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+# METANORM: LEARNING TO NORMALIZE FEW-SHOT BATCHES ACROSS DOMAINS
+
+Yingjun $\mathbf { D } \mathbf { u } ^ { 1 }$ , Xiantong $\mathbf { Z } \mathbf { h e n } ^ { 1 , 2 }$ , Ling Shao2, Cees G. M. Snoek1 1AIM Lab, University of Amsterdam 2Inception Institute of Artificial Intelligence
+
+# ABSTRACT
+
+Batch normalization plays a crucial role when training deep neural networks. However, batch statistics become unstable with small batch sizes and are unreliable in the presence of distribution shifts. We propose MetaNorm, a simple yet effective meta-learning normalization. It tackles the aforementioned issues in a unified way by leveraging the meta-learning setting and learns to infer adaptive statistics for batch normalization. MetaNorm is generic, flexible and model-agnostic, making it a simple plug-and-play module that is seamlessly embedded into existing meta-learning approaches. It can be efficiently implemented by lightweight hypernetworks with low computational cost. We verify its effectiveness by extensive evaluation on representative tasks suffering from the small batch and domain shift problems: few-shot learning and domain generalization. We further introduce an even more challenging setting: few-shot domain generalization. Results demonstrate that MetaNorm consistently achieves better, or at least competitive, accuracy compared to existing batch normalization methods.
+
+# 1 INTRODUCTION
+
+Batch normalization (Ioffe & Szegedy, 2015) is crucial for training neural networks, and with its variants, e.g., layer normalization (Ba et al., 2016), group normalization (Wu & He, 2018) and instance normalization (Ulyanov et al., 2016), has thus become an essential part of the deep learning toolkit (Bjorck et al., 2018; Luo et al., 2018a; Yang et al., 2019; Jia et al., 2019; Luo et al., 2018b; Summers & Dinneen, 2020). Batch normalization helps stabilize the distribution of internal activations when a model is being trained. Given a mini-batch $\boldsymbol { B }$ , the normalization is conducted along each individual feature channel for 2D convolutional neural networks. During training, the batch normalization moments are calculated as follows:
+
+$$
+\mu _ { B } = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } a _ { i } , \sigma _ { B } ^ { 2 } = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } ( a _ { i } - \mu _ { B } ) ^ { 2 } ,
+$$
+
+where $a _ { i }$ indicates the $i$ -th element of the $M$ activations in the batch, $M = | \boldsymbol { B } | \times H \times W$ , in which $H$ and $W$ are the height and width of the feature map in each channel. We can now apply the normalization statistics to each activation:
+
+$$
+a _ { i } ^ { \prime }  \mathbf { B N } ( a _ { i } ) \equiv \gamma \hat { a } _ { i } + \beta , \quad \mathrm { w h e r e , } \quad \hat { a } _ { i } = \frac { a _ { i } - \mu _ { \mathcal { B } } } { \sqrt { \sigma _ { \mathcal { B } } ^ { 2 } + \epsilon } } ,
+$$
+
+where $\gamma$ and $\beta$ are parameters learned during training, $\epsilon$ is a small scalar to prevent division by 0, and operations between vectors are element-wise. At test time, the standard practice is to normalize activations using the moving average over mini-batch means $\mu _ { B }$ and variance $\bar { \sigma } _ { B } ^ { 2 }$ . Batch normalization is based on an implicit assumption that the samples in the dataset are independent and identically distributed. However, this assumption does not hold in challenging settings like few-shot learning and domain generalization. In this paper, we strive for batch normalization when batches are of small size and suffer from distributions shifts between source and target domains.
+
+Batch normalization for few-shot learning and domain generalization problems have so far been considered separately, predominantly in a meta-learning setting. For few-shot meta-learning (Finn et al., 2017; Gordon et al., 2019), most existing methods rely critically on transductive batch normalization, except those based on prototypes (Snell et al., 2017; Allen et al., 2019; Zhen et al., 2020a). However, the nature of transductive learning restricts its application due to the requirement to sample from the test set. To address this issue, Bronskill et al. (2020) proposes TaskNorm, which leverages other statistics from both layer and instance normalization. As a non-transductive normalization approach, it achieves impressive performance and outperforms conventional batch normalization (Ioffe & Szegedy, 2015). However, its performance is not always performing better than transductive batch normalization. Meanwhile, domain generalization (Muandet et al., 2013; Balaji et al., 2018; Li et al., 2017a;b) suffers from distribution shifts from training to test, which makes it problematic to directly apply statistics calculated from a seen domain to test data from unseen domains (Wang et al., 2019; Seo et al., 2019). Recent works deal with this problem by learning a domain specific normalization (Chang et al., 2019; Seo et al., 2019) or a transferable normalization in place of existing normalization techniques (Wang et al., 2019). We address the batch normalization challenges for few-shot classification and domain generalization in a unified way by learning a new batch normalization under the meta-learning setting.
+
+We propose MetaNorm, a simple but effective meta-learning normalization. We leverage the metalearning setting and learn to infer normalization statistics from data, instead of applying direct calculations or blending various normalization statistics. MetaNorm is a general batch normalization approach, which is model-agnostic and serves as a plug-and-play module that can be seamlessly embedded into existing meta-learning approaches. We demonstrate its effectiveness for few-shot classification and domain generalization, where it learns task-specific statistics from limited data samples in the support set for each few-shot task; and it can also learn to generate domain-specific statistics from the seen source domains for unseen target domains. We verify the effectiveness of MetaNorm by extensive evaluation on few-shot classification and domain generalization tasks. For few-shot classification, we experiment with representative gradient, metric and model-based meta-learning approaches on fourteen benchmark datasets. For domain generalization, we evaluate the model on three widely-used benchmarks for cross-domain visual object classification. Last but not least, we introduce the challenging new task of few-shot domain generalization, which combines the challenges of both few-shot learning and domain generalization. The experimental results demonstrate the benefit of MetaNorm compared to existing batch normalizations.
+
+# 2 RELATED WORKS
+
+Transductive Batch Normalization For conventional batch normalization under supervised settings, i.i.d. assumptions about the data distribution imply that estimating moments from the training set will provide appropriate normalization statistics for test data. However, in the meta-learning scenario data points are only assumed to be i.i.d. within a specific task. Therefore, it is critical to select the moments when batch normalization is applied to support and query set data points during meta training and meta testing. Hence, in the recent meta-learning literature the running moments are no longer used for normalization at meta-test time, but instead replaced with support/query set statistics. These statistics are used for normalization, both at meta-train and meta-test time. This approach is referred to as transductive batch normalization (TBN) (Bronskill et al., 2020). Competitive meta-learning methods (e.g., Gordon et al., 2019; Finn et al., 2017; Zhen et al., 2020b) rely on TBN to achieve state-of-the-art performance. However, there are two critical problems with TBN. First, TBN is sensitive to the distribution over the query set used during meta-training, and as such is less generally applicable than non-transductive learning. Second, TBN uses extra information for multiple test samples, compared to non-transductive batch normalization at prediction time, which could be problematic as we are not guaranteed to have a set of test samples available during training in practical applications. In contrast, MetaNorm is a non-transductive normalization. It generates statistics from the support set only, without relying on query samples, making it more practical.
+
+Meta Batch Normalization To address the problem of transductive batch normalization and improve conventional batch normalization, meta-batch normalization (MetaBN) was introduced (Triantafillou et al., 2020; Bronskill et al., 2020). In MetaBN, the support set alone is used to compute the normalization statistics for both the support and query sets at both meta-training and meta-test time. MetaBN is non-transductive since the normalization of a test input does not depend on other test inputs in the query set. However, Bronskill et al. (2020) observe that MetaBN performs less well for small-sized support sets. This leads to high variance in moment estimates, which is similar to the difficulty of using batch normalization with small-batch training (Wu & He, 2018). To address this issue, Bronskill et al. (2020) proposed TaskNorm, which learns to combine statistics from both layer normalization and instance normalization, with a lending parameter to be learned at meta-train time. As a non-transductive normalization, TaskNorm achieves impressive performance, outperforming conventional batch normalization. However, it can not always perform better than transductive batch normalization. TaskNorm indicates non-transductive batch normalization estimates proper normalization statistics by involving learning in the normalization process. We also propose to learn batch normalization within the meta-learning framework, but instead of employing a learnable combination of existing normalization statistics, we directly learn to infer statistics from data. At meta-train time, the model learns to acquire the ability to generate statistics only from the support set and at meta-test time we directly apply the model to infer statistics for new tasks.
+
+Batch Normalization for Domain Adaptation and Domain Generalization Domain adaption suffers from a distribution shift between source and target domains, which makes it sub-optimal to directly apply batch normalization (Bilen & Vedaldi, 2017). Li et al. (2016) proposed adaptive batch normalization to increase the generalization ability of a deep neural network. By modulating the statistical information of all batch normalization layers in the neural network, it achieves deep adaptation effects for domain-adaptive tasks. Nado et al. (2020) noted the possibility of accessing small unlabeled batches of the shifted data just before prediction time. To improve model accuracy and calibration under covariate shift, they proposed prediction-time batch normalization. Since the activation statistics obtained during training do not reflect statistics of the test distribution, when testing in an out-of-distribution environment, Schneider et al. (2020) proposed estimating the batch statistics on the corrupted images. Kaku et al. (2020) demonstrated that standard non-adaptive feature normalization fails to correctly normalize the features of convolutional neural networks on held-out data where extraneous variables take values not seen during training. Learning domain-specific batch normalization has been explored (Chang et al., 2019; Wang et al., 2019). Wang et al. (2019) introduced transferable normalization, TransNorm, which normalizes the feature representations from source and target domain separately using domain-specific statistics. Along a similar vein, Chang et al. (2019) proposed a domain-specific batch normalization layer, which consists of two branches, each in charge of a single domain exclusively. The hope is that, through the normalization, the feature representation will become domain invariant. Nevertheless, these normalization methods are specifically designed for domain adaptation tasks, where data from target domains are available, though often unlabelled. This makes them inapplicable to domain generalization tasks where data from target domains are inaccessible at training time. Seo et al. (2019) proposed learning to optimize domain specific normalization for domain generalization tasks. Under the meta-learning settings, a mixture of different normalization techniques is optimized for each domain, where the mixture weights are learned specifically for different domains. Instead of combining different normalization statistics, MetaNorm learns from data to generate adaptive statistics specific to each domain. Moreover, we introduce an even more challenging setting, i.e., few-shot domain generalization, which combines the challenges of few-shot classification and domain generalization.
+
+Conditional Batch Normalization de Vries et al. (2017) proposed conditional batch normalization to modulate visual processing by predicting the scalars $\gamma$ and $\beta$ of the batch normalization conditioned on the language from an early processing stage. Conditional batch normalization has also been applied to align different data distributions for domain adaptation (Li et al., 2016). Oreshkin et al. (2018) applies conditional batch normalization to metric-based models for the few-shot classification task. Tseng et al. (2020) proposed a learning-to-learn method to optimize the hyper-parameters of the feature-wise transformation layers by conditional batch normalization for cross-domain classification. Unlike conditional batch normalization, we use extra data (the query set) to generate normalization statistics under the meta-learning setting, rather than the scalars.
+
+# 3 METHODOLOGY
+
+We view finding appropriate statistics for batch normalization as a density estimation problem. We need to infer the distribution parameters, such as, $\mu$ and $\sigma$ when a Gaussian distribution is presumed, as in existing batch normalization approaches. The motivation behind MetaNorm is to leverage the meta-learning setting and learn from data to generate adaptive normalization statistics. MetaNorm is generic and model-agnostic, addressing batch normalization in a unified way for different settings by minimizing the KL divergence, which is a common metric to measure the difference between two
+
+probability distributions:
+
+$$
+D _ { \mathrm { K L } } \big [ q _ { \phi } ( m ) | p _ { \theta } ( m ) \big ] ,
+$$
+
+where $m$ is a random variable that represents the distribution of activations, $p _ { \theta } ( m )$ and $q _ { \phi } ( m )$ are defined as Gaussian distributions with different implementations depending on the task of interest, e.g., few-shot classification or domain generalization. We leverage the amortized inference technique (Kingma & Welling, 2013) and implement this by inference networks. To be more specific, for each individual channel in each $\ell$ convolutional layer, we infer the moments $\mu$ and $\sigma$ by $f _ { \mu } ^ { \ell } ( \cdot )$ and $f _ { \sigma } ^ { \ell } ( \cdot )$ , respectively, which are realized as multi-layer perceptrons and we call hypernetworks (Ha et al., 2016). Hypernetworks use one network to generate the weights for another network. Our hypernetworks generate the statistics from data by using amortization techniques.
+
+We simply incorporate the $D _ { \mathrm { K L } }$ term into the optimization of the existing model with the cross-entropy loss $\mathcal { L } _ { \mathrm { C E } }$ , resulting in a general loss function as follows:
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathrm { C E } } - \lambda D _ { \mathrm { K L } } \big [ q _ { \phi } ( m ) | p _ { \theta } ( m ) \big ]
+$$
+
+where $\lambda > 0$ is a regularization hyper-parameter.
+
+MetaNorm for Few-Shot Classification In the few-shot classification scenario, we define the $\mathcal { C }$ -way $K$ -shot problem using the episodic formulation from (Vinyals et al., 2016). Each task $\mathcal { T } _ { i }$ is a classification problem sampled from a task distribution $p ( \tau )$ . The tasks are divided into a training meta-set ${ \mathcal { T } } ^ { t r }$ , validation meta-set ${ \mathcal { T } } ^ { \nu a l }$ , and test meta-set $\mathcal { T } ^ { t e s t }$ , each with a disjoint set of target classes (i.e., a class seen during testing is not seen during training). The validation meta-set is used for model selection, and the testing meta-set is used only for final evaluation. Each task instance $\mathcal { T } _ { i } \sim p \left( \mathcal { T } \right)$ is composed of a support set $s$ and a query set $\mathcal { Q }$ , and only contains $N$ classes randomly selected from the appropriate meta-set.
+
+We aim to infer statistics from the support set that better match the query set. Therefore, we adopt a straightforward criterion for the inference:
+
+$$
+D _ { \mathrm { K L } } \big [ q _ { \phi } ( m | S ) | | p _ { \theta } ( m | Q ) \big ] ,
+$$
+
+where we define $q ( m | S ) { = } N ( \mu _ { S } , \sigma _ { S } )$ and $p ( m | Q ) { = } { \mathcal { N } } ( \mu _ { Q } , \sigma _ { Q } )$ , which are the distributions inferred from the support and query sets in a few-shot learning task. By minimizing the KL term in conjunction with the prime objective of a meta-learning algorithm, we are able to find the appropriate statistics from limited data samples for batch normalization. The KL term adheres to a closed form, which makes it easy to implement and computationally efficient. The $p ( m | Q )$ can be estimated by directly calculating statistics using the query set, which however performs inferior to inference by optimization. We note the inference from the query set only happens during meta-training time and we use the learned inference network to generate normalization statistics at meta-test time for a test task using its support set.
+
+To infer $\mu _ { S }$ , we deploy an inference function $f _ { \mu } ^ { \ell } ( \cdot )$ that takes activations of a sample as input, and the outputs from all samples are then averaged as the final $\mu _ { S }$ :
+
+$$
+\mu _ { S } = \frac { 1 } { | S | } \sum _ { i = 1 } ^ { | S | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { i } ) ,
+$$
+
+where $\mathbf { a } _ { i } \in \mathbb { R } ^ { w \times h }$ is the flattened vector of the activation map of the $i$ -th sample in the support set, $w$ is the width of activations, and $h$ is the height of the activation map. To infer $\sigma _ { S }$ , we use the obtained $\mu _ { S }$ and deploy a separate inference function $f _ { \sigma } ^ { \ell } ( \cdot )$ :
+
+$$
+\sigma _ { S } = \frac { 1 } { | S | } \sum _ { i = 1 } ^ { | S | } f _ { \sigma } ^ { \ell } \big ( ( \mathbf { a } _ { i } - \mu _ { S } ) ^ { 2 } \big ) .
