><td>NA</td><td>0.60(0.53,0.68)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>Surv-BART</td><td>352.07</td><td>77.89</td><td>0.0(0.0,0.0)</td><td>(0.706,0.686)</td><td>0.001</td><td>(0.398,0)</td></tr><tr><td>AFT-Weibull</td><td>367.92</td><td>133.93</td><td>0.47(0.47,0.47)</td><td>(0.21,0.267)</td><td>6.209</td><td>(0.707,0.729)</td></tr><tr><td>AFT-log-Normal</td><td>377.76</td><td>157.64</td><td>0.47(0.47,0.47)</td><td>(0.675, 0.556)</td><td>6.971</td><td>(0.707,0.729)</td></tr><tr><td>SR</td><td>369.47</td><td>88.55</td><td>0.38(0.330.65)</td><td>(0.791,0.744)</td><td>0</td><td>(0.985,1.027)</td></tr><tr><td>CSA (proposed)</td><td>358.72</td><td>0.8</td><td>0.45(0.39,0.65)</td><td>(0.787,0.767)</td><td>0.131</td><td>(0.985,1.026)</td></tr><tr><td>CSA-INFO (proposed)</td><td>344.3</td><td>31.19</td><td>0.53(0.41,0.67)</td><td>(0.78,0.764)</td><td>0.13</td><td>(0.999,1.029)</td></tr></table>
+
+Bias from $S ^ { \prime } ( t )$ can be reduced by considering more complex function approximations for $S ( t )$ , e.g., polynomial or spline. For the nonparametric estimation of $S ( t )$ we leverage the model-free population point-estimate-based nonparametric Kaplan-Meier (Kaplan & Meier, 1958) estimator of the survival function $\hat { S } ^ { \mathrm { P K M } } ( t )$ in Chapfuwa et al. (2020) to marginalize both factual and counterfactual predictions given covariates $x$ . The approximated hazard ratio, $\mathrm { \hat { H R } } ( t )$ , is thus obtained by combining the approximations $\hat { S } _ { a } ^ { \mathrm { P K M } } ( t )$ and $m _ { a }$ . A similar formulation for the conditional, $\mathrm { H } \mathrm { \hat { R } } ( t | x )$ , can also be derived. See the SM for full details on the evaluation or $\mathrm { \hat { H R } } ( t )$ and $\mathrm { H } \mathrm { \hat { R } } ( t | x )$ . Note that for some AFT- or CoxPH-based parametric formulations, $\mathrm { H R } ( t | x )$ , can be readily evaluated because $f ( t _ { a } | x )$ and $S ( t _ { a } | x )$ are available in closed form.
+
+In the experiments, we will use $\mathrm { H R } ( t )$ to compare different approaches against results reported in RCTs (see Tables 1 and 3). Further, we will use $\mathrm { H R } ( t | x )$ to illustrate stratified treatment effects (see Figure 2). Note that though a neural-based survival recommender system (Katzman et al., 2018) has been previously used to estimate $\operatorname { H R } ( t | x )$ , their approach does not account for confounding or informative censoring thus it is susceptible to bias.
+
+Precision in Estimation of Heterogeneous Effect (PEHE) A general individualized estimation error is formulated as $\epsilon _ { \mathrm { P E H E } } = \sqrt { \mathbb { E } _ { X } [ ( \mathrm { I T E } ( x ) - \mathrm { I } \hat { \mathrm { T E } } ( x ) ) ^ { 2 } ] }$ , where $\operatorname { I T E } ( x )$ is the ground truth, $\mathrm { I } \hat { \mathrm { T E } } ( x ) = \mathbb { E } _ { T } \left[ \gamma \left( T _ { 1 } \right) - \gamma \left( T _ { 0 } \right) \vert X = x \right]$ and $\gamma ( \cdot )$ is a deterministic transformation. In our experiments, $\gamma ( \cdot )$ is the average over samples from $\dot { t _ { a } } \sim p _ { h , \Phi } ( T _ { A } | X = x )$ . Alternative estimands, e.g., thresholding survival times $\gamma ( T _ { A } ) = I \{ T _ { A } > \tau \}$ , can also be considered as described above.
+
+Average Treatment Effect (ATE) The population treatment effect estimation error is defined as $\epsilon _ { \mathrm { A T E } } = | \mathrm { A T E } - \mathrm { A \hat { T } E } |$ , where $\mathrm { A T E } = \mathbb { E } _ { X } [ \mathrm { I T E } ( x ) ]$ (ground truth) and $\mathrm { A } \hat { \mathrm { T E } } = \mathbb { E } _ { X } [ \hat { \mathrm { I T E } } ( x ) ]$ .
+
+# 5 EXPERIMENTS
+
+We describe the baselines and datasets that will be used to evaluate the proposed counterfactual survival analysis methods (CSA and CSA-INFO). Pytorch code including the new semi-synthetic dataset (see below) will be made publicly available. Throughout the experiments, we use the standard $\mathrm { H R } ( t )$ for CoxPH based methods and (10) for all others. The bound in (5) is sensitive to $\alpha$ , thus we propose approximating proxy counterfactual outcomes $\{ Y _ { \mathrm { C F } } , \delta _ { \mathrm { C F } } \}$ for the validation set, according to the covariate Euclidean nearest-neighbour (NN) from the training set. We select the $\alpha$ that minimizes the validation loss $\mathcal { L } = \mathcal { L } _ { \mathrm { F } } + \mathcal { L } _ { \mathrm { C F } }$ from the set $( 0 , 0 . 1 , 1 , 1 0 , 1 0 0 )$ .
+
+Baselines We consider the following competitive baseline approaches: $( i )$ propensity weighted CoxPH (Schemper et al., 2009; Buchanan et al., 2014; Rosenbaum & Rubin, 1983); $( i i )$ IPM (5) regularized AFT (log-Normal and Weibull) models; $( i i i )$ an IPM (5) regularized deterministic semisupervised regression (SR) model with accuracy objective from (Chapfuwa et al., 2018), as a contrast for the proposed stochastic predictors (CSA and CSA-INFO); and $( i v )$ survival Bayesian additive regression trees (Surv-BART) (Sparapani et al., 2016). For CoxPH, we consider three normalized weigh2009), $\begin{array} { r } { \mathrm { I P W } _ { i } = \frac { a _ { i } } { \hat { e } _ { i } } + \frac { 1 - a _ { i } } { 1 - \hat { e } _ { i } } ; i i ) } \end{array}$ $( i )$ weilappi ng (IPW) (Horvitz & Thompson, 1952; Cao et al.,weights (OW) (Crump et al., 2006; Li et al., 2018), $\mathrm { O W } _ { i } = a _ { i } \cdot \left( 1 - \hat { e } _ { i } \right) + \left( 1 - a _ { i } \right) \cdot \hat { e } _ { i }$ $i i i$   
+linear logistic model $\hat { e } _ { i } = \sigma ( x _ { i } ; w )$ , is used as an approximation, $\boldsymbol { \hat { e } } _ { i }$ , to the unknown propensity score $P ( A = { \bar { 1 } } | X = x )$ . See the SM for a details of the baselines.
+
+Table 3: Performance comparisons on FRAMINGHAM data, with $9 5 \%$ $\mathrm { H R } ( t )$ confidence interval. Test set NN assignment of $y _ { \mathrm { C F } }$ and $\delta _ { \mathrm { C F } }$ yields biased $\mathrm { H R ( t ) } = 1 . 2 3 _ { ( 1 . 1 7 , 1 . 2 5 ) }$ , while previous large scale longitudinal RCT studies estimated $\mathrm { H R } ( \mathrm { t } ) = 0 . 7 5$ (0.64,0.88) (Yusuf et al., 2016).   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Causal HR(t)</td><td rowspan="2">C-Index (A=0, A=1)</td><td rowspan="2">Factual Mean COV</td><td rowspan="2">C-Slope (A=0,A=1)</td></tr><tr><td></td></tr><tr><td>CoxPH-Uniform</td><td>1.69(1.38,2.07)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-IPW</td><td>1.09(0.76,1.57)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-OW</td><td>0.88(0.73,1.08)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>Surv-BART</td><td>14.99(14.9,14.9e8)</td><td>(0.629,0.630)</td><td>0.003</td><td>(0.232,0.084)</td></tr><tr><td>AFT-Weibull</td><td>1.09(1.09,1.09)</td><td>(0.734,0.395)</td><td>8.609</td><td>(0.857,0.89)</td></tr><tr><td>AFT-log-Normal</td><td>1.55(1.46,1.55)</td><td>(0.68,0.56)</td><td>10.415</td><td>(0.979,0.732)</td></tr><tr><td>SR</td><td>0.58(0.53,0.71)</td><td>(0.601, 0.57)</td><td>0</td><td>(0.491,0.63)</td></tr><tr><td>CSA(proposed)</td><td>1.04(1.00,1.09)</td><td>(0.763,0.728)</td><td>0.161</td><td>(0.891,0.81)</td></tr><tr><td>CSA-INFO (proposed)</td><td>0.81(0.77,0.83)</td><td>(0.752,0.651)</td><td>0.156</td><td>(0.907, 0.881)</td></tr></table>
+
+Datasets We consider the following datasets: (i) FRAMINGHAM, is an EHR-based longitudinal cardiovascular cohort study that we use to evaluate the effect of statins on future coronary heart disease outcomes (Benjamin et al., 1994); $( i i )$ ACTG, is a longitudinal RCT study comparing monotherapy with Zidovudine or Didanosine with combination therapy in HIV patients (Hammer et al., 1996); and $( i i i )$ ACTG
+
+Table 2: Summary statistics of the datasets.   
+
+<table><tr><td></td><td>FRAMINGHAM</td><td>ACTG</td><td>ACTG-SYNTHETIC</td></tr><tr><td>Events (%)</td><td>26.0</td><td>26.9</td><td>48.9</td></tr><tr><td>Treatment (%)</td><td>10.4</td><td>49.5</td><td>55.9</td></tr><tr><td>N</td><td>3,435</td><td>1,054</td><td>2,139</td></tr><tr><td>p</td><td>32</td><td>23</td><td>23</td></tr><tr><td>Missing (%)</td><td>0.23</td><td>1.41</td><td>1.38</td></tr><tr><td>tmax (days)</td><td>7,279</td><td>1,231</td><td>1,313</td></tr></table>
+
+SYNTHETIC, is a semi-synthetic dataset based on ACTG covariates. We simulate potential outcomes according to a Gompertz-Cox distribution (Bender et al., 2005) with selection bias from a simple logistic model for $\bar { P ( A = 1 | X = x ) }$ and AFT-based censoring mechanism. The generative process is detailed in the SM. Table 2 summarizes the datasets according to $( i )$ covariates of size $p$ ; $( i i )$ proportion of non-censored events, treated units, and missing entries in the $N \times p$ covariate matrix; and $( i i i )$ time range $t _ { \mathrm { m a x } }$ for both censored and non-censored events. Missing entries are imputed with median or mode if continuous or categorical, respectively.
+
+Quantitative Results Experimental results for two data-sets in Tables 1 and 3, illustrate that AFTbased methods are high variance, inferior in calibration and C-Index than accuracy-based methods (SR, CSA, CSA-INFO). Surv-BART is the least calibrated but low variance method. CSA-INFO and CSA outperform all methods across all factual metrics, whereas CSA-INFO is better calibrated, low variance but slightly lower C-Index than CSA. Note that we fit CoxPH using the entire dataset; since it does not support counterfactual inference, we do not present factual metrics. By properly adjusting for both informative censoring and selection bias, CSA-INFO significantly outperforms all methods in treatment effect estimation according to $\mathrm { H R } ( t )$ and $\epsilon _ { \mathrm { P E H E } }$ , across non-RCT datasets, while remaining comparable to AFT-Weibull on the RCT dataset (see the SM). Further, RCT-based results on ACTG data in the SM illustrate comparable $\mathrm { H R } ( t )$ across all models except for AFT-log-Normal and Surv-BART, which overestimate, and SR, which underestimates risk. For non-RCT datasets (ACTG-SYNTHENTIC and FRAMINGHAM), CoxPH-OW has a clear advantage over all CoxPH based methods, mostly credited to the well-behaved bounded propensity weights $\in [ 0 , 1 ]$ . Interestingly, the FRAMINGHAM observational data exhibits a common paradox, where without proper adjustment of selection and censoring bias, naive approaches would result in a counter-intuitive treatment effect from statins. However, there is severe confounding from covariates such as age, BMI, diabetes, CAD, PAD, MI, stroke, etc., that influence both treatment likelihood and survival time. Table 3, demonstrates that CSA-INFO is clearly the best performing approach. Specifically, its $\mathrm { H R } ( t )$ , reverses the biased observational treatment effect, to demonstrate positive treatment from statins, which is consistent with prior large RCT longitudinal findings (Yusuf et al., 2016).
+
+Qualitative Results Figure 2a demonstrates that CSA-INFO matches the ground truth population hazard, $\mathrm { H R } ( t )$ , better than alternative methods on ACTG-SYNTHETIC data. See the SM for ACTG and FRAMINGHAM. Figure 2b shows sub-population log hazard ratios for four patient clusters obtained via hierarchical clustering on the individual log hazard ratios, $\log \mathrm { H R } ( t | x )$ , of the test set of FRAMINGHAM data. Interestingly, these clusters stratify treatment effects into: positive (2), negative (1 and 3), and neutral (4) sub-populations. Moreover, the estimated density of median $\log \mathrm { H } \bar { \mathrm { R } } ( t | x )$ values in Figure $_ { 2 \mathrm { c } }$ illustrates that nearly $7 0 \%$ of the testing set individuals have $\log \mathrm { H R } ( t | x ) < 0$ , thus may benefit from taking statins. Further, we isolated the extreme top and bottom quantiles, $\mathrm { H R } ( t | x ) < 0 . 0 2 4$ and $\mathrm { H R } ( \bar { t } | x ) > 1 . 9 1 6$ , respectively, of the median $\log \mathrm { \bar { H R } } ( t | x )$ values for the test set of FRAMINGHAM, as shown in Figure 2c. After comparing their covariates, we found that individuals with the following characteristics may benefit from taking statins: young, male, diabetic, without prior history (CAD, PAD, stroke or MI), high BMI, cholesterol, triglycerides, fasting glucose, and low high-density lipoprotein. There seem to be consensus that diabetics and high-cholesterol patients benefit from statins (Cheung et al., 2004; Wilt et al., 2004). See SM for additional results.
+
+![](images/4c89c16905f48a1bdcc35b1ade4f0f866e4e151f0c05e3f1b2a559591a62f777.jpg)  
+Figure 2: (a) Inferred population $\mathrm { H R } ( t )$ compared against ground truth (EMP) on ACTG-SYNTHETIC data. CSA-INFO-based (b) cluster-specific average l $\operatorname { \Pi } ^ { } \operatorname { \Pi } ^ { } \operatorname { \mathrm { ) g } } \operatorname { \mathrm { { H R } } } ( t | x )$ curves and (c) estimated density of median $\log \mathrm { H R } ( t | x )$ values on the test set of the FRAMINGHAM dataset. Clusters assignment were obtained via hierarchical clustering of individualized $\log \mathrm { H R } ( t | x )$ traces.
+
+# 6 CONCLUSIONS
+
+We have proposed a unified counterfactual inference framework for survival analysis. Our approach adjusts for bias from two unknown sources, namely, confounding due to covariate dependent selection bias and censoring (informative or non-informative). Relative to competitive alternatives, we demonstrate superior performance for both survival-outcome prediction and treatment-effect estimation, across three diverse datasets, including a semi-synthetic dataset which we introduce. Moreover, we formulate a model-free nonparametric hazard ratio metric for comparing treatment effects or leveraging prior randomized real-world experiments in longitudinal studies.
+
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+        "text": "Balanced representation learning methods have been applied successfully to counterfactual inference from observational data. However, approaches that account for survival outcomes are relatively limited. Survival data are frequently encountered across diverse medical applications, i.e., drug development, risk profiling, and clinical trials, and such data are also relevant in fields like manufacturing (for equipment monitoring). When the outcome of interest is time-to-event, special precautions for handling censored events need to be taken, as ignoring censored outcomes may lead to biased estimates. We propose a theoretically grounded unified framework for counterfactual inference applicable to survival outcomes. Further, we formulate a nonparametric hazard ratio metric for evaluating average and individualized treatment effects. Experimental results on real-world and semi-synthetic datasets, the latter which we introduce, demonstrate that the proposed approach significantly outperforms competitive alternatives in both survival-outcome predictions and treatment-effect estimation. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Survival analysis or time-to-event studies focus on modeling the time of a future event, such as death or failure, and investigate its relationship with covariates or predictors of interest. Specifically, we may be interested in the causal effect of a given intervention or treatment on survival time. A typical question may be: will a given therapy increase the chances of survival of an individual or population? Such causal inquiries on survival outcomes are common in the fields of epidemiology and medicine (Robins, 1986; Hammer et al., 1996; Yusuf et al., 2016). As an important current example, the COVID-19 pandemic is creating a demand for methodological development to address such questions, specifically, when evaluating the effectiveness of a potential vaccine or therapeutic outside randomized controlled trial settings. ",
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+        "text": "Traditional causal survival analysis is typically carried out in the context of a randomized controlled trial (RCT), where the treatment assignment is controlled by researchers. Though they are the gold standard for causal inference, RCTs are usually long-term engagements, expensive and limited in sample size. Alternatively, the availability of observational data with comprehensive information about patients, such as electronic health records (EHRs), constitutes a more accessible but also more challenging source for estimating causal effects (Häyrinen et al., 2008; Jha et al., 2009). Such observational data may be used to augment and verify an RCT, after a particular treatment is approved and in use (Gombar et al., 2019; Frankovich et al., 2011; Longhurst et al., 2014). Moreover, the wealth of information from observational data also allows for the estimation of the individualized treatment effect (ITE), namely, the causal effect of an intervention at the individual level. In this work, we develop a novel framework for counterfactual time-to-event prediction to estimate the ITE for survival or time-to-event outcomes from observational data. ",
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+        "text": "Estimating the causal effect for survival outcomes in observational data manifests two principal challenges. First, the treatment assignment mechanism is not known a priori. Therefore, there may be variables, known as confounders, affecting both the treatment and survival time, which lead to selection bias (Bareinboim & Pearl, 2012), i.e., that the distributions across treatment groups are not the same. In this work, we focus on selection biases due to confounding, but other sources may also be considered. For instance, patients who are severely ill are likely to receive more aggressive therapy, however, their health status may also inevitably influence survival. Traditional survival analysis neglects such bias, leading to incorrect causal estimation. Second, the exact time-to-event is not always observed, i.e., sometimes we only know that an event has not occurred up to a certain point in time. This is known as the censoring problem. Moreover, censoring might be informative depending on the characteristics of the individuals and their treatment assignments, thus proper adjustment is required for accurate causal estimation (Cole & Hernán, 2004; Díaz, 2019). ",
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+        "text": "Traditional causal survival-analysis approaches typically model the effect of the treatment or covariates (not time or survival) in a parametric manner. Two commonly used models are the Cox proportional hazards (CoxPH) model (Cox, 1972) and the accelerated failure time (AFT) model (Wei, 1992), which presume a linear relationship between the covariates and survival probability. Further, proper weighting for each individual has been employed to account for confounding bias from these models (Austin, 2007; 2014; Hernán et al., 2005). For instance, probability weighting schemes that account for both selection bias and covariate dependent censoring have been considered for adjusted survival curves (Cole & Hernán, 2004; Díaz, 2019). Moreover, such probability weighting schemes have been applied to causal survival-analysis under time-varying treatment and confounding (Robins, 1986; Hernán et al., 2000). See van der Laan & Robins (2003); Tsiatis (2007); Van der Laan & Rose (2011); Hernán & Robins (2020) for an overview. Such linear specification makes these models interpretable but compromises their flexibility, and makes it difficult to adapt them for high-dimensional data or to capture complex interactions among covariates. Importantly, these methods lack a counterfactual prediction mechanism, which is key for ITE estimation (see Section 2). ",
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+        "text": "Fortunately, recent advances in machine learning, such as representation learning or generative modeling, have enabled causal inference methods to handle high-dimensional data and to characterize complex interactions effectively. For instance, there has been recent interest in tree-based (Chipman et al., 2010; Wager & Athey, 2018) and neural-network-based (Shalit et al., 2017; Zhang et al., 2020) approaches. For pre-specified time-horizons, the nonparametric Random Survival Forest (RSF) (Ishwaran et al., 2008) and Bayesian Additive regression trees (BART) (Chipman et al., 2010) have been extended to causal survival analysis. RSF has been applied to causal survival forests with weighted bootstrap inference (Shen et al., 2018; Cui et al., 2020) while a BART is extended to account for survival outcomes in Surv-BART (Sparapani et al., 2016), and AFT-BART (Henderson et al., 2020). See Hu et al. (2020) for an extensive investigation of the causal survival tree-based methods. Alternatively, when estimating the ITE, neural-network-based methods propose to regularize the transformed covariates or representations for an individual to have balanced distributions across treatment groups, thus accounting for the confounding bias and improving ITE prediction. However, most approaches employing representation learning techniques for counterfactual inference deal with continuous or binary outcomes, instead of time-to-event outcomes with censoring (informative or non-informative). Hence, a principled generalization to the context of counterfactual survival analysis is needed. ",
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+        "type": "text",
+        "text": "In this work we leverage balanced (latent) representation learning to estimate ITE via counterfactual prediction of survival outcomes in observational studies. We develop a framework to predict event times from a low-dimensional transformation of the original covariate space. To address the specific challenges associated with counterfactual survival analysis, we make the following contributions: ",
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+        "text": "• We develop an optimization objective incorporating adjustments for informative censoring, as well as a balanced regularization term bounding the generalization error for ITE prediction. For the latter, we repurpose a recently proposed bound (Shalit et al., 2017) for our time-to-event scenario.   \n• We propose a generative model for event times to relax restrictive survival linear and parametric assumptions, thus allowing for more flexible modeling. Our approach can also provide nonparametric uncertainty quantification for ITE predictions.   \n• We provide survival-specific evaluation metrics, including a new nonparametric hazard ratio estimator, and discuss how to perform model selection for survival outcomes. The proposed model demonstrates superior performance relative to the commonly used baselines in real-world and semi-synthetic datasets.   \n• We introduce a survival-specific semi-synthetic dataset and demonstrate an approach for leveraging prior randomized experiments in longitudinal studies for model validation. ",
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+        "img_path": "images/482e9df6a7b5d031a444403a4be41d381d10c980808568d8cbf42f012f3882e6.jpg",
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+            "Figure 1: (a) Illustration of the proposed counterfactual survival analysis (CSA). Covariates $X =$ $x$ are mapped into latent representation $r$ via deterministic mapping $r \\ = \\ \\Phi ( x )$ . The potential outcomes are sampled from $\\bar { t } _ { a } \\sim p ( T _ { A } | X = x )$ for $A = a$ via stochastic mapping $h _ { A } ( r , \\tilde { \\epsilon } )$ , where randomness is induced with a flow-based transformation, $\\tilde { \\epsilon }$ , of a simple distribution $p ( \\epsilon )$ , i.e., uniform or Gaussian. (b) and (c) show the proposed causal graphs for non-informative and informative censoring, respectively. "
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+        "type": "text",
+        "text": "2 PROBLEM FORMULATION ",
+        "text_level": 1,
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+        "text": "We first introduce the basic setup for performing causal survival analysis in observational studies. Suppose we have $N$ units, with $N _ { 1 }$ units being treated and $N _ { 0 }$ in the control group $( N = N _ { 1 } + N _ { 0 } )$ . For each unit (individual), we have covariates $X$ , which can be heterogeneous, e.g., a mixture of categorical and continuous covariates which, in the context of medicine, may include labs, vitals, procedure codes, etc. We also have a treatment indicator $A$ , where $A = 0$ for the controls and $A = 1$ for the treated, as well as the outcome (event) of interest $T$ . Under the potential-outcomes framework (Rubin, 2005), let $T _ { 0 }$ and $T _ { 1 }$ be the potential event times for a given subject under control and treatment, respectively. In practice we only observe one realization of the potential outcomes, i.e., the factual outcome $T = T _ { A }$ , while the counterfactual outcome $T _ { 1 - A }$ is unobserved. ",
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+        "type": "text",
+        "text": "In survival analysis, the problem becomes more difficult because we do not always observe the exact event time for each individual, but rather the time up to which we are certain that the event has not occurred; specifically, we have a (right) censoring problem, most likely due to the loss of follow-up. We denote the censoring time as $C$ and censoring indicator as $\\delta \\in \\{ 0 , 1 \\}$ . The actual observed time is $Y = \\operatorname* { m i n } ( T _ { A } , C )$ , i.e., the outcome is observed (non-censored) if $T _ { A } < C$ and $\\delta = 1$ . ",
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+    {
+        "type": "text",
+        "text": "In this work, we are interested in the expected difference between the $T _ { 1 }$ and $T _ { 0 }$ conditioned on $X$ for a given unit (individual), which is commonly known as the individualized treatment effect (ITE). Specifically, we wish to perform inference on the conditional distributions of $T _ { 1 }$ and $T _ { 0 }$ , i.e., $p ( T _ { 1 } | X )$ and $p ( T _ { 0 } | X )$ , respectively, as shown in Figure 1a. In practice, we observe $N$ realizations of $( Y , \\delta , X , A )$ for observed time, censoring indicator, covariates and treatment indicator, respectively; hence, from an observational study the dataset takes the form $\\mathcal { D } = \\{ ( y _ { i } , \\delta _ { i } , x _ { i } , a _ { i } ) \\} _ { i = 1 } ^ { N }$ . Below, we discuss several common choices of estimands in survival analysis. ",
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+    {
+        "type": "text",
+        "text": "Estimands of Interest We begin by considering survival analysis in the absence of an intervening treatment choice, $A$ . Let $F ( t | x ) \\triangleq P ( T \\leq t | X = x )$ be the cumulative distribution function of the event (failure) time, $t$ , given a realization of the covariates, $x$ . Survival analysis is primarily concerned with characterization of the survival function conditioned on covariates $S ( t | x ) \\triangleq 1 - F ( t | x )$ , and the hazard function or risk score, $\\lambda ( t | x )$ , defined below. $S ( t | x )$ is a monotonically decreasing function indicating the probability of survival up to time $t$ . The hazard function measures the instantaneous probability of the event occurring between $\\{ t , t + \\Delta t \\}$ given $T > t$ and $\\Delta t  0$ . From standard definitions (Kleinbaum & Klein, 2010), the relationship between cumulative and hazard function is formulated as ",
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+        "img_path": "images/89816b6b4ab5614eea0ade34840cd19c6b9c6faa995393a64ef37e851560bce4.jpg",
+        "text": "$$\n\\lambda ( t | x ) = \\operatorname* { l i m } _ { d t  0 } { \\frac { P ( t < T < t + d t | X = x ) } { P ( T > t | X = x ) d t } } = - { \\frac { d \\log S ( t | x ) } { d t } } = { \\frac { f ( t | x ) } { S ( t | x ) } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "From (1) we see that $f ( t | x ) \\triangleq P ( T = t | X = x ) = \\lambda ( t | x ) S ( t | x )$ , is the conditional event time density function (Kleinbaum $\\&$ Klein, 2010). ",
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+        "type": "text",
+        "text": "Given the binary treatment $A$ , we are interested in its impact on the survival time. For ITE estimation, we are also interested in the difference between the two potential outcomes $T _ { 1 } , T _ { 0 }$ . Let $S _ { A } ( t | x )$ and $\\lambda _ { A } ( t | x )$ denote the survival and hazard functions for the potential outcomes $T _ { A }$ , i.e., $T _ { 1 }$ and $T _ { 0 }$ . Several common estimands of interest include (Zhao et al., 2012; Trinquart et al., 2016): difference in expected lifetime: $\\begin{array} { r } { \\mathrm { { I T E } } ( t , x ) = \\int _ { 0 } ^ { t _ { \\operatorname* { m a x } } } \\{ S _ { 1 } ( t | x ) - S _ { 0 } ( t | x ) \\} \\mathrm { d } t = \\mathbb { E } \\{ T _ { 1 } - T _ { 0 } | X = x \\} . } \\end{array}$ , difference in survival function: $\\mathrm { I T E } ( t , x ) = S _ { 1 } ( t | x ) - S _ { 0 } ( t | x )$ , and hazard ratio: $\\mathrm { I T E } ( t , x ) = \\lambda _ { 1 } ( t | x ) / \\lambda _ { 0 } ( t | x )$ . The inference difficulties associated with the above estimands from observational data are two-fold. First, there are confounders affecting both the treatment assignment and outcomes, which stem from selection bias, i.e., the treatment and control covariate distributions are not necessarily the same. Also, we do not have direct knowledge of the conditional treatment assignment mechanism, i.e., $P ( A = a | X = x )$ , also known as the propensity score. Let ⊥⊥ denote statistical independence. For estimands to be identifiable from observational data, we make two assumptions: (i) $\\{ T _ { 1 } , T _ { 0 } \\} \\perp \\perp A | X$ , i.e., no unobserved confounders or ignorability, and $( i i )$ overlap in the covariate support $0 < P ( A =$ $1 | X = x ) < 1$ almost surely if $p ( X = x ) > 0$ . Second, the censoring mechanism is also unknown and may lead to bias without proper adjustment. We consider two censoring mechanisms in our work, $( i )$ conditionally independent or informative censoring: $T \\perp \\perp C | X , A$ , and $( i i )$ random or non-informative censoring: $T \\perp \\perp C$ . Note that for informative censoring, we also have to consider potential censoring times $C _ { 1 }$ and $C _ { 0 }$ and their conditionals $p ( C _ { 1 } | X )$ and $p ( C _ { 0 } | X )$ , respectively. Figure 1 shows causal graphs illustrating these modeling assumptions. ",
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+        "text": "3 MODELING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To overcome the above challenges and adjust for observational biases, we propose a unified framework for counterfactual survival analysis (CSA). Specifically, we repurpose the counterfactual bound in Shalit et al. (2017) for our time-to-event scenario and introduce a nonparametric approach for stochastic survival outcome predictions. Below we formulate a theoretically grounded and unified approach for estimating $( i )$ the encoder function $r = \\Phi ( x )$ , which deterministically maps covariates $x$ to their corresponding latent representation $r \\in  { \\mathbb { R } } ^ { d }$ , and $( i i )$ two stochastic time-to-event generative functions, $h _ { A } ( \\cdot )$ , to implicitly draw samples from both potential outcome conditionals $\\bar { t } _ { a } \\sim p _ { h , \\Phi } ( T _ { A } | X = x )$ , for $A = \\{ 1 , 0 \\}$ , and where $t _ { a }$ indicates the sample from $p _ { h , \\Phi } ( T _ { A } | X = x )$ is for $A = a$ . Further, we formulate a general extension that accounts for informative censoring by introducing two stochastic censoring generative functions, $\\nu _ { A } ( \\cdot )$ , to draw samples for potential censoring times $c _ { a } \\sim p _ { \\nu , \\Phi } ( C _ { A } | X = x )$ . The model-specifying functions, $\\{ h _ { A } ( \\cdot ) , \\nu _ { A } ( \\cdot ) , \\Phi ( \\cdot ) \\}$ , are parameterized via neural networks. See the Supplementary Material (SM) for details. Figure 1a summarizes our modeling approach. ",
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+        "type": "text",
+        "text": "Accounting for selection bias We wish to estimate the potential outcomes, i.e., event times, which are sampled by distributions parameterized by functions $\\{ h _ { A } ( \\cdot ) , \\Phi ( \\cdot ) \\}$ , i.e., ",
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+        "text": "$$\n\\begin{array} { c } { { t \\sim p _ { h , \\Phi } ( T | X = x , A = a ) } } \\\\ { { { \\it t } _ { a } \\sim p _ { h , \\Phi } ( T _ { a } | X = x ) } } \\end{array}\n$$",
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+        "type": "text",
+        "text": "We obtain (3) from (2) via the strong ignorability assumption, i.e., $\\{ T _ { 0 } , T _ { 1 } \\}$ ⊥⊥ $A | X$ (consistent with the causal graphs in Figure 1b and 1c) and $0 < P ( A = \\mathsf { \\bar { a } } | X \\doteq x ) < 1 .$ , and the consistency assumption, i.e., $T = T _ { A } | A = a$ . A similar argument can be made for informative censoring based on Figure 1c, so we can also write $c _ { a } \\sim p _ { \\nu , \\Phi } ( C _ { A } | X = x )$ . Given (3), model functions $\\{ h _ { A } ( \\cdot ) , \\Phi ( \\cdot ) \\bar  \\}$ and $\\nu _ { A } ( \\cdot )$ for informative censoring can be learned by leveraging standard statistical optimization approaches, that minimize a loss hypothesis $\\mathcal { L }$ given samples from the empirical distribution $( y , \\delta , x , a ) \\sim p ( Y , \\delta , X , A )$ , i.e., from dataset $\\mathcal { D }$ . Specifically, we write $\\mathcal { L } = \\mathbb { E } _ { ( y , \\delta , x , a ) \\sim p ( Y , \\delta , X , A ) } \\left[ \\ell _ { h , \\Phi } ( t _ { a } , y , \\delta ) \\right]$ , where $\\ell _ { h , \\Phi } ( t _ { a } , y , \\delta )$ is a loss function that measures the agreement of $t _ { a } \\sim p _ { h , \\Phi } ( T _ { A } | X = x )$ (and $c _ { a } \\sim p _ { \\nu , \\Phi } ( C _ { A } | X = x )$ for informative censoring) with ground truth $\\{ y , \\delta \\}$ , the observed time and censoring indicator, respectively. ",
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+        "type": "text",
+        "text": "For some parametric formulations of event time distribution $p _ { h , \\Phi } ( T _ { A } | X = x )$ , e.g., exponential, Weibull, log-Normal, etc., and provided the censoring mechanism is non-informative, $- \\ell _ { h , \\Phi } ( t _ { a } , y , \\delta )$ is the closed form log likelihood. Specifically, $- \\ell _ { h , \\Phi } ( t _ { a } , y , \\delta ) \\triangleq \\log p _ { h , \\Phi } ( T _ { a } | X = x ) = \\delta$ · $\\log f _ { h , \\Phi } ( t _ { a } | x ) + ( 1 - \\bar { \\delta } ) \\cdot \\log S _ { h , \\Phi } ( t _ { a } | x )$ , which implies that the conditional event time density and survival functions can be calculated in closed form from transformations $\\{ h _ { A } ( \\cdot ) , \\Phi ( \\cdot ) \\}$ of $x$ . See the SM for parametric examples of $\\mathcal { L }$ accounting for informative censoring. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We further define the expected loss for a given realization of covariates $x$ and treatment assignment $a$ over observed times $y$ (censored and non-censored), and the censoring indicator $\\delta$ as $\\zeta _ { h , \\Phi } ( x , a ) \\triangleq$ $\\mathbb { E } _ { ( y , \\delta , x ) \\sim p ( Y , \\delta | X ) } \\ell _ { h , \\Phi } ( t _ { a } , y , \\delta )$ as in Shalit et al. (2017). For a given subject with covariates $x$ and treatment assignment $a$ , we wish to minimize both the factual and counterfactual losses, $\\mathcal { L } _ { \\mathrm { F } }$ and $\\mathcal { L } _ { \\mathrm { C F } }$ respectively, by decomposing $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { F } } + \\mathcal { L } _ { \\mathrm { C F } }$ as follows ",
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+    {
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+        "img_path": "images/141da49929ef287744dea443052c2291ced1fb2cbc977e821917e72659db41f1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { F } } = \\mathbb { E } _ { ( x , a ) \\sim p ( A , X ) } \\zeta _ { h , \\Phi } ( x , a ) , \\quad \\mathcal { L } _ { \\mathrm { C F } } = \\mathbb { E } _ { ( x , a ) \\sim p ( 1 - A , X ) } \\zeta _ { h , \\Phi } ( x , a ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Let $u \\triangleq P ( A = 1 )$ denote the marginal probability of treatment assignment. We can readily decompose the losses in (4) according to treatment assignments. The decomposed factual $\\mathcal { L } _ { \\mathrm { F } } =$ $u \\cdot \\mathcal { L } _ { \\mathrm { F } } ^ { A = 1 } + \\left( 1 - u \\right) \\cdot \\mathcal { L } _ { \\mathrm { F } } ^ { A = 0 }$ , and similarly, the decomposed counterfactual $\\mathcal { L } _ { \\mathrm { C F } } = \\dot { ( } 1 - u ) \\cdot \\mathcal { L } _ { \\mathrm { C F } } ^ { A = 1 } + u$ $\\mathcal { L } _ { \\mathrm { C F } } ^ { A = 0 }$ F CF . In practice, only factual outcomes are observed, hence, for a non-randomized non-controlled experiment, we cannot obtain an unbiased estimate of $\\mathcal { L } _ { \\mathrm { C F } }$ from data due to selection bias (or confounding). Therefore, we bound $\\mathcal { L } _ { \\mathrm { C F } }$ and $\\mathcal { L }$ below following Shalit et al. (2017). ",
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+        "type": "text",
+        "text": "Corollary 1 Assume $\\Phi ( \\cdot )$ is an invertible map, and $\\alpha ^ { - 1 } \\zeta _ { h , \\Phi } ( x , a ) \\in G$ , where $G$ is a family of functions, $p _ { \\Phi } ^ { A = a } \\triangleq p _ { \\Phi } ( R | A = a )$ is the latent distribution for group $A = a$ , and $\\alpha > 0$ is a constant. Then, we have: ",
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+        "img_path": "images/370f28aa2b443822fd375b3828179499d1816978110177146821204aef6fa36a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { C F } } \\leq ( 1 - u ) \\cdot \\mathcal { L } _ { \\mathrm { F } } ^ { A = 1 } + u \\cdot \\mathcal { L } _ { \\mathrm { F } } ^ { A = 0 } + \\alpha \\cdot \\mathrm { I P M } _ { G } ( p _ { \\Phi } ^ { A = 1 } , p _ { \\Phi } ^ { A = 0 } ) } \\\\ & { \\quad \\mathcal { L } \\leq \\mathcal { L } _ { \\mathrm { F } } ^ { A = 1 } + \\mathcal { L } _ { \\mathrm { F } } ^ { A = 0 } + \\alpha \\cdot \\mathrm { I P M } _ { G } ( p _ { \\Phi } ^ { A = 1 } , p _ { \\Phi } ^ { A = 0 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The integral probability metric (IPM) (Müller, 1997; Sriperumbudur et al., 2012) measures the distance between two probability distributions $p$ and $q$ defined over $M$ , i.e., the latent space of $R$ . Formally, $\\begin{array} { r } { \\mathrm { I P M } _ { G } ( p , q ) \\triangleq \\operatorname* { s u p } _ { g \\in G } \\vert \\int _ { M } g ( m ) \\left( p ( m ) - q ( m ) \\right) d m \\vert } \\end{array}$ , where $g : m  \\mathbb { R }$ , represents a class of real-valued bounded measurable functions on $M$ (Shalit et al., 2017). Therefore, model functions $\\{ h _ { a } ( \\cdot ) , \\Phi ( \\cdot ) \\}$ can be learned by minimizing the upper bound in (5) consisting of $( i )$ only factual losses under both treatment assignments and $( i i )$ an IPM regularizer enforcing latent distributional equivalence between the treatment groups. Note that if the data originates from a RCT it follows (by construction) that $\\mathrm { I P M } _ { G } ( p _ { \\Phi } ^ { A = 1 } , p _ { \\Phi } ^ { \\tilde { A = 0 } } ) \\stackrel { . } { = } 0$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Accounting for censoring bias Below we formulate an approach for estimating functions $h _ { A } ( \\cdot )$ and $\\nu _ { A } ( \\cdot )$ for synthesizing (sampling) non-censored $t _ { a } \\sim \\stackrel { \\sim } { p } _ { h , \\Phi } ( T _ { A } | X = x )$ and censored $c _ { a } \\sim$ $p _ { \\nu , \\Phi } ( C _ { A } | X = x )$ times, respectively. While some parametric assumptions for $p _ { h , \\Phi } ( T _ { A } | X = x )$ yield easy-to-evaluate closed forms for $S _ { h , \\Phi } ( t _ { a } | x )$ that can be used as likelihood for censored observations, they are restrictive, and have been shown to generate unrealistic high variance samples (Chapfuwa et al., 2018). So motivated, we seek a nonparametric likelihood-based approach that can model a flexible family of distributions, with an easy-to-sample approach for event times $t _ { a } \\sim$ $p _ { h , \\Phi } ( T _ { a } | X = x )$ . We model the event time generation process with a source of randomness, $p ( \\epsilon )$ , e.g. Gaussian or uniform, which is obtained from a neural-network-based nonlinear transformation. In the experiments we use a planar flow formulation parameterized by $\\{ U _ { h } , W _ { h } , b _ { h } \\}$ (Rezende & Mohamed, 2015), however, other specifications can also be used. Note that Miscouridou et al. (2018) has previously leveraged normalizing flows for survival analysis, however, our approach is very different in that it focuses on formulating $\\romannumeral 1$ ) a counterfactual survival analysis framework that accounts for informative or non-informative censoring mechanisms and confounding, and $_ { i i }$ ) model event times as a continuous variable instead of discretizing them. Specifically, we transform the source of randomness, $\\epsilon$ , using a single layer specification as follows ",
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+        "type": "equation",
+        "img_path": "images/5d0fe636d034b8d80f53f9ef96d49f1ce924a440335a5cca19c9ccfdb29ad614.jpg",
+        "text": "$$\n{ \\widetilde { \\epsilon } } _ { h } = \\epsilon + U _ { h } \\operatorname { t a n h } ( W _ { h } \\epsilon + b _ { h } ) , \\quad \\epsilon \\sim \\operatorname { U n i f o r m } ( 0 , 1 ) , t _ { a } = h _ { A } ( r , { \\widetilde { \\epsilon } } _ { h } ) , \\quad r = \\Phi ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\{ U _ { h } , W _ { h } \\} \\in \\mathbb { R } ^ { d \\times d }$ , $\\{ b _ { h } , \\epsilon \\} \\in \\mathbb { R } ^ { d }$ , $d$ is the dimensionality of the normalizing flow; each component of $\\epsilon$ is drawn independently from Uniform $( 0 , 1 )$ , and $\\tilde { \\epsilon } _ { h }$ may be viewed as a skip connection with stochasticity in $\\epsilon$ . Further, $h _ { A } ( r , \\tilde { \\epsilon } _ { h } )$ and $\\Phi ( x )$ are time-to-event generative and encoding functions, respectively, parameterized as neural networks. For simplicity, the dimensions of $r$ and $\\epsilon$ are set to $d$ , however, they can be set independently if desired. In practice, we are interested in generating realistic event-time samples; therefore, we account for both censored and non-censored observations by adopting the objective from Chapfuwa et al. (2018), formulated as ",
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+    {
+        "type": "equation",
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+        "text": "$$\n\\mathcal { L } _ { \\mathrm { F } } ^ { \\mathrm { C S A } } \\triangleq \\mathbb { E } _ { ( y , \\delta , x , a ) \\sim p ( Y , \\delta , X , A ) , \\epsilon \\sim p ( \\epsilon ) } \\left[ \\delta \\cdot \\left( \\left| y - t _ { a } \\right| \\right) + \\left( 1 - \\delta \\right) \\cdot \\left( \\operatorname* { m a x } ( 0 , y - t _ { a } ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where the first term encourages sampled event times $t _ { a }$ to be close to $y$ , the ground truth for observed events, i.e., $\\delta = 1$ , while penalizing $t _ { a }$ for being smaller than the censoring time when $\\delta = 0$ . Further, the expectation is taken over samples (a minibatch) from empirical distribution $p ( Y , \\delta , X , A )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Informative censoring We model informative censoring similar to (7) but mirroring the censoring indicators to encourage accurate censoring time samples $c _ { a }$ for $\\delta = 0$ , while penalizing $c _ { a }$ for being smaller than $y$ for $\\delta \\ : = \\ : 1$ (observed events). Specifically, we set an independent source of randomness like in (6) but parameterized by $\\{ U _ { \\nu } , W _ { \\nu } , b _ { \\nu } \\}$ and censoring generative functions $\\nu _ { A } ( r , \\tilde { \\epsilon } _ { \\nu } )$ , parameterized as neural networks, where $c _ { a } \\sim p _ { \\nu , \\Phi } ( C _ { A } | X = x )$ formulated as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/30f94ae655fc0f98b5a994dfdb2c983552b9b03f08d4eae0320432c33861540e.jpg",
+        "text": "$$\n\\ell _ { c } ( \\nu , \\Phi ) = \\mathbb { E } _ { ( y , \\delta , x , a ) \\sim p ( y , \\delta , X , A ) , \\epsilon \\sim p ( \\epsilon ) } \\left[ ( 1 - \\delta ) \\cdot ( | y - c _ { a } | ) + \\delta \\cdot ( \\operatorname* { m a x } ( 0 , y - c _ { a } ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Further, we introduce an additional time-order-consistency loss that enforces the correct order of the observed time relative to the censoring indicator, i.e., $c _ { a } < t _ { a }$ if $\\delta = 0$ and $t \\sb a < c \\sb a$ if $\\delta = 1$ , thus ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a8c95490dd905c65beac4f71a501b50581c5ec62609efa122fed7c912a3ba02.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { T C } } ( h , \\nu , \\Phi ) = \\mathbb { E } _ { ( \\delta , x , a ) \\sim p ( \\delta , X , A ) , \\epsilon \\sim p ( \\epsilon ) } \\left[ \\delta \\cdot ( \\operatorname* { m a x } ( 0 , t _ { a } - c _ { a } ) ) + ( 1 - \\delta ) \\cdot ( \\operatorname* { m a x } ( 0 , c _ { a } - t _ { a } ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that $\\ell _ { \\mathrm { T C } } ( h , \\nu , \\Phi )$ does not depend on the observed event times but only on the censoring indicators. Finally, we write the consolidated CSA loss for informative censoring (CSA-INFO) by aggregating (7), (8) and (9) as $\\mathcal { L } _ { \\mathrm { F } } ^ { \\mathrm { C S A - I N F O } } \\triangleq \\mathcal { L } _ { \\mathrm { F } } ^ { \\mathrm { C S A } } + \\ell _ { \\mathrm { c } } + \\ell _ { \\mathrm { T C } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Learning Model functions $\\{ h _ { A } ( \\cdot ) , \\Phi ( \\cdot ) , \\nu _ { A } ( \\cdot ) \\}$ are learned by minimizing the bound (5), via stochastic gradient descent on minibatches from $\\mathcal { D }$ , with $\\mathcal { L } _ { \\mathrm { F } } ^ { \\mathrm { C S A } }$ for non-informative censoring and $\\mathcal { L } _ { \\mathrm { F } } ^ { \\mathrm { C S A - I N F O } }$ for informative censoring. Further, for the IPM regularization loss in (5), we optimize the dual formulation of the Wasserstein distance, via the regularized optimal transport (Villani, 2008; Cuturi, 2013). Consequently, we only require $\\alpha ^ { - 1 } \\zeta _ { h , \\Phi } ( \\bar { x , } a )$ to be 1-Lipschitz (Shalit et al., 2017) and $\\alpha$ is selected by grid search on the validation set using only factual data (details below). ",
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+    {
+        "type": "text",