+$$
+
+It is worth mentioning that we actually use each sample to infer the statistics and take the average of all inferred statistics as the final normalization statistics. This enables us to fully exploit the samples to generate more accurate statistics.
+
+Note that the inference functions $f _ { \mu } ^ { \ell } ( \cdot )$ and $f _ { \sigma } ^ { \ell } ( \cdot )$ are shared by different channels in the same layer and we will learn $L$ pairs of those functions if we have $L$ convolutional layers in the meta-learning model. They are parameterized by feed-forward multiple layer perception networks, which we call hypernetworks. Using these hypernetworks, we generate support moments $( \mu _ { S } , \sigma _ { S } )$ and query moments $( \mu _ { Q } , \sigma _ { Q } )$ from the support and query sets, which are used for calculating the KL term in Eq. (5) for optimization during meta-training time. At meta-training time, we apply the statistics inferred from the support set for normalization of both support and query samples:
+
+$$
+a ^ { \prime } = \gamma \left( \frac { a - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ,
+$$
+
+where $\gamma$ and $\beta$ are jointly learned with parameters of the hypernetworks at meta-training time and directly applied at meta-test time, as in conventional batch normalization. At meta-test time, given a test task, we use hypernetworks that take the support set as input to generate normalization statistics directly used for the query set.
+
+MetaNorm for Domain Generalization In the domain generalization scenario, we adopt the metalearning setting from (Li et al., 2018a; Balaji et al., 2018; Du et al., 2020), and divide a dataset into the source domains used for training and the target domains held out for testing. At meta-training time, data in the source domains is episodically divided into sets of meta-source $\mathcal { D } ^ { s }$ and meta-target $\mathcal { D } ^ { t }$ domains.
+
+In a similar vein to few-shot classification, we would like to learn to acquire the ability to generate domain-specific statistics from a single example, which can then be applied to unseen domains. We assume we can generate reasonable normalization statistics by using only one sample from the new domain, because, intuitively, a single sample already carries sufficient domain information. We use a single example and all the examples in the same domain to infer the domain-specific statistics and minimize the KL term:
+
+$$
+D _ { \mathrm { K L } } \big [ q _ { \phi } ( m | \mathbf { a } _ { i } ) | | p _ { \theta } ( m | \mathcal { D } ^ { s } \backslash \mathbf { a } _ { i } ) \big ] ,
+$$
+
+where we define $q ( m | \mathbf { a } _ { i } ) { = } { \mathcal { N } } ( { \boldsymbol { \mu } } _ { a } , { \boldsymbol { \sigma } } _ { a } )$ , and $p ( m | \mathcal { D } ^ { s } \backslash \mathbf { a } _ { i } ) { = } \mathcal { N } ( \mu _ { D } , \sigma _ { D } )$ , which are implemented in a similar way as Eq. (6) and Eq. (7), and ${ \bf a } _ { i }$ is an example from its own domain $\mathcal { D } ^ { s }$ . In both the meta-source and meta-target domains, each example is normalized using the statistics generated by itself, like in Eq. (8), in which we make $\gamma$ and $\beta$ shared across all domains. The minimization of the KL term in Eq. (9) is to encourage the model to generate domain-specific statistics for normalization from only a single example. This enables us to generate domain-specific statistics on target domains that are never seen at meta-training time.
+
+In practice, we take the sum of all samples in all source domains as follows:
+
+$$
+\sum _ { i } ^ { | { \mathcal D } ^ { s } | } \sum _ { j } ^ { J } D _ { \mathrm { K L } } \big [ q _ { \phi } ( m | { \bf a } _ { i } ) | | p _ { \theta } ( m | { \mathcal D } _ { j } ^ { s } \backslash { \bf a } _ { i } ) \big ] ,
+$$
+
+where $\mathcal { D } _ { j } ^ { s }$ denotes the $j$ -th of $J$ meta-source domains. The inference networks are first at metatraining time learned and then directly used as examples from the target domain at meta-test time. Note that on the meta-target domain we do not apply the KL term; instead, we simply rely on each example to generate its statistics for normalization.
+
+MetaNorm for Few-Shot Domain Generalization We introduce an even more challenging setting, i.e., few-shot domain generalization, that combines the challenges of both few-shot classification and domain generalization. Specifically, we aim to learn a model from a set of classification tasks, each of which has only a few samples in a support set for training and test the model on tasks in a query set, which are in a different domain from the support set. Like few-shot classification, the label space is not shared between training and testing. Cross-domain few-shot learning has been explored recently by Tseng et al. (2020) and Guo et al. (2020). However, the setting of our few-shot domain generalization is different and considered to be more challenging, as the support and query set are from different domains in the meta-test stage and the target domain is also unseen throughout the training stage. An example for the few-shot domain generalization setting is provided in Figure 1.
+
+We divide a dataset into the source domains $\boldsymbol { S }$ used for training and the target domains $\tau$ held out for testing. During training time, data in the source domains $s$ is episodically divided into sets of meta-train $D ^ { s }$ and meta-test $\mathcal { D } ^ { t }$ domains. We sample $\mathcal { C }$ -way $k$ -shot data as the support set from each meta-source domain $\mathcal { D } ^ { s }$ , where $k$ is the number of labelled examples for each of the $\mathcal { C }$ classes. We sample $\mathcal { C }$ classes from the meta-test $\mathcal { D } ^ { t }$ domain as the query set. At test time, we sample $\mathcal { C }$ -way $k$ -shot data as the support set from each of the source domains $s$ . The model learned at meta-training time is then fine-tuned on few-shot tasks samples from the source domains and tested on the target domain $\tau$ . To learn the normalization statistics, we minimize the following KL term:
+
+![](images/a12c6a25225a9d04ad0464f882d29e7db657982f95f2e9a30d6b4b5ef7ab8f22.jpg)  
+Figure 1: Illustration of the novel few-shot domain generalization scenario using the 5-way, 1-shot setting. The training set in the upper box contains the meta-source domains $\mathcal { D } ^ { s }$ and the meta-target domain $\mathcal { D } ^ { t }$ , which are from different domains. Each training task contains meta-source domains with five different classes and one example of each meta-source domain, and more than four examples for evaluation in the meta-target domain. The test set is defined in the same way but with all source domains $s$ covering classes not present in any of the datasets in the training set, and more than four examples are used for evaluation in the target domain $\tau$ .
+
+$$
+\sum _ { i } ^ { | { \mathcal { D } } ^ { s } | } D _ { \mathrm { K L } } [ q _ { \phi } ( m | { \bf a } _ { i } ) | | p _ { \theta } ( m | { \mathcal { D } } ^ { s } ) ] ,
+$$
+
+where ${ \bf a } _ { i }$ is the activation associated with each sample from the meta-source domain $\mathcal { D } ^ { s }$ . Likewise, $q ( m | \mathbf { a } _ { i } )$ and $p ( m | \mathcal { D } ^ { s } )$ are also defined as factorized Gaussian distributions. We also adopt $\gamma$ and $\beta$ , which are shared across tasks and jointly learned. MetaNorm learns to acquire the ability to generate proper statistics for itself, and applies it to the samples in the meta-target domain.
+
+# 4 EXPERIMENTAL RESULTS
+
+We conduct an extensive set of experiments on a total of 17 datasets containing more than 15 million images. We use three representative approaches to meta-learning as our base models, i.e., MAML (Finn et al., 2017), ProtoNets (Snell et al., 2017), and VERSA (Gordon et al., 2019), which can verify our MetaNorm is generic, flexible and model-agnostic, making it a simple plug-and-play module that is seamlessly embedded into existing meta-learning approaches. We further compare different normalization methods: transductive batch normalization (TBN), “example” that denotes testing with one example at a time by using TBN, “class” that denotes testing with one class at a time by using TBN, w/o BN which is not using batch normalization, CBN which is using conventional batch normalization, RN (Nichol et al., 2018), MetaBN (Bronskill et al., 2020), TaskNorm-L (Bronskill et al., 2020), and TaskNorm-I (Bronskill et al., 2020). All details about datasets and implementation settings are provided in the appendix. More experimental results, including convergence analysis, are also provided in the appendix. Our code will be publicly released. 1
+
+Effect of KL Term We first conduct ablation studies that measure the effectiveness of MetaNorm. The key of MetaNorm is the introduced KL term for learning to learn statistics. We test the performance of MetaNorm without the KL term by directly using the statistics generated from data.
+
+Table 1: Effect of KL Term in MetaNorm for few-shot classification with MAML (Finn & Levine, 2018) on miniImageNet and domain generalization on PACS with ResNet-18. More few-shot classification results with ProtoNets (Snell et al., 2017) and VERSA (Gordon et al., 2019), as well as domain generalization results on Office-Home are provided in the appendix. Best performing methods and any other runs within the $9 5 \%$ confidence margin in bold. The KL term is crucial.
+
+![](images/263fcc7035b7f5105f1b2577ddbf1ede5433bef8be0206960a025a0ba9572f69.jpg)  
+Figure 2: Impact of Target Set Size. The performance increases for larger target sets and plateaus at around 125 for few-shot classification on miniImageNet and around 256 for domain generalization on PACS. TBN here is based on VERSA. MetaNorm generates proper normalization statistics with a reasonable batch size.
+
+In this case, we also use the hypernetworks to generate the moments, $\mu$ and $\sigma$ by simply removing the KL term in the objective function. In Table 1 we present results for few-shot classification on miniImageNet (Vinyals et al., 2016) and for domain generalization on PACS (Li et al., 2017a). The performance of MetaNorm without KL degrades significantly. This is expected, as without the KL term the generation process of normalization statistics lacks direct supervision from the target distribution, resulting in improper statistics.
+
+Impact of Target Set Size The other key parameter in MetaNorm is the size of the target set; that is, the number $| \mathcal { Q } |$ of samples in the query set (in few-shot classification) and the number $| \mathcal { D } ^ { s } |$ of samples in each domain (in domain generalization). This parameter is important when learning normalization statistics because we use the statistics generated by the target set as the ‘ground truth’. We evaluate its impact on the performance of MetaNorm in Figure 2. The experimental results show that TBN is not affected by the target size, both in the 5-way, 1-shot and 5 way, 5-shot tasks. MetaNorm performance rises as the size of the target set increases and plateaus at a reasonable size. In the few-shot setting, the performance reaches its peak at a size of about 125, which is slightly larger than the standard size of 75, while in the domain generalization setting, the performance plateaus at a size of about 128. This demonstrates that we are able to generate proper statistics with the mini-batch gradient descent optimization. In scenarios demanding a very small target set size, we could leverage image synthesis techniques to generate more samples for the targets sets.
+
+Sensitivity to Algorithm We evaluate MetaNorm using the MAML (Finn et al., 2017), ProtoNets (Snell et al., 2017) and VERSA (Gordon et al., 2019) algorithms, which are representative gradient, metric and model based meta-learning approaches for few-shot classification. These experiments are conducted on the Omniglot and miniImageNet datasets under different settings. The comparison results on miniImageNet are summarized in Table 2 and the results on Omniglot are provided in the appendix. For all three meta-learning approaches under all settings, MetaNorm consistently achieves comparable performance both to the non-transductive and transductive normalization methods. Being non-transductive, TaskNorm can achieve impressive performance on all the tasks, but its performance is not always better than transductive batch normalization. MetaNorm achieves comparable performance to transductive batch normalization, especially under the 5-way-1-shot setting, which is challenging since only a few examples are available to generate statistics. Notice that, MetaNorm performs well with the standard query set size $| \mathcal { Q } |$ of 75 (15 per category). It is slightly better than non-transductive TaskNorm and comparable with TBN. MetaNorm achieves its best performance with a query size $| \mathcal { Q } |$ of 125 (25 per category), only slightly larger than the standard size of 75. This demonstrates the benefit of leveraging meta-learning by MetaNorm for batch normalization. We conclude that MetaNorm is general and serves as a plug-and-play module for existing meta-learning models to improve their performance.
+
+Table 2: Sensitivity to Algorithm. Few-shot results on miniImageNet using different algorithms. Results on Omniglot are provided in the appendix. Best performing methods and any other runs within the $9 5 \%$ confidence margin in bold. Transductive results indicated above dashed line. MetaNorm is a consistent top-performer, regardless of the meta-learning algorithm.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">ProtoNets†</td><td colspan="2">MAML†</td><td colspan="2">VERSA†</td></tr><tr><td>5-way, 1-shot</td><td>5-way,5-shot</td><td>5-way,1-shot</td><td>5-way,5-shot</td><td>5-way,1-shot</td><td>5-way, 5-shot</td></tr><tr><td>TBN</td><td>45.9 ±0.6</td><td>65.5 ± 0.9</td><td>45.5 ±1.8</td><td>59.7 ± 0.9</td><td>53.4 ±1.8</td><td>67.3 ±0.9</td></tr><tr><td>example</td><td>43.9 ± 1.9</td><td>60.1 ± 0.8</td><td>26.9 ± 1.5</td><td>30.3 ±0.7</td><td>44.1 ± 1.7</td><td>60.3 ± 0.7</td></tr><tr><td>class</td><td>43.1 ± 1.8</td><td>59.8±0.8</td><td>26.9 ± 1.5</td><td>27.2 ±0.6</td><td>43.8 ± 1.8</td><td>59.7 ±0.6</td></tr><tr><td>w/BN</td><td>44.1 ±0.5</td><td>60.1 ±0.6</td><td>34.7 ±1.5</td><td>51.3±0.8</td><td>48.1 ±1.5</td><td>63.8 ±0.6</td></tr><tr><td>CBN (Ioffe &amp; Szegedy,2015)</td><td>47.8 ± 0.6</td><td>66.7 ± 0.5</td><td>20.1 ± 0.0</td><td>20.2 ± 0.2</td><td>45.7 ± 1.4</td><td>60.7 ± 0.8</td></tr><tr><td>RN (Nichol et al., 2018)</td><td>39.7 ± 0.5</td><td>63.1 ± 0.5</td><td>40.7 ± 1.7</td><td>57.6 ± 0.9</td><td></td><td></td></tr><tr><td>MetaBN (Bronskill et al., 2020)</td><td>42.6 ± 0.6</td><td>64.6 ± 0.5</td><td>41.6 ± 1.6</td><td>58.6 ± 0.9</td><td>50.1 ± 1.7</td><td>65.8 ± 0.9</td></tr><tr><td>TaskNorm-L (Bronskill et al.,2020)</td><td>47.5 ± 0.6</td><td>65.3 ± 0.5</td><td>42.0 ± 1.7</td><td>58.1 ± 0.9</td><td>52.1 ± 1.6</td><td>66.1 ± 0.7</td></tr><tr><td>TaskNorm-I (Bronskill et al., 2020)</td><td>43.2 ± 0.6</td><td>63.9 ±0.5</td><td>42.4 ± 1.7</td><td>58.7 ±0.9</td><td>52.9 ± 1.7</td><td>66.5 ±0.8</td></tr><tr><td>MetaNorm (|Q|=75)</td><td>47.3 ± 0.6</td><td>65.4 ± 0.5</td><td>44.7 ± 1.5</td><td>59.6 ± 0.8</td><td>52.7 ± 1.6</td><td>67.5 ± 0.8</td></tr><tr><td>MetaNorm (|Q|= 125)</td><td>48.1 ± 0.6</td><td>65.9 ± 0.9</td><td>46.8 ± 1.6</td><td>60.1 ± 0.8</td><td>53.7 ± 1.6</td><td>68.1 ± 0.8</td></tr></table>
+
+† Results for MAML and ProtoNets (except w/o BN) provided by (Bronskill et al., 2020), and VERSA with TBN provided by (Gordon et al., 2019). All other results based on our re-implementations.