+        "text": "4 METRICS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We propose a comprehensive evaluation approach that accounts for both factual and causal metrics. Factual survival outcome predictions are evaluated according to standard survival metrics that measure diverse performance characteristics, such as concordance index (C-Index) (Harrell Jr et al., 1984), mean coefficient of variation (COV) and calibration slope (C-slope) (Chapfuwa et al., 2020). See the SM for more details on these metrics. For causal metrics, defined below, we introduce a nonparametric hazard ratio (HR) between treatment outcomes, and adopt the conventional precision in estimation of heterogeneous effect (PEHE) and average treatment effect (ATE) performance metrics (Hill, 2011). Note that PEHE and ATE require ground truth counterfactual event times, which is only possible in (semi-)synthetic data. For HR, we compare our findings with those independently reported in the literature from gold-standard RCT data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Nonparametric Hazard Ratio In a medical setting, the population hazard ratio $\\mathrm { H R } ( t )$ between treatment groups is considered informative thus has been widely used in drug development and RCT (Yusuf et al., 2016; Mihaylova et al., 2012). For example, $\\mathrm { H R } ( t ) < 1 , > 1$ , or $\\approx 1$ indicate population positive, negative and neutral treatment effects at time $t$ , respectively. Moreover, $\\mathrm { H R } ( t )$ naturally accounts for both censored and non-censored outcomes. Standard approaches for computing $\\mathrm { H R } ( { \\dot { t } } )$ rely on the restrictive proportional hazard assumption from CoxPH (Cox, 1972), which is constituted as a semi-parametric linear model $\\lambda ( t | a ) = \\lambda _ { \\mathrm { b } } \\bar { ( } t ) \\exp ( a \\beta )$ . However, the constant covariate (time independent) effect is often violated in practice (see Figure 2b). For CoxPH, the marginal HR between treatment and control can be obtained from regression coefficient $\\beta$ learned via maximum likelihood without the need for specifying the baseline hazard $\\begin{array} { r } { \\mathrm {  ~ \\cdot ~ } _ { \\mathrm { b } } ( t ) \\colon \\mathrm { H R } _ { \\mathrm { C o x P H } } ( t ) = \\frac { \\lambda ( t | a = 1 ) } { \\lambda ( t | a = 0 ) } = \\exp ( \\beta ) } \\end{array}$ . So motivated, we propose a nonparametric, model-free approach for computing $\\mathrm { H R } ( t )$ , in which we do not assume a parametric form for the event time distribution or the proportional hazard assumption from CoxPH. This approach only relies on samples from the conditional event time density functions, $f ( t _ { 1 } | x )$ and $f ( t _ { 0 } | x )$ , via $t _ { a } = h _ { A } ( \\cdot )$ from (6). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 1 We define the nonparametric marginal Hazard Ratio and its approximation, $\\mathrm { \\hat { H R } } ( t )$ , as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d9ca3513b198afb6432afe6930b65739952ee5cac99ad41dbc5cf652e4ac4216.jpg",
+        "text": "$$\n\\mathrm { H R } ( t ) = \\frac { \\lambda _ { 1 } ( t ) } { \\lambda _ { 0 } ( t ) } = \\frac { S _ { 0 } ( t ) } { S _ { 1 } ( t ) } \\cdot \\frac { S _ { 1 } ^ { \\prime } ( t ) } { S _ { 0 } ^ { \\prime } ( t ) } , \\quad \\mathrm { H R } ( t ) = \\frac { \\hat { S } _ { 0 } ^ { \\mathrm { P K M } } ( t ) } { \\hat { S } _ { 1 } ^ { \\mathrm { P K M } } ( t ) } \\cdot \\frac { m _ { 1 } ( t ) } { m _ { 0 } ( t ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where for $\\mathrm { H R } ( t )$ we leveraged (1) to obtain (10) and $S ^ { \\prime } ( t ) \\triangleq d S ( t ) / d t$ . The nonparametric assumption for $S ( t )$ makes the computation of $S ^ { \\prime } ( t )$ challenging. Provided that $S ( t )$ is a monotonically decreasing function, for simplicity, we fit a linear function $S ( t ) = m \\cdot t + c .$ , and set $S ^ { \\prime } ( t ) \\approx m$ . Note that the linear model is only used for estimating $S ^ { \\prime } ( t )$ from the nonparametric estimation of $S ( t )$ . ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/5d3d824a74b40f44971099aa53bc084369e33021afa3cae740ca983e4448a9de.jpg",
+        "table_caption": [
+            "Table 1: Performance comparisons on ACTG-SYNTHETIC data, with $9 5 \\%$ $\\mathrm { H R } ( t )$ confidence interval. The ground truth, test set, hazard ratio is $\\mathrm { H R ( t ) } = 0 . 5 2 _ { ( 0 . 3 9 , 0 . 7 1 ) }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Causal</td><td colspan=\"2\">Factual</td><td rowspan=\"2\">C-Slope (A=0,A=1)</td></tr><tr><td>EPEHE</td><td>EATE</td><td>HR(t)</td><td>C-Index (A=0,A=1)</td><td>Mean COV</td></tr><tr><td>CoxPH-Uniform</td><td>NA</td><td>NA</td><td>0.97(0.86,1.09)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-IPW</td><td>NA</td><td>NA</td><td>0.48(0.03,7.21)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-OW</td><td>NA</td><td>NA</td><td>0.60(0.53,0.68)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>Surv-BART</td><td>352.07</td><td>77.89</td><td>0.0(0.0,0.0)</td><td>(0.706,0.686)</td><td>0.001</td><td>(0.398,0)</td></tr><tr><td>AFT-Weibull</td><td>367.92</td><td>133.93</td><td>0.47(0.47,0.47)</td><td>(0.21,0.267)</td><td>6.209</td><td>(0.707,0.729)</td></tr><tr><td>AFT-log-Normal</td><td>377.76</td><td>157.64</td><td>0.47(0.47,0.47)</td><td>(0.675, 0.556)</td><td>6.971</td><td>(0.707,0.729)</td></tr><tr><td>SR</td><td>369.47</td><td>88.55</td><td>0.38(0.330.65)</td><td>(0.791,0.744)</td><td>0</td><td>(0.985,1.027)</td></tr><tr><td>CSA (proposed)</td><td>358.72</td><td>0.8</td><td>0.45(0.39,0.65)</td><td>(0.787,0.767)</td><td>0.131</td><td>(0.985,1.026)</td></tr><tr><td>CSA-INFO (proposed)</td><td>344.3</td><td>31.19</td><td>0.53(0.41,0.67)</td><td>(0.78,0.764)</td><td>0.13</td><td>(0.999,1.029)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Bias from $S ^ { \\prime } ( t )$ can be reduced by considering more complex function approximations for $S ( t )$ , e.g., polynomial or spline. For the nonparametric estimation of $S ( t )$ we leverage the model-free population point-estimate-based nonparametric Kaplan-Meier (Kaplan & Meier, 1958) estimator of the survival function $\\hat { S } ^ { \\mathrm { P K M } } ( t )$ in Chapfuwa et al. (2020) to marginalize both factual and counterfactual predictions given covariates $x$ . The approximated hazard ratio, $\\mathrm { \\hat { H R } } ( t )$ , is thus obtained by combining the approximations $\\hat { S } _ { a } ^ { \\mathrm { P K M } } ( t )$ and $m _ { a }$ . A similar formulation for the conditional, $\\mathrm { H } \\mathrm { \\hat { R } } ( t | x )$ , can also be derived. See the SM for full details on the evaluation or $\\mathrm { \\hat { H R } } ( t )$ and $\\mathrm { H } \\mathrm { \\hat { R } } ( t | x )$ . Note that for some AFT- or CoxPH-based parametric formulations, $\\mathrm { H R } ( t | x )$ , can be readily evaluated because $f ( t _ { a } | x )$ and $S ( t _ { a } | x )$ are available in closed form. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the experiments, we will use $\\mathrm { H R } ( t )$ to compare different approaches against results reported in RCTs (see Tables 1 and 3). Further, we will use $\\mathrm { H R } ( t | x )$ to illustrate stratified treatment effects (see Figure 2). Note that though a neural-based survival recommender system (Katzman et al., 2018) has been previously used to estimate $\\operatorname { H R } ( t | x )$ , their approach does not account for confounding or informative censoring thus it is susceptible to bias. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Precision in Estimation of Heterogeneous Effect (PEHE) A general individualized estimation error is formulated as $\\epsilon _ { \\mathrm { P E H E } } = \\sqrt { \\mathbb { E } _ { X } [ ( \\mathrm { I T E } ( x ) - \\mathrm { I } \\hat { \\mathrm { T E } } ( x ) ) ^ { 2 } ] }$ , where $\\operatorname { I T E } ( x )$ is the ground truth, $\\mathrm { I } \\hat { \\mathrm { T E } } ( x ) = \\mathbb { E } _ { T } \\left[ \\gamma \\left( T _ { 1 } \\right) - \\gamma \\left( T _ { 0 } \\right) \\vert X = x \\right]$ and $\\gamma ( \\cdot )$ is a deterministic transformation. In our experiments, $\\gamma ( \\cdot )$ is the average over samples from $\\dot { t _ { a } } \\sim p _ { h , \\Phi } ( T _ { A } | X = x )$ . Alternative estimands, e.g., thresholding survival times $\\gamma ( T _ { A } ) = I \\{ T _ { A } > \\tau \\}$ , can also be considered as described above. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Average Treatment Effect (ATE) The population treatment effect estimation error is defined as $\\epsilon _ { \\mathrm { A T E } } = | \\mathrm { A T E } - \\mathrm { A \\hat { T } E } |$ , where $\\mathrm { A T E } = \\mathbb { E } _ { X } [ \\mathrm { I T E } ( x ) ]$ (ground truth) and $\\mathrm { A } \\hat { \\mathrm { T E } } = \\mathbb { E } _ { X } [ \\hat { \\mathrm { I T E } } ( x ) ]$ . ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We describe the baselines and datasets that will be used to evaluate the proposed counterfactual survival analysis methods (CSA and CSA-INFO). Pytorch code including the new semi-synthetic dataset (see below) will be made publicly available. Throughout the experiments, we use the standard $\\mathrm { H R } ( t )$ for CoxPH based methods and (10) for all others. The bound in (5) is sensitive to $\\alpha$ , thus we propose approximating proxy counterfactual outcomes $\\{ Y _ { \\mathrm { C F } } , \\delta _ { \\mathrm { C F } } \\}$ for the validation set, according to the covariate Euclidean nearest-neighbour (NN) from the training set. We select the $\\alpha$ that minimizes the validation loss $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { F } } + \\mathcal { L } _ { \\mathrm { C F } }$ from the set $( 0 , 0 . 1 , 1 , 1 0 , 1 0 0 )$ . ",
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+    {
+        "type": "text",
+        "text": "Baselines We consider the following competitive baseline approaches: $( i )$ propensity weighted CoxPH (Schemper et al., 2009; Buchanan et al., 2014; Rosenbaum & Rubin, 1983); $( i i )$ IPM (5) regularized AFT (log-Normal and Weibull) models; $( i i i )$ an IPM (5) regularized deterministic semisupervised regression (SR) model with accuracy objective from (Chapfuwa et al., 2018), as a contrast for the proposed stochastic predictors (CSA and CSA-INFO); and $( i v )$ survival Bayesian additive regression trees (Surv-BART) (Sparapani et al., 2016). For CoxPH, we consider three normalized weigh2009), $\\begin{array} { r } { \\mathrm { I P W } _ { i } = \\frac { a _ { i } } { \\hat { e } _ { i } } + \\frac { 1 - a _ { i } } { 1 - \\hat { e } _ { i } } ; i i ) } \\end{array}$ $( i )$ weilappi ng (IPW) (Horvitz & Thompson, 1952; Cao et al.,weights (OW) (Crump et al., 2006; Li et al., 2018), $\\mathrm { O W } _ { i } = a _ { i } \\cdot \\left( 1 - \\hat { e } _ { i } \\right) + \\left( 1 - a _ { i } \\right) \\cdot \\hat { e } _ { i }$ $i i i$   \nlinear logistic model $\\hat { e } _ { i } = \\sigma ( x _ { i } ; w )$ , is used as an approximation, $\\boldsymbol { \\hat { e } } _ { i }$ , to the unknown propensity score $P ( A = { \\bar { 1 } } | X = x )$ . See the SM for a details of the baselines. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 3: Performance comparisons on FRAMINGHAM data, with $9 5 \\%$ $\\mathrm { H R } ( t )$ confidence interval. Test set NN assignment of $y _ { \\mathrm { C F } }$ and $\\delta _ { \\mathrm { C F } }$ yields biased $\\mathrm { H R ( t ) } = 1 . 2 3 _ { ( 1 . 1 7 , 1 . 2 5 ) }$ , while previous large scale longitudinal RCT studies estimated $\\mathrm { H R } ( \\mathrm { t } ) = 0 . 7 5$ (0.64,0.88) (Yusuf et al., 2016). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Causal HR(t)</td><td rowspan=\"2\">C-Index (A=0, A=1)</td><td rowspan=\"2\">Factual Mean COV</td><td rowspan=\"2\">C-Slope (A=0,A=1)</td></tr><tr><td></td></tr><tr><td>CoxPH-Uniform</td><td>1.69(1.38,2.07)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-IPW</td><td>1.09(0.76,1.57)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>CoxPH-OW</td><td>0.88(0.73,1.08)</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>Surv-BART</td><td>14.99(14.9,14.9e8)</td><td>(0.629,0.630)</td><td>0.003</td><td>(0.232,0.084)</td></tr><tr><td>AFT-Weibull</td><td>1.09(1.09,1.09)</td><td>(0.734,0.395)</td><td>8.609</td><td>(0.857,0.89)</td></tr><tr><td>AFT-log-Normal</td><td>1.55(1.46,1.55)</td><td>(0.68,0.56)</td><td>10.415</td><td>(0.979,0.732)</td></tr><tr><td>SR</td><td>0.58(0.53,0.71)</td><td>(0.601, 0.57)</td><td>0</td><td>(0.491,0.63)</td></tr><tr><td>CSA(proposed)</td><td>1.04(1.00,1.09)</td><td>(0.763,0.728)</td><td>0.161</td><td>(0.891,0.81)</td></tr><tr><td>CSA-INFO (proposed)</td><td>0.81(0.77,0.83)</td><td>(0.752,0.651)</td><td>0.156</td><td>(0.907, 0.881)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Datasets We consider the following datasets: (i) FRAMINGHAM, is an EHR-based longitudinal cardiovascular cohort study that we use to evaluate the effect of statins on future coronary heart disease outcomes (Benjamin et al., 1994); $( i i )$ ACTG, is a longitudinal RCT study comparing monotherapy with Zidovudine or Didanosine with combination therapy in HIV patients (Hammer et al., 1996); and $( i i i )$ ACTG",
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+    {
+        "type": "table",
+        "img_path": "images/662567b84ec6b8412fd1a2eb1f6e306f029c8bde26af3d28de506fa5a1d065e3.jpg",
+        "table_caption": [
+            "Table 2: Summary statistics of the datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>FRAMINGHAM</td><td>ACTG</td><td>ACTG-SYNTHETIC</td></tr><tr><td>Events (%)</td><td>26.0</td><td>26.9</td><td>48.9</td></tr><tr><td>Treatment (%)</td><td>10.4</td><td>49.5</td><td>55.9</td></tr><tr><td>N</td><td>3,435</td><td>1,054</td><td>2,139</td></tr><tr><td>p</td><td>32</td><td>23</td><td>23</td></tr><tr><td>Missing (%)</td><td>0.23</td><td>1.41</td><td>1.38</td></tr><tr><td>tmax (days)</td><td>7,279</td><td>1,231</td><td>1,313</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "SYNTHETIC, is a semi-synthetic dataset based on ACTG covariates. We simulate potential outcomes according to a Gompertz-Cox distribution (Bender et al., 2005) with selection bias from a simple logistic model for $\\bar { P ( A = 1 | X = x ) }$ and AFT-based censoring mechanism. The generative process is detailed in the SM. Table 2 summarizes the datasets according to $( i )$ covariates of size $p$ ; $( i i )$ proportion of non-censored events, treated units, and missing entries in the $N \\times p$ covariate matrix; and $( i i i )$ time range $t _ { \\mathrm { m a x } }$ for both censored and non-censored events. Missing entries are imputed with median or mode if continuous or categorical, respectively. ",
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+    {
+        "type": "text",
+        "text": "Quantitative Results Experimental results for two data-sets in Tables 1 and 3, illustrate that AFTbased methods are high variance, inferior in calibration and C-Index than accuracy-based methods (SR, CSA, CSA-INFO). Surv-BART is the least calibrated but low variance method. CSA-INFO and CSA outperform all methods across all factual metrics, whereas CSA-INFO is better calibrated, low variance but slightly lower C-Index than CSA. Note that we fit CoxPH using the entire dataset; since it does not support counterfactual inference, we do not present factual metrics. By properly adjusting for both informative censoring and selection bias, CSA-INFO significantly outperforms all methods in treatment effect estimation according to $\\mathrm { H R } ( t )$ and $\\epsilon _ { \\mathrm { P E H E } }$ , across non-RCT datasets, while remaining comparable to AFT-Weibull on the RCT dataset (see the SM). Further, RCT-based results on ACTG data in the SM illustrate comparable $\\mathrm { H R } ( t )$ across all models except for AFT-log-Normal and Surv-BART, which overestimate, and SR, which underestimates risk. For non-RCT datasets (ACTG-SYNTHENTIC and FRAMINGHAM), CoxPH-OW has a clear advantage over all CoxPH based methods, mostly credited to the well-behaved bounded propensity weights $\\in [ 0 , 1 ]$ . Interestingly, the FRAMINGHAM observational data exhibits a common paradox, where without proper adjustment of selection and censoring bias, naive approaches would result in a counter-intuitive treatment effect from statins. However, there is severe confounding from covariates such as age, BMI, diabetes, CAD, PAD, MI, stroke, etc., that influence both treatment likelihood and survival time. Table 3, demonstrates that CSA-INFO is clearly the best performing approach. Specifically, its $\\mathrm { H R } ( t )$ , reverses the biased observational treatment effect, to demonstrate positive treatment from statins, which is consistent with prior large RCT longitudinal findings (Yusuf et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Qualitative Results Figure 2a demonstrates that CSA-INFO matches the ground truth population hazard, $\\mathrm { H R } ( t )$ , better than alternative methods on ACTG-SYNTHETIC data. See the SM for ACTG and FRAMINGHAM. Figure 2b shows sub-population log hazard ratios for four patient clusters obtained via hierarchical clustering on the individual log hazard ratios, $\\log \\mathrm { H R } ( t | x )$ , of the test set of FRAMINGHAM data. Interestingly, these clusters stratify treatment effects into: positive (2), negative (1 and 3), and neutral (4) sub-populations. Moreover, the estimated density of median $\\log \\mathrm { H } \\bar { \\mathrm { R } } ( t | x )$ values in Figure $_ { 2 \\mathrm { c } }$ illustrates that nearly $7 0 \\%$ of the testing set individuals have $\\log \\mathrm { H R } ( t | x ) < 0$ , thus may benefit from taking statins. Further, we isolated the extreme top and bottom quantiles, $\\mathrm { H R } ( t | x ) < 0 . 0 2 4$ and $\\mathrm { H R } ( \\bar { t } | x ) > 1 . 9 1 6$ , respectively, of the median $\\log \\mathrm { \\bar { H R } } ( t | x )$ values for the test set of FRAMINGHAM, as shown in Figure 2c. After comparing their covariates, we found that individuals with the following characteristics may benefit from taking statins: young, male, diabetic, without prior history (CAD, PAD, stroke or MI), high BMI, cholesterol, triglycerides, fasting glucose, and low high-density lipoprotein. There seem to be consensus that diabetics and high-cholesterol patients benefit from statins (Cheung et al., 2004; Wilt et al., 2004). See SM for additional results. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4c89c16905f48a1bdcc35b1ade4f0f866e4e151f0c05e3f1b2a559591a62f777.jpg",
+        "image_caption": [
+            "Figure 2: (a) Inferred population $\\mathrm { H R } ( t )$ compared against ground truth (EMP) on ACTG-SYNTHETIC data. CSA-INFO-based (b) cluster-specific average l $\\operatorname { \\Pi } ^ { } \\operatorname { \\Pi } ^ { } \\operatorname { \\mathrm { ) g } } \\operatorname { \\mathrm { { H R } } } ( t | x )$ curves and (c) estimated density of median $\\log \\mathrm { H R } ( t | x )$ values on the test set of the FRAMINGHAM dataset. Clusters assignment were obtained via hierarchical clustering of individualized $\\log \\mathrm { H R } ( t | x )$ traces. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We have proposed a unified counterfactual inference framework for survival analysis. Our approach adjusts for bias from two unknown sources, namely, confounding due to covariate dependent selection bias and censoring (informative or non-informative). Relative to competitive alternatives, we demonstrate superior performance for both survival-outcome prediction and treatment-effect estimation, across three diverse datasets, including a semi-synthetic dataset which we introduce. Moreover, we formulate a model-free nonparametric hazard ratio metric for comparing treatment effects or leveraging prior randomized real-world experiments in longitudinal studies. ",
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parse/train/IhiU6AJYpDs/IhiU6AJYpDs.md

@@ -0,0 +1,415 @@
+# Online Robust Reinforcement Learning with Model Uncertainty
+
+Yue Wang   
+University at Buffalo   
+Buffalo, NY 14228   
+ywang294@buffalo.edu
+
+Shaofeng Zou University at Buffalo Buffalo, NY 14228 szou3@buffalo.edu
+
+# Abstract
+
+Robust reinforcement learning (RL) is to find a policy that optimizes the worstcase performance over an uncertainty set of MDPs. In this paper, we focus on model-free robust RL, where the uncertainty set is defined to be centering at a misspecified MDP that generates a single sample trajectory sequentially, and is assumed to be unknown. We develop a sample-based approach to estimate the unknown uncertainty set, and design robust Q-learning algorithm (tabular case) and robust TDC algorithm (function approximation setting), which can be implemented in an online and incremental fashion. For the robust Q-learning algorithm, we prove that it converges to the optimal robust Q function, and for the robust TDC algorithm, we prove that it converges asymptotically to some stationary points. Unlike the results in [Roy et al., 2017], our algorithms do not need any additional conditions on the discount factor to guarantee the convergence. We further characterize the finite-time error bounds of the two algorithms, and show that both the robust Qlearning and robust TDC algorithms converge as fast as their vanilla counterparts (within a constant factor). Our numerical experiments further demonstrate the robustness of our algorithms. Our approach can be readily extended to robustify many other algorithms, e.g., TD, SARSA, and other GTD algorithms.
+
+# 1 Introduction
+
+Existing studies on Markov decision process (MDP) and reinforcement learning (RL) [Sutton and Barto, 2018] mostly rely on the crucial assumption that the environment on which a learned policy will be deployed is the same one that was used to generate the policy, which is often violated in practice – e.g., the simulator may be different from the true environment, and the MDP may evolve over time. Due to such model deviation, the actual performance of the learned policy can significantly degrade. To address this problem, the framework of robust MDP was formulated in [Bagnell et al., 2001, Nilim and El Ghaoui, 2004, Iyengar, 2005], where the transition kernel of the MDP is not fixed and lies in an uncertainty set, and the goal is to learn a policy that performs well under the worst-case MDP in the uncertainty set. In [Bagnell et al., 2001, Nilim and El Ghaoui, 2004, Iyengar, 2005], it was assumed that the uncertainty set is known beforehand, i.e., model-based approach, and dynamic programming can be used to find the optimal robust policy.
+
+The model-based approach, however, requires a model of the uncertainty set known beforehand, and needs a large memory to store the model when the state and action spaces are large, which make it less applicable for many practical scenarios. This motivates the study in this paper, model-free robust RL with model uncertainty, which is to learn a robust policy using a single sample trajectory from a misspecified MDP, e.g., a simulator and a similar environment in which samples are easier to collect than in the target environment where the policy is going to be deployed. The major challenge lies in that the transition kernel of the misspecified MDP is not given beforehand, and thus, the uncertainty set and the optimal robust policy need to be learned simultaneously using sequentially observed data from the misspecified MDP. Moreover, robust RL learns the value function of the worst-case MDP in the uncertainty set which is different from the misspecified MDP that generates samples. This is similar to the off-policy learning, which we refer to as the "off-transition-kernel" setting. Therefore, the learning may be unstable and could diverge especially when function approximation is used [Baird, 1995].
+
+In this paper, we develop a model-free approach for robust RL with model uncertainty. Our major contributions in this paper are summarized as follows.
+
+• Motivated by empirical studies of adversarial training in RL [Huang et al., 2017, Kos and Song, 2017, Lin et al., 2017, Pattanaik et al., 2018, Mandlekar et al., 2017] and the $R$ -contamination model in robust detection (called $\epsilon$ -contamination model in [Huber, 1965]), we design the uncertainty set using the $R$ -contamination model (see (4) for the details). We then develop an approach to estimate the unknown uncertainty set using only the current sample, which does not incur any additional memory cost. Unlike the approach in [Roy et al., 2017], where the uncertainty set is relaxed to one not depending on the misspecified MDP that generates samples so that an online algorithm can be constructed, our approach does not need to relax the uncertainty set.
+
+• We develop a robust Q-learning algorithm for the tabular case, which can be implemented in an online and incremental fashion, and has the same memory cost as the vanilla Q-learning algorithm. We show that our robust Q-learning algorithm converges asymptotically, and further characterize its finite-time error bound. Unlike the results in [Roy et al., 2017] where a stringent condition on the discount factor (which is due to the relaxation of the uncertainty set, and prevents the use of a discount factor close to 1 in practice) is needed to guarantee the convergence, our algorithm converges without the need of such condition. Furthermore, our robust Q-learning algorithm converges as fast as the vanilla Q-learning algorithm [Li et al., 2020] (within a constant factor), while being robust to model uncertainty.
+
+• We generalize our approach to the case with function approximation (for large state/action space). We investigate the robust policy evaluation problem, i.e., evaluate a given policy under the worstcase MDP in the uncertainty set. As mentioned before, the robust RL problem is essentially "off-transition-kernel", and therefore non-robust methods with function approximation may diverge [Baird, 1995] (also see our experiments). We develop a novel extension of the gradient TD (GTD) method [Maei et al., 2010, Maei, 2011, Sutton et al., 2008] to robust RL. Our approach introduces a novel smoothed robust Bellman operator to construct the smoothed mean-squared projected robust Bellman error (MSPRBE). Using our uncertainty set design and online sample-based estimation, we develop a two time-scale robust TDC algorithm. We further characterize its convergence and finite-time error bound.
+
+• We conduct numerical experiments to validate the robustness of our approach. In our experiments, our robust Q-learning algorithm achieves a much higher reward than the vanilla Q-learning algorithm when being trained on a misspecified MDP; and our robust TDC algorithm converges much faster than the vanilla TDC algorithm, and the vanilla TDC algorithm may even diverge.
+
+# 1.1 Related Work
+
+Model-Based Robust MDP. The framework of robust MDP was investigated in [Iyengar, 2005, Nilim and El Ghaoui, 2004, Bagnell et al., 2001, Satia and Lave Jr, 1973, Wiesemann et al., 2013], where the transition kernel is assumed to be in some uncertainty set, and the problem can be solved by dynamic programming. This approach was further extended to the case with function approximation in [Tamar et al., 2014]. However, these studies are model-based, which assume beforehand knowledge of the uncertainty set. In this paper, we investigate the model-free setting, where the uncertainty set is a set of MDPs centered around some unknown Markov transition kernel from which a single sample trajectory can be sequentially observed.
+
+Adversarial Robust RL. It was shown in [Iyengar, 2005] that the robust MDP problem is equivalent to a zero-sum game between the agent and the nature. Motivated by this fact, the adversarial training approach, where an adversary perturbs the state transition, was studied in [Vinitsky et al., 2020, Pinto et al., 2017, Abdullah et al., 2019, Hou et al., 2020, Rajeswaran et al., 2017, Atkeson and Morimoto, 2003, Morimoto and Doya, 2005]. This method relies on a simulator, where the state transition can be modified in an arbitrary way. Another approach is to modify the current state through adversarial samples, which is more heuristic, e.g., [Huang et al., 2017, Kos and Song, 2017, Lin et al., 2017, Pattanaik et al., 2018, Mandlekar et al., 2017]. Despite the empirical success of these approaches, theoretical performance guarantees, e.g., convergence to the optimal robust policy and convergence rate, are yet to be established. The main difference lies in that during the training, our approach does not need to manipulate the state transition of the MDP. More importantly, we develop the asymptotic convergence to the optimal robust policy and further characterize the finite-time error bound. In [Lim et al., 2013], the scenario where some unknown parts of the state space can have arbitrary transitions while other parts are purely stochastic was studied. Adaptive algorithm to adversarial behavior was designed, and its regret bound is shown to be similar to the purely stochastic case. In [Zhang et al., 2020a], the robust adversarial RL problem for the special linear quadratic case was investigated.
+
+Model-free Robust RL. In [Roy et al., 2017, Badrinath and Kalathil, 2021] model-free RL with model uncertainty was studied, where in order to construct an algorithm that can be implemented in an online and incremental fashion, the uncertainty set was firstly relaxed by dropping the dependency on the misspecified MDP that generates the samples (centroid of the uncertainty set). Such a relaxation is pessimistic since the relaxed uncertainty set is not centered at the misspecified MDP anymore (which is usually similar to the target MDP), making the robustness to the relaxed uncertainty set not well-justified. Such a relaxation will further incur a stringent condition on the discounted factor to guarantee the convergence, which prevents the use of a discount factor close to 1 in practice. Moreover, only asymptotic convergence was established in [Roy et al., 2017]. In this paper, we do not relax the uncertainty set, and instead propose an online approach to estimate it. Our algorithms converge without the need of the condition on the discount factor. We also provides finite-time error bounds for our algorithms. The multi-agent RL robust to reward uncertainty was investigated in [Zhang et al., 2020b], where the reward uncertainty set is known, but the transition kernel is fixed.
+
+Finite-time Error Bound for RL Algorithms. For the tubular case, Q-learning has been studied intensively, e.g., in [Even-Dar et al., 2003, Beck and Srikant, 2012, Qu and Wierman, 2020, Li et al., 2020, Wainwright, 2019, Li et al., 2021]. TD with function approximation were studied in [Dalal, Gal and Szörényi, Balázs and Thoppe, Gugan and Mannor, Shie, 2018, Bhandari et al., 2018, Srikant and Ying, 2019, Cai et al., 2019, Sun et al., 2020]. Q-learning and SARSA with linear function approximation were investigated in [Zou et al., 2019, Chen et al., 2019]. The finite-time error bounds for the gradient TD algorithms [Maei et al., 2010, Sutton et al., 2009, Maei et al., 2010] were further developed recently in [Dalal et al., 2018, Liu et al., 2015, Gupta et al., 2019, Xu et al., 2019, Dalal et al., 2020, Kaledin et al., 2020, Ma et al., 2020, Wang and Zou, 2020, Ma et al., 2021, Doan, 2021]. There are also finite-time error bounds on the policy gradient methods and actor critic methods, e.g., [Wang et al., 2020, Yang et al., 2019, Kumar et al., 2019, Qiu et al., 2019, Wu et al., 2020, Cen et al., 2020, Bhandari and Russo, 2019, Agarwal et al., 2021, Mei et al., 2020]. We note that these studies are for the non-robust RL algorithms, and in this paper, we design robust RL algorithms, and characterize their finite-time error bounds.
+
+# 2 Preliminaries
+
+Markov Decision Process. An MDP can be characterized by a tuple $( \mathcal { S } , \mathcal { A } , \mathsf { P } , c , \gamma )$ , where S and $\mathcal { A }$ are the state and action spaces, $\mathsf { P } = \{ p _ { s } ^ { a } \in \varDelta _ { | \mathcal { S } | } , a \in \mathcal { A } , s \in \mathcal { S } \}$ is the transition kernel1, $c$ is the cost function, and $\gamma \in [ 0 , 1 )$ is the discount factor. Specifically, $p _ { s } ^ { a }$ denotes the distribution of the next state if taking action $a$ at state $s$ . Let $p _ { s } ^ { a } = \{ p _ { s , s ^ { \prime } } ^ { a } \} _ { s ^ { \prime } \in \mathcal S }$ , where $p _ { s , s ^ { \prime } } ^ { a }$ denotes the probability that the environment transits to state $s ^ { \prime }$ if taking action $a$ at state $s$ . The cost of taking action $a$ at state $s$ is given by $c ( s , a )$ . A stationary policy $\pi$ is a mapping from S to a distribution over $\mathcal A$ . At each time $t$ , an agent takes an action $A _ { t } \in \mathcal A$ at state $S _ { t } \in \mathcal S$ . The environment then transits to the next state $S _ { t + 1 }$ with probability $p _ { S _ { t } , S _ { t + 1 } } ^ { A _ { t } }$ , and the agent receives cost given by $c ( S _ { t } , A _ { t } )$ . The value function of a policy by follo $\pi$ staing $s \in \mathcal { S }$ fined as the expected accumulatand the goal is to find the policy discounted costthat minimizes $\pi \colon \bar { \mathbb { E } } \left[ \sum _ { t = 0 } ^ { \infty } \dot { \gamma } ^ { t } c ( S _ { t } , A _ { t } ) | S _ { 0 } = s , \pi \right]$ $\pi$   
+$s \in \mathcal { S }$
+
+Robust Markov Decision Process. In the robust case, the transition kernel is not fixed and lies in some uncertainty set. Denote the transition kernel at time $t$ by $\mathsf { P } _ { t }$ , and let $\kappa = ( \mathsf { P } _ { 0 } , \mathsf { P } _ { 1 } , \ldots )$ , where $\mathsf { P } _ { t } \in \mathbf { P } , \forall t \ge 0$ , and $\mathbf { P }$ is the uncertainty set of the transition kernel. The sequence $\kappa$ can be viewed as the policy of the nature, and is adversarially chosen by the nature [Bagnell et al., 2001, Nilim and
+
+El Ghaoui, 2004, Iyengar, 2005]. Define the robust value function of a policy $\pi$ as the worst-case expected accumulated discounted cost following a fixed policy $\pi$ over all transition kernels in the uncertainty set:
+
+$$
+V ^ { \pi } ( s ) = \operatorname* { m a x } _ { \kappa } \mathbb { E } _ { \kappa } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } c ( S _ { t } , A _ { t } ) | S _ { 0 } = s , \pi \right] ,
+$$
+
+where $\mathbb { E } _ { \kappa }$ denotes the expectation when the state transits according to $\kappa$ . Similarly, define the robust action-value function for a policy $\begin{array} { r } { \pi \colon Q ^ { \pi } ( s , a ) = \operatorname* { m a x } _ { \kappa } \mathbb { E } _ { \kappa } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } c ( S _ { t } , A _ { t } ) | \dot { S _ { 0 } } = s , A _ { 0 } = a , \pi \right] } \end{array}$ . The goal of robust RL is to find the optimal robust policy $\pi ^ { * }$ that minimizes the worst-case accumulated discounted cost:
+
+$$
+\pi ^ { * } = \arg \operatorname* { m i n } _ { \pi } V ^ { \pi } ( s ) , \forall s \in \mathcal { S } .
+$$
+
+We also denote $V ^ { \pi ^ { * } }$ and $Q ^ { \pi ^ { * } }$ by $V ^ { * }$ and $Q ^ { * }$ , respectively, and $\begin{array} { r } { V ^ { * } ( s ) = \operatorname* { m i n } _ { a \in \mathcal { A } } Q ^ { * } ( s , a ) } \end{array}$ .
+
+Note that a transition kernel is a collection of conditional distributions. Therefore, the uncertainty set $\mathbf { P }$ of the transition kernel can be equivalently written as a collection of $\mathcal { P } _ { s } ^ { a }$ for all $s \in \mathcal { S } , a \in \mathcal { A }$ , where $\mathcal { P } _ { s } ^ { a }$ is a set of conditional distributions $p _ { s } ^ { a }$ over the state space S. Denote by $\sigma _ { \mathcal { P } } ( v ) \triangleq \operatorname* { m a x } _ { p \in \mathcal { P } } ( p ^ { \top } v )$ the support function of vector $v$ over a set of probability distributions $\mathcal { P }$ . For robust MDP, the following robust analogue of the Bellman recursion was provided in [Nilim and El Ghaoui, 2004, Iyengar, 2005].
+
+Theorem 1. [Nilim and El Ghaoui, 2004] The following perfect duality condition holds for all $s \in \mathcal { S }$ :
+
+$$
+\operatorname* { m i n } _ { \pi } \operatorname* { m a x } _ { \kappa } \mathbb { E } _ { \kappa } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } c ( S _ { t } , A _ { t } ) \big | \pi , S _ { 0 } = s \right] = \operatorname* { m a x } _ { \kappa } \operatorname* { m i n } _ { \pi } \mathbb { E } _ { \kappa } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } c ( S _ { t } , A _ { t } ) \big | \pi , S _ { 0 } = s \right] .
+$$
+
+The optimal robust value function $V ^ { * }$ satisfies $\begin{array} { r } { V ^ { * } ( s ) = \operatorname* { m i n } _ { a \in \mathcal { A } } \bigl ( c ( s , a ) + \gamma \sigma _ { \mathscr { P } _ { s } ^ { a } } ( V ^ { * } ) \bigr ) } \end{array}$ , and the optimal robust action-value function $Q ^ { * }$ satisfies $Q ^ { * } ( s , a ) = c ( s , a ) + \gamma \sigma _ { \mathscr P _ { s } ^ { a } } ( V ^ { * } )$ .
+
+Define the robust Bellman operator $\mathbf { T }$ by $\begin{array} { r } { \mathbf { T } Q ( s , a ) = c ( s , a ) + \gamma \sigma _ { \mathcal { P } _ { s } ^ { a } } ( \operatorname* { m i n } _ { a \in \mathcal { A } } Q ( s , a ) ) } \end{array}$ . It was shown in [Nilim and El Ghaoui, 2004, Iyengar, 2005] that $\mathbf { T }$ is a contraction and its fixed point is the optimal robust $Q ^ { * }$ . When the uncertainty set is known, so that $\sigma _ { \mathcal P _ { s } ^ { a } }$ can be computed exactly, $V ^ { * }$ and $Q ^ { * }$ can be solved by dynamic programming [Iyengar, 2005, Nilim and El Ghaoui, 2004].
+
+# 3 R-Contamination Model For Uncertainty Set Construction
+
+In this section, we construct the uncertainty set using the $R$ -contamination model.
+
+Let $\mathsf { P } = \{ p _ { s } ^ { a } , s \in \mathscr { S } , a \in \mathscr { A } \}$ be the centroid of the uncertainty set, i.e., the transition kernel that generates the sample trajectory, and $\mathsf { P }$ is unknown. For example, P can be the simulator at hand, which may not be exactly accurate; and $\mathsf { P }$ can be the transition kernel of environment 1, from which we can take samples to learn a policy that will be deployed in a similar environment 2. The goal is to learn a policy using samples from $\mathsf { P }$ that performs well when applied to a perturbed MDP from P.
+
+Motivated by empirical studies of adversarial training in RL [Huang et al., 2017, Kos and Song, 2017, Lin et al., 2017, Pattanaik et al., 2018, Mandlekar et al., 2017] and the $R$ -contamination model in robust detection [Huber, 1965], we use the $R$ -contamination model to define the uncertainty set:
+
+$$
+\begin{array} { r } { \mathfrak { P } _ { s } ^ { a } = \big \{ ( 1 - R ) p _ { s } ^ { a } + R q | q \in \mathcal { A } _ { | \mathcal { S } | } \big \} , s \in \mathcal { S } , a \in \mathcal { A } , \mathrm { f o r ~ s o m e ~ } 0 \le R \le 1 . } \end{array}
+$$
+
+Here, $p _ { s } ^ { a }$ is the centroid of the uncertainty set $\mathcal { P } _ { s } ^ { a }$ at $( s , a )$ , which is unknown, and $R$ is the design parameter of the uncertainty set, which measures the size of the uncertainty set, and is assumed to be known in the algorithm. We then let $\mathbf { P } = \bigotimes _ { s \in \mathcal { S } , a \in \mathcal { A } } \mathcal { P } _ { s } ^ { a }$ .
+
+Remark 1. $R$ -contamination model is closely related to other uncertainty set models like total variation and $K L$ -divergence. It can be shown that $R$ -contamination set certered at $p$ is a subset of total variation ball : $\{ ( 1 - R ) p + R q | q \in \varDelta _ { | \mathcal { S } | } \} \subset \{ q \in \varDelta _ { | \mathcal { S } | } | d _ { T V } ( p , q ) \leq R \}$ . Hence the total variation uncertainty set is less conservative than our $R$ -contamination uncertainty set. KL-divergence moreover can be related to total variation using Pinsker’s inequality, i.e., $\begin{array} { r } { d _ { T V } ( p , q ) \leq \sqrt { \frac 1 2 d _ { K L } ( p , q ) } } \end{array}$ .
+
+# 4 Tabular Case: Robust Q-Learning
+
+In this section, we focus on the tabular case with finite state and action spaces. We focus on the asynchronous setting where a single sample trajectory is available with Markovian noise. We will develop an efficient approach to estimate the unknown uncertainty set $\mathbf { P }$ , and further the support function $\sigma _ { \mathcal { P } _ { s } ^ { a } } ( \cdot )$ , and then design our robust Q-learning algorithm.
+
+We propose an efficient and data-driven approach to estimate the unknown $p _ { s } ^ { a }$ and thus the unknown uncertainty set $\mathcal { P } _ { s } ^ { a }$ for any $s \in \mathcal { S }$ and $a \in { \mathcal { A } }$ . Specifically, denote the sample at $t$ -th time step by $O _ { t } = ( s _ { t } , a _ { t } , s _ { t + 1 } )$ . We then use $O _ { t }$ to obtain the maximum likelihood estimate (MLE) $\hat { p } _ { t } \triangleq \mathbb { 1 } _ { s _ { t + 1 } }$ of the transition kernel $p _ { s _ { t } } ^ { a _ { t } }$ , where $\mathbb { 1 } _ { s _ { t + 1 } }$ is a probability distribution taking probability 1 at $s _ { t + 1 }$ and 0 at other states. This is an unbiased estimate of the transition kernel $p _ { s _ { t } } ^ { a _ { t } }$ conditioning on $S _ { t } = s _ { t }$ and $A _ { t } = a _ { t }$ . We then design a sample-based estimate $\hat { \mathcal { P } } _ { t } \triangleq \left\{ ( 1 - R ) \hat { p } _ { t } + R q | q \in \Delta _ { | \mathcal { S } | } \right\}$ of the uncertainty set $\mathcal { P } _ { s _ { t } } ^ { a _ { t } }$ . Using the sample-based uncertainty set $\hat { \mathcal { P } } _ { t }$ , we construct the following robust Q-learning algorithm in Algorithm 1. For any $t$ , $\sigma _ { \hat { \mathcal { P } } _ { t } } ( V _ { t } )$ can be easily computed: $\sigma _ { \hat { \Phi } _ { t } } ( V _ { t } ) =$
+
+# Algorithm 1 Robust Q-Learning
+
+Initialization: $T$ , $Q _ { 0 } ( s , a )$ for all $( s , a ) \in \mathcal { S } \times \mathcal { A }$ , behavior policy $\pi _ { b }$ , $s _ { 0 }$ , step size $\alpha _ { t }$
+
+1: for $t = 0 , 1 , 2 , . . . , T - 1$ do   
+2: Choose $a _ { t }$ according to $\pi _ { b } ( \cdot | s _ { t } )$   
+3: Observe $s _ { t + 1 }$ and $c _ { t }$   
+4: $\begin{array} { r } { V _ { t } ( s )  \operatorname* { m i n } _ { a \in \mathcal { A } } Q _ { t } ( s , a ) , \forall s \in \mathfrak { s } } \end{array}$   
+5: Qt+1( $s _ { t } , a _ { t } ) \gets ( 1 - \alpha _ { t } ) Q _ { t } ( s _ { t } , a _ { t } ) + \alpha _ { t } ( c _ { t } + \gamma \sigma _ { \hat { \mathcal P } _ { t } } ( V _ { t } ) )$   
+6: $Q _ { t + 1 } ( s , a ) \gets Q _ { t } ( s , a )$ for $( s , a ) \neq ( s _ { t } , a _ { t } )$   
+7: end for
+
+Output: $Q _ { T }$
+
+$\begin{array} { r } { R \operatorname* { m a x } _ { s \in \mathcal { S } } V _ { t } ( s ) + ( 1 - R ) V _ { t } ( s _ { t + 1 } ) } \end{array}$ . Hence the update in Algorithm 1 (line 5) can be written as
+
+$$
+Q _ { t + 1 } ( s _ { t } , a _ { t } ) \gets ( 1 - \alpha _ { t } ) Q _ { t } ( s _ { t } , a _ { t } ) + \alpha _ { t } ( c _ { t } + \gamma R \operatorname* { m a x } _ { s \in \mathcal { S } } V _ { t } ( s ) + \gamma ( 1 - R ) V _ { t } ( s _ { t + 1 } ) ) .
+$$
+
+Compared to the model-based approach, our approach is model-free. It does not require the prior knowledge of the uncertainty set, i.e., the knowledge of $p _ { s } ^ { a } , \forall s \in \mathcal { S } , a \in \mathcal { A }$ . Furthermore, the memory requirement of our algorithm is $| \mathcal { S } | \times | \mathcal { A } |$ (used to store the Q-table), and unlike the modelbased approach it does not need a table of size $| \dot { \mathcal { S } } | ^ { 2 } | \mathcal { A } |$ to store $p _ { s } ^ { a } , \forall s \in \mathcal { S } , a \in \mathcal { A }$ , which could be problematic if the state space is large. Moreover, our algorithm does not involve a relaxation of the uncertainty set like the one in [Roy et al., 2017], which will incur a stringent condition on the discount factor to guarantee the convergence. As will be shown below, the convergence of our Algorithm 1 does not require any condition on the discount factor.
+
+We show in the following theorem that the robust Q-learning algorithm converges asymptotically to the optimal robust action-value function $Q ^ { * }$ .
+
+Theorem 2. (Asymptotic Convergence) If step sizes $\alpha _ { t }$ satisfy that $\textstyle \sum _ { t = 0 } ^ { \infty } \alpha _ { t } = \infty$ and $\textstyle \sum _ { t = 0 } ^ { \infty } \alpha _ { t } ^ { 2 } < \infty$ then $Q _ { t } \to Q ^ { * }$ as with probability .
+
+To further establish the finite-time error bound for our robust Q-learning algorithm in Algorithm 1, we make the following assumption that is commonly used in the analysis of vanilla Q-learning.
+
+Assumption 1. The Markov chain induced by the behavior policy $\pi _ { b }$ and the transition kernel $p _ { s } ^ { a } , \forall s \in \mathcal { S } , a \in \mathcal { A }$ is uniformly ergodic.
+
+Let $\mu _ { \pi _ { b } }$ denote the stationary distribution over $\mathcal { S } \times \mathcal { A }$ induced by $\pi _ { b }$ and $p _ { s } ^ { a } , \forall s \in \mathcal { S } , a \in \mathcal { A }$ . We then further define $\begin{array} { r } { \mu _ { \mathrm { m i n } } = \operatorname* { m i n } _ { ( s , a ) \in \mathcal { S } \times \mathcal { A } } \mu _ { \pi _ { b } } ( s , a ) } \end{array}$ . This quantity characterizes how many samples are needed to visit every state-action pair sufficiently often. Define the following mixing time of the induced Markov chain: $\begin{array} { r } { t _ { \mathrm { m i x } } = \operatorname* { m i n } \left. t : \operatorname* { m a x } _ { s \in \mathcal { S } } \mathsf { \bar { d } _ { \mathrm { T V } } } ( \mu _ { \pi } , P ( s _ { t } = \cdot | s _ { 0 } = s ) ) \breve { \leq } \frac { 1 } { 4 } \right. } \end{array}$ , where $d _ { \mathrm { T V } }$ is the total variation distance.
+
+The following theorem establishes the finite-time error bound of our robust Q-learning algorithm.
+
+Theorem 3. (Finite-Time Error Bound) There exist some positive constants $c _ { 0 }$ and $c _ { 1 }$ such that for any $\delta < 1$ , any $\begin{array} { r } { \epsilon < \frac { 1 } { 1 - \gamma } } \end{array}$ , any $T$ satisfying
+
+$$
+T \geq c _ { 0 } \left( \frac { 1 } { \mu _ { m i n } ( 1 - \gamma ) ^ { 5 } \epsilon ^ { 2 } } + \frac { t _ { m i x } } { \mu _ { m i n } ( 1 - \gamma ) } \right) \log { \left( \frac { T | \mathcal { S } | | \mathcal { A } | } { \delta } \right) } \log { \left( \frac { 1 } { \epsilon ( 1 - \gamma ) ^ { 2 } } \right) } ,
+$$
+
+and step size αt = $\begin{array} { r } { \alpha _ { t } = \frac { c _ { 1 } } { \log \left( \frac { T | \mathcal { S } | | \mathcal { A } | } { \delta } \right) } \operatorname* { m i n } \left( \frac { 1 } { t _ { m i x } } , \frac { \epsilon ^ { 2 } ( 1 - \gamma ) ^ { 4 } } { \gamma ^ { 2 } } \right) , \forall t \geq 0 } \end{array}$ we have with probability at least $1 - 6 \delta$ , $\| Q _ { T } - Q ^ { * } \| _ { \infty } \leq 3 \epsilon$ .