+
+Sensitivity to Dataset We evaluate MetaNorm on a demanding few-shot classification challenge called Meta-Dataset (Triantafillou et al., 2020), which is composed of thirteen image classification datasets (eight for training, five testing). To compare with previous work, we perform experiments with ProtoNets and report the results in Table 3. All thirteen per-dataset results can be found in the appendix. MetaNorm achieves high performance in terms of average rank, with highest accuracy on eight of the thirteen datasets. MetaNorm outperforms transductive batch normalization on eleven datasets. It achieves comparable performance with transductive batch normalization on Omniglot and MNIST, which are relatively less challenging. Moreover, MetaNorm performs better than TaskNorm on seven of the thirteen datasets. We conclude that MetaNorm is effective, outperforming alternative normalizations for most datasets.
+
+Table 3: Sensitivity to Dataset. Few-shot classification on Meta-Dataset using ProtoNets. MetaNorm performs best overall.   
+
+<table><tr><td>Wins</td><td>Rank</td></tr><tr><td>IN 1</td><td>10.61</td></tr><tr><td>CBN /</td><td>9.11</td></tr><tr><td>LN 3</td><td>8.19</td></tr><tr><td>TaskNorm-r</td><td>7.88</td></tr><tr><td>BRN 1</td><td>6.23</td></tr><tr><td>TBN 2</td><td>4.81</td></tr><tr><td>MetaBN 4</td><td>4.78</td></tr><tr><td>RN 3</td><td>4.73</td></tr><tr><td>TaskNorm-L 4</td><td>4.19</td></tr><tr><td>TaskNorm-I 6</td><td>3.07</td></tr><tr><td>MetaNorm 10</td><td>2.35</td></tr></table>
+
+Sensitivity to Domains For this experiment we adopt two widely-used benchmarks for domain generalization of visual object recognition, i.e., PACS (Li et al., 2017a) and Office-Home (Venkateswara et al., 2017). Detailed descriptions on the experimental settings and implementations are provided in the appendix. For fair comparison with prior methods (Balaji et al., 2018; Li et al., 2018b; Seo et al., 2019), we employ ResNet-18 as the backbone network in all experiments. As shown in Table 4, MetaNorm achieves the best performance on PACS and Office-Home in terms of average accuracy. On PACS, MetaNorm consistently outperforms other normalization approaches including domain-specific normalization (Seo et al., 2019), on all four domains. It is worth mentioning that the baseline normalization uses the statistics from the source domains for the batch normalization of the target domain. As expected, the baseline method produces relatively poor performance on most domains, since the source domains cannot provide proper statistics for target domains due to the distribution shift. We have also done an experiment using standard batch normalization. In the training stage, we compute the ground truth statistics using all the test data on the meta-target domain $\mathcal { D } ^ { t }$ instead of using inferred statistics $p ( m | \mathcal { D } ^ { s } \backslash \mathbf { a } _ { i } )$ . MetaNorm is still better on most domains and on average. This is reasonable because ground truth statistics from the test data do not necessarily reflect the true data distribution. The experimental results demonstrate MetaNorm can generate reasonable normalization statistics from only one sample in its domain. We conclude that MetaNorm is effective for domain generalization.
+
+Table 4: Sensitivity to Domains. Performance comparison on domain generalization. MetaNorm consistently achieves the best performance among all normalization methods.   
+
+<table><tr><td rowspan="2"></td><td colspan="5">PACS</td><td colspan="5">Office-Home</td></tr><tr><td>Photo</td><td>Art</td><td>Cartoon</td><td>Sketch</td><td>Mean</td><td>Art</td><td>Clipart</td><td>Product</td><td>Real-World</td><td>Mean</td></tr><tr><td>Baseline</td><td>95.87</td><td>78.47</td><td>70.41</td><td>70.68</td><td>78.86</td><td>58.71</td><td>44.20</td><td>71.75</td><td>73.19</td><td>61.96</td></tr><tr><td> IBN (Pan et al., 2018)</td><td>92.04</td><td>75.29</td><td>72.95</td><td>77.42</td><td>79.43</td><td>55.41</td><td>44.82</td><td>68.28</td><td>71.95</td><td>60.09</td></tr><tr><td>DSBN (Chang et al., 2019)</td><td>95.51</td><td>78.61</td><td>66.17</td><td>70.15</td><td>77.61</td><td>59.04</td><td>45.02</td><td>72.67</td><td>71.98</td><td>62.18</td></tr><tr><td> SN (Luo et al., 2018a)</td><td>93.47</td><td>82.50</td><td>76.80</td><td>80.77</td><td>83.38</td><td>54.10</td><td>44.97</td><td>64.54</td><td>71.40</td><td>58.75</td></tr><tr><td>DSON (Seo et al., 2019)</td><td>95.87</td><td>84.67</td><td>77.65</td><td>82.23</td><td>85.11</td><td>59.37</td><td>45.70</td><td>71.84</td><td>74.68</td><td>62.90</td></tr><tr><td>Ground truth statistics</td><td>95.78</td><td>85.17</td><td>78.15</td><td>82.91</td><td>85.50</td><td>59.35</td><td>46.12</td><td>72.77</td><td>75.08</td><td>63.33</td></tr><tr><td> MetaNorm</td><td>95.99</td><td>85.01</td><td>78.63</td><td>83.17</td><td>85.70</td><td>59.77</td><td>45.98</td><td>73.13</td><td>75.29</td><td>63.55</td></tr></table>
+
+Few-Shot Domain Generalization In our final experiment, we adopt the DomainNet dataset (Peng et al., 2019) and introduce a new, more challenging setting to evaluate the performance for few-shot domain generalization. Detailed descriptions on the dataset and experimental settings are provided in the appendix. We conduct the experiments with the MAML and ProtoNets algorithms under both 5-way 1-shot and 5-way 5-shot settings, and the results are reported in Table 5. We implement transductive batch normalization, MetaBN and the variants of TaskNorm for direct comparison. Under both settings, our MetaNorm produces the best performance and surpasses the transductive batch normalization by large margins of up to $4 . 0 \%$ on the challenging 5-way 1-shot setting with MAML. MetaNorm also achieves better results than the non-transductive TaskNorm approaches. At the same time, with ProtoNet our MetaNorm again consistently delivers the best performance and surpasses both transductive and non-transductive normalizations, The performance on the challenging few-shot domain generalization scenario with different meta-learning algorithms again demonstrates the effectiveness of MetaNorm in handling the challenges of batch normalization for small batches and across domains.
+
+Table 5: Few-Shot Domain Generalization. Comparison with different normalizations using MAML and ProtoNets on the Few-shot DomainNet dataset. Best performing methods and any other runs within the $9 5 \%$ confidence margin denoted in bold. Reported results use “Painting” as the target domain, all based on our implementations. MetaNorm consistently achieves top performance.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">MAML</td><td colspan="2">ProtoNets</td></tr><tr><td>5-way,1-shot</td><td>5-way,5-shot</td><td>5-way,1-shot</td><td>5-way, 5-shot</td></tr><tr><td>TBN</td><td>28.7 ± 1.8</td><td>49.3 ± 0.8</td><td>27.9 ± 1.8</td><td>47.1 ± 0.8</td></tr><tr><td>w/o BN</td><td>23.5 ± 1.7</td><td>45.4 ± 0.7</td><td>23.8 ± 1.8</td><td>45.9 ± 0.7</td></tr><tr><td>CBN</td><td>20.0 ± 0.0</td><td>20.1 ± 0.2</td><td>28.4 ± 1.8</td><td>47.9 ± 0.7</td></tr><tr><td>MetaBN</td><td>24.7 ± 1.6</td><td>46.1 ± 0.8</td><td>25.1 ± 1.8</td><td>46.1 ± 0.8</td></tr><tr><td>TaskNorm-L</td><td>26.9 ± 1.7</td><td>47.4 ± 0.8</td><td>29.5 ± 1.6</td><td>48.3 ± 0.8</td></tr><tr><td>TaskNorm-I</td><td>27.5 ± 1.6</td><td>48.8± 0.6</td><td>26.8 ± 1.8</td><td>46.9 ± 0.7</td></tr><tr><td> MetaNorm</td><td> 32.7 ± 1.7</td><td> 51.9 ± 0.9</td><td> 30.7 ± 1.8</td><td> 49.1 ± 0.9</td></tr></table>
+
+# 5 CONCLUSION
+
+In this paper we present MetaNorm, a meta-learning based batch normalization. MetaNorm tackles the challenging scenarios where the batch size is too small to produce sufficient statistics or when training statistics are not directly applicable to test data due to a domain shift. MetaNorm learns to learn adaptive statistics that are specific to tasks or domains. It is generic and model-agnostic, which enables it to be used with various meta-learning algorithms for different applications. We evaluate MetaNorm on two well-known existing tasks, i.e., few-shot classification and domain generalization, and we also introduce the challenging evaluation scenario of few-shot domain generalization that addresses the small batch and distribution shift problems simultaneously. An extensive evaluation on 17 datasets reveals that MetaNorm consistently achieves results that are better, or at least competitive, compared to other normalization approaches, verifying its effectiveness as a new meta-learning based batch normalization approach.
+
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+
+# A ALGORITHMS DESCRIPTIONS
+
+In this Appendix we provide the detailed MetaNorm algorithm descriptions to conduct batch normalization for few-shot classification (Algorithm 1), domain generalization (Algorithm 2) and few-shot domain generalization (Algorithm 3). The dataflow of the implementation is shown in Figure 3.
+
+# Algorithm 1 MetaNorm for Few-Shot Classification
+
+Meta-train: Input values of $a$ over support set $\mathbf { a } _ { S , i }$ and query set $\mathbf { a } Q , i$ ;   
+$\gamma$ , $\beta \gets$ Initialize parameters.
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { S } = \frac { 1 } { | S | } \sum _ { i = 1 } ^ { | S | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { S , i } ) ; \mu _ { Q } = \frac { 1 } { | Q | } \sum _ { i = 1 } ^ { | Q | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { Q , i } ) ; } \\ { \displaystyle \sigma _ { S } = \frac { 1 } { | S | } \sum _ { i = 1 } ^ { | S | } f _ { \sigma } ^ { \ell } \left( ( \mathbf { a } _ { S , i } - \mu _ { S } ) ^ { 2 } \right) ; \sigma _ { Q } = \frac { 1 } { | Q | } \sum _ { i = 1 } ^ { | Q | } f _ { \sigma } ^ { \ell } \left( ( \mathbf { a } _ { Q , i } - \mu _ { Q } ) ^ { 2 } \right) ; } \\ { \displaystyle a _ { S , i } ^ { \prime } = \gamma \left( \frac { a _ { S , i } - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ; a _ { Q , i } ^ { \prime } = \gamma \left( \frac { a _ { Q , i } - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ; } \\ { \displaystyle \mathcal { L } _ { \mathrm { K L } } = D _ { \mathrm { K L } } \left[ N ( \mu _ { S } , \sigma _ { S } ) | | N ( \mu _ { Q } , \sigma _ { Q } ) \right] } \\ { \mathrm { r e t u r n } a _ { S , i } = \mathbf { M e t a N o r m } ( \mathbf { a } _ { S , i } ) ; a _ { Q , i } ^ { \prime } = \mathbf { M e t a N o r m } ( \mathbf { a } _ { Q , i } ) ; \mathcal { L } _ { K L } } \end{array}
+$$
+
+Meta-test: Input values of $a$ over support set $\mathbf { a } _ { S , i }$ and query set $\mathbf { a } _ { Q , i }$
+
+$$
+\begin{array} { r l } & { \mu _ { S } = \frac { 1 } { | S | } \displaystyle \sum _ { i = 1 } ^ { | S | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { S , i } ) ; } \\ & { \sigma _ { S } = \frac { 1 } { | S | } \displaystyle \sum _ { i = 1 } ^ { | S | } f _ { \sigma } ^ { \ell } \left( ( \mathbf { a } _ { S , i } - \mu _ { S } ) ^ { 2 } \right) ; } \\ & { a _ { Q , i } ^ { \prime } = \gamma \left( \frac { \mathbf { a } _ { Q , i } - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ; } \\ & { \mathrm { { r e t u r n } } a _ { Q , i } ^ { \prime } = \mathbf { M e t a N o r m } ( a _ { Q , i } } \end{array}
+$$
+
+# Algorithm 2 MetaNorm for Domain Generalization
+
+<table><tr><td>Train: Input values of α over meta-source domain as,i and meta-target domain aT,i; γ,β← Initialize parameters.</td></tr><tr><td>|S|</td></tr><tr><td>μs,i= f（as,i）; μs= f(as,i）;</td></tr><tr><td>S σS,i = f(as,i - μs,i)²）;σs = f（(as,i-μs）²）； M</td></tr><tr><td>μT,i = f²(aT,i); OT,i = f＆((aT,i - μT,i)²);</td></tr><tr><td>aT,i =γ aT,i-μT,i +β;</td></tr><tr><td>V+e LKL=DkL[N(μs,),σs,))lIN(μs,σs)]</td></tr><tr><td>return aT,i =MetaNorm(aT,i); LKL</td></tr><tr><td>Test: Input values of a over test domain ai;</td></tr><tr><td>μi=f(ai)；</td></tr><tr><td>Oi=f((ai-μi）²）；</td></tr><tr><td>a=γ ai-μi +β；</td></tr></table>
+
+# B DATASETS
+
+We conduct an extensive set of experiments on a total of 17 datasets containing more than 15 million images. All dataset details and settings are provided in this Appendix.
+
+miniImageNet. The miniImageNet is originally proposed in (Vinyals et al., 2016) and has been widely used for evaluating few-shot learning algorithms. It consists of 60,000 color images from 100
+
+# Algorithm 3 MetaNorm for Few-Shot Domain Generalization
+
+Meta-train: Input values of $a$ over meta-source domain set $\mathbf { a } _ { S , i }$ and meta-target domain set $\mathbf { a } _ { Q , i }$ $\gamma$ , $\beta \gets$ Initialize parameters.