+
+From the theorem, we can see that to guarantee an $\epsilon$ -accurate estimate, a sample size $\begin{array} { r } { \tilde { \mathcal { O } } ( \frac { 1 } { \mu _ { \mathrm { m i n } } ( 1 - \gamma ) ^ { 5 } \epsilon ^ { 2 } } + } \end{array}$ tmixµmin(1−γ) ) (up to some logarithmic terms) is needed. This complexity matches with the one for the vanilla Q-learning in [Li et al., 2020] (within a constant factor), while our algorithm also guarantees robustness to MDP model uncertainty. Our algorithm design and analysis can be readily extended to robustify TD and SARSA. The variance-reduction technique [Wainwright, 2019] can also be combined with our robust Q-learning algorithm to further improve the dependency on $( 1 - \gamma )$ .
+
+# 5 Function Approximation: Robust TDC
+
+In this section, we investigate the case where the state and action spaces can be large or even continuous. A popular approach is to approximate the value function using a parameterized function, e.g., linear function and neural network. In this section, we focus on the case with linear function approximation to illustrate the main idea of designing robust RL algorithms. Our approach can be extended to non-linear (smooth) function approximation using techniques in, e.g., [Cai et al., 2019, Bhatnagar et al., 2009, Wai et al., 2019, Wang et al., 2021].
+
+We focus on the problem of robust policy evaluation, i.e., estimate the robust value function $V ^ { \pi }$ defined in (1) for a given policy $\pi$ under the worst-case MDP transition kernel in the uncertainty set. Note that for robust RL with model uncertainty, any policy evaluation problem can be viewed as "off-transition-kernel", as it is to evaluate the value function under the worst-case MDP using samples from a different MDP. Since the TD algorithm with function approximation may diverge under off-policy training [Baird, 1995] and importance sampling cannot be applied here due to unknown transition kernel, in this paper we generalize the GTD method [Maei et al., 2010, Maei, 2011] to the robust setting.
+
+Let $\left\{ \phi ^ { ( i ) } : \mathbb { S } \to \mathbb { R } , i = 1 , \ldots , N \right\}$ be a set of $N$ fixed base functions, where $N \ll | \mathcal S | | \mathcal A |$ . In particular, we approximate the robust value function using a linear combination of $\begin{array} { r } { \sum _ { i = 1 } ^ { N } \theta ^ { i } \phi _ { s } ^ { ( i ) } = \phi _ { s } ^ { \top } \theta } \end{array}$ , where $\boldsymbol { \theta } \in \mathbb { R } ^ { N }$ is the weight vector. $\phi ^ { ( i ) }$ ’s: $V _ { \theta } ( s ) =$
+
+Define the following robust Bellman operator for a given policy $\pi$ :
+
+$$
+\begin{array} { r l } & { \mathrm { \bf T } _ { \pi } V ( s ) \triangleq \mathbb E _ { A \sim \pi ( \cdot | s ) } [ c ( s , A ) + \gamma \sigma _ { \mathcal { P } _ { s } ^ { A } } ( V ) ] } \\ & { \quad \quad \quad = \mathbb E _ { A \sim \pi ( \cdot | s ) } \left[ c ( s , A ) + \gamma ( 1 - R ) \displaystyle \sum _ { s ^ { \prime } \in \mathcal { S } } p _ { s , s ^ { \prime } } ^ { A } V ( s ^ { \prime } ) + \gamma R \operatorname* { m a x } _ { s ^ { \prime } \in \mathcal { S } } V ( s ^ { \prime } ) \right] . } \end{array}
+$$
+
+We then define the mean squared projected robust Bellman error (MSPRBE) as
+
+$$
+\begin{array} { r } { \mathrm { M S P R B E } ( \theta ) = \| \mathbf { I I T } _ { \pi } V _ { \theta } - V _ { \theta } \| _ { \mu _ { \pi } } ^ { 2 } , } \end{array}
+$$
+
+where $\begin{array} { r } { \| v \| _ { \mu _ { \pi } } ^ { 2 } = \int v ^ { 2 } ( s ) \mu _ { \pi } ( d s ) } \end{array}$ , $\mu _ { \pi }$ is the stationary distribution induced by $\pi$ , and $\mathbf { I I }$ is a projection onto the linear function space w.r.t. $\| \cdot \| _ { \mu _ { \pi } }$ . We will develop a two time-scale gradient-based approach to minimize the MSPRBE. However, it can be seen that $\operatorname* { m a x } _ { s } V _ { \theta } ( s )$ in (7) is not smooth in $\theta$ , which is troublesome in both algorithm design and analysis. To solve this issue, we introduce the following smoothed robust Bellman operator $\hat { \mathbf { T } } _ { \pi }$ by smoothing the max with a LSE(LogSumExp):
+
+$$
+\hat { \mathbf { T } } _ { \pi } V ( s ) = \mathbb { E } _ { A \sim \pi ( \cdot | s ) } \left[ c ( s , A ) + \gamma ( 1 - R ) \sum _ { s ^ { \prime } \in \mathcal { S } } p _ { s , s ^ { \prime } } ^ { A } V ( s ^ { \prime } ) + \gamma R \cdot \mathrm { L S E } ( V ) \right] ,
+$$
+
+where $\begin{array} { r } { \mathrm { L S E } ( V ) = \frac { \log \left( \sum _ { s } e ^ { \varrho V ( s ) } \right) } { \varrho } } \end{array}$ is the LogSumExp w.r.t. $V$ with a parameter $\varrho > 0$ . Note that when $\varrho \to \infty$ , the smoothed robust Bellman operator $\hat { \mathbf { T } } _ { \pi } \to \mathbf { T } _ { \pi }$ . The LSE operator can also be replaced by some other operator that approximates the max operator and is smooth, e.g., mellow-max [Asadi and Littman, 2017]. In the following, we first show that the fixed point of $\hat { \mathbf { T } } _ { \pi }$ exists for any $\varrho$ , and the fixed points converge to the one of $\mathbf { T } _ { \pi }$ for large $\varrho$ .
+
+Theorem 4. (1). For any $\varrho$ , $\hat { \mathbf { T } } _ { \pi }$ has a fixed point.
+
+(2). Let $V _ { 1 }$ and $V _ { 2 }$ be the fixed points of $\hat { \mathbf { T } } _ { \pi }$ and $\mathbf { T } _ { \pi }$ , respectively. Then
+
+$$
+\| V _ { 1 } - V _ { 2 } \| _ { \infty } \leq \frac { \gamma R } { 1 - \gamma } \frac { \log | \mathcal { S } | } { \varrho }  0 , a s \varrho  \infty .
+$$
+
+We then denote by $J ( \theta )$ the smoothed MSPRBE with the LSE operator, and the goal is:
+
+$$
+\operatorname* { m i n } _ { \theta } J ( \theta ) = \operatorname* { m i n } _ { \theta } \Big \| \mathbf { H } \hat { \mathbf { T } } _ { \pi } V _ { \theta } - V _ { \theta } \Big \| _ { \mu _ { \pi } } ^ { 2 } .
+$$
+
+# 5.1 Algorithm Development
+
+In the following, we develop the robust TDC algorithm to solve the problem in (11). We will first derive the gradient of the smoothed MSPRBE, $J ( \theta )$ , and then design a two time-scale update rule using the weight doubling trick in [Sutton et al., 2009] to solve the double sampling problem. Define $\delta _ { s , a , s ^ { \prime } } ( \theta ) \ \triangleq \ c ( s , a ) + \gamma ( 1 - R ) V _ { \theta } ( s ^ { \prime } ) + \gamma R \mathrm { L S E } ( V _ { \theta } ) - V _ { \theta } ( s )$ , where $\operatorname { L S E } ( V _ { \theta } )$ is the LogSumExp function w.r.t. $V _ { \theta } = \theta ^ { \top } \phi$ . Denote by $C \triangleq \mathbb { E } _ { \mu _ { \pi } } \left[ \phi _ { S } ^ { \intercal } \phi _ { S } \right]$ . Then, $\mathbb { E } _ { \mu } [ \delta _ { S , A , S ^ { \prime } } ( \theta ) \phi _ { S } ] \ = \ \varPhi ^ { \top } D \left( \hat { \mathbf { T } } _ { \pi } V _ { \theta } - V _ { \theta } \right)$ , where $\textit { D } = \ \mathrm { d i a g } ( \mu _ { \pi } ( s _ { 1 } ) , \mu _ { \pi } ( s _ { 2 } ) , . . . , \mu _ { \pi } ( s _ { | \mathcal { S } | } ) )$ and $\boldsymbol { \varPhi } = ( \phi _ { s _ { 1 } } , \phi _ { s _ { 2 } } , . . . , \phi _ { s _ { | \mathcal { S } | } } ) ^ { \top } \in \dot { \mathbb { R } } ^ { | \mathcal { S } | \times N }$ . We know that $\begin{array} { r } { \pmb { \Pi } ^ { \top } D \pmb { \Pi } = D ^ { \top } \pmb { \varPhi } ( \pmb { \varPhi } ^ { \top } D \pmb { \varPhi } ) ^ { - 1 } \pmb { \varPhi } ^ { \top } D } \end{array}$ from [Maei, 2011]. Hence we have
+
+$$
+\begin{array} { r } { \boldsymbol J ( \theta ) = \Big \| \mathbf { \hat { I } } \mathbf { \hat { T } } _ { \boldsymbol \pi } V _ { \theta } - V _ { \theta } \Big \| _ { \mu _ { \boldsymbol \pi } } ^ { 2 } = \mathbb { E } _ { \mu _ { \boldsymbol \pi } } [ \delta _ { S , A , S ^ { \prime } } ( \theta ) \phi _ { S } ] ^ { \top } \boldsymbol C ^ { - 1 } \mathbb { E } _ { \mu _ { \boldsymbol \pi } } [ \delta _ { S , A , S ^ { \prime } } ( \theta ) \phi _ { S } ] . } \end{array}
+$$
+
+Then, its gradient can be written as:
+
+$$
+\begin{array} { l } { \displaystyle - \frac { 1 } { 2 } \nabla J ( \theta ) = - \mathbb { E } _ { \mu _ { \pi } } \big [ ( \nabla \delta _ { S , A , S ^ { \prime } } ( \theta ) ) \phi _ { S } \big ] ^ { \top } C ^ { - 1 } \mathbb { E } _ { \mu _ { \pi } } \big [ \delta _ { S , A , S ^ { \prime } } ( \theta ) \phi _ { S } \big ] } \\ { = \mathbb { E } _ { \mu _ { \pi } } \big [ \delta _ { S , A , S ^ { \prime } } ( \theta ) \phi _ { S } \big ] - \gamma \mathbb { E } _ { \mu _ { \pi } } \bigg [ \bigg ( ( 1 - R ) \phi _ { S ^ { \prime } } + R \cdot \nabla \mathrm { L S E } ( V _ { \theta } ) \bigg ) \phi _ { S } ^ { \top } \bigg ] \omega ( \theta ) , } \end{array}
+$$
+
+where $\omega ( \theta ) = C ^ { - 1 } \mathbb { E } _ { \mu _ { \pi } } [ \delta _ { S , A , S ^ { \prime } } ( \theta ) \phi _ { S } ]$ . It can be seen that to obtain an unbiased estimate of $\nabla J ( \theta )$ , two independent samples are needed as there exists a multiplication of two expectations, which is not applicable when there is only one sample trajectory. We then utilize the weight doubling trick in [Sutton et al., 2009], and design the robust TDC algorithm in Algorithm 2. Specifically, we introduce a fast time scale to estimate $\omega ( \theta )$ , and a slow time scale to estimate $\nabla J ( \theta )$ . Denote the projection by $\Pi _ { K } ( x ) \triangleq \arg \operatorname* { m i n } _ { \| y \| \leq K } \| y - x \|$ for any $\boldsymbol { x } \in \mathbb { R } ^ { N }$ . Our robust TDC algorithm in Algorithm 2 can be implemented in an online and incremental fashion. If the uncertainty set becomes a singleton, i.e., $R = 0$ , then Algorithm 2 reduces to the vanilla TDC algorithm.
+
+# 5.2 Finite-Time Error Bound of Robust TDC
+
+Unlike the vanilla TDC algorithm, $J ( \theta )$ here is non-convex. Therefore, we are interested in the convergence to stationary points, i.e., the rate of $\| \nabla J ( \theta ) \|  0$ . We first make some standard assumptions which are commonly used in RL algorithm analysis, e.g., [Wang and Zou, 2020, Kaledin et al., 2020, Xu et al., 2019, Srikant and Ying, 2019, Bhandari et al., 2018].
+
+Assumption 2 (Bounded feature). $\| \phi _ { s } \| _ { 2 } \leq 1 , \forall s \in \mathcal { S }$ .
+
+Assumption 3 (Bounded cost function). $| c ( s , a ) | \leq c _ { \mathrm { m a x } } , \forall s \in \mathcal { S }$ and $a \in { \mathcal { A } }$ .
+
+Assumption 4 (Problem solvability). The matrix $C = \mathbb { E } _ { \mu _ { \pi } } [ \phi _ { S } \phi _ { S } ^ { \top } ]$ is non-singular with $\lambda > 0$ being its smallest eigenvalue.
+
+Input: $T , \alpha , \beta , \varrho , \phi _ { i }$ for $i = 1 , . . . , N$ , projection radius $K$   
+Initialization: θ0,w0, s0   
+1: Choose $W \sim \mathrm { U n i f o r m } ( 0 , 1 , . . . , T - 1 )$   
+2: for $t = 0 , 1 , 2 , . . . , W - 1$ do   
+3: Take action according to $\pi ( \cdot | s _ { t } )$ and observe $s _ { t + 1 }$ and $c _ { t }$   
+4: $\phi _ { t }  \phi _ { s _ { t } }$   
+7: 5: 6: $\begin{array} { r l } & { \ddot { \delta _ { t } } ( \theta _ { t } ) \gets c _ { t } + \gamma ( 1 - R ) V _ { \theta _ { t } } ( s _ { t + 1 } ) + \gamma R \frac { \log ( \sum _ { s } e ^ { \varrho \theta ^ { \top } \phi _ { s } } ) } { \varrho } - V _ { \theta _ { t } } ( s _ { t } ) } \\ & { \theta _ { t + 1 } \gets \Pi _ { K } \left( \theta _ { t } + \alpha \left( \delta _ { t } ( \theta _ { t } ) \phi _ { t } - \gamma \bigg ( ( 1 - R ) \phi _ { t + 1 } + R \sum _ { s \in \mathcal { S } } \left( \frac { e ^ { \sigma V _ { \theta } ( s ) } \phi _ { s } } { \sum _ { j \in \mathcal { S } } e ^ { \sigma V _ { \theta } ( j ) } } \right) \bigg ) \phi _ { t } ^ { \top } \omega _ { t } \right) \right) } \\ & { \omega _ { t + 1 } \gets \Pi _ { K } ( \omega _ { t } + \beta ( \delta _ { t } ( \theta _ { t } ) - \phi _ { t } ^ { \top } \omega _ { t } ) \phi _ { t } ) } \end{array}$
+
+8: end for
+
+Output: $\theta _ { W }$
+
+Assumption 5 (Geometric uniform ergodicity). There exist some constants $m > 0$ and $\rho \in ( 0 , 1 )$ such that for any $t > 0$ , $\begin{array} { r } { \operatorname* { m a x } _ { s \in \mathcal { S } } d _ { T V } ( \bar { \mathbb { P } } ( s _ { t } | s _ { 0 } = s ) , \mu _ { \pi } ) \leq m \rho ^ { t } } \end{array}$ .
+
+In the following theorem, we characterize the finite-time error bound for the convergence of our robust TDC algorithm. Here we only provide the order of the bounds in terms of $T$ . The explicit bounds can be found in (129) in Appendix D.3.
+
+Theorem 5. Consider the following step-sizes: $\begin{array} { r } { \beta = \mathcal { O } \left( \frac { 1 } { T ^ { b } } \right) } \end{array}$ , and $\begin{array} { r } { \alpha = \mathcal { O } \left( \frac { 1 } { T ^ { a } } \right) } \end{array}$ , where $\begin{array} { r } { \frac { 1 } { 2 } < a \leq 1 } \end{array}$ and $0 < b \leq a$ . Then we have that
+
+$$
+\mathbb { E } [ \| \nabla J ( \theta _ { W } ) \| ^ { 2 } ] = \odot \left( \frac { 1 } { T \alpha } + \alpha \log ( 1 / \alpha ) + \frac { 1 } { T \beta } + \beta \log ( 1 / \beta ) \right) ,
+$$
+
+If we further let $a = b = 0 . 5$ , then $\begin{array} { r } { \mathbb { E } [ \| \nabla J ( \theta _ { W } ) \| ^ { 2 } ] = \odot \left( \frac { \log T } { \sqrt { T } } \right) . } \end{array}$
+
+The robust TDC has a matching complexity with the vanilla TDC with non-linear function approximation [Wang et al., 2021], but provides the additional robustness to model uncertainty. It does not need to relax the uncertainty set like in [Roy et al., 2017], and our convergence results do not need a condition on the discount factor.
+
+# 6 Experiments
+
+# 6.1 Robust Q-Learning
+
+In this section, we compare our robust Q-learning with the vanilla non-robust Q-learning. We use OpenAI gym framework [Brockman et al., 2016], and consider two different problems: Frozen lake and Cart-Pole. One more example of the taxi problem is given in the appendix. To demonstrate the robustness, the policy is learned in a perturbed MDP, and is then tested on the true unperturbed MDP. Specifically, during the training, we set a probability $p$ such that after the agent takes an action, with probability $p$ , the state transition is uniformly over S, and with probability $1 - p$ the state transition is according to the true unperturbed transition kernel. The behavior policy for all the experiments below is set to be a uniform distribution over the action space given any state, i.e., $\begin{array} { r } { \pi _ { b } ( a | s ) ^ { \bullet } = \frac { 1 } { | \mathcal { A } | } } \end{array}$ for any $s \in \mathcal { S }$ and $a \in { \mathcal { A } }$ . We then evaluate the performance of the obtained policy in the unperturbed environment. At each time t, the policy we evaluate is the greedy-policy w.r.t. the current estimate of the Q-function, i.e., $\pi _ { t } ( s ) = \arg \operatorname* { m a x } _ { a } Q _ { t } ( s , a )$ . A Monte-Carlo method with horizon 100 is used to evaluate the accumulated discounted reward of the learned policy on the unperturbed MDP. We take the average over 30 trajectories. More details are provided in the appendix.
+
+In Figure 1 and Figure 2, we plot the accumulated discounted reward of both algorithms under different $p$ and $R$ for both problems. The upper and lower envelopes of the curves correspond to the 95 and 5 percentiles of the 30 trajectories, respectively. It can be seen that overall our robust Q-learning algorithm achieves a much higher reward than the vanilla Q-learning. This demonstrates the robustness of our robust Q-learning algorithm to model uncertainty. Moreover, as $p$ and $R$ getting larger, i.e., as the MDP that we learn the policy deviates from the MDP we test the policy, the advantage of our robust Q-learning algorithm is getting more significant compared to the vanilla Q-learning algorithm.
+
+![](images/f6e7e9e2735f3d2f3cb6a78235a2c13acfd85270e0532cc9f312127eab2de53d.jpg)  
+Figure 1: FrozenLake-v0: robust Q-learning v.s. non-robust Q-learning.
+
+![](images/a43d2928aff120beef997e93a1a33000db2a33693b2a692ee8c4a435bdabda59.jpg)  
+Figure 2: CartPole-v0: robust Q-learning v.s. non-robust Q-learning.
+
+# 6.2 Robust TDC with Linear Function Approximation
+
+In this section we compare our robust TDC with the vanilla non-robust TDC with linear function approximation on the $4 \times 4$ Frozen Lake problem. The problem setting is the same as the one in Section 6.1. More details about the experiment setup are provided in the appendix.
+
+We implement the two algorithms using samples from the perturbed MDP both for 30 times, and obtain 30 sequences of $\{ \theta _ { t } ^ { i } \} _ { t = 1 } ^ { \infty }$ , $i = 1 , 2 , . . . , 3 0$ . We then compute the squared gradient norm $\| \nabla J ( \theta ) \| ^ { 2 }$ on the true unperturbed MDP, and see whether $\{ \theta _ { t } ^ { i } \} _ { t = 1 } ^ { \infty }$ converges to some stationary points on the true unperturbed MDP. In Fig. 3, we plot the average squared gradient norm $\| \nabla J ( \theta ) \| ^ { 2 }$ for different $p$ and $R$ . The upper and lower envelops are the 95 and 5 percentiles of the 30 curves. It can be seen that our robust TDC converges much faster than vanilla TDC, and as the model mismatch between the training and test MDPs enlarges, the vanilla TDC may diverge (Fig. 3(c)), while our robust TDC still converges to some stationary point. Also, the robust TDC has a much smaller variance, which indicates a much stable behavior under model uncertainty.
+
+![](images/88a93ae259a06e7bc8fa9f4ad18dbea7ab033fd1a750f80210fcc36ef5a44b84.jpg)  
+Figure 3: FrozenLake-v0: $\| \nabla J ( \theta ) \| ^ { 2 }$ of robust TDC and non-robust TDC.
+
+# 6.3 Comparison with The Adversarial Training Approach
+
+We also compare our robust Q-learning with Robust Adversarial Reinforcement Learning (RARL) in [Pinto et al., 2017]. To apply their algorithm to our problem setting, we model the nature as an adversarial player, and its goal is to minimize the reward that the agent receives. The action space $\mathcal { A } _ { a d }$ of the nature is set to be the state space $\mathcal { A } _ { a d } \triangleq \mathcal { S }$ . Then the perturbed training environment can be viewed as an adversarial model: both the agent and the adversary take actions $a _ { a } , a _ { a d }$ , then the environment will transit to state $a _ { a d }$ with probability $R$ and transit following the unperturbed MDP $p _ { s } ^ { a _ { a } }$ with probability $1 - R$ . The goal of the maximize its accumulated reward, while the goal of the natural is to minimize it.
+
+Following the RARL algorithm [Pinto et al., 2017], in each iteration of the training, we first fix the adversarial policy and use Q-learning to optimize the agent’s policy and obtain the Q-table $Q _ { t }$ . Then we fix the agent’s policy and optimize the adversarial policy.
+
+After each training iteration, we test the performance of the greedy policies w.r.t. Q-tables obtained from robust Q-learning and RARL. The testing environment is set to be the worst-case, i.e., after the agent takes an action, the environment transits to the state which has the minimal value function $( \mathrm { a r g m i n } _ { s \in \mathcal { S } } V _ { t } ( s ) )$ with probability $p$ . We plot the accumulated discounted rewards of both algorithms against number of training iterations under different parameters. We set $\alpha = 0 . 2$ and $\gamma = 0 . 9$ . It can be seen from Fig. 4 that our robust Q-learning achieves a higher accumulative reward, and thus is more robust that the RARL algorithm in [Pinto et al., 2017]. Also our robust Q-learning is more stable during training, i.e., the variance is smaller.
+
+![](images/05ae19de592c7a368e9a827c529316238af525b6bd10c6a6b56e811a7ad5cc4c.jpg)  
+Figure 4: Taxi-v3: robust Q-learning v.s. RARL.
+
+# 7 Conclusion
+
+In this paper, we develop a novel approach for solving model-free robust RL problems with model uncertainty. Our algorithms can be implemented in an online and incremental fashion, do not require additional memory than their non-robust counterparts. We theoretically proved the convergence of our algorithms under no additional assumption on the discount factor, and further characterized their finite-time error bounds, which match with their non-robust counterparts (within a constant factor). Our approach can be readily extended to robustify TD, SARSA and other GTD algorithms.
+
+Limitations: It is also of future interest to investigate robustness to reward uncertainty, and other types of uncertainty sets, e.g., ones defined by KL divergence, Wasserstein distance and total variation. Negative societal impact: To the best of the authors’ knowledge, this study does not have any potential negative impact on the society.
+
+# 8 Acknowledgment
+
+The work of Y. Wang and S. Zou was supported by the National Science Foundation under Grants CCF-2106560 and CCF- 2007783.