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { S } = \frac { 1 } { | S | } \sum _ { i = 1 } ^ { | S | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { S , i } ) ; \mu _ { Q } = \frac { 1 } { | Q | } \sum _ { i = 1 } ^ { | Q | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { Q , i } ) ; } \\ { \displaystyle \sigma _ { S } = \frac { 1 } { | S | } \sum _ { i = 1 } ^ { | S | } f _ { \sigma } ^ { \ell } ( ( \mathbf { a } _ { S , i } - \mu _ { S } ) ^ { 2 } ) ; \sigma _ { Q } = \frac { 1 } { | Q | } \sum _ { i = 1 } ^ { | Q | } f _ { \sigma } ^ { \ell } \big ( ( \mathbf { a } _ { Q , i } - \mu _ { Q } ) ^ { 2 } \big ) ; } \\ { \displaystyle a _ { S , i } ^ { \prime } = \gamma \left( \frac { a _ { S , i } - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ; a _ { Q , i } ^ { \prime } = \gamma \left( \frac { a _ { Q , i } - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ; } \\ { \displaystyle C _ { \mathrm { K L } } = D _ { \mathrm { K L } } \left[ N ( \mu _ { S } , \sigma _ { S } ) | | N ( \mu _ { Q } , \sigma _ { Q } ) \right] } \\ { \mathrm { r e t u r n } a _ { S , i } ^ { \prime } = \mathbf { M e t a N o r m } ( \mathbf { a } _ { S , i } ) ; a _ { Q , i } ^ { \prime } = \mathbf { M e t a N o r m } ( \mathbf { a } _ { Q , i } ) ; \mathcal { L } _ { K L } } \end{array}
+$$
+
+Meta-test: Input values of $a$ over support set $\mathbf { a } _ { S , i }$ and query set $\mathbf { a } _ { Q , i }$
+
+$$
+\begin{array} { r l } & { \mu _ { S } = \frac { 1 } { | \mathcal { S } | } \displaystyle \sum _ { i = 1 } ^ { | S | } f _ { \mu } ^ { \ell } ( \mathbf { a } _ { S , i } ) ; } \\ & { \sigma _ { S } = \frac { 1 } { | \mathcal { S } | } \displaystyle \sum _ { i = 1 } ^ { | \mathcal { S } | } f _ { \sigma } ^ { \ell } \left( ( \mathbf { a } _ { S , i } - \mu _ { S } ) ^ { 2 } \right) ; } \\ & { a _ { Q , i } ^ { \ell } = \gamma \left( \frac { \mathbf { a } _ { Q , i } - \mu _ { S } } { \sqrt { \sigma _ { S } ^ { 2 } + \epsilon } } \right) + \beta ; } \\ & { \mathrm { r e t u r n } a _ { Q , i } ^ { \prime } = \mathbf { M e t a N o r m } ( a _ { Q , \ell } } \end{array}
+$$
+
+![](images/a2c8a4ae2a779c11be8835af390b83997d35a8cee32961d66364322ab1025b8d.jpg)  
+Figure 3: The dataflow of the implementation for few-shot learning. “N” indicates support size, “M” indicates query size, “C” indicates the channel of activations, “W” indicates the width of activations, “H” indicates the height of activations.
+
+classes with 600 examples per class. The images have dimensions of $8 4 \times 8 4$ pixels. We follow the train/val/ test split introduced in (Ravi & Larochelle, 2017), which uses 64 classes for meta-training, 16 classes for meta-validation, and the remaining 20 classes for meta-testing.
+
+Omniglot. Omniglot (Lake et al., 2015) is a few-shot learning dataset consisting of 1,623 handwritten characters (each with 20 instances) derived from 50 alphabets. We follow the pre-processing and training procedure defined in (Vinyals et al., 2016). We resize images to $2 8 \times 2 8$ . The training, validation and test sets consist of a random split of 1,100, 100, and 423 characters.
+
+PACS (Li et al., 2017a) contains a total of 9,991 images of the size $2 2 4 \times 2 2 4$ from 4 domains, i.e., photo, art-painting, cartoon and sketch, which demonstrate huge domain gaps. Images are from 7 object classes, i.e., dog, elephant, giraffe, guitar, horse, house, and person. We follow the “leave-one-out” protocol in (Li et al., 2017a; 2018b; Carlucci et al., 2019), where the model is trained on any three of the four domains, which we call source domains, and tested on the last (target) domain. The train-val-test splits are the same as in (Li et al., 2017a).
+
+Office-Home (Venkateswara et al., 2017) also has 4 domains: art, product, clipart and real-world. For each domain, the dataset contains images of 65 object categories found typically in office and home settings. We use the same experimental protocol as for PACS.
+
+DomainNet (Peng et al., 2019) contains 6 distinct domains, i.e., clipart, infograph, painting, quickdraw, real, and sketch for 345 categories. The categories are from 24 divisions, which are: Furniture, Mammal, Tool, Cloth, Electricity, Building, Office, Human Baby, Road Transportation, Food, Nature,
+
+Cold Blooded, Music, Fruit, Sport, Tree, Bird, Vegetable, Shape, Kitchen, Water Transportation, Sky Transportation, Insect, Others.
+
+Meta-Dataset (Triantafillou et al., 2020) is composed of ten (eight train, two test) existing image classification datasets. These are: ILSVRC-2012 (ImageNet, (Russakovsky et al., 2015)), Omniglot (Lake et al., 2015), Aircraft (Maji et al., 2013), CUB-200-2011 (Birds, (Wah et al., 2011)), Describable Textures (Cimpoi et al., 2014), Quick Draw, Fungi, VGG Flowr (Nilsback & Zisserman, 2008), Traffic Signs (Houben et al., 2013) and MSCOCO (Lin et al., 2014). Each episode generated in Meta-Dataset uses classes from a single dataset. Two of these datasets, Traffic Signs and MSCOCO, are fully reserved for evaluation, it means no classes from these sets participate in the training set. Except for Traffic Signs and MSCOCO, the remaining datasets contribute some classes to each of training, validation and test splits of classes. There are about 14 million images in total in Meta-Dataset.
+
+# C FEW-SHOT DOMAINNET
+
+To construct Few-shot DomainNet, we chose 200 random classes from DomainNet and used 140 for training, 20 for validation and the last 40 for testing. Note that the last 40 object classes were never seen during training. The dataset consists of 200,000 colour images of size $8 4 \times 8 4$ with each of the 200 classes having 1,000 examples. Please see Table 6, Table 7 and Table 8 for training, validation, and test classes.
+
+Table 6: Training classes of Few-shot DomainNet   
+
+<table><tr><td rowspan=1 colspan=1>Furniture: bathtub,ceiling fan, couch, fence, hot tub,mailbox</td></tr><tr><td rowspan=1 colspan=1>Mammal: tiger, rhinoceros,bat,cat, lion, panda</td></tr><tr><td rowspan=1 colspan=1>Tool: anvil, basket, broom, sword, pliers</td></tr><tr><td rowspan=1 colspan=1>Cloth: belt, camouflage, eyeglasses, crown, bowtie</td></tr><tr><td rowspan=1 colspan=1>Electricity: calculator, computer,camera, cooler, dishwasher</td></tr><tr><td rowspan=1 colspan=1>Building: bridge, jail, pool, tent, castle</td></tr><tr><td rowspan=1 colspan=1>Offce: alarm clock, binoculars, backpack,book, bandage</td></tr><tr><td rowspan=1 colspan=1>Human Body:arm,ear,face,beard, elbow, finger, brain,eye,foot, knee</td></tr><tr><td rowspan=1 colspan=1>Road Transportation: ambulance,bus motorbike, bicycle, train</td></tr><tr><td rowspan=1 colspan=1>Food: birthday cake,cookie,hot dog,peanut, sandwich,bread, donut, pizza,steak,lollipop</td></tr><tr><td rowspan=1 colspan=1>Nature: beach, lightning,ocean, river, sun, cloud,moon,rain, tornado</td></tr><tr><td rowspan=1 colspan=1>Cold Blooded: crab, frog, crocodile,lobster, fish, octopus,shark</td></tr><tr><td rowspan=1 colspan=1>Music: cello, guitar, saxophone, violin, clarinet, harp, trombone</td></tr><tr><td rowspan=1 colspan=1>Fruit: apple, banana, blackberry, blueberry, grapes, pear</td></tr><tr><td rowspan=1 colspan=1>Sport: baseball,baseball bat, basketball,snorkel, yoga,tennis racquet</td></tr><tr><td rowspan=1 colspan=1>Tree: bush, grass,cactus, tree, flower</td></tr><tr><td rowspan=1 colspan=1>Bird: bird,owl</td></tr><tr><td rowspan=1 colspan=1>Vegetable: asparagus, broccoli, carrot, mushroom, onion</td></tr><tr><td rowspan=1 colspan=1>Shape: circle,hexagon</td></tr><tr><td rowspan=1 colspan=1>Kitchen: fork,frying pan,hourglass,knife,lighter</td></tr><tr><td rowspan=1 colspan=1>Water Transportation: aircraft carrier, canoe, cruise ship, submarine</td></tr><tr><td rowspan=1 colspan=1>Sky Transportation: airplane, helicopter</td></tr><tr><td rowspan=1 colspan=1>Insect: ant, bee</td></tr><tr><td rowspan=1 colspan=1>Others: angel,cannon, dragon, mermaid,stop sign, snowman, feather</td></tr></table>
+
+Table 7: Validation classes of Few-shot DomainNet   
+
+<table><tr><td rowspan=1 colspan=1>Furniture: stairs,ladder</td></tr><tr><td rowspan=1 colspan=1> Mammal: monkey</td></tr><tr><td rowspan=1 colspan=1>Tool: paint can</td></tr><tr><td rowspan=1 colspan=1>Cloth: purse, t-shirt</td></tr><tr><td rowspan=1 colspan=1>Electricity: radio</td></tr><tr><td rowspan=1 colspan=1>Building: pond</td></tr><tr><td rowspan=1 colspan=1>Office: nail</td></tr><tr><td rowspan=1 colspan=1>Human Body: skull, tooth</td></tr><tr><td rowspan=1 colspan=1>Road Transportation: firetruck</td></tr><tr><td rowspan=1 colspan=1>Food: -</td></tr><tr><td rowspan=1 colspan=1>Nature: star, hurricane</td></tr><tr><td rowspan=1 colspan=1>Cold Blooded: sea turtle</td></tr><tr><td rowspan=1 colspan=1>Music: -</td></tr><tr><td rowspan=1 colspan=1>Fruit: strawberry</td></tr><tr><td rowspan=1 colspan=1>Sport: hockey stick</td></tr><tr><td rowspan=1 colspan=1>Tree: -</td></tr><tr><td rowspan=1 colspan=1>Bird: penguin</td></tr><tr><td rowspan=1 colspan=1>Vegetable: -</td></tr><tr><td rowspan=1 colspan=1>Shape: -</td></tr><tr><td rowspan=1 colspan=1>Kitchen: wine botle</td></tr><tr><td rowspan=1 colspan=1>Water Transportation: -</td></tr><tr><td rowspan=1 colspan=1>Sky Transportation: -</td></tr><tr><td rowspan=1 colspan=1>Insect: -</td></tr><tr><td rowspan=1 colspan=1>Others: teddy-bear</td></tr></table>
+
+# D IMPLEMENTATION DETAILS.
+
+In the few-shot learning task, MAML and ProtoNets use a simple CNN containing 4 convolutional layers, each of which is a $3 { \times } 3$ convolution with 32 filters, followed by MetaNorm, a ReLU nonlinearity, and finally a $2 \times 2$ max-pooling. VERSA uses a CNN containing 5 convolutional layers, each of which is a $3 { \times } 3$ convolution with 64 filters, followed by MetaNorm, a ReLU non-linearity, and finally a $2 \times 2$ max-pooling. In the domain generalization task, we rely on ResNet-18 as backbone for fair comparison with previous work. Each convolutional layer is followed by MetaNorm. The hypernetwork is a 3-layer MLP with 128 units per layer and rectifier nonlinearities. We implemented all models in the Tensorflow framework and tested on an NVIDIA Tesla V100. All code will be available at: https://github.com/YDU-AI/MetaNorm.
+
+# D.1 MAML EXPERIMENTS
+
+For MAML experiments, we used the codebase by Finn (Finn, 2017). We use the Adam optimizer with default parameters, and a meta batch size of 4 tasks. The number of test episodes is set as 600. The number of training iterations is 60,000. We set $\lambda { = } 0 . 0 0 1$ . The other hyper-parameters we use are the default MAML parameters. No early stopping was used. We used the first-order approximation of MAML for the experiments.
+
+Table 8: Test classes of Few-shot DomainNet   
+
+<table><tr><td rowspan=1 colspan=1>Furniture: teapot, toothpaste, stove,umbrella</td></tr><tr><td rowspan=1 colspan=1>Mammal: mouse,</td></tr><tr><td rowspan=1 colspan=1>Tool: bucket, paint can</td></tr><tr><td rowspan=1 colspan=1>Cloth: sweater, shoe, flip flops</td></tr><tr><td rowspan=1 colspan=1>Electricity: television,stereo, toaster, flashlight</td></tr><tr><td rowspan=1 colspan=1>Building: waterslide, garden</td></tr><tr><td rowspan=1 colspan=1>Office: map, clock,calendar, scissors</td></tr><tr><td rowspan=1 colspan=1>Human Body: finger, nose, toe</td></tr><tr><td rowspan=1 colspan=1>Road Transportation: bulldozer</td></tr><tr><td rowspan=1 colspan=1>Food: peanut</td></tr><tr><td rowspan=1 colspan=1>Nature: mountain, sun</td></tr><tr><td rowspan=1 colspan=1>Cold Blooded: lobster, scorpion</td></tr><tr><td rowspan=1 colspan=1>Music: harp</td></tr><tr><td rowspan=1 colspan=1>Fruit: pineapple</td></tr><tr><td rowspan=1 colspan=1>Sport: soccer ball, hockey stick</td></tr><tr><td rowspan=1 colspan=1>Tree: house plant, leaf</td></tr><tr><td rowspan=1 colspan=1>Bird: swan</td></tr><tr><td rowspan=1 colspan=1>Vegetable: string bean</td></tr><tr><td rowspan=1 colspan=1>Shape: squiggle</td></tr><tr><td rowspan=1 colspan=1>Kitchen: -</td></tr><tr><td rowspan=1 colspan=1>Water Transportation: -</td></tr><tr><td rowspan=1 colspan=1>Sky Transportation: -</td></tr><tr><td rowspan=1 colspan=1>Insect: -</td></tr><tr><td rowspan=1 colspan=1>Others: feather, snowman</td></tr></table>
+
+# D.2 PROTONETS EXPERIMENTS
+
+For ProtoNets, we used the codebase by Fatir (Fatir, 2018). For miniImageNet, we used the following ProtoNets options: a learning rate of 0.001, 60,000 training iterations, 200 validation episodes, 600 test episodes and $\lambda { = } 0 . 0 0 0 1$ . We choose the units of hidden layers and $\lambda$ by cross-validation. For Meta-Dataset, we reproduce the code provided by CNAPS (Requeima et al., 2019) with TensorFlow. We simply replace its normalization method with our MetaNorm method and add the KL term to the final loss. We are consistent with the dataset configuration and follow the training process as specified in (Triantafillou et al., 2020). The number of training iterations is 80,000. We use a constant learning rate of 0.0001. We set $\lambda { = } 0 . 0 0 1$ . We follow TaskNorm’s (Bronskill et al., 2020) options: they do not use feature adaptation, and allow updates pre-trained feature extractor weights during meta-training stage.
+
+# D.3 VERSA EXPERIMENTS
+
+For VERSA, we used the codebase by Gordon (Gordon, 2019). For the 5-way 5-shot model, we train using the setting of 8 tasks per batch for 100,000 iterations and use a constant learning rate of 0.0001, $\lambda { = } 0 . 0 0 1$ . For the 5-way 1-shot model, we train with the setting of 8 tasks per batch for 150,000 iterations and use a constant learning rate of 0.00025, $\lambda { = } 0 . 0 1$ . We set validation episodes as 200, and test episodes as 600. The units of hidden layers and $\lambda$ were chosen by cross-validation.
+
+Table 9: Inference function $f _ { \mu } ^ { l } ( \cdot )$   
+
+<table><tr><td>Output size</td><td>Layers</td></tr><tr><td>w×h</td><td>Input flattened vector of the activation map</td></tr><tr><td>128</td><td>fully connected, ELU</td></tr><tr><td>128</td><td>fully connected, ELU</td></tr><tr><td>w×h</td><td>fully connected to μ</td></tr></table>
+
+Table 10: Inference function $f _ { \sigma } ^ { l } ( \cdot )$   
+
+<table><tr><td>Output size</td><td>Layers</td></tr><tr><td>w×h</td><td>Input flattened vector of the activation map and μ</td></tr><tr><td>128</td><td>fully connected, ELU</td></tr><tr><td>128</td><td>fully connected, ELU</td></tr><tr><td>w×h</td><td>fully connected to o</td></tr></table>
+
+# E EXTRA RESULTS FOR EFFECT OF KL TERM
+
+In this Appendix we consider extra results for the ablation on measuring the effect of the KL term. We report results for few-shot classification on miniImageNet with ProtoNets (Snell et al., 2017) and VERSA (Gordon et al., 2019) in Table 11. We also report domain generalization results on Office-Home in Table 12. In all cases the KL term is crucial.
+
+Table 11: Effect of KL Term in MetaNorm for few-shot classification on miniImageNet with ProtoNets and VERSA. Best performing methods and any other runs within $9 5 \%$ confidence margin denoted in bold.   
+
+<table><tr><td rowspan="3">MetaNorm</td><td colspan="2">ProtoNets</td><td colspan="2">VERSA</td></tr><tr><td>5-way,1-shot</td><td>5-way,5-shot</td><td>5-way,1-shot</td><td>5-way, 5-shot</td></tr><tr><td>w/o KL</td><td>40.1 ± 1.6</td><td>58.7 ± 0.8</td><td>48.7 ± 1.6</td><td>64.3 ± 0.8</td></tr><tr><td>w/ KL</td><td>48.1 ± 1.6</td><td>65.9 ± 0.9</td><td>53.7 ± 1.6</td><td>68.1 ± 0.8</td></tr></table>
+
+Table 12: Effect of KL Term in MetaNorm for domain generalization on Office-Home.   
+
+<table><tr><td rowspan="2">MetaNorm</td><td colspan="5">Office-Home</td></tr><tr><td>Art</td><td>Clipart</td><td>Product</td><td>Real-World</td><td> Mean</td></tr><tr><td>w/o KL</td><td>51.25</td><td>39.27</td><td>69.75</td><td>68.19</td><td>57.12</td></tr><tr><td>w/ KL</td><td>59.77</td><td>45.98</td><td>73.13</td><td>75.29</td><td>63.55</td></tr></table>
+
+# F SENSITIVITY TO ALGORITHM ON OMNIGLOT
+
+The experiments on Omniglot for few-shot classification under the meta-learning settings of MAML, VERSA and ProtoNets are reported in Tables 13, 14 and 15. MetaNorm consistently outperforms both transductive and non-transductive normalization approaches.
+
+# G SENSITIVITY TO DATASET
+
+The complete set of results for each of the thirteen datasets in Meta-Dataset are provided in Table 16.
+
+Table 13: Sensitivity to Algorithm. Few-shot results on Omniglot using MAML. Best performing methods and any other runs within the $9 5 \%$ confidence margin in bold. Transductive results indicated above dashed line.   
+
+<table><tr><td rowspan="2"></td><td colspan="4">Omniglot</td></tr><tr><td>5-way, 1-shot</td><td>5-way, 5-shot</td><td>20-way,1-shot</td><td>20-way, 5-shot</td></tr><tr><td>TBN</td><td>98.4 ± 0.7</td><td>99.2 ± 0.2</td><td>90.9 ± 0.5</td><td>96.6 ± 0.2</td></tr><tr><td>example</td><td>21.6 ± 1.3</td><td>22.0 ± 0.5</td><td>3.7±0.2</td><td>5.5±0.2</td></tr><tr><td>class</td><td>21.6 ± 1.3</td><td>23.2 ± 0.5</td><td>3.7 ±0.2</td><td>14.5 ± 0.3</td></tr><tr><td>w/oBN</td><td>92.6 ± 0.9</td><td>90.7 ± 0.1</td><td>84.3 ± 0.4</td><td>91.7±0.2</td></tr><tr><td>CBN (Ioffe &amp; Szegedy, 2015)</td><td>20.1 ±0.0</td><td>20.0±0.0</td><td>5.0±0.0</td><td>5.0±0.0</td></tr><tr><td>RN (Nichol et al., 2018)</td><td>92.6 ± 0.9</td><td>98.2 ±0.2</td><td>89.0 ± 0.6</td><td>96.8 ± 0.2</td></tr><tr><td>MetaBN (Bronskill et al., 2020)</td><td>91.8 ±0.9</td><td>98.1 ± 0.3</td><td>89.6 ± 0.5</td><td>96.4 ± 0.2</td></tr><tr><td>TaskNorm-L (Bronskill et al., 2020)</td><td>94.0 ±0.8</td><td>98.0±0.3</td><td>89.6 ± 0.5</td><td>96.4 ±0.2</td></tr><tr><td>TaskNorm-I (Bronskill et al., 2020)</td><td>94.4 ±0.8</td><td>98.6± 0.2</td><td>90.0 ± 0.5</td><td>96.3 ±0.2</td></tr><tr><td>MetaNorm</td><td>98.8 ± 0.5</td><td>99.3 ± 0.2</td><td>91.3 ± 0.5</td><td>97.1 ± 0.2</td></tr></table>
+
+Results (except w/o BN and our MetaNorm) provided by (Bronskill et al., 2020).
+
+Table 14: Sensitivity to Algorithm. Few-shot results on Omniglot using VERSA. Best performing methods and any other runs within the $9 5 \%$ confidence margin in bold. Transductive results indicated above dashed line.   
+
+<table><tr><td rowspan="2"></td><td colspan="4">Omniglot</td></tr><tr><td>5-way, 1-shot</td><td>5-way, 5-shot</td><td>20-way, 1-shot</td><td>20-way, 5-shot</td></tr><tr><td>TBN</td><td>99.7 ± 0.2</td><td>99.8 ± 0.2</td><td>97.7 ± 0.2</td><td>98.8 ± 0.1</td></tr><tr><td>example</td><td>94.9 ± 0.2</td><td>95.1 ± 0.3</td><td>92.9 ± 0.2</td><td>95.9 ± 0.2</td></tr><tr><td>class</td><td>94.3 ± 0.3</td><td>94.8 ± 0.1</td><td>91.8 ± 0.3</td><td>95.1 ± 0.4</td></tr><tr><td>w/oBN</td><td>95.6 ± 0.7</td><td>96.5 ± 0.1</td><td>93.1 ± 0.3</td><td>96.3±0.2</td></tr><tr><td>CBN (Ioffe &amp; Szegedy, 2015)</td><td>94.3 ± 0.3</td><td>95.7 ± 0.0</td><td>92.7 ± 0.2</td><td>95.2 ± 0.3</td></tr><tr><td>MetaBN (Bronskill et al., 2020)</td><td>96.7 ± 0.3</td><td>98.1 ± 0.3</td><td>95.8 ± 0.2</td><td>97.1 ± 0.2</td></tr><tr><td>TaskNorm-L (Bronskill et al., 2020)</td><td>97.9 ± 0.3</td><td>99.2 ±0.2</td><td>96.1 ± 0.2</td><td>98.0±0.2</td></tr><tr><td>TaskNorm-I (Bronskill et al., 2020)</td><td>98.3 ±0.2</td><td>99.5 ± 0.2</td><td>96.7 ± 0.2</td><td>98.1 ± 0.1</td></tr><tr><td>MetaNorm</td><td>99.8 ± 0.1</td><td>99.9 ± 0.1</td><td>97.9 ± 0.2</td><td>98.8 ± 0.2</td></tr></table>
+
+† Results of TBN provided by (Gordon et al., 2019). All other results based on our re-implementations.
+
+Table 15: Sensitivity to Algorithm. Few-shot results on Omniglot using ProtoNets. Best performing methods and any other runs within $9 5 \%$ confidence margin denoted in bold. Transductive results indicated above dashed line.   
+
+<table><tr><td rowspan="2"></td><td colspan="4">Omniglot†</td></tr><tr><td>5-way, 1-shot</td><td>5-way, 5-shot</td><td>20-way, 1-shot</td><td>20-way, 5-shot</td></tr><tr><td>TBN</td><td>98.4 ± 0.2</td><td>99.6 ± 0.2</td><td>94.5 ± 0.2</td><td>98.6 ± 0.1</td></tr><tr><td>example</td><td>98.4 ± 0.2</td><td>99.5 ± 0.2</td><td>94.3 ± 0.2</td><td>98.5 ± 0.1</td></tr><tr><td>class</td><td>98.4 ± 0.2</td><td>99.3 ± 0.2</td><td>94.2 ± 0.2</td><td>98.4 ± 0.1</td></tr><tr><td>w/o BN</td><td>94.6±0.7</td><td>95.5 ±0.1</td><td>91.7 ±0.3</td><td>94.3±0.2</td></tr><tr><td>CBN (Ioffe &amp; Szegedy, 2015)</td><td>98.5 ± 0.2</td><td>99.6 ± 0.1</td><td>94.5 ± 0.2</td><td>98.6 ± 0.1</td></tr><tr><td>RN (Bronskill et al., 2020)</td><td>98.0±0.2</td><td>99.6 ± 0.1</td><td>94.1 ± 0.2</td><td>98.6 ± 0.1</td></tr><tr><td>MetaBN (Bronskill et al., 2020)</td><td>98.4 ± 0.2</td><td>99.6 ± 0.1</td><td>94.5 ± 0.2</td><td>98.6 ± 0.1</td></tr><tr><td>TaskNorm-L (Bronskill et al., 2020)</td><td>98.6 ±0.2</td><td>99.6 ± 0.1</td><td>95.0 ± 0.2</td><td>98.7 ± 0.1</td></tr><tr><td>TaskNorm-I (Bronskill et al., 2020)</td><td>98.4 ± 0.2</td><td>99.6 ± 0.2</td><td>93.4 ± 0.2</td><td>98.6 ± 0.1</td></tr><tr><td> MetaNorm</td><td>98.9 ± 0.2</td><td>99.7 ± 0.2</td><td>95.8 ± 0.2</td><td>98.9 ± 0.2</td></tr></table>
+
+† Results (except w/o BN and our MetaNorm) provided by (Bronskill et al., 2020).
+
+![](images/e9f9057e57e5aa2fd5e997a1561de84cd0681c96954abcbb96ae4c929560b3be.jpg)  
+Figure 4: Training loss. Results of using the ProtoNets algorithm on miniImageNet with respect to training loss versus iterations. Our MetaNorm achieves fastest training convergence.
+
+# H TRAINING SPEED
+
+We plot the training loss versus training iterations by using the ProtoNets algorithm in Figure 4. MetaNorm achieves fastest training convergence. From Table 2 and Figure 4, MetaNorm achieves best classification accuracy and training efficiency, which demonstrates the benefit of leveraging meta-learning by MetaNorm for batch normalization.
+
+<table><tr><td>48 3 8 10</td><td>3 10 4 4 5 4 30 3 10 10 10 10 二</td></tr><tr><td>CIIRIIO 4108 311</td><td>6000 444111 31171 6 404 4 王 干0&#x27;os</td></tr><tr><td>8049 30325 CITIIII</td><td>801&#x27;09 S05515 80505 8079 800£9 80419 80699 80 干9&#x27;29 L&#x27;0 干 1&#x27;29</td></tr><tr><td>40424 LSINW</td><td>9&#x27;0&#x27;98 406:16 406 0418 403356 03636 404155 406:16 533335 10  1:5</td></tr><tr><td></td><td></td></tr><tr><td>WSCCCC 33730</td><td>33999 333580 60500 60 33130 33983 33410 33770 333585 3.0 57.69</td></tr><tr><td>srs rrrer 80009</td><td></td></tr><tr><td></td><td>80565 8&#x27;0干&#x27;S9 60王169 80L09 80干0&#x27;99 8079 806.9 20029 L&#x27;08&#x27;S9 9&#x27;0 干 1&#x27;89</td></tr><tr><td>GEeeeoe 90188</td><td></td></tr><tr><td></td><td>908 90558 808 80469 20448 90098 L&#x27;Oi18 0678 903738 10 33.38</td></tr><tr><td>4494 5</td><td></td></tr><tr><td></td><td>45.14 333531 218.6 44.94 4594 441 44514 4 9.61 90 干 119</td></tr><tr><td>mrarm 8&#x27;00&#x27;SL</td><td></td></tr><tr><td>01179</td><td>805 8080L 14144 80 9&#x27;0 干 9&#x27;LL 804 L0454 1454 L&#x27;0SLL</td></tr><tr><td>L0LS</td><td></td></tr><tr><td>019 Jses</td><td></td></tr><tr><td></td><td>20509 90 干 L&#x27;99 5008 5085S 54355 L0169 L0干8&#x27;09 80459 L&#x27;06S9</td></tr><tr><td>BPpg</td><td></td></tr><tr><td>60579 60干969 600&#x27;69</td><td></td></tr><tr><td></td><td>8&#x27;0 干90/</td></tr><tr><td></td><td>31110 51411 60干9&#x27;89 809&#x27;89 6.4 747 60干889 6001</td></tr><tr><td>90388 Aarat</td><td></td></tr><tr><td>10448 Z00L</td><td></td></tr><tr><td></td><td>10691 900&#x27;SL 90干608 908 9096L 90718 933338 10148</td></tr><tr><td></td><td></td></tr><tr><td>L0主1&#x27;68 90L06 nomiii</td><td></td></tr><tr><td></td><td>90王806 80458 9°0806 90406 90706 $0 干 8&#x27;06 L09&#x27;88 9&#x27;0 干906</td></tr><tr><td>JTSSSC</td><td></td></tr><tr><td>45 410</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>305630 3.53750 41511 0 410 451 1 3.33320</td></tr><tr><td> gge ss NN7  L 1 67 UT GE 0TN</td><td>eir eg TrTrnrr rnn rrenerggree</td></tr></table>
+
+sign indicates the 95% confiden
+
+Table 17: Effect of number of units of hidden layers in MetaNorm for few-shot classification with MAML (Finn & Levine, 2018) on miniImageNet. The $\pm$ sign indicates the $9 5 \%$ confidence interval over tasks. We achieve best results with 128 units of hidden layers.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">MAML</td></tr><tr><td>5-way, 1-shot</td><td>5-way, 5-shot</td></tr><tr><td>n=64</td><td>44.3 ± 1.5</td><td>58.1 ± 0.8</td></tr><tr><td>n = 128</td><td>46.8 ± 1.6</td><td>60.1 ± 0.8</td></tr><tr><td>n = 256</td><td>46.2 ± 1.6</td><td>59.8 ± 0.8</td></tr><tr><td>n = 512</td><td>45.9 ± 1.5</td><td>59.5 ± 0.9</td></tr><tr><td>n = 1024</td><td>44.9 ± 1.5</td><td>58.7 ±0.8</td></tr></table>

parse/train/BkM3ibZRW/BkM3ibZRW.md

@@ -0,0 +1,438 @@
+# ADVERSARIALLY REGULARIZED AUTOENCODERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+While autoencoders are a key technique in representation learning for continuous structures, such as images or wave forms, developing general-purpose autoencoders for discrete structures, such as text sequence or discretized images, has proven to be more challenging. In particular, discrete inputs make it more difficult to learn a smooth encoder that preserves the complex local relationships in the input space. In this work, we propose an adversarially regularized autoencoder (ARAE) with the goal of learning more robust discrete-space representations. ARAE jointly trains both a rich discrete-space encoder, such as an RNN, and a simpler continuous space generator function, while using generative adversarial network (GAN) training to constrain the distributions to be similar. This method yields a smoother contracted code space that maps similar inputs to nearby codes, and also an implicit latent variable GAN model for generation. Experiments on text and discretized images demonstrate that the GAN model produces clean interpolations and captures the multimodality of the original space, and that the autoencoder produces improvements in semi-supervised learning as well as state-of-the-art results in unaligned text style transfer task using only a shared continuous-space representation.