+
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parse/train/NiM9Q7Z95z/NiM9Q7Z95z.md

@@ -0,0 +1,479 @@
+# Secure Quantized Training for Deep Learning
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We have implemented training of neural networks in secure multi-party computa  
+2 tion (MPC) using quantization commonly used in said setting. To the best of our   
+3 knowledge, we are the first to present training of MNIST purely implemented in   
+4 MPC that comes within one percent of accuracy of training using plaintext compu  
+5 tation. We found that training with MPC is possible, but it takes more epochs and   
+6 achieves a lower accuracy than the usual CPU/GPU computation. More concretely,   
+7 we have trained a network with two convolution and two dense layers to $9 8 . 5 \%$   
+8 accuracy in 150 epochs. This took a day in our MPC implementation.
+
+# 9 1 Introduction
+
+10 Secure multi-party computation (MPC) is a cryptographic technique that allows a set of parties to   
+11 compute a public output on private inputs without revealing the inputs or any intermediate results.   
+12 This makes it a potential solution to federated learning where the sample data stays private and only   
+13 the model or even only inference results are revealed.   
+14 Imagine a set of healthcare providers holding sensitive patient data. MPC allows them to collabo  
+15 ratively train a model. This model could then either be released or even kept private for inference   
+16 using MPC again. See Figure for an illustration. A more conceptual example is the well-known   
+17 millionaires’ problem where two people want to find out who is richer without revealing their wealth.   
+18 There is clearly a difference between the one bit of information desired and the full figures.   
+19 There has been a sustained interest in applying secure computation to machine learning and neural   
+20 networks going back to at least Barni et al. [2006]. More recent advantages in practical MPC have   
+21 led to an increased effort in implementing both inference and training.   
+22 A number of works such as Mohassel and Zhang [2017], Mohassel and Rindal [2018], Wagh et al.   
+23 [2019], Wagh et al. [2021] implement neural network training with MPC at least in parts. However,   
+24 they either give accuracy figures below $9 5 \%$ or figures that have been obtained using plaintext training.   
+25 For the latter case, the works do not clarify how close the computation for plaintext training matches   
+26 the lower precision and other differences in the MPC setting. Agrawal et al. [2019] claim a higher   
+27 accuracy in a comparable setting for a convolutional neural network with more channels than we   
+28 use. However, they have only implemented dense layers, and we achieve comparable accuracy to   
+29 them with only dense layers. All works use quantization in the sense that a fractional number $x$ is   
+30 represented as $\lfloor x \cdot 2 ^ { - f } \rceil$ . This makes addition considerably faster in the secure computation setting   
+31 because it reduces to integer addition. Furthermore, some of the works suggest to replace the softmax   
+32 function that uses exponentiation with a ReLU-based replacement. Keller and Sun [2020] have found   
+33 that this softmax replacement deteriorates accuracy in dense neural networks to the extent that it does   
+34 not justify the performance gains.   
+35 The concurrent work of Tan et al. [2021] gives some figures on the learning curve when run using   
+36 secure computation. However, they stop at five epochs for MNIST training where they achieve $94 \%$   
+37 accuracy whereas we present the figures up to 150 epochs and $9 8 . 5 \%$ accuracy. Furthermore, their   
+38 choice of fixed-point precision 20 is considerably below 32, which we found to be optimal. We also   
+39 found that our implementation is $40 \%$ faster than theirs. Note that we use the CPU of one AWS   
+40 c5.9xlarge instance per party whereas Tan et al. use one NVIDIA Tesla V100 GPU per party. We   
+41 believe this somewhat counter-intuitive result comes from MPC heavily relying on communication,   
+42 which is an aspect where GPUs do not have an advantage over CPUs.   
+43 In this paper, we present an extensible framework for implementing deep learning training based on   
+44 MP-SPDZ by Keller [2020], a framework for multi-party computation.1 Similar to TensorFlow and   
+45 PyTorch, our approach allows representing deep learning models as succession of layers. We then   
+46 use this implementation to obtain accuracy figures for MNIST training by utilizing the MP-SPDZ   
+47 emulator, which allows to run the plaintext equivalent of secure computation, that is, the same   
+48 algorithms with the same precision. Finally, we run one of the most promising instantiation in real   
+49 secure computation in order to benchmark it confirm the result from the plaintext emulator.   
+50 There are a number of projects that integrate secure computation directly into popular machine   
+51 learning frameworks such as CrypTen by Gunning et al. [2019], PySyft by Ryffel et al. [2018], and   
+52 TF Encrypted by Dahl et al. [2018]. Our approach differs from all of them by running the protocol as   
+53 native CPU code (implemented using $\mathrm { C } { + + }$ ). This allows for much faster execution. For example,   
+54 CrypTen provides an MNIST training example (mpc_autograd_cnn) that takes over one minute   
+55 to run one epoch with 100 samples on one machine. In comparison, our implementation takes 11   
+56 minutes to run one epoch with the full dataset of 60,000 samples.   
+57 Another line of work (e.g., Quoc et al. [2021]) uses trusted execution environments that provide   
+58 computation outside the reach of the operating system. This is a different security model to multi-party   
+59 computation that works with distributing the information among several entities.   
+60 The paper is structured as follows: After introducing the basics of the protocol we use in Section 2,   
+61 we will explain the mathematical building blocks in Section 3 and their use in the context of deep   
+62 learning in Section 4. Finally, we will present our implementation in Section 5 and our experimental   
+63 results for MNIST in Section 6.
+
+![](images/bb0f136189595323f73d65ddc0b25170a7b7ffb1f9107b9e37395cb5ee7bab1c.jpg)  
+Figure 1: Outsourced computation: Data holders (on the left) secret-share their data to a number of computing parties (on the right), who then return the desired the result (e.g., a model or inference results on further queries). All communication except outputs are secret-shared and thus secure if no two computing parties collude.
+
+# 64 2 An Efficient Secure Multi-Party Computation Protocol
+
+There is a wide range of MPC protocols with a variety of security properties (see Keller [2020] for an overview). In this paper we focus on the setting of three-party computation with one semi-honest corruption. This means that out of the three parties two are expected to behave honestly, i.e., they follow the protocol and keep their view of the protocol secret, and one party is expected to follow the protocol but might try extract information from their view. The reason for choosing this setting is that it allows an efficient MPC protocol while still allowing secure outsourced computation. The concrete protocol we use goes back to Benaloh and Leichter [1990] with further aspects by Araki et al. [2016], Mohassel and Rindal [2018], and Eerikson et al. [2020]. We summarize the core protocol below. The mathematical building blocks in the next section mostly use the aspects below.
+
+74 Secret sharing All intermediate values in our protocol are stored using replicated secret sharing. A   
+75 secret value $x$ is a represented as a random sum $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ , and party $P _ { i }$ holds $( x _ { i - 1 } , x _ { i + 1 } )$   
+76 where the indices are computed modulo three. Clearly, each party is missing one value to compute   
+77 the sum. On the other hand, each pair of parties hold all necessary to reconstruct the secret. For a   
+78 uniformly random generation of shares, the computation domain has to be finite. Most commonly,   
+79 this domain is defined by integer computation modulo a number. We use $2 ^ { k }$ for $k$ being a multiple 64   
+80 and 2 as the moduli. The first case corresponds to an extension of 64-bit arithmetic found on most   
+81 processors. We will refer to the two settings as arithmetic and binary secret sharing throughout the   
+82 paper.   
+83 Input sharing The secret sharing scheme implies a protocol to share inputs where the inputting   
+84 party samples the shares and distributes them accordingly. Eerikson et al. [2020] have proposed a   
+85 more efficient protocol where the inputting party only needs to send one value instead of two pairs of   
+86 values. If $P _ { i }$ would like to input $x$ , $x _ { i }$ is set to zero, and $x _ { i - 1 }$ is generated with a pseudo-random   
+87 generator using a key previously shared between $P _ { i }$ and $P _ { i + 1 }$ . $P _ { i }$ can compute $x _ { i + 1 } = x - x _ { i - 1 }$ and   
+88 send it to $P _ { i - 1 }$ . While the resulting secret sharing is not entirely random, the fact that $P _ { i }$ already   
+89 knows $x$ makes randomizing $x _ { i }$ obsolete.   
+90 Addition The commutative nature of addition allows to add secret sharings without communication.   
+91 More concretely, secret sharings $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ and $y = y _ { 0 } + y _ { 1 } + y _ { 2 }$ imply the secret sharing   
+92 $x + y = ( x _ { 0 } + y _ { 0 } ) + ( x _ { 1 } + y _ { 1 } ) + ( x _ { 2 } + y _ { 2 } )$ .
+
+93 Multiplication The product of $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ and $y = y _ { 0 } + y _ { 1 } + y _ { 2 }$ is
+
+$$
+{ \begin{array} { r l } & { x \cdot y = ( x _ { 0 } + x _ { 1 } + x _ { 2 } ) \cdot ( y _ { 0 } + y _ { 1 } + y _ { 2 } ) } \\ & { \qquad = ( x _ { 0 } y _ { 0 } + x _ { 0 } y _ { 1 } + x _ { 1 } y _ { 0 } ) + ( x _ { 1 } y _ { 1 } + x _ { 1 } y _ { 2 } + x _ { 1 } y _ { 1 } ) + ( x _ { 2 } y _ { 2 } + x _ { 2 } y _ { 0 } + x _ { 0 } y _ { 2 } ) . } \end{array} }
+$$
+
+94 Each of the brackets only contains shares known by one of the parties. They can thus compute an   
+95 additive secret sharing (one summand per party) of the product. However, every party only holding   
+96 one share does not satisfy the replication requirement for further multiplications. It is not secure   
+97 for every party to pass their value on to another party because the summands are not distributed   
+98 randomly. This can be fixed by rerandomization: Let $x y = z _ { 0 } + z _ { 1 } + z _ { 2 }$ where $z _ { i }$ is know to $P _ { i }$ .   
+99 Every party $P _ { i }$ computes $z _ { i } ^ { \prime } = z _ { i } + r _ { i , i + 1 } - r _ { i - 1 , i }$ where $r _ { i , i + 1 }$ is generated with a pseudo-random   
+100 generator using a key pre-shared between $P _ { i }$ and $P _ { i + 1 }$ . The resulting sum $x y = \hat { z } _ { 0 } ^ { \prime } + z _ { 1 } ^ { \prime } + z _ { 2 } ^ { \prime }$ is   
+101 pseudo-random, and it is thus secure for $P _ { i }$ to send $z _ { i } ^ { \prime }$ to $P _ { i + 1 }$ in order to create a replicated secret   
+102 sharing $\left( ( x y ) _ { i - 1 } , ( x y ) _ { i + 1 } \right) = \left( z _ { i } ^ { \prime } , z _ { i - 1 } ^ { \prime } \right)$ .
+
+# 103 3 Secure Computation Building Blocks
+
+104 In this section, we will discuss how to implement computation with MPC with a focus on how   
+105 it differs from computation on CPUs or GPUs. Most of the techniques below are already known   
+106 individually. To the best of our knowledge however, we are the first to put them together in an efficient   
+107 and extensible framework for secure computation of deep learning training.   
+108 Domain conversion Recall we that we use computation modulo $2 ^ { k }$ for $k$ being a multiple of 64   
+109 as well as 1. Given that the main operations are just addition and multiplication in the respective   
+110 domain, it is desirable to compute integer arithmetic in the large domain but operations with a   
+111 straight-forward binary circuit modulo two. There has been a long-running interest in this going   
+112 back to least Kolesnikov et al. [2013]. We mainly rely on the approach proposed by Mohassel and   
+113 114 $\{ x _ { 0 } ^ { ( i ) } \} _ { j = 0 } ^ { k - 1 }$ 018] and Araki et al. [20the bit decomposition of $x _ { 0 }$ . Recal, that is, $x _ { 0 } ^ { ( i ) } \in \{ 0 , 1 \}$ $x \in 2 ^ { k }$ is shand $\begin{array} { r } { x _ { 0 } = \sum _ { i = 0 } ^ { k - 1 } x _ { 0 } ^ { ( i ) } 2 ^ { i } } \end{array}$ $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ . Now letlf-evident   
+115 that $x _ { 0 } ^ { ( i ) } = x _ { 0 } ^ { ( i ) } + 0 + 0$ is a valid secret sharing modulo two (albeit not a random one). Furthermore,   
+116 every party holding $x _ { 0 }$ can generate $x _ { 0 } ^ { ( i ) }$ . It is therefore possible for the parties to generate a secret   
+117 sharing modulo two of a single share modulo $2 ^ { k }$ . Repeating this for all shares and the computing   
+118 the addition as a binary circuit allows the parties to generate a secret sharing modulo two from a   
+119 secret sharing modulo $\bar { 2 } ^ { k }$ . Conversion in the other direction can be achieved using a similar technique   
+120 or using daBits as described by Rotaru and Wood [2019]. In the following we will use the term   
+121 mixed-circuit computation for any technique that works over both computation domains.   
+122 Quantization While Aliasgari et al. [2013] showed that it is possible to implement floating-point   
+123 computation, the cost is far higher than integer computation. It is therefore common to represent   
+124 fractional numbers using quantization (also called fixed-point representation) as suggested by Catrina   
+125 and Saxena [2010]. A real number $x$ is represented as $\bar { \boldsymbol { x } } = \lfloor \boldsymbol { x } \cdot \mathbf { \bar { 2 } } ^ { f } \rceil$ where $f$ is an integer specifying   
+126 the precision. The linearity of the representation allows to compute addition by simply adding the   
+127 representing integers. Multiplication however requires to adjust the result because it will have twice   
+128 the precision: $( x \cdot 2 ^ { f } ) \cdot ( y \cdot \dot { 2 } ^ { f } ) = x y \cdot 2 ^ { 2 f }$ . There are two ways to rectify this:
+
+• An obvious correction would be to shift the result by $f$ bits after adding $2 ^ { f - 1 }$ to the integer representation. This ensures rounding to the nearest number possible in the representation, with the tie being broken by rounding up. Dalskov et al. [2021] presented an efficient implementation of the truncation using mixed-circuit computation.
+
+• However, Catrina and Saxena have found that in the context of secure computation it is more efficient to use probabilistic truncation. This method rounds up or down probabilistically depending on the input. For example, probabilistically rounding 0.75 to an integer would see it rounded of up with probability 0.75 and down with probability 0.25. The probabilistic truncation is an effect of the fact that the operation involves the truncation of a randomized value, that is the computation of $\left\lfloor ( x + r ) / 2 ^ { m } \right\rfloor$ for a random $m$ -bit value $r$ . It is easy to see that
+
+$$
+\lfloor ( x + r ) / 2 ^ { m } \rfloor = \left\{ \begin{array} { l l } { \lfloor x / 2 ^ { m } \rfloor } & { ( x \bmod 2 ^ { m } + r ) < 2 ^ { m } } \\ { \lfloor x / 2 ^ { m } \rfloor + 1 } & { ( x \bmod 2 ^ { m } + r ) \ge 2 ^ { m } . } \end{array} \right.
+$$
+
+Therefore, the larger $( x \bmod 2 ^ { m } )$ ) is, the more likely the latter condition is true. Dalskov et al. [2020] present an efficient protocol in our security model.
+
+142 Our quantization scheme is related to quantized neural networks (see e.g. Hubara et al. [2016]).   
+143 However, our consideration is not to compress the model, but to improve the computational speed   
+144 and save communication cost.   
+145 Dot products Dot products are an essential building block of linear computation such as matrix   
+146 multiplication. In the light of quantization, it is possible to reduce the usage of truncation by deferring   
+147 after the summation. In other words, the dot product in the integer representations is computed before   
+148 truncating. This not only reduces the truncation error, it is also more efficient because the truncation   
+149 is the most expensive part in quantized secure multiplication. Similarly, our protocol allows to defer   
+150 the communication needed for multiplication. Let $\vec { x }$ and $\vec { y }$ be two vectors where the elements are   
+151 secret shared, that is, $\{ x ^ { ( i ) } \} = x _ { 0 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } + x _ { 2 } ^ { ( i ) }$ and similarly for $\boldsymbol y ^ { ( i ) }$ . The inner product then is
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { i } x ^ { ( i ) } \cdot y ^ { ( i ) } = \sum _ { i } ( x _ { 0 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } + x _ { 2 } ^ { ( i ) } ) \cdot ( y _ { 0 } ^ { ( i ) } + y _ { 1 } ^ { ( i ) } + y _ { 2 } ^ { ( i ) } ) } \\ & { \quad \quad \quad \quad = \displaystyle \sum _ { i } ( x _ { 0 } ^ { ( i ) } y _ { 0 } ^ { ( i ) } + x _ { 0 } ^ { ( i ) } y _ { 1 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } y _ { 0 } ^ { ( i ) } ) + \sum _ { i } ( x _ { 1 } ^ { ( i ) } y _ { 1 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } y _ { 2 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } y _ { 1 } ^ { ( i ) } ) } \\ & { \quad \quad \quad \quad + \displaystyle \sum _ { i } ( x _ { 2 } ^ { ( i ) } y _ { 2 } ^ { ( i ) } + x _ { 2 } ^ { ( i ) } y _ { 0 } ^ { ( i ) } + x _ { 0 } ^ { ( i ) } y _ { 2 } ^ { ( i ) } ) . } \end{array}
+$$
+
+152 The three sums in the last term can be compute locally by one party each before applying the same   
+153 protocol as for a single multiplication.   
+154 Comparisons Arithmetic secret sharing does not allow to access the individual bits directly. It   
+155 is therefore not straightforward to compute comparisons such as “less than”. There is a long line   
+156 of literature on how to achieve this going back to at least Damgård et al. [2006]. More recently,   
+157 most attention has been given to combine the power of arithmetic and binary secret sharing in   
+158 order to combine the best of worlds. One possibility to do so is to plainly convert to the binary   
+159 domain and compute the comparison circuit there. In our concrete implementation we use the more   
+160 efficient approach by Mohassel and Rindal [2018]. It starts by taking the difference of the two   
+161 inputs. Computing the comparison then reduces to comparing to zero, which in turn is equivalent   
+162 to extracting the most significant bit as it indicates the sign. The latter is achieved by converting   
+163 the shares locally to bit-wise sharing of the arithmetic shares, which sum up to the secret value. It   
+164 remains to compute the sum of the binary shares in order to come up with the most significant bit.   
+165 Oblivious Selection Plain secure computation does not allow branching because the parties would   
+166 need to be aware which branch is followed. Conditional assignment can be implemented as follows   
+167 however. If $b \in \{ 0 , 1 \}$ denotes the condition, $x + b \cdot ( y - x )$ is either $x$ or $y$ depending on $b$ . If the   
+168 condition is available in binary secret sharing but $x$ and $y$ in arithmetic secret sharing, $b$ has to be   
+169 converted to the latter. This can be done using a daBit as introduced by Rotaru and Wood [2019],   
+170 which is a secret random bit shared both in arithmetic and binary. It allows to mask a bit in one world   
+171 by XORing it. The result is then revealed and the masking is undone in the other world.   
+172 Division Catrina and Saxena [2010] have shown how to implement quantized division using the   
+173 algorithm by Goldschmidt [1964]. It mainly uses arithmetic and the probabilistic truncation already   
+174 explained. In addition, the initial approximation requires a full bit decomposition as described above.   
+175 The error of the output depends on the error in the multiplications used for Goldschmidt’s iteration,   
+176 which compounds in particular when using probabilistic truncation. Due to the nature of secure   
+177 computation, the result of division by zero is undefined. One could obtain a secret failure bit by   
+178 testing the divisor to zero. However, we found that not to be necessary in our algorithm. This is   
+179 because we only use division by secret value only for the softmax function where the it is guaranteed   
+180 to strictly positive.   
+181 Logarithm Computation logarithm with any public base can be reduced to logarithm to base two   
+182 using $\log _ { x } y = \log _ { 2 } y \cdot \log _ { x } 2$ . Aly and Smart [2019] have proposed computing $y = a \cdot 2 ^ { b }$ where   
+183 $a \in [ 0 . 5 , 1 )$ and $b \in \mathbb { Z }$ . This then allows to compute $\log _ { 2 } y = \log _ { 2 } a + b$ . Given the restricted range   
+184 of $a$ , $\log _ { 2 } a$ can be approximated using a division of polynomials. Numerical stability and input range   
+185 control are less of an issue here because we only use logarithm for the loss computation, which does   
+186 not influence the training.   
+187 Exponentiation By using $x ^ { y } = 2 ^ { y \log _ { 2 } x }$ , any exponentiation can be reduced to exponentiation   
+188 with base two. Aly and Smart [2019] have shown how to compute $2 ^ { a } = 2 ^ { \lfloor a \rfloor } \cdot 2 ^ { a - \lfloor a \rfloor }$ by computing   
+189 the two exponents using bit decomposition and the second factor using a polynomial approximation.   
+190 Regarding the first factor, if $\begin{array} { r } { b = \sum _ { i \geq 0 } b _ { i } 2 ^ { i } } \end{array}$ is an integer with $b _ { i } \in \{ 0 , 1 \}$ ,
+
+$$
+2 ^ { b } = 2 ^ { \sum _ { i \geq 0 } b _ { i } 2 ^ { i } } = \prod _ { i \geq 0 } 2 ^ { b _ { i } 2 ^ { i } } = \prod _ { i \geq 0 } ( 1 + b _ { i } \cdot ( 2 ^ { 2 ^ { i } } - 1 ) ) .
+$$
+
+191 As with division the numerical stability depends on the truncation used for multiplication.
+
+Inverse square root Aly and Smart [2019] have proposed to compute square root using Goldschmidt and Raphson-Newton iterations. We could combine this with the division above. However, Lu et al. [2020] have proposed a more direct computation that avoids running two successive iterations.
+
+Uniformly random fractional number Limiting ourselves to intervals of the form $[ x , x + 2 ^ { e } ]$ for a potentially negative integer $e$ , we can reduce the problem to generate a random $( f + { \dot { e } } )$ -bit number where $f$ is the fixed-point precision. Recall that we represent a fractional number $x$ as $\lfloor x \cdot 2 ^ { - f } \rceil$ Generating a random $n$ -bit number is straight-forward using random bits, which in our protocol can be generated as presented by Damgård et al. [2019]. In the context of our protocol however, Dalskov et al. [2021] have a presented a more efficient approach that involves mixed-circuit computation.
+
+Communication cost Table 1 show the total communication cost of some of the building blocks in our protocol for $f = 3 2$ . This setting mandates the modulus $2 ^ { 1 2 8 }$ because the division protocol requires bit length $4 f$ .
+
+# 4 Machine Learning Building Blocks
+
+In this section, we will use the building blocks in Section 3 to construct high-level computational modules for deep learning.
+
+Fully connected layers Both forward and back-propagation of fully connected layers can be seen as matrix multiplications and thus can be implemented using dot products. A particular challenge in secure computation is to compute a number of outputs in parallel in order to save communication
+
+Table 1: Communication cost of select computation for $f = 3 2$ and integer modulus $2 ^ { 1 2 8 }$   
+
+<table><tr><td></td><td>Bits</td></tr><tr><td>Integer multiplication</td><td>384</td></tr><tr><td>Probabilistic truncation Nearest truncation</td><td>1,536 4,462</td></tr><tr><td>Comparison</td><td>1,369</td></tr><tr><td>Division (prob. truncation)</td><td>29,866</td></tr><tr><td>Division (nearest truncation)</td><td>57,798</td></tr><tr><td>Exponentiation (prob. truncation)</td><td>77,684</td></tr><tr><td>Exponentiation (nearest truncation)</td><td></td></tr><tr><td></td><td>171,638</td></tr><tr><td>Invert square root (prob. truncation) Invert square root (nearest truncation)</td><td>20,073 27,699</td></tr></table>
+
+10 rounds. We solve this by having a dedicated infrastructure in our implementation that computes all dot   
+11 products for a matrix multiplication in a single batch, thus combining all necessary communication.
+
+2D convolution layers Similar to fully connected layers, 2D convolution and its corresponding gradient can be implemented using only dot products, and we again compute several output values in parallel.
+
+215 Rectified Linear Unit (ReLU) ReLU Nair and Hinton [2010] is defined as follows:
+
+$$
+{ \mathsf { R e L U } } ( x ) : = { \left\{ \begin{array} { l l } { x , } & { { \mathrm { i f ~ } } x > 0 } \\ { 0 . } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+
+216 It can thus be implemented as a comparison followed by an oblivious selection. For back-propagation,   
+217 it is advantageous to reuse the comparison results from forward propagation due to the relatively high   
+218 cost in secure computation. Note that the comparison results are stored in secret-shared form and   
+219 thus there is no reduction in security.   