+
+# 1 INTRODUCTION
+
+Recent work on regularized autoencoders, such as variational (Kingma & Welling, 2014; Rezende et al., 2014) and denoising (Vincent et al., 2008) variants, has shown significant progress in learning smooth representations of complex, high-dimensional continuous data such as images. These codespace representations facilitate the ability to apply smoother transformations in latent space in order to produce complex modifications of generated outputs, while still remaining on the data manifold.
+
+Unfortunately, learning similar latent representations of discrete structures, such as text sequences or discretized images, remains a challenging problem. Initial work on VAEs for text has shown that optimization is difficult, as the decoder can easily degenerate into a unconditional language model (Bowman et al., 2015b). Recent work on generative adversarial networks (GANs) for text has mostly focused on getting around the use of discrete structures either through policy gradient methods (Che et al., 2017; Hjelm et al., 2017; Yu et al., 2017) or with the Gumbel-Softmax distribution (Kusner & Hernandez-Lobato, 2016). However, neither approach can yet produce robust representations directly.
+
+A major difficulty of discrete autoencoders is mapping a discrete structure to a continuous code vector while also smoothly capturing the complex local relationships of the input space. Inspired by recent work combining pretrained autoencoders with deep latent variable models, we propose to target this issue with an adversarially regularized autoencoder (ARAE). Specifically we jointly train a discrete structure encoder and continuous space generator, while constraining the two models with a discriminator to agree in distribution. This approach allows us to utilize a complex encoder model, such as an RNN, and still constrain it with a very flexible, but more limited generator distribution. The full model can be then used as a smoother discrete structure autoencoder or as a latent variable GAN model where a sample can be decoded, with the same decoder, to a discrete output. Since the system produces a single continuous coded representation—in contrast to methods that act on each RNN state—it can easily be further regularized with problem-specific invariants, for instance to learn to ignore style, sentiment or other attributes for transfer tasks.
+
+Experiments apply ARAE to discretized images and sentences, and demonstrate that the key properties of the model. Using the latent variable model (ARAE-GAN), the model is able to generate varied samples that can be quantitatively shown to cover the input spaces and to generate consistent image and sentence manipulations by moving around in the latent space via interpolation and offset vector arithmetic. Using the discrete encoder, the model can be used in a semi-supervised setting to give improvement in a sentence inference task. When the ARAE model is trained with task-specific adversarial regularization, the model improves the current best results on sentiment transfer reported in Shen et al. (2017) and produces compelling outputs on a topic transfer task using only a single shared code space. All outputs are listed in the Appendix 9 and code is available at (removed for review).
+
+# 2 RELATED WORK
+
+In practice unregularized autoencoders often learn a degenerate identity mapping where the latent code space is free of any structure, so it is necessary to apply some method of regularization. A popular approach is to regularize through an explicit prior on the code space and use a variational approximation to the posterior, leading to a family of models called variational autoencoders (VAE) (Kingma & Welling, 2014; Rezende et al., 2014). Unfortunately VAEs for discrete text sequences can be challenging to train—for example, if the training procedure is not carefully tuned with techniques like word dropout and KL annealing (Bowman et al., 2015b), the decoder simply becomes a language model and ignores the latent code (although there has been some recent successes with convolutional models (Semeniuta et al., 2017; Yang et al., 2017)). One possible reason for the difficulty in training VAEs is due to the strictness of the prior (usually a spherical Gaussian) and/or the parameterization of the posterior. There has been some work on making the prior/posterior more flexible through explicit parameterization (Rezende & Mohamed, 2015; Kingma et al., 2016; Chen et al., 2017). A notable technique is adversarial autoencoders (AAE) (Makhzani et al., 2015) which attempt to imbue the model with a more flexible prior implicitly through adversarial training. In AAE framework, the discriminator is trained to distinguish between samples from a fixed prior distribution and the input encoding, thereby pushing the code distribution to match the prior. While this adds more flexibility, it has similar issues for modeling text sequences and suffers from mode-collapse in our experiments. Our approach has similar motivation, but notably we do not sample from a fixed prior distribution—our ‘prior’ is instead parameterized through a flexible generator. Nonetheless, this view (which has been observed by various researchers (Tran et al., 2017; Mescheder et al., 2017; Makhzani & Frey, 2017)) provides an interesting connection between VAEs and GANs.
+
+The success of GANs on images have led many researchers to consider applying GANs to discrete data such as text. Policy gradient methods are a natural way to deal with the resulting non-differentiable generator objective when training directly in discrete space (Glynn, 1987; Williams, 1992). When trained on text data however, such methods often require pre-training/co-training with a maximum likelihood (i.e. language modeling) objective (Che et al., 2017; Yu et al., 2017; Li et al., 2017). This precludes there being a latent encoding of the sentence, and is also a potential disadvantage of existing language models (which can otherwise generate locally-coherent samples). Another direction of work has been through reparameterizing the categorical distribution with the Gumbel-Softmax trick (Jang et al., 2017; Maddison et al., 2017)—while initial experiments were encouraging on a synthetic task (Kusner & Hernandez-Lobato, 2016), scaling them to work on natural language is a challenging open problem. There has also been a flurry of recent, related approaches that work directly with the soft outputs from a generator (Gulrajani et al., 2017; Sai Rajeswar, 2017; Shen et al., 2017; Press et al., 2017). For example, Shen et al. (Shen et al., 2017) exploits adversarial loss for unaligned style transfer between text by having the discriminator act on the RNN hidden states and using the soft outputs at each step as input to an RNN generator, utilizing the Professor-forcing framework (Lamb et al., 2016). Our approach instead works entirely in code space and does not require utilizing RNN hidden states directly.
+
+# 3 BACKGROUND
+
+Discrete Structure Autoencoders Define $\mathcal { X } = \mathcal { V } ^ { n }$ to be a set of discrete structures where $\nu$ is a vocabulary of symbols and $\mathbb { P } _ { x }$ to be a distribution over this space. For instance, for binarized images $\mathcal { V } = \{ 0 , 1 \}$ and $n$ is the number of pixels, while for sentences $\nu$ is the vocabulary and $n$ is the sentence length. A discrete autoencoder consists of two parameterized functions: a deterministic encoder function $\mathrm { e n c } _ { \phi } : \mathcal { X } \mapsto \mathcal { C }$ with parameters $\phi$ that maps from input to code space and a conditional decoder distribution $p _ { \psi } ( \mathbf { x } \mid \mathbf { c } )$ over structures $\mathcal { X }$ with parameters $\psi$ . The parameters are trained on a cross-entropy reconstruction loss:
+
+$$
+{ \mathcal { L } } _ { \mathrm { r e c } } ( \phi , \psi ) = - \log p _ { \psi } ( \mathbf { x } \mid \mathrm { e n c } _ { \phi } ( \mathbf { x } ) )
+$$
+
+The choice of the encoder and decoder parameterization is specific to the structure of interest, for example we use RNNs for sequences. We use the notation, $\hat { \mathbf { x } } = \arg \operatorname* { m a x } _ { \mathbf { x } } p _ { \psi } ( \mathbf { x } \mid \mathrm { e n c } _ { \phi } ( \mathbf { x } ) )$ for the (approximate) decoder mode. When $\mathbf { x } = { \hat { \mathbf { x } } }$ the autoencoder is said to perfectly reconstruct x.
+
+Generative Adversarial Networks GANs are a class of parameterized implicit generative models (Goodfellow et al., 2014). The method approximates drawing samples from a true distribution $\mathbf { c } \sim \mathbb { P } _ { r }$ by instead employing a latent variable $\mathbf { z }$ and a parameterized deterministic generator function $\tilde { \mathbf { c } } = g _ { \theta } ( \mathbf { z } )$ to produce samples $\tilde { \mathbf { c } } \sim \mathbb { P } _ { g }$ . Initial work on GANs minimizes the Jensen-Shannon divergence between the distributions. Recent work on Wasserstein GAN (WGAN) (Arjovsky et al., 2017), replaces this with the Earth-Mover (Wasserstein-1) distance.
+
+GAN training utilizes two separate models: a generator $g _ { \boldsymbol \theta } ( \mathbf { z } )$ maps a latent vector from some easy-to-sample source distribution to a sample and a critic/discriminator $f _ { w } ( { \bf c } )$ aims to distinguish real data and generated samples from $g _ { \theta }$ . Informally, the generator is trained to fool the critic, and the critic to tell real from generated. WGAN training uses the following min-max optimization over generator parameters $\theta$ and critic parameters $w$ ,
+
+$$
+\operatorname* { m i n } _ { \theta } \operatorname* { m a x } _ { w \in \mathcal { W } } \mathbb { E } _ { \mathbf { c } \sim \mathbb { P } _ { r } } [ f _ { w } ( \mathbf { c } ) ] - \mathbb { E } _ { \tilde { \mathbf { c } } \sim \mathbb { P } _ { g } } [ f _ { w } ( \tilde { \mathbf { c } } ) ] ,
+$$
+
+where $f _ { w } : { \mathcal { C } } \mapsto \mathbb { R }$ denotes the critic function, c˜ is obtained from the generator, $\tilde { \mathbf { c } } = g _ { \theta } ( \mathbf { z } )$ , and $\mathbb { P } _ { r }$ and $\mathbb { P } _ { g }$ are real and generated distributions. If the critic parameters $w$ are restricted to an 1-Lipschitz function set $\mathcal { W }$ , this term correspond to minimizing Wasserstein-1 distance $W ( \mathbb { P } _ { r } , \mathbb { P } _ { g } )$ . We use a naive approximation to enforce this property by weight-clipping, i.e. $w = [ - \epsilon , \epsilon ] ^ { d }$ (Arjovsky et al., 2017).
+
+# 4 MODEL: ADVERSARIALLY REGULARIZED AUTOENCODER
+
+Ideally, a discrete autoencoder should be able to reconstruct $x$ from $c .$ , but also smoothly assign similar codes $c$ and $c ^ { \prime }$ to similar $x$ and $x ^ { \prime }$ . For continuous autoencoders, this property can be enforced directly through explicit regularization. For instance, contractive autoencoders (Rifai et al., 2011) regularize their loss by the functional smoothness of enc $\phi$ . However, this criteria does not apply when inputs are discrete and we lack even a metric on the input space. How can we enforce that similar discrete structures map to nearby codes?
+
+Adversarially regularized autoencoders target this issue by learning a parallel continuous-space generator with a restricted functional form to act as a smoother reference encoding. The joint objective regularizes the autoencoder to constrain the discrete encoder to agree in distribution with its continuous counterpart:
+
+$$
+\begin{array} { r l } { \underset { \phi , \psi , \theta } { \operatorname* { m i n } } } & { { } \mathcal { L } _ { \mathrm { r e c } } ( \phi , \psi ) + \lambda ^ { ( 1 ) } W ( \mathbb { P } _ { r } , \mathbb { P } _ { g } ) } \end{array}
+$$
+
+Above $W$ is the Wasserstein-1 distance between $\mathbb { P } _ { r }$ the distribution of codes from the discrete encoder model (enc ${ } _ { \phi } ( x )$ where $x \sim \mathbb { P } ( x ) )$ and $\mathbb { P } _ { g }$ is the distribution of codes from the continuous generator model $( g _ { \boldsymbol { \theta } } ( z )$ for some $z$ , e.g. $z \sim \mathcal { N } ( 0 , I ) )$ . To approximate Wasserstein-1 term, the $W$ function includes an embedded critic function which is optimized adversarially to the encoder and generator as described in the background. The full model is shown in Figure 1.
+
+To train the model, we use a block coordinate descent to alternate between optimizing different parts of the model: (1) the encoder and decoder to minimize reconstruction loss, (2) the WGAN critic function to approximate the $W$ term, (3) the encoder and generator to adversarially fool the critic to minimize $W$ :
+
+$$
+\begin{array} { r l r l r l } { { 1 } ) \underset { \phi , \psi } { \mathrm { m i n } } } & { \quad } & & { \mathcal { L } _ { \mathrm { r e c } } ( \phi , \psi ) } & & { } \\ { 2 ) \underset { w \in \mathcal { W } } { \mathrm { m i n } } } & { \quad } & & { \mathcal { L } _ { \mathrm { c r i } } ( w ) = } & { \quad } & & { \underset { w \in \mathcal { W } } { \mathrm { m a x } } \quad \mathbb { E } _ { \mathbf { x } \sim \mathbb { P } _ { x } } \left[ f _ { w } ( \mathbf { e n c } _ { \phi } ( \mathbf { x } ) ) \right] - \mathbb { E } _ { \widetilde { \mathbf { c } } \sim \mathbb { P } _ { g } } \left[ f _ { w } ( \widetilde { \mathbf { c } } ) \right] } \\ { 3 ) \underset { \phi , \theta } { \mathrm { m i n } } } & { \quad } & & { \mathcal { L } _ { \mathrm { e n c s } } ( \phi , \theta ) = } & { \quad } & & { \underset { \phi , \theta } { \mathrm { m i n } } } & { \quad \mathbb { E } _ { \mathbf { x } \sim \mathbb { P } _ { x } } \left[ f _ { w } ( \mathbf { e n c } _ { \phi } ( \mathbf { x } ) ) \right] - \mathbb { E } _ { \widetilde { \mathbf { c } } \sim \mathbb { P } _ { g } } \left[ f _ { w } ( \widetilde { \mathbf { c } } ) \right] } \end{array}
+$$
+
+The full training algorithm is shown in Algorithm 1.
+
+![](images/ca36a5089dd4385655adadbef891cf50e331699403af8d1054ab6e0dad78b413.jpg)  
+Figure 1: ARAE architecture. The model can be used as an autoencoder, where a structure $\mathbf { x }$ is encoded and decoded to produce $\hat { \bf x }$ , and as a GAN (ARAE-GAN), where a sample $\mathbf { z }$ is passed though a generator $g _ { \theta }$ to produce a code vector, which is similarly decoded to $\tilde { \mathbf { x } }$ . The critic function $f _ { w }$ is only used at training to help approximate $W$ .
+
+# Algorithm 1 ARAE Training
+
+<table><tr><td>for number of training iterations do (1) Train the autoencoder for reconstruction [Lrec(Φ,)].</td><td></td><td></td></tr><tr><td>Sample {x()}m1~P andcomputecode-vectors c(i) =en(x().</td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>(2) Train the critic [Lcri(w)] (Repeat k times)</td><td></td><td></td></tr><tr><td>Sample {x)1~P and {z@1~N(0,I).</td><td></td><td></td></tr><tr><td>Compute code-vectors c(i)=en(x()and c()= ge(z().</td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>(3) Train the generator and encoder adversarially to critic [Lencs(,0)]</td><td></td><td></td></tr><tr><td>Sample {x)ym1~P and {z)}1~N(0,I)</td><td></td><td></td></tr><tr><td>Compute code-vectors c(i) = enc(x(i)) and c(i) = ge(z(i).</td><td></td><td></td></tr><tr><td>Backpropagate adversariallossm∑1 f(c(i))-∑=1 fu(c()and update.</td><td></td><td></td></tr></table>
+
+Extension: Code Space Transfer One benefit of the ARAE framework is that it compresses the input to a single code vector. This framework makes it ideal for manipulating discrete objects while in continuous code space. For example, consider the problem of unaligned transfer, where we want to change an attribute of a discrete input without supervised examples, e.g. to change the topic or sentiment of a sentence. First, we extend the decoder to condition on a transfer variable denoting this attribute $\mathbf { y }$ which is known during training, to learn $p _ { \psi } ( \mathbf { x } \mid \mathbf { c } , y )$ . Next, we train the code space to be invariant to this attribute, to force it to be learned fully by the decoder. Specifically, we further regularize the code space to map similar $x$ with different attribute labels $y$ near enough to fool a code space attribute classifier, i.e.:
+
+$$
+\operatorname* { m i n } _ { \phi , \psi , \theta } \quad \mathcal { L } _ { \mathrm { r e c } } ( \phi , \psi ) + \lambda ^ { ( 1 ) } W ( \mathbb { P } _ { r } , \mathbb { P } _ { g } ) - \lambda ^ { ( 2 ) } \mathcal { L } _ { \mathrm { c l a s s } } ( \phi , u )
+$$
+
+where $\mathcal { L } _ { \mathrm { c l a s s } } ( \phi , u )$ is the loss of a classifier $p _ { u } ( y \mid \mathbf { c } )$ from code space to labels (in our experiments we always set $\lambda ^ { ( 2 ) } = 1 \AA$ ). To incorporate this additional regularization, we simply add two more gradient update steps: (2b) training a classifier to discriminate codes, and (3b) adversarially training the encoder to fool this classifier. The algorithm is shown in Algorithm 2. Note that similar technique has been introduced in other domains, notably in images (Lample et al., 2017) and video modeling (Denton $\&$ Birodkar, 2017).