+220 Max pooling Similar to ReLU, max pooling can be reduced to comparison and oblivious selection.   
+221 In secure computation, it saves communication rounds if the process uses a balanced tree rather than   
+222 iterating over all input values of one maximum computation. For back-propagation it again pays off   
+223 to the store intermediate results from forward propagation, again in secret-shared form.   
+224 Softmax and cross entropy loss This combination requires computing the following gradient for   
+225 back-propagation:
+
+$$
+\nabla i : = \frac { \partial \ell } { \partial x _ { i } } = \frac { \partial } { \partial x _ { i } } \Big ( - \sum _ { k } y _ { k } \cdot x _ { k } + \log \sum _ { j } e ^ { x _ { j } } \Big ) = - y _ { i } + \frac { e ^ { x _ { i } } } { \sum _ { j } e ^ { x _ { j } } } ,
+$$
+
+226 where $y _ { i }$ denotes an element of the ground truth as a one-hot vector, and $x _ { i }$ denotes the output of the   
+227 last layer.   
+228 On the right hand side of eq. (1), the values in the denominator are potentially large due to the use   
+229 of the exponential. This is prone to numerical overflow in our quantized representation because the   
+230 latter puts relatively strict limits on the values. We therefore optimize the computation by computing   
+231 the maximum of the input values:
+
+$$
+m = \operatorname* { m a x } _ { j } ( \{ x _ { j } \} ) .
+$$
+
+232 Then we compute
+
+$$
+{ \frac { e ^ { x _ { i } - m } } { \sum _ { j } e ^ { x _ { j } - m } } } = { \frac { e ^ { x _ { i } } e ^ { - m } } { ( \sum _ { j } e ^ { x _ { j } } ) e ^ { - m } } } = { \frac { e ^ { x _ { i } } } { \sum _ { j } e ^ { x _ { j } } } } .
+$$
+
+233 All the exponents on the left-most term are at most zero, and thus the dividend is at most one and the   
+234 divisor is at most the number of labels (which is 10 in MNIST). The same technique can be used to   
+235 compute the sigmoid activation function, as sigmoid(x) = 11+exp(−x) $\begin{array} { r } { \mathrm { \begin{array} { r } { \mathrm { i d } ( x ) = \frac { 1 } { 1 + \exp ( - x ) } = \frac { \exp ( 0 ) } { \exp ( 0 ) + \exp ( x ) } } } \end{array} } \end{array}$ is a special   
+236 case of softmax.   
+237 Stochastic gradient descent The model parameter update in SGD only involves basic arithmetic:   
+238 $\begin{array} { r } { \theta _ { j }  \theta _ { j } - \frac { \gamma } { B } \sum _ { i = 1 } ^ { B } \bigtriangledown _ { i j } } \end{array}$ where $\theta _ { j }$ is the parameter indexed by $j , B$ is the mini-batch size, $\gamma > 0$ is   
+239 the learning rate, and $\nabla i j$ is the gradient of the loss with respect to the ’th sample in the mini-batch   
+240 and the parameter $\theta _ { j }$ . In order to tackle the limited precision with quantization, we defer dividing by   
+241 the batch size to the model update. This means that we do not divide the gradient value by the batch   
+242 size when computing them as described in the previous paragraph. Instead, we divide the model   
+243 update by the batch size. Since we use a batch size that is a power of two (128), it is sufficient to use   
+244 probabilistic truncation instead of full-blown division. This saves both time and decreases the error.
+
+Adam The main difference to SGD in terms of basic computational operations is the additional use of an inverse square root. We again defer the division by the batch size to just before the model update.
+
+Parameter initialization We use the Glorot initialization by Glorot and Bengio [2010]. Besides basic operations, it mainly involves generating a uniformly random fractional value in a given interval.
+
+# 250 5 Implementation
+
+51 We built our implementation on MP-SPDZ by Keller [2020]. MP-SPDZ not only implements a   
+52 range of MPC protocols, it also comes with a high-level library containing the building blocks in   
+253 Section 3. MP-SPDZ already featured capabilities to train dense neural networks as well as inference   
+254 for convolutional neural networks. We have added backward propagation for a number of layer types,   
+255 including 2D convolution. Furthermore, we have corrected a bug in the backward propagation for   
+256 dense layers.
+
+MP-SPDZ allows implementing the computation in Python code, which is then compiled a specific bytecode. This code can be execute by a virtual machine executing the actual secure computation. The process allows to optimize the computation in the context of MPC.
+
+The framework also features an emulator that executes the exact computation that could be done securely in the clear. This allowed us to collect the accuracy figures in the next section at a lower cost.
+
+262 It is licensed under a BSD-style license, which allows to extend the code.
+
+# 63 6 MNIST
+
+For a concrete measurement of accuracy and running times, we have implemented training for the well-known MNIST dataset by LeCun et al. [2010]. We work mainly with the models that have been used by Wagh et al. [2019] with secure computation, and we will reuse their numbering (A–D). The models contain up to four linear layers. Network C is a convolutional neural network going back to the seminal work by LeCun et al. [1998] whereas the others are simpler networks that have been proposed by works on secure computation such as Mohassel and Zhang [2017], Liu et al. [2017], and Riazi et al. [2018]. We present the networks as Keras code in the supplemental material.
+
+Figure 2 shows the results for various quantization precisions and and the two rounding options. We have used SGD with learning rate 0.01, batch size 128, and the usual MNIST training/test split. $f = 6 4$ is the best option with probabilistic rounding, improving on both $f = 1 6$ and $f = 3 2$ Furthermore, nearest rounding performs worse that probabilistic for $f = 1 6$ and $f = 3 2$ . Due to the high cost, we only ran $f = 3 2$ with probabilistic rounding several times. The range is indicate by the shaded area. We focus on $f = 3 2$ because it offers the faster convergence.
+
+Figure 3 then shows the result with a variety of optimizers. While increasing the learning rate for SGD leads to a lower stability, Adam exposes a smoother learning learning curve albeit not a faster process.
+
+280 Finally, Figures 4 shows our results for all networks used by Wagh et al. [2019]. As one would expect,   
+281 the most sophisticated network performs best. Somewhat surprisingly, however, Network A (without   
+282 convolutional layers) performs better than the simpler networks containing convolutional layers.
+
+![](images/3a38cf90fbe641b15f4921c674ee686f614e2a42472767f16d74facf9f248ace.jpg)  
+Figure 2: Loss and accuracy for network C and precision options when running SGD with rate 0.01.
+
+![](images/ca865aaa95abbc18690e028b3ca52b226046ffa5e278339ad08eb557c5491c91.jpg)  
+Figure 3: Loss and accuracy for network C with various optimizer options, $f = 3 2$ , and probabilistic truncation.
+
+Resources We ran the emulator on AWS c5.9xlarge instances. One epoch takes a few second to several minutes depending on the model. Overall, we estimate that we have used a few weeks worth of computing time including experiments not included here because of bugs in the code.
+
+# 6.1 Secure computation
+
+In order to verify our emulation results, we have run Network C with precision $f \ = \ 3 2$ and probabilistic rounding in our actual multi-party computation protocol. We could verify that it converges on $9 8 . 5 \%$ accuracy at 150 epochs, taking 20 hours. Table 2 compares our result to previous works in a LAN setting. Note that Wagh et al. [2019] and Wagh et al. [2021] give accuracy figures. From personal communication with the authors and the fact that the source repository for the latter work2 says that their “code has not run end-to-end training”, we derive our assessment that their figures do not reflect the secure computation.
+
+![](images/700a37281e56f21023ee263453c7785f933f7176bea7c10046675acc845cf598.jpg)  
+Figure 4: Loss and accuracy for various networks, $f = 3 2$ , and probabilistic truncation.
+
+Table 2: Comparison to previous work in the LAN setting. $( ^ { * } )$ Mohassel and Zhang [2017] and Agrawal et al. [2019] use a different security model and are thus incomparable. We include them for completeness. Two numbers refer to online and offline time. Accuracy N/A means that the accuracy figures were not given or computed in a way that does not reflect the secure computation.   
+
+<table><tr><td>Network</td><td></td><td>Epoch time (s)</td><td>Acc. (# epochs)</td><td>Precision(f)</td></tr><tr><td rowspan="7">A</td><td>Mohassel and Zhang [2017]</td><td>283/19333*</td><td>93.4% (15)</td><td>13</td></tr><tr><td>Mohassel and Rindal [2018]</td><td>180</td><td>94.0% (15)</td><td>N/A</td></tr><tr><td>Agrawal et al. [2019]</td><td>31392*</td><td>95.0% (10)</td><td>N/A</td></tr><tr><td>Wagh et al. [2019]</td><td>247</td><td>N/A</td><td>13</td></tr><tr><td>Wagh et al. [2021]</td><td>41</td><td>N/A</td><td>13</td></tr><tr><td>Ours</td><td>31</td><td>97.9% (15)</td><td>16</td></tr><tr><td>Ours</td><td>50</td><td>97.7% (15)</td><td>32</td></tr><tr><td rowspan="4">B</td><td>Wagh et al. [2019]</td><td>4176</td><td>N/A</td><td>13</td></tr><tr><td>Wagh et al. [2021]</td><td>101</td><td>N/A</td><td>13</td></tr><tr><td>Ours</td><td>144</td><td>93.6% (15)</td><td>16</td></tr><tr><td>Ours</td><td>249</td><td>94.7% (15)</td><td>32</td></tr><tr><td rowspan="5">C</td><td>Wagh et al. [2019]</td><td>7188</td><td>N/A</td><td>13</td></tr><tr><td>Wagh et al. [2021]</td><td>891</td><td>N/A</td><td>13</td></tr><tr><td>Tan et al. [2021]</td><td>1036</td><td>94.0% (5)</td><td>20</td></tr><tr><td>Ours</td><td>344</td><td>94.9% (5)</td><td>16</td></tr><tr><td>Ours</td><td>643</td><td>93.8% (5)</td><td>32</td></tr><tr><td rowspan="3">D</td><td>Mohassel and Rindal [2018]</td><td>234</td><td>N/A</td><td>N/A</td></tr><tr><td>Ours</td><td>41</td><td>96.8% (15)</td><td>16</td></tr><tr><td>Ours</td><td>68</td><td>96.8% (15)</td><td>32</td></tr></table>
+
+# 294 7 Conclusions
+
+295 We have presented an implementation of deep learning training purely in multi-party computation   
+296 with extensive results on the accuracy. We have found that the lower precision of MPC computation   
+297 increases the error considerably. We only have considered one particular implementation of more   
+298 complex computation such as division and exponentiation, which are crucial to the learning process   
+299 as part of softmax. Future work might consider different approximations of these building blocks.
+
+#
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 7.   
+(c) Did you discuss any potential negative societal impacts of your work? [No] We only work with MNIST, and we believe that the potential negative impacts associated to it are very low.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include all code in the supplemental material, and the references include the URL for MNIST.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 6.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Only in Figure 2.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 6.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 6.   
+(b) Did you mention the license of the assets? [Yes] See the references and and Section 5.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Our code is included the supplemental material.   
+(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]

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+        "text": "Secure Quantized Training for Deep Learning ",
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+        "text": "Abstract ",
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+        "text": "1 We have implemented training of neural networks in secure multi-party computa  \n2 tion (MPC) using quantization commonly used in said setting. To the best of our   \n3 knowledge, we are the first to present training of MNIST purely implemented in   \n4 MPC that comes within one percent of accuracy of training using plaintext compu  \n5 tation. We found that training with MPC is possible, but it takes more epochs and   \n6 achieves a lower accuracy than the usual CPU/GPU computation. More concretely,   \n7 we have trained a network with two convolution and two dense layers to $9 8 . 5 \\%$   \n8 accuracy in 150 epochs. This took a day in our MPC implementation. ",
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+        "text": "9 1 Introduction ",
+        "text_level": 1,
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+        "text": "10 Secure multi-party computation (MPC) is a cryptographic technique that allows a set of parties to   \n11 compute a public output on private inputs without revealing the inputs or any intermediate results.   \n12 This makes it a potential solution to federated learning where the sample data stays private and only   \n13 the model or even only inference results are revealed.   \n14 Imagine a set of healthcare providers holding sensitive patient data. MPC allows them to collabo  \n15 ratively train a model. This model could then either be released or even kept private for inference   \n16 using MPC again. See Figure for an illustration. A more conceptual example is the well-known   \n17 millionaires’ problem where two people want to find out who is richer without revealing their wealth.   \n18 There is clearly a difference between the one bit of information desired and the full figures.   \n19 There has been a sustained interest in applying secure computation to machine learning and neural   \n20 networks going back to at least Barni et al. [2006]. More recent advantages in practical MPC have   \n21 led to an increased effort in implementing both inference and training.   \n22 A number of works such as Mohassel and Zhang [2017], Mohassel and Rindal [2018], Wagh et al.   \n23 [2019], Wagh et al. [2021] implement neural network training with MPC at least in parts. However,   \n24 they either give accuracy figures below $9 5 \\%$ or figures that have been obtained using plaintext training.   \n25 For the latter case, the works do not clarify how close the computation for plaintext training matches   \n26 the lower precision and other differences in the MPC setting. Agrawal et al. [2019] claim a higher   \n27 accuracy in a comparable setting for a convolutional neural network with more channels than we   \n28 use. However, they have only implemented dense layers, and we achieve comparable accuracy to   \n29 them with only dense layers. All works use quantization in the sense that a fractional number $x$ is   \n30 represented as $\\lfloor x \\cdot 2 ^ { - f } \\rceil$ . This makes addition considerably faster in the secure computation setting   \n31 because it reduces to integer addition. Furthermore, some of the works suggest to replace the softmax   \n32 function that uses exponentiation with a ReLU-based replacement. Keller and Sun [2020] have found   \n33 that this softmax replacement deteriorates accuracy in dense neural networks to the extent that it does   \n34 not justify the performance gains.   \n35 The concurrent work of Tan et al. [2021] gives some figures on the learning curve when run using   \n36 secure computation. However, they stop at five epochs for MNIST training where they achieve $94 \\%$   \n37 accuracy whereas we present the figures up to 150 epochs and $9 8 . 5 \\%$ accuracy. Furthermore, their   \n38 choice of fixed-point precision 20 is considerably below 32, which we found to be optimal. We also   \n39 found that our implementation is $40 \\%$ faster than theirs. Note that we use the CPU of one AWS   \n40 c5.9xlarge instance per party whereas Tan et al. use one NVIDIA Tesla V100 GPU per party. We   \n41 believe this somewhat counter-intuitive result comes from MPC heavily relying on communication,   \n42 which is an aspect where GPUs do not have an advantage over CPUs.   \n43 In this paper, we present an extensible framework for implementing deep learning training based on   \n44 MP-SPDZ by Keller [2020], a framework for multi-party computation.1 Similar to TensorFlow and   \n45 PyTorch, our approach allows representing deep learning models as succession of layers. We then   \n46 use this implementation to obtain accuracy figures for MNIST training by utilizing the MP-SPDZ   \n47 emulator, which allows to run the plaintext equivalent of secure computation, that is, the same   \n48 algorithms with the same precision. Finally, we run one of the most promising instantiation in real   \n49 secure computation in order to benchmark it confirm the result from the plaintext emulator.   \n50 There are a number of projects that integrate secure computation directly into popular machine   \n51 learning frameworks such as CrypTen by Gunning et al. [2019], PySyft by Ryffel et al. [2018], and   \n52 TF Encrypted by Dahl et al. [2018]. Our approach differs from all of them by running the protocol as   \n53 native CPU code (implemented using $\\mathrm { C } { + + }$ ). This allows for much faster execution. For example,   \n54 CrypTen provides an MNIST training example (mpc_autograd_cnn) that takes over one minute   \n55 to run one epoch with 100 samples on one machine. In comparison, our implementation takes 11   \n56 minutes to run one epoch with the full dataset of 60,000 samples.   \n57 Another line of work (e.g., Quoc et al. [2021]) uses trusted execution environments that provide   \n58 computation outside the reach of the operating system. This is a different security model to multi-party   \n59 computation that works with distributing the information among several entities.   \n60 The paper is structured as follows: After introducing the basics of the protocol we use in Section 2,   \n61 we will explain the mathematical building blocks in Section 3 and their use in the context of deep   \n62 learning in Section 4. Finally, we will present our implementation in Section 5 and our experimental   \n63 results for MNIST in Section 6. ",
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+        "text": "64 2 An Efficient Secure Multi-Party Computation Protocol ",
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+        "text": "There is a wide range of MPC protocols with a variety of security properties (see Keller [2020] for an overview). In this paper we focus on the setting of three-party computation with one semi-honest corruption. This means that out of the three parties two are expected to behave honestly, i.e., they follow the protocol and keep their view of the protocol secret, and one party is expected to follow the protocol but might try extract information from their view. The reason for choosing this setting is that it allows an efficient MPC protocol while still allowing secure outsourced computation. The concrete protocol we use goes back to Benaloh and Leichter [1990] with further aspects by Araki et al. [2016], Mohassel and Rindal [2018], and Eerikson et al. [2020]. We summarize the core protocol below. The mathematical building blocks in the next section mostly use the aspects below. ",
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+        "text": "74 Secret sharing All intermediate values in our protocol are stored using replicated secret sharing. A   \n75 secret value $x$ is a represented as a random sum $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ , and party $P _ { i }$ holds $( x _ { i - 1 } , x _ { i + 1 } )$   \n76 where the indices are computed modulo three. Clearly, each party is missing one value to compute   \n77 the sum. On the other hand, each pair of parties hold all necessary to reconstruct the secret. For a   \n78 uniformly random generation of shares, the computation domain has to be finite. Most commonly,   \n79 this domain is defined by integer computation modulo a number. We use $2 ^ { k }$ for $k$ being a multiple 64   \n80 and 2 as the moduli. The first case corresponds to an extension of 64-bit arithmetic found on most   \n81 processors. We will refer to the two settings as arithmetic and binary secret sharing throughout the   \n82 paper.   \n83 Input sharing The secret sharing scheme implies a protocol to share inputs where the inputting   \n84 party samples the shares and distributes them accordingly. Eerikson et al. [2020] have proposed a   \n85 more efficient protocol where the inputting party only needs to send one value instead of two pairs of   \n86 values. If $P _ { i }$ would like to input $x$ , $x _ { i }$ is set to zero, and $x _ { i - 1 }$ is generated with a pseudo-random   \n87 generator using a key previously shared between $P _ { i }$ and $P _ { i + 1 }$ . $P _ { i }$ can compute $x _ { i + 1 } = x - x _ { i - 1 }$ and   \n88 send it to $P _ { i - 1 }$ . While the resulting secret sharing is not entirely random, the fact that $P _ { i }$ already   \n89 knows $x$ makes randomizing $x _ { i }$ obsolete.   \n90 Addition The commutative nature of addition allows to add secret sharings without communication.   \n91 More concretely, secret sharings $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ and $y = y _ { 0 } + y _ { 1 } + y _ { 2 }$ imply the secret sharing   \n92 $x + y = ( x _ { 0 } + y _ { 0 } ) + ( x _ { 1 } + y _ { 1 } ) + ( x _ { 2 } + y _ { 2 } )$ . ",
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+        "text": "93 Multiplication The product of $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ and $y = y _ { 0 } + y _ { 1 } + y _ { 2 }$ is ",
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+        "text": "$$\n{ \\begin{array} { r l } & { x \\cdot y = ( x _ { 0 } + x _ { 1 } + x _ { 2 } ) \\cdot ( y _ { 0 } + y _ { 1 } + y _ { 2 } ) } \\\\ & { \\qquad = ( x _ { 0 } y _ { 0 } + x _ { 0 } y _ { 1 } + x _ { 1 } y _ { 0 } ) + ( x _ { 1 } y _ { 1 } + x _ { 1 } y _ { 2 } + x _ { 1 } y _ { 1 } ) + ( x _ { 2 } y _ { 2 } + x _ { 2 } y _ { 0 } + x _ { 0 } y _ { 2 } ) . } \\end{array} }\n$$",
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+        "text": "94 Each of the brackets only contains shares known by one of the parties. They can thus compute an   \n95 additive secret sharing (one summand per party) of the product. However, every party only holding   \n96 one share does not satisfy the replication requirement for further multiplications. It is not secure   \n97 for every party to pass their value on to another party because the summands are not distributed   \n98 randomly. This can be fixed by rerandomization: Let $x y = z _ { 0 } + z _ { 1 } + z _ { 2 }$ where $z _ { i }$ is know to $P _ { i }$ .   \n99 Every party $P _ { i }$ computes $z _ { i } ^ { \\prime } = z _ { i } + r _ { i , i + 1 } - r _ { i - 1 , i }$ where $r _ { i , i + 1 }$ is generated with a pseudo-random   \n100 generator using a key pre-shared between $P _ { i }$ and $P _ { i + 1 }$ . The resulting sum $x y = \\hat { z } _ { 0 } ^ { \\prime } + z _ { 1 } ^ { \\prime } + z _ { 2 } ^ { \\prime }$ is   \n101 pseudo-random, and it is thus secure for $P _ { i }$ to send $z _ { i } ^ { \\prime }$ to $P _ { i + 1 }$ in order to create a replicated secret   \n102 sharing $\\left( ( x y ) _ { i - 1 } , ( x y ) _ { i + 1 } \\right) = \\left( z _ { i } ^ { \\prime } , z _ { i - 1 } ^ { \\prime } \\right)$ . ",
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+        "text": "103 3 Secure Computation Building Blocks ",
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+        "text": "104 In this section, we will discuss how to implement computation with MPC with a focus on how   \n105 it differs from computation on CPUs or GPUs. Most of the techniques below are already known   \n106 individually. To the best of our knowledge however, we are the first to put them together in an efficient   \n107 and extensible framework for secure computation of deep learning training.   \n108 Domain conversion Recall we that we use computation modulo $2 ^ { k }$ for $k$ being a multiple of 64   \n109 as well as 1. Given that the main operations are just addition and multiplication in the respective   \n110 domain, it is desirable to compute integer arithmetic in the large domain but operations with a   \n111 straight-forward binary circuit modulo two. There has been a long-running interest in this going   \n112 back to least Kolesnikov et al. [2013]. We mainly rely on the approach proposed by Mohassel and   \n113 114 $\\{ x _ { 0 } ^ { ( i ) } \\} _ { j = 0 } ^ { k - 1 }$ 018] and Araki et al. [20the bit decomposition of $x _ { 0 }$ . Recal, that is, $x _ { 0 } ^ { ( i ) } \\in \\{ 0 , 1 \\}$ $x \\in 2 ^ { k }$ is shand $\\begin{array} { r } { x _ { 0 } = \\sum _ { i = 0 } ^ { k - 1 } x _ { 0 } ^ { ( i ) } 2 ^ { i } } \\end{array}$ $x = x _ { 0 } + x _ { 1 } + x _ { 2 }$ . Now letlf-evident   \n115 that $x _ { 0 } ^ { ( i ) } = x _ { 0 } ^ { ( i ) } + 0 + 0$ is a valid secret sharing modulo two (albeit not a random one). Furthermore,   \n116 every party holding $x _ { 0 }$ can generate $x _ { 0 } ^ { ( i ) }$ . It is therefore possible for the parties to generate a secret   \n117 sharing modulo two of a single share modulo $2 ^ { k }$ . Repeating this for all shares and the computing   \n118 the addition as a binary circuit allows the parties to generate a secret sharing modulo two from a   \n119 secret sharing modulo $\\bar { 2 } ^ { k }$ . Conversion in the other direction can be achieved using a similar technique   \n120 or using daBits as described by Rotaru and Wood [2019]. In the following we will use the term   \n121 mixed-circuit computation for any technique that works over both computation domains.   \n122 Quantization While Aliasgari et al. [2013] showed that it is possible to implement floating-point   \n123 computation, the cost is far higher than integer computation. It is therefore common to represent   \n124 fractional numbers using quantization (also called fixed-point representation) as suggested by Catrina   \n125 and Saxena [2010]. A real number $x$ is represented as $\\bar { \\boldsymbol { x } } = \\lfloor \\boldsymbol { x } \\cdot \\mathbf { \\bar { 2 } } ^ { f } \\rceil$ where $f$ is an integer specifying   \n126 the precision. The linearity of the representation allows to compute addition by simply adding the   \n127 representing integers. Multiplication however requires to adjust the result because it will have twice   \n128 the precision: $( x \\cdot 2 ^ { f } ) \\cdot ( y \\cdot \\dot { 2 } ^ { f } ) = x y \\cdot 2 ^ { 2 f }$ . There are two ways to rectify this: ",
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+        "text": "• An obvious correction would be to shift the result by $f$ bits after adding $2 ^ { f - 1 }$ to the integer representation. This ensures rounding to the nearest number possible in the representation, with the tie being broken by rounding up. Dalskov et al. [2021] presented an efficient implementation of the truncation using mixed-circuit computation. ",