+
+# 5 METHODS AND ARCHITECTURES
+
+We experiment with three different ARAE models: (1) an autoencoder for discretized images trained on the binarized version of MNIST, (2) an autoencoder for text sequences trained using the Stanford Natural Language Inference (SNLI) corpus (Bowman et al., 2015a), and (3) an autoencoder trained
+
+# Algorithm 2 ARAE Transfer Extension
+
+<table><tr><td>[Each loop additionally:] (2b) Train the code classifier [minu Lclass(,u)]</td></tr><tr><td></td></tr><tr><td> Sample {x()}m=1 ~ Px,lookup y(),and computecode-vectors c(i) = enc(x(i).</td></tr><tr><td></td></tr><tr><td>(3b) Train the encoder adversarially to code classifier [max Lclass(,u)]</td></tr><tr><td>Sample {x()}m1P,lookupy(),ndcomputecode-vectors c(i) =ec(x(i).</td></tr><tr><td></td></tr></table>
+
+for text transfer (Section 6.2) based on the Yelp and Yahoo datasets for unaligned sentiment and topic transfer. All three models utilize the same generator architecture, $g _ { \theta }$ . The generator architecture uses a low dimensional $\mathbf { z }$ with a Gaussian prior $p ( \mathbf { z } ) = \mathcal { N } ( 0 , \mathbf { I } )$ , and maps it to c. Both the critic $f _ { w }$ and the generator $g _ { \theta }$ are parameterized as feed-forward MLPs.
+
+The image model uses fully-connected NN to autoencode binarized images. Here $\mathcal { X } = \{ 0 , 1 \} ^ { n }$ where $n$ is the image size. The encoder used is a feed-forward MLP network mapping from $\{ 0 , 1 \} ^ { n } \mapsto \mathbb { R } ^ { m }$ , ${ \mathrm { e n c } } _ { \phi } ( \mathbf { x } ) = \mathbf { M L P } ( \mathbf { x } ; \phi ) = \mathbf { c }$ . The decoder predicts each pixel in $\mathbf { x }$ as a parameterized logistic regression, $\begin{array} { r } { p _ { \psi } ( \mathbf { x } \mid \mathbf { c } ) = \prod _ { j = 1 } ^ { n } \sigma ( \mathbf { h } ) ^ { x _ { j } } ( 1 - \sigma ( \mathbf { h } ) ) ^ { 1 - x _ { j } } } \end{array}$ where $\mathbf { h } = \mathbf { M L P } ( \mathbf { c } ; \psi )$ .
+
+The text model uses a recurrent neural network (RNN) for both the encoder and decoder. Here $\mathcal { X } = \mathcal { V } ^ { n }$ where $n$ is the sentence length and $\nu$ is the vocabulary of the underlying language. Define an RNN as a parameterized recurrent function $\mathbf { h } _ { j } = \mathrm { R N N } ( x _ { j } , \mathbf { h } _ { j - 1 } ; \phi )$ for $j = 1 \ldots n$ (with $\mathbf { h } _ { 0 } = \mathbf { 0 } .$ ) that maps a discrete input structure $\mathbf { x }$ to hidden vectors $\mathbf { h } _ { 1 } \ldots . . . \mathbf { h } _ { n }$ . For the encoder, we define $\mathbf { e n c } _ { \phi } ( \mathbf { x } ) = \mathbf { h } _ { n } = \mathbf { c }$ . For decoding we feed $\mathbf { c }$ as an additional input to the decoder RNN at each time step, i.e. $\tilde { \mathbf { h } } _ { j } = \mathrm { R N N } ( x _ { j } , \tilde { \mathbf { h } } _ { j - 1 } , \mathbf { c } ; \psi )$ , and further calculate the distribution over $\nu$ at each time step via softmax, $\begin{array} { r } { p _ { \psi } ( \mathbf { x } \mid \mathbf { \bar { c } } ) = \prod _ { j = 1 } ^ { n } \mathrm { s o f t m a x } ( \mathbf { W } \tilde { \mathbf { h } } _ { j } + \mathbf { b } ) _ { x _ { j } } } \end{array}$ where $\mathbf { W }$ and $\mathbf { b }$ are parameters (part of $\psi$ ). Finding the most likely sequence $\tilde { \bf x }$ under this distribution is intractable, but it is possible to approximate it using greedy search or beam search. In our experiments we use an LSTM architecture (Hochreiter & Schmidhuber, 1997) for both the encoder/decoder and decode using greedy search. The text transfer model uses the same architecture as the text model but extends it with a code space classifier $p ( y | \mathbf { c } )$ which is modeled using an MLP and trained to minimize cross-entropy.
+
+Our baselines utilize a standard autoencoder (AE) and the cross-aligned autoencoder (Shen et al., 2017) for transfer. Note that in both our ARAE and standard AE experiments, the encoded code from the encoder is normalized to lie on the unit sphere, and the generated code is bounded to lie in $( - 1 , 1 ) ^ { n }$ by the tanh function at output layer. We additionally experimented with the sequence VAE introduced by Bowman et al. (2015b) and the adversarial autoencoder (AAE) model (Makhzani et al., 2015) on the SNLI dataset. However despite extensive parameter tuning we found that neither model was able to learn meaningful latent representations—the VAE simply ignored the latent code and the AAE experienced mode-collapse and repeatedly generated the same samples. The Appendix 12 includes detailed descriptions of the hyperparameters, model architecture, and training regimes.
+
+# 6 EXPERIMENTS
+
+Our experiments consider three aspects of the model. First we measure the empirical impact of regularization on the autoencoder. Next we apply the discrete autoencoder to two applications, unaligned style transfer and semi-supervised learning. Finally we employ the learned generator network as an implicit latent variable model (ARAE-GAN) over discrete sequences.
+
+# 6.1 IMPACT OF REGULARIZATION ON DISCRETE ENCODING
+
+Our main goal for ARAE is to regularize the model produce a smoother encoder by requiring the distribution from the encoder to match the distribution from the continuous generator over a simple latent variable. To examine this claim we consider two basic statistical properties of the code space during training of the text model on SNLI, shown in Figure 2. On the left, we see that the $\ell 2$ norm of c and code c˜ converge quickly in ARAE training. The encoder code is always restricted to be on the unit sphere, and the generated code c˜ quickly learns to match it. The middle plot shows the convergence of the trace of the covariance matrix between the generator and the encoder as training progresses. We find that variance of the encoder and the generator match after several epochs. To check the smoothness of the model, for both ARAE/AE, we take a sentence and calculate the average cosine similarity of 100 randomly-selected sentences that had an edit-distance of at most 5 to the original sentence. We do this for 250 sentences and calculate the mean of the average cosine similarity. Figure 2 (right) shows that the cosine similarity of nearby sentences is quite high for the ARAE than in the case for the AE. Edit-distance is not an ideal proxy for similarity in sentences, but it is often a sufficient condition.
+
+![](images/cdd4cb1a96e120505ce485a5de0b9d938f8ffd9e8488c2dff6d54dbadd0d3010.jpg)  
+Figure 2: Left: \`2 norm of encoder code c and generator code c˜ during ARAE training. The encoder c is normalized by the model, whereas the generator learns to match this as training progresses. Middle: Sum of the dimension-wise variances of the encoder codes $\mathbb { P } _ { r }$ and generator codes $\mathbb { P } _ { g }$ compared to that of the standard AE. Right: Average cosine similarity of nearby sentences (edit-distance wise) for the ARAE and AE.
+
+Table 1: Left. Reconstruction error (negative log-likelihood averaged over sentences) of the original sentence from a corrupted sentence. Here $k$ is the number of swaps performed on the original sentence. Right. Samples generated from AE and ARAE where the input is noised by swapping words.   
+
+<table><tr><td>k</td><td>AE</td><td>ARAE</td><td>Original Noised</td><td>A woman wearing sunglasses .</td><td>Original Noised</td><td>They have been swimming . been have They swimming.</td></tr><tr><td>0</td><td>1.06</td><td>2.19</td><td>AE</td><td>A woman sunglasses wearing A woman sunglasses wearing sunglasses .</td><td>AE</td><td>been have been swimming.</td></tr><tr><td>1</td><td>4.51</td><td>4.07</td><td>ARAE</td><td>A woman wearing sunglasses .</td><td>ARAE</td><td>Children have been swimming .</td></tr><tr><td>2</td><td>6.61</td><td>5.39</td><td>Original</td><td>Pets galloping down the street .</td><td>Original</td><td>The child is sleeping .</td></tr><tr><td>3</td><td>9.14</td><td>6.86</td><td>Noised</td><td>Pets down the galloping street .</td><td>Noised</td><td>child The is sleeping.</td></tr><tr><td></td><td></td><td></td><td>AE</td><td>Pets riding the down galloping .</td><td>AE</td><td>The child is sleeping is .</td></tr><tr><td>4</td><td>9.97</td><td>7.47</td><td>ARAE</td><td>Pets congregate down the street near a ravine .</td><td>ARAE</td><td>The child is sleeping.</td></tr></table>
+
+Finally an ideal representation should be robust to small changes of the input around the training examples in code space (Rifai et al., 2011). We can test this property by feeding a noised input to the encoder and (i) calculating the score given to the original input, and (ii) checking the reconstructions. Table 1 (right) shows an experiment for text where we add noise by permuting $k$ words in each sentence. We observe that the ARAE is able to map a noised sentence to a natural sentence, (though not necessarily the denoised sentence). Table 1 (left) shows empirical results for these experiments. We obtain the reconstruction error (i.e. negative log likelihood) of the original (non-noised) sentence under the decoder, utilizing the noised code. We find that when $k = 0$ (i.e. no swaps), the regular AE better reconstructs the input as expected. However, as we increase the number of swaps and push the input further away from the data manifold, the ARAE is more likely to produce the original sentence. We note that unlike denoising autoencoders which require a domain-specific noising function (Hill et al., 2016; Vincent et al., 2008), the ARAE is not explicitly trained to denoise an input, but learns to do so as a byproduct of adversarial regularization.
+
+# 6.2 APPLICATIONS OF DISCRETE AUTOENCODER
+
+Unaligned Text Transfer A smooth autoencoder combined with low reconstruction error should make it possible to more robustly manipulate discrete objects through code space without dropping off the data manifold. To test this hypothesis, we experimented with two unaligned text transfer tasks. For these tasks, we attempt to change one attribute of a sentence without aligned examples of this change. To perform this transfer, we learn a code space that can represent an input that is agnostic to this attribute, and a decoder that can incorporate the attribute (as described in Section 4). We experiment with unaligned transfer of sentiment on the Yelp corpus and topic on the Yahoo corpus (Zhang et al., 2015).
+
+Table 2: Experiments on sentiment transfer. Left shows the automatic metrics (Transfer/BLEU/PPL/Reverse PPL) while right shows human evaluation metrics (Transfer/Similarity/Naturalness). Cross-Aligned AE is from Shen et al. (2017)   
+
+<table><tr><td></td><td colspan="4">Automatic Evaluation</td><td colspan="3">Human Evaluation</td></tr><tr><td>Model</td><td>Transfer</td><td>BLEU</td><td>PPL</td><td>Reverse PPL</td><td>Transfer</td><td>Similarity</td><td>Naturalness</td></tr><tr><td>Cross-Aligned AE</td><td>77.1%</td><td>17.75</td><td>65.9</td><td>124.2</td><td>57%</td><td>3.8</td><td>2.7</td></tr><tr><td>AE</td><td>59.3%</td><td>37.28</td><td>31.9</td><td>68.9</td><td>-</td><td>1</td><td>-</td></tr><tr><td>ARAE, 入(1</td><td>73.4%</td><td>31.15</td><td>29.7</td><td>70.1</td><td>-</td><td>-</td><td>1</td></tr><tr><td>ARAE,X</td><td>81.8%</td><td>20.18</td><td>27.7</td><td>77.0</td><td>74%</td><td>3.7</td><td>3.8</td></tr></table>
+
+<table><tr><td></td><td>Positive = Negative</td><td></td><td>Negative=Positive</td></tr><tr><td></td><td>great indoor mall</td><td></td><td>hell no ！</td></tr><tr><td>ARAE</td><td>no smoking mall.</td><td>ARAE</td><td>hell great！</td></tr><tr><td>Cross-AE</td><td>terrible outdoor urine .</td><td>Cross-AE</td><td>incredible pork ！</td></tr><tr><td></td><td>it has a great atmosphere,with wonderful service .</td><td></td><td>small,smokey,dark and rude management .</td></tr><tr><td>ARAE</td><td>it has no taste,with a complete jerk .</td><td>ARAE</td><td>small, intimate,and cozy friendly staff .</td></tr><tr><td>Cross-AE</td><td>it has a great horrible food and run out service .</td><td>Cross-AE</td><td>great,,,chips and wine .</td></tr><tr><td></td><td>we came on the recommendation of a bellboy and the food was amazing .</td><td></td><td>the people who ordered off the menu did n&#x27;t seem to do much better .</td></tr><tr><td>ARAE</td><td>we came on the recommendation and the food was a joke .</td><td>ARAE</td><td>the people who work there are super friendly and the menu is good</td></tr><tr><td>Cross-AE</td><td>we went on the car of the time and the chicken was awful .</td><td>Cross-AE</td><td>the place,one of the office is always worth you do a business</td></tr></table>
+
+Table 3: Sentiment transfer results. Original sentence and transferred output (from ARAE and the Cross-Aligned AE) of 6 randomly-drawn examples.
+
+For sentiment we follow the same setup as Shen et al. (2017) and split the Yelp corpus into two sets of unaligned positive and negative reviews. We train an ARAE as an autoencoder with two separate decoders, one for positive and one for negative sentiment, and incorporate adversarial training of the encoder to remove sentiment information from the code space. We test by encoding in sentences of one class and decoding, greedily, with the opposite decoder.
+
+Our evaluation is based on four automatic metrics, shown in Table 2: (i) Transfer: measuring how successful the model is at transferring sentiment based on an automatic classifier (we use the fastText library (Joulin et al., 2016)). (ii) BLEU: measuring the consistency between the transferred text and the original. We expect the model to maintain as much information as possible and transfer only the style; (iii) Perplexity: measuring the fluency of the generated text; (iv) Reverse Perplexity: measuring the extent to which the generations are representative of the underlying data distribution.1 Both perplexity numbers are obtained by training an RNN language model.