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+        "text": "• However, Catrina and Saxena have found that in the context of secure computation it is more efficient to use probabilistic truncation. This method rounds up or down probabilistically depending on the input. For example, probabilistically rounding 0.75 to an integer would see it rounded of up with probability 0.75 and down with probability 0.25. The probabilistic truncation is an effect of the fact that the operation involves the truncation of a randomized value, that is the computation of $\\left\\lfloor ( x + r ) / 2 ^ { m } \\right\\rfloor$ for a random $m$ -bit value $r$ . It is easy to see that ",
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+        "text": "$$\n\\lfloor ( x + r ) / 2 ^ { m } \\rfloor = \\left\\{ \\begin{array} { l l } { \\lfloor x / 2 ^ { m } \\rfloor } & { ( x \\bmod 2 ^ { m } + r ) < 2 ^ { m } } \\\\ { \\lfloor x / 2 ^ { m } \\rfloor + 1 } & { ( x \\bmod 2 ^ { m } + r ) \\ge 2 ^ { m } . } \\end{array} \\right.\n$$",
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+        "text": "Therefore, the larger $( x \\bmod 2 ^ { m } )$ ) is, the more likely the latter condition is true. Dalskov et al. [2020] present an efficient protocol in our security model. ",
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+        "text": "142 Our quantization scheme is related to quantized neural networks (see e.g. Hubara et al. [2016]).   \n143 However, our consideration is not to compress the model, but to improve the computational speed   \n144 and save communication cost.   \n145 Dot products Dot products are an essential building block of linear computation such as matrix   \n146 multiplication. In the light of quantization, it is possible to reduce the usage of truncation by deferring   \n147 after the summation. In other words, the dot product in the integer representations is computed before   \n148 truncating. This not only reduces the truncation error, it is also more efficient because the truncation   \n149 is the most expensive part in quantized secure multiplication. Similarly, our protocol allows to defer   \n150 the communication needed for multiplication. Let $\\vec { x }$ and $\\vec { y }$ be two vectors where the elements are   \n151 secret shared, that is, $\\{ x ^ { ( i ) } \\} = x _ { 0 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } + x _ { 2 } ^ { ( i ) }$ and similarly for $\\boldsymbol y ^ { ( i ) }$ . The inner product then is ",
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+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { i } x ^ { ( i ) } \\cdot y ^ { ( i ) } = \\sum _ { i } ( x _ { 0 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } + x _ { 2 } ^ { ( i ) } ) \\cdot ( y _ { 0 } ^ { ( i ) } + y _ { 1 } ^ { ( i ) } + y _ { 2 } ^ { ( i ) } ) } \\\\ & { \\quad \\quad \\quad \\quad = \\displaystyle \\sum _ { i } ( x _ { 0 } ^ { ( i ) } y _ { 0 } ^ { ( i ) } + x _ { 0 } ^ { ( i ) } y _ { 1 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } y _ { 0 } ^ { ( i ) } ) + \\sum _ { i } ( x _ { 1 } ^ { ( i ) } y _ { 1 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } y _ { 2 } ^ { ( i ) } + x _ { 1 } ^ { ( i ) } y _ { 1 } ^ { ( i ) } ) } \\\\ & { \\quad \\quad \\quad \\quad + \\displaystyle \\sum _ { i } ( x _ { 2 } ^ { ( i ) } y _ { 2 } ^ { ( i ) } + x _ { 2 } ^ { ( i ) } y _ { 0 } ^ { ( i ) } + x _ { 0 } ^ { ( i ) } y _ { 2 } ^ { ( i ) } ) . } \\end{array}\n$$",
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+        "text": "152 The three sums in the last term can be compute locally by one party each before applying the same   \n153 protocol as for a single multiplication.   \n154 Comparisons Arithmetic secret sharing does not allow to access the individual bits directly. It   \n155 is therefore not straightforward to compute comparisons such as “less than”. There is a long line   \n156 of literature on how to achieve this going back to at least Damgård et al. [2006]. More recently,   \n157 most attention has been given to combine the power of arithmetic and binary secret sharing in   \n158 order to combine the best of worlds. One possibility to do so is to plainly convert to the binary   \n159 domain and compute the comparison circuit there. In our concrete implementation we use the more   \n160 efficient approach by Mohassel and Rindal [2018]. It starts by taking the difference of the two   \n161 inputs. Computing the comparison then reduces to comparing to zero, which in turn is equivalent   \n162 to extracting the most significant bit as it indicates the sign. The latter is achieved by converting   \n163 the shares locally to bit-wise sharing of the arithmetic shares, which sum up to the secret value. It   \n164 remains to compute the sum of the binary shares in order to come up with the most significant bit.   \n165 Oblivious Selection Plain secure computation does not allow branching because the parties would   \n166 need to be aware which branch is followed. Conditional assignment can be implemented as follows   \n167 however. If $b \\in \\{ 0 , 1 \\}$ denotes the condition, $x + b \\cdot ( y - x )$ is either $x$ or $y$ depending on $b$ . If the   \n168 condition is available in binary secret sharing but $x$ and $y$ in arithmetic secret sharing, $b$ has to be   \n169 converted to the latter. This can be done using a daBit as introduced by Rotaru and Wood [2019],   \n170 which is a secret random bit shared both in arithmetic and binary. It allows to mask a bit in one world   \n171 by XORing it. The result is then revealed and the masking is undone in the other world.   \n172 Division Catrina and Saxena [2010] have shown how to implement quantized division using the   \n173 algorithm by Goldschmidt [1964]. It mainly uses arithmetic and the probabilistic truncation already   \n174 explained. In addition, the initial approximation requires a full bit decomposition as described above.   \n175 The error of the output depends on the error in the multiplications used for Goldschmidt’s iteration,   \n176 which compounds in particular when using probabilistic truncation. Due to the nature of secure   \n177 computation, the result of division by zero is undefined. One could obtain a secret failure bit by   \n178 testing the divisor to zero. However, we found that not to be necessary in our algorithm. This is   \n179 because we only use division by secret value only for the softmax function where the it is guaranteed   \n180 to strictly positive.   \n181 Logarithm Computation logarithm with any public base can be reduced to logarithm to base two   \n182 using $\\log _ { x } y = \\log _ { 2 } y \\cdot \\log _ { x } 2$ . Aly and Smart [2019] have proposed computing $y = a \\cdot 2 ^ { b }$ where   \n183 $a \\in [ 0 . 5 , 1 )$ and $b \\in \\mathbb { Z }$ . This then allows to compute $\\log _ { 2 } y = \\log _ { 2 } a + b$ . Given the restricted range   \n184 of $a$ , $\\log _ { 2 } a$ can be approximated using a division of polynomials. Numerical stability and input range   \n185 control are less of an issue here because we only use logarithm for the loss computation, which does   \n186 not influence the training.   \n187 Exponentiation By using $x ^ { y } = 2 ^ { y \\log _ { 2 } x }$ , any exponentiation can be reduced to exponentiation   \n188 with base two. Aly and Smart [2019] have shown how to compute $2 ^ { a } = 2 ^ { \\lfloor a \\rfloor } \\cdot 2 ^ { a - \\lfloor a \\rfloor }$ by computing   \n189 the two exponents using bit decomposition and the second factor using a polynomial approximation.   \n190 Regarding the first factor, if $\\begin{array} { r } { b = \\sum _ { i \\geq 0 } b _ { i } 2 ^ { i } } \\end{array}$ is an integer with $b _ { i } \\in \\{ 0 , 1 \\}$ , ",
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+        "text": "$$\n2 ^ { b } = 2 ^ { \\sum _ { i \\geq 0 } b _ { i } 2 ^ { i } } = \\prod _ { i \\geq 0 } 2 ^ { b _ { i } 2 ^ { i } } = \\prod _ { i \\geq 0 } ( 1 + b _ { i } \\cdot ( 2 ^ { 2 ^ { i } } - 1 ) ) .\n$$",
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+        "type": "text",
+        "text": "191 As with division the numerical stability depends on the truncation used for multiplication. ",
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+        "text": "Inverse square root Aly and Smart [2019] have proposed to compute square root using Goldschmidt and Raphson-Newton iterations. We could combine this with the division above. However, Lu et al. [2020] have proposed a more direct computation that avoids running two successive iterations. ",
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+        "text": "Uniformly random fractional number Limiting ourselves to intervals of the form $[ x , x + 2 ^ { e } ]$ for a potentially negative integer $e$ , we can reduce the problem to generate a random $( f + { \\dot { e } } )$ -bit number where $f$ is the fixed-point precision. Recall that we represent a fractional number $x$ as $\\lfloor x \\cdot 2 ^ { - f } \\rceil$ Generating a random $n$ -bit number is straight-forward using random bits, which in our protocol can be generated as presented by Damgård et al. [2019]. In the context of our protocol however, Dalskov et al. [2021] have a presented a more efficient approach that involves mixed-circuit computation. ",
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+        "text": "Communication cost Table 1 show the total communication cost of some of the building blocks in our protocol for $f = 3 2$ . This setting mandates the modulus $2 ^ { 1 2 8 }$ because the division protocol requires bit length $4 f$ . ",
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+        "text": "4 Machine Learning Building Blocks ",
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+        "text": "In this section, we will use the building blocks in Section 3 to construct high-level computational modules for deep learning. ",
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+        "text": "Fully connected layers Both forward and back-propagation of fully connected layers can be seen as matrix multiplications and thus can be implemented using dot products. A particular challenge in secure computation is to compute a number of outputs in parallel in order to save communication ",
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+        "table_caption": [
+            "Table 1: Communication cost of select computation for $f = 3 2$ and integer modulus $2 ^ { 1 2 8 }$ "
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+        "table_body": "<table><tr><td></td><td>Bits</td></tr><tr><td>Integer multiplication</td><td>384</td></tr><tr><td>Probabilistic truncation Nearest truncation</td><td>1,536 4,462</td></tr><tr><td>Comparison</td><td>1,369</td></tr><tr><td>Division (prob. truncation)</td><td>29,866</td></tr><tr><td>Division (nearest truncation)</td><td>57,798</td></tr><tr><td>Exponentiation (prob. truncation)</td><td>77,684</td></tr><tr><td>Exponentiation (nearest truncation)</td><td></td></tr><tr><td></td><td>171,638</td></tr><tr><td>Invert square root (prob. truncation) Invert square root (nearest truncation)</td><td>20,073 27,699</td></tr></table>",
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+        "text": "10 rounds. We solve this by having a dedicated infrastructure in our implementation that computes all dot   \n11 products for a matrix multiplication in a single batch, thus combining all necessary communication. ",
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+        "text": "2D convolution layers Similar to fully connected layers, 2D convolution and its corresponding gradient can be implemented using only dot products, and we again compute several output values in parallel. ",
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+        "text": "215 Rectified Linear Unit (ReLU) ReLU Nair and Hinton [2010] is defined as follows: ",
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+        "text": "$$\n{ \\mathsf { R e L U } } ( x ) : = { \\left\\{ \\begin{array} { l l } { x , } & { { \\mathrm { i f ~ } } x > 0 } \\\\ { 0 . } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$",
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+        "type": "text",
+        "text": "216 It can thus be implemented as a comparison followed by an oblivious selection. For back-propagation,   \n217 it is advantageous to reuse the comparison results from forward propagation due to the relatively high   \n218 cost in secure computation. Note that the comparison results are stored in secret-shared form and   \n219 thus there is no reduction in security.   \n220 Max pooling Similar to ReLU, max pooling can be reduced to comparison and oblivious selection.   \n221 In secure computation, it saves communication rounds if the process uses a balanced tree rather than   \n222 iterating over all input values of one maximum computation. For back-propagation it again pays off   \n223 to the store intermediate results from forward propagation, again in secret-shared form.   \n224 Softmax and cross entropy loss This combination requires computing the following gradient for   \n225 back-propagation: ",
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+        "img_path": "images/eae777e4bc355b0a5c52d65a7001eadc3f0665c189d84dfb531be701ff493e40.jpg",
+        "text": "$$\n\\nabla i : = \\frac { \\partial \\ell } { \\partial x _ { i } } = \\frac { \\partial } { \\partial x _ { i } } \\Big ( - \\sum _ { k } y _ { k } \\cdot x _ { k } + \\log \\sum _ { j } e ^ { x _ { j } } \\Big ) = - y _ { i } + \\frac { e ^ { x _ { i } } } { \\sum _ { j } e ^ { x _ { j } } } ,\n$$",
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+        "type": "text",
+        "text": "226 where $y _ { i }$ denotes an element of the ground truth as a one-hot vector, and $x _ { i }$ denotes the output of the   \n227 last layer.   \n228 On the right hand side of eq. (1), the values in the denominator are potentially large due to the use   \n229 of the exponential. This is prone to numerical overflow in our quantized representation because the   \n230 latter puts relatively strict limits on the values. We therefore optimize the computation by computing   \n231 the maximum of the input values: ",
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+        "text": "$$\nm = \\operatorname* { m a x } _ { j } ( \\{ x _ { j } \\} ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "232 Then we compute ",
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+        "text": "$$\n{ \\frac { e ^ { x _ { i } - m } } { \\sum _ { j } e ^ { x _ { j } - m } } } = { \\frac { e ^ { x _ { i } } e ^ { - m } } { ( \\sum _ { j } e ^ { x _ { j } } ) e ^ { - m } } } = { \\frac { e ^ { x _ { i } } } { \\sum _ { j } e ^ { x _ { j } } } } .\n$$",
+        "text_format": "latex",
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+        "text": "233 All the exponents on the left-most term are at most zero, and thus the dividend is at most one and the   \n234 divisor is at most the number of labels (which is 10 in MNIST). The same technique can be used to   \n235 compute the sigmoid activation function, as sigmoid(x) = 11+exp(−x) $\\begin{array} { r } { \\mathrm { \\begin{array} { r } { \\mathrm { i d } ( x ) = \\frac { 1 } { 1 + \\exp ( - x ) } = \\frac { \\exp ( 0 ) } { \\exp ( 0 ) + \\exp ( x ) } } } \\end{array} } \\end{array}$ is a special   \n236 case of softmax.   \n237 Stochastic gradient descent The model parameter update in SGD only involves basic arithmetic:   \n238 $\\begin{array} { r } { \\theta _ { j }  \\theta _ { j } - \\frac { \\gamma } { B } \\sum _ { i = 1 } ^ { B } \\bigtriangledown _ { i j } } \\end{array}$ where $\\theta _ { j }$ is the parameter indexed by $j , B$ is the mini-batch size, $\\gamma > 0$ is   \n239 the learning rate, and $\\nabla i j$ is the gradient of the loss with respect to the ’th sample in the mini-batch   \n240 and the parameter $\\theta _ { j }$ . In order to tackle the limited precision with quantization, we defer dividing by   \n241 the batch size to the model update. This means that we do not divide the gradient value by the batch   \n242 size when computing them as described in the previous paragraph. Instead, we divide the model   \n243 update by the batch size. Since we use a batch size that is a power of two (128), it is sufficient to use   \n244 probabilistic truncation instead of full-blown division. This saves both time and decreases the error. ",
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+    {
+        "type": "text",
+        "text": "Adam The main difference to SGD in terms of basic computational operations is the additional use of an inverse square root. We again defer the division by the batch size to just before the model update. ",
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+    {
+        "type": "text",
+        "text": "Parameter initialization We use the Glorot initialization by Glorot and Bengio [2010]. Besides basic operations, it mainly involves generating a uniformly random fractional value in a given interval. ",
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+    {
+        "type": "text",
+        "text": "250 5 Implementation ",
+        "text_level": 1,
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+        "text": "51 We built our implementation on MP-SPDZ by Keller [2020]. MP-SPDZ not only implements a   \n52 range of MPC protocols, it also comes with a high-level library containing the building blocks in   \n253 Section 3. MP-SPDZ already featured capabilities to train dense neural networks as well as inference   \n254 for convolutional neural networks. We have added backward propagation for a number of layer types,   \n255 including 2D convolution. Furthermore, we have corrected a bug in the backward propagation for   \n256 dense layers. ",
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+    {
+        "type": "text",
+        "text": "MP-SPDZ allows implementing the computation in Python code, which is then compiled a specific bytecode. This code can be execute by a virtual machine executing the actual secure computation. The process allows to optimize the computation in the context of MPC. ",
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+    {
+        "type": "text",
+        "text": "The framework also features an emulator that executes the exact computation that could be done securely in the clear. This allowed us to collect the accuracy figures in the next section at a lower cost. ",
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+        "type": "text",
+        "text": "262 It is licensed under a BSD-style license, which allows to extend the code. ",
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+        "type": "text",
+        "text": "63 6 MNIST ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For a concrete measurement of accuracy and running times, we have implemented training for the well-known MNIST dataset by LeCun et al. [2010]. We work mainly with the models that have been used by Wagh et al. [2019] with secure computation, and we will reuse their numbering (A–D). The models contain up to four linear layers. Network C is a convolutional neural network going back to the seminal work by LeCun et al. [1998] whereas the others are simpler networks that have been proposed by works on secure computation such as Mohassel and Zhang [2017], Liu et al. [2017], and Riazi et al. [2018]. We present the networks as Keras code in the supplemental material. ",
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+    {
+        "type": "text",
+        "text": "Figure 2 shows the results for various quantization precisions and and the two rounding options. We have used SGD with learning rate 0.01, batch size 128, and the usual MNIST training/test split. $f = 6 4$ is the best option with probabilistic rounding, improving on both $f = 1 6$ and $f = 3 2$ Furthermore, nearest rounding performs worse that probabilistic for $f = 1 6$ and $f = 3 2$ . Due to the high cost, we only ran $f = 3 2$ with probabilistic rounding several times. The range is indicate by the shaded area. We focus on $f = 3 2$ because it offers the faster convergence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 then shows the result with a variety of optimizers. While increasing the learning rate for SGD leads to a lower stability, Adam exposes a smoother learning learning curve albeit not a faster process. ",
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+        "type": "text",
+        "text": "280 Finally, Figures 4 shows our results for all networks used by Wagh et al. [2019]. As one would expect,   \n281 the most sophisticated network performs best. Somewhat surprisingly, however, Network A (without   \n282 convolutional layers) performs better than the simpler networks containing convolutional layers. ",
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+        "image_caption": [
+            "Figure 2: Loss and accuracy for network C and precision options when running SGD with rate 0.01. "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/ca865aaa95abbc18690e028b3ca52b226046ffa5e278339ad08eb557c5491c91.jpg",
+        "image_caption": [
+            "Figure 3: Loss and accuracy for network C with various optimizer options, $f = 3 2$ , and probabilistic truncation. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Resources We ran the emulator on AWS c5.9xlarge instances. One epoch takes a few second to several minutes depending on the model. Overall, we estimate that we have used a few weeks worth of computing time including experiments not included here because of bugs in the code. ",
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+    {
+        "type": "text",
+        "text": "6.1 Secure computation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to verify our emulation results, we have run Network C with precision $f \\ = \\ 3 2$ and probabilistic rounding in our actual multi-party computation protocol. We could verify that it converges on $9 8 . 5 \\%$ accuracy at 150 epochs, taking 20 hours. Table 2 compares our result to previous works in a LAN setting. Note that Wagh et al. [2019] and Wagh et al. [2021] give accuracy figures. From personal communication with the authors and the fact that the source repository for the latter work2 says that their “code has not run end-to-end training”, we derive our assessment that their figures do not reflect the secure computation. ",
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+    {
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+        "img_path": "images/700a37281e56f21023ee263453c7785f933f7176bea7c10046675acc845cf598.jpg",
+        "image_caption": [
+            "Figure 4: Loss and accuracy for various networks, $f = 3 2$ , and probabilistic truncation. "
+        ],
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+        "table_caption": [
+            "Table 2: Comparison to previous work in the LAN setting. $( ^ { * } )$ Mohassel and Zhang [2017] and Agrawal et al. [2019] use a different security model and are thus incomparable. We include them for completeness. Two numbers refer to online and offline time. Accuracy N/A means that the accuracy figures were not given or computed in a way that does not reflect the secure computation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td></td><td>Epoch time (s)</td><td>Acc. (# epochs)</td><td>Precision(f)</td></tr><tr><td rowspan=\"7\">A</td><td>Mohassel and Zhang [2017]</td><td>283/19333*</td><td>93.4% (15)</td><td>13</td></tr><tr><td>Mohassel and Rindal [2018]</td><td>180</td><td>94.0% (15)</td><td>N/A</td></tr><tr><td>Agrawal et al. [2019]</td><td>31392*</td><td>95.0% (10)</td><td>N/A</td></tr><tr><td>Wagh et al. [2019]</td><td>247</td><td>N/A</td><td>13</td></tr><tr><td>Wagh et al. [2021]</td><td>41</td><td>N/A</td><td>13</td></tr><tr><td>Ours</td><td>31</td><td>97.9% (15)</td><td>16</td></tr><tr><td>Ours</td><td>50</td><td>97.7% (15)</td><td>32</td></tr><tr><td rowspan=\"4\">B</td><td>Wagh et al. [2019]</td><td>4176</td><td>N/A</td><td>13</td></tr><tr><td>Wagh et al. [2021]</td><td>101</td><td>N/A</td><td>13</td></tr><tr><td>Ours</td><td>144</td><td>93.6% (15)</td><td>16</td></tr><tr><td>Ours</td><td>249</td><td>94.7% (15)</td><td>32</td></tr><tr><td rowspan=\"5\">C</td><td>Wagh et al. [2019]</td><td>7188</td><td>N/A</td><td>13</td></tr><tr><td>Wagh et al. [2021]</td><td>891</td><td>N/A</td><td>13</td></tr><tr><td>Tan et al. [2021]</td><td>1036</td><td>94.0% (5)</td><td>20</td></tr><tr><td>Ours</td><td>344</td><td>94.9% (5)</td><td>16</td></tr><tr><td>Ours</td><td>643</td><td>93.8% (5)</td><td>32</td></tr><tr><td rowspan=\"3\">D</td><td>Mohassel and Rindal [2018]</td><td>234</td><td>N/A</td><td>N/A</td></tr><tr><td>Ours</td><td>41</td><td>96.8% (15)</td><td>16</td></tr><tr><td>Ours</td><td>68</td><td>96.8% (15)</td><td>32</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "294 7 Conclusions ",
+        "text_level": 1,
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+        "text": "295 We have presented an implementation of deep learning training purely in multi-party computation   \n296 with extensive results on the accuracy. We have found that the lower precision of MPC computation   \n297 increases the error considerably. We only have considered one particular implementation of more   \n298 complex computation such as division and exponentiation, which are crucial to the learning process   \n299 as part of softmax. Future work might consider different approximations of these building blocks. ",
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+        "text": "00 References   \n301 N. Agrawal, A. S. Shamsabadi, M. J. Kusner, and A. Gascón. QUOTIENT: Two-party secure neural   \n302 network training and prediction. In L. Cavallaro, J. Kinder, X. Wang, and J. Katz, editors, ACM   \n303 CCS 2019, pages 1231–1247. ACM Press, Nov. 2019. doi: 10.1145/3319535.3339819.   \n304 M. Aliasgari, M. Blanton, Y. Zhang, and A. Steele. Secure computation on floating point numbers.   \n305 In NDSS 2013. The Internet Society, Feb. 2013.   \n306 A. Aly and N. P. Smart. Benchmarking privacy preserving scientific operations. In R. H. Deng,   \n307 V. Gauthier-Umaña, M. Ochoa, and M. Yung, editors, ACNS 19, volume 11464 of LNCS, pages   \n308 509–529. Springer, Heidelberg, June 2019. doi: 10.1007/978-3-030-21568-2_25.   \n309 T. Araki, J. Furukawa, Y. Lindell, A. Nof, and K. Ohara. High-throughput semi-honest secure   \n310 three-party computation with an honest majority. In E. R. Weippl, S. Katzenbeisser, C. Kruegel,   \n311 A. C. Myers, and S. 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