+
+We additionally perform human evaluations on the cross-aligned AE and our best ARAE model. We randomly select 1000 sentences (500/500 positive/negative), obtain the corresponding transfers from both models, and ask Amazon Mechanical Turkers to evaluate the sentiment (Positive/Neutral/Negative) and naturalness (1-5, 5 being most natural) of the transferred sentences. We create a separate task in which we show the Turkers the original and the transferred sentences, and ask them to evaluate the similarity based on sentence structure (1-5, 5 being most similar). We explicitly ask the Turkers to disregard sentiment in their similarity assessment.
+
+In addition to comparing against the cross-aligned AE of Shen et al. (2017), we also compare against a vanilla AE trained without adversarial regularization. For ARAE, we experimented with different $\lambda ^ { ( 1 ) }$ weighting on the adversarial loss (see section 4) with $\lambda _ { a } ^ { ( 1 ) } = 1 , \lambda _ { b } ^ { ( 1 ) } = 1 0$ . We generally set $\lambda ^ { ( 2 ) } = 1$ . Experimentally the adversarial regularization enhances transfer and perplexity, but tends to make the transferred text less similar to the original, compared to the AE. Some randomly selected sentences are shown in figure 6 and more samples are shown available in Appendix 9.
+
+The same method can be applied to other style transfer tasks, for instance the more challenging Yahoo QA data (Zhang et al., 2015). For Yahoo we chose 3 relatively distinct topic classes for transfer: Science & Math, Entertainment & Music, and Politics & Government. As the dataset contains both
+
+questions and answers, we separated our experiments into titles (questions) and replies (answers). The qualitative results are showed in table 4. See Appendix 9 for additional generation examples.   
+Table 4: Random samples from Yahoo topic transfer. Note the first row is from ARAE trained on titles while the following ones are from replies.   
+
+<table><tr><td></td><td colspan="2">Original Science</td><td colspan="2">Original Music</td><td>Original Politics</td></tr><tr><td></td><td>what is an event horizon with regards to black holes ？</td><td></td><td>do you know a website that you can find people who want to join bands ?</td><td></td><td>republicans :would you vote for a cheney /satan ticket in 2008 ？</td></tr><tr><td>Music</td><td>what is your favorite sitcom with adam sandler ?</td><td>Science</td><td>do you know a website that can help me with sci- ence ？</td><td>Science</td><td>guys : how would you solve this question ?</td></tr><tr><td>Politics</td><td>what is an event with black people ?</td><td>Politics</td><td>do you think that you can find a person who is in prison ？</td><td>Music</td><td>guys : would you rather be a good movie ?</td></tr><tr><td>50ml.</td><td>take lml of hcl（concentrated ）and dilute it to</td><td></td><td>all three are fabulous artists,with just incredible talent！！</td><td></td><td>4 years of an idiot in office + electing the idiot again = ？</td></tr><tr><td>Music</td><td>take em to you and shout it to me</td><td>Science</td><td>all three are genetically bonded with water,but just as many substances ,are capable of producing</td><td>Science</td><td>4 years of an idiot in the office of science ?</td></tr><tr><td>Politics</td><td>take bribes to islam and it will be punished .</td><td>Politics</td><td>a special case . all three are competing with the government, just as far as i can.</td><td>Music</td><td>4 ）&lt;unk&gt; in an idiot ,the idiot is the best of the two points ever !</td></tr><tr><td>of the other .</td><td> just multiply the numerator of one fraction by that</td><td></td><td>but there are so many more ican &amp;apos;t think of</td><td></td><td>anyone who doesnt have a billion dollars for all the publicity cant win .</td></tr><tr><td>Music</td><td>just multiply the fraction of the other one that</td><td>Science</td><td>but there are so many more of the number of ques-</td><td>Science</td><td>anyone who doesnt have a decent chance is the</td></tr><tr><td>Politics</td><td>&amp;apos;s just like it. just multiply the same fraction of other countries.</td><td>Politics</td><td>tions. but there are so many more of the can i think of today.</td><td>Music</td><td>same for all the other . anyone who doesnt have a lot of the show for the publicity.</td></tr></table>
+
+Semi-Supervised Training We further utilize ARAE in a standard AE setup for semi-supervised training. We experiment on a natural language inference task, shown in Table 5 (right). We use $2 2 . 2 \%$ , $1 0 . 8 \%$ and $5 . 2 5 \%$ of the original labeled training data, and use the rest of the training set for unlabeled training. The labeled set is randomly picked. The full SNLI training set contains $5 4 3 \mathrm { k }$ sentence pairs, and we use supervised sets of 120k, $5 9 \mathrm { k }$ and 28k sentence pairs respectively for the three settings. As a baseline we use an AE trained on the additional data, similar to the setting explored in Dai & Le (2015). For ARAE we use the subset of unsupervised data of length $< 1 5$ , which roughly includes $6 5 5 \mathrm { k }$ single sentences (due to the length restriction, this is a subset of 715k sentences that were used for AE training). As observed by Dai & Le (2015), training on unlabeled data with an AE objective improves upon a model just trained on labeled data. Training with adversarial regularization provides further gains.
+
+# .3 A LATENT VARIABLE MODEL FOR DISCRETE STRUCTURES
+
+After training, an ARAE can also be used as an implicit latent variable model controlled by $\mathbf { z }$ and the generator $g _ { \theta }$ , which we refer to as ARAE-GAN. While models of this form have been widely used for generation in other modalities, they have been less effective for discrete structures. In this section, we attempt to measure the effectiveness of this induced discrete GAN.
+
+A common test for a GANs ability mimic the true distribution $\mathbb { P } _ { r }$ is to train a simple model on generated samples from $\mathbb { P } _ { g }$ . While there are pitfalls of this evaluation (Theis et al., 2016), it provides a starting point for text modeling. Here we generate $1 0 0 \mathrm { k }$ samples from (i) ARAE-GAN, (ii) an $\mathsf { A E } ^ { 2 }$ , (iii) a RNN LM trained on the same data, and (iv) the real training set (samples from the models are
+
+2To “sample” from an AE we fit a multivariate Gaussian to the code space after training and generate code vectors from this Gaussian to decode back into sentence space.   
+
+<table><tr><td>Data for LM</td><td>Reverse PPL</td></tr><tr><td>Real data</td><td>27.4</td></tr><tr><td>LM samples</td><td>90.6</td></tr><tr><td>AE samples</td><td>97.3</td></tr><tr><td>ARAE-GAN samples</td><td>82.2</td></tr></table>
+
+<table><tr><td>Model</td><td>Medium</td><td>Small</td><td>Tiny</td></tr><tr><td>Supervised Encoder</td><td>65.9%</td><td>62.5%</td><td>57.9%</td></tr><tr><td>Semi-Supervised AE</td><td>68.5%</td><td>64.6%</td><td>59.9%</td></tr><tr><td>Semi-Supervised ARAE</td><td>70.9%</td><td>66.8%</td><td>62.5%</td></tr></table>
+
+Table 5: Left. Semi-Supervised accuracy on the natural language inference (SNLI) test set, respectively using $2 2 . 2 \%$ (medium), $1 0 . 8 \%$ (small), $5 . 2 5 \%$ (tiny) of the supervised labels of the full SNLI training set (rest used for unlabeled AE training). Right. Perplexity (lower is better) of language models trained on the synthetic samples from a GAN/AE/LM, and evaluated on real data (Reverse PPL).
+
+#
+
+A man is on the corner in a sport area .   
+A man is on corner in a road all .   
+A lady is on outside a racetrack .   
+A lady is outside on a racetrack .   
+A lot of people is outdoors in an urban setting .   
+A lot of people is outdoors in an urban setting .   
+A lot of people is outdoors in an urban setting . A man is on a ship path with the woman .   
+A man is on a ship path with the woman .   
+A man is passing on a bridge with the girl .   
+A man is passing on a bridge with the girl .   
+A man is passing on a bridge with the girl .   
+A man is passing on a bridge with the dogs .   
+A man is passing on a bridge with the dogs .
+
+A man in a cave is used an escalator .
+
+A man in a cave is used an escalator A man in a cave is used chairs A man in a number is used many equipment A man in a number is posing so on a big rock . People are posing in a rural area . People are posing in a rural area.
+
+Figure 3: Sample interpolations from the ARAE-GAN. Constructed by linearly interpolating in the latent space and decoding to the output space. Word changes are highlighted in black. Results of the ARAE. The top block shows output generation of the decoder taking fake hidden codes generated by the GAN; the bottom block shows sample interpolation results.   
+
+<table><tr><td>Transform</td><td>Match %</td><td>Prec</td></tr><tr><td>walking</td><td>85</td><td>79.5</td></tr><tr><td>man</td><td>92</td><td>80.2</td></tr><tr><td>two</td><td>86</td><td>74.1</td></tr><tr><td>dog</td><td>88</td><td>77.0</td></tr><tr><td>standing</td><td>89</td><td>79.3</td></tr><tr><td>several</td><td>70</td><td>67.0</td></tr></table>
+
+A man in a tie is sleeping and clapping on balloons .   
+A person is standing in the air beneath a criminal .   
+The jewish boy is trying to stay out of his skateboard .   
+The people works in a new uniform studio .   
+Some child head a playing plastic with drink .   
+A baby workers is watching steak with the water .   
+The people shine or looks into an area .   
+The boy ’s babies is wearing a huge factory .   
+A women are walking outside near a man .   
+The dogs are sleeping in front of the dinner .   
+A side child listening to a piece with steps playing on a table .   
+Two children are working in red shirt at the cold field . ⇒walking A man in a tie is clapping and walking dogs .   
+⇒walking A person is walking in the air beneath a pickup .   
+⇒man The jewish man is trying to stay out of his horse .   
+⇒man A man works in a new studio uniform .   
+⇒Two Two children playing a head with plastic drink .   
+⇒Two Two workers watching baby steak with the grass .   
+⇒dog The dog arrives or looks into an area .   
+⇒dog The dog ’s babies is wearing a huge ears .   
+⇒standing Three women are standing near a man walking .   
+⇒standing Two dogs are standing in front of the dinner .   
+⇒Several Several child playing a guitar on side with a table .   
+⇒Several Several children working in red shirt are cold at the field .
+
+![](images/7c4347f7aebd494f7ebcd51df16ea27ea9fe42effbe8ace90a650f1c0ba63ee4.jpg)  
+Figure 4: Left. Quantitative evaluation of transformations. Match $\%$ refers to the $\%$ of samples where at least one decoder samples (per 100) had the desired transformation in the output, while Prec. measures the average precision of the output against the original sentence. Right. Examples (out of 100 decoder samples per sentence) where the offset vectors produced successful transformations of the original sentence. See Appendix 11 for full methodology.
+
+shown in Appendix 10). All models are of the same size to allow for fair comparison. We train an RNN language model on generated samples and evaluate on held-out data to calculate the reverse perplexity. As can be seen from Table 5, training on real data (understandably) outperforms training on generated data by a large margin. Surprisingly however, we find that a language model trained on ARAE-GAN data performs slightly better than one trained on LM-generated/AE-generated data. We further found that the reverse PPL of an AAE (Makhzani et al., 2015) was quite high (980) due to mode-collapse.
+
+Another property of GANs (and VAEs) is that the Gaussian form of $\mathbf { z }$ induces the ability to smoothly interpolate between outputs by exploiting the structure of the latent space. While language models may provide a better estimate of the underlying probability space, constructing this style of interpolation would require combinatorial search, which makes this a useful feature of text GANs. We experiment with this property by sampling two points $\mathbf { z } _ { 0 }$ and $\mathbf { z } _ { 1 }$ from $p ( \mathbf { z } )$ and constructing intermediary points ${ \bf z } _ { \lambda } = \lambda { \bf \bar { z } } _ { 1 } + ( 1 - \lambda ) { \bf z } _ { 0 }$ . For each we generate the argmax output $\tilde { \mathbf { x } } _ { \lambda }$ . The samples are shown in Figure 3 (left) for text and in Figure 3 (right) for a discretized MNIST ARAE-GAN.
+
+A final intriguing property of image GANs is the ability to move in the latent space via offset vectors (similar to the case with word vectors (Mikolov et al., 2013)). For example, Radford et al. (Radford et al., 2016) observe that when the mean latent vector for “men with glasses” is subtracted from the mean latent vector for “men without glasses” and applied to an image of a “woman without glasses”, the resulting image is that of a “woman with glasses”. To experiment with this property we generate 1 million sentences from the ARAE-GAN and compute vector transforms in this space to attempt to change main verbs, subjects and modifier (details in Appendix 11). Some examples of successful transformations are shown in Figure 4 (right). Quantitative evaluation of the success of the vector transformations is given in Figure 4 (left).
+
+# 7 CONCLUSION
+
+We present adversarially regularized autoencoders, as a simple approach for training a discrete structure autoencoder jointly with a code-space generative adversarial network. The model learns a improved autoencoder as demonstrated by semi-supervised experiments and improvements on text transfer experiments. It also learns a useful generative model for text that exhibits a robust latent space, as demonstrated by natural interpolations and vector arithmetic. We do note that (as has been frequently observed when training GANs) our model seemed to be quite sensitive to hyperparameters. Finally, while many useful models for text generation already exist, text GANs provide a qualitatively different approach influenced by the underlying latent variable structure. We envision that such a framework could be extended to a conditional setting, combined with other existing decoding schemes, or used to provide a more interpretable model of language.
+
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+
+# 8 APPENDIX: OPTIMALITY PROPERTY
+
+One can interpret the ARAE framework as a dual pathway network mapping two distinct distributions into a similar one; $\mathrm { e n c } _ { \phi }$ and $g _ { \theta }$ both output code vectors that are kept similar in terms of Wasserstein distance as measured by the critic. We provide the following proposition showing that under our parameterization of the encoder and the generator, as the Wasserstein distance converges, the encoder distribution $( \mathbf { c } \sim \mathbb { P } _ { r } $ ) converges to the generator distribution $( \tilde { \mathbf { c } } \sim \mathbb { P } _ { g } )$ ), and further, their moments converge.
+
+This is ideal since under our setting the generated distribution is simpler than the encoded distribution, because the input to the generator is from a simple distribution (e.g. spherical Gaussian) and the generator possesses less capacity than the encoder. However, it is not so simple that it is overly restrictive (e.g. as in VAEs). Empirically we observe that the first and second moments do indeed converge as training progresses (Section 6.1).
+
+Proposition 1. Let $\mathbb { P }$ be a distribution on a compact set $\chi$ , and $( \mathbb { P } _ { n } ) _ { n \in N }$ be a sequence of distributions on $\chi$ . Further suppose that $W ( \mathbb { P } _ { n } , \mathbb { P } ) \to 0$ . Then the following statements hold:
+
+(i) $\mathbb { P } _ { n } \sim \mathbb { P }$ (i.e. convergence in distribution).
+
+(ii) All moments converge, i.e. for all $k > 1 , k \in \mathbb { N } ,$ ,
+
+$$
+\mathbb { E } _ { X \sim \mathbb { P } _ { n } } \bigg [ \prod _ { i = 1 } ^ { d } X _ { i } ^ { p _ { i } } \bigg ]  \mathbb { E } _ { X \sim \mathbb { P } } \bigg [ \prod _ { i = 1 } ^ { d } X _ { i } ^ { p _ { i } } \bigg ]
+$$
+
+for all $p _ { 1 } , \ldots , p _ { d }$ such that $\textstyle \sum _ { i = 1 } ^ { d } p _ { i } = k$
+
+Proof. (i) has been proved in Villani (2008) Theorem 6.9.
+
+For (ii), using The Portmanteau Theorem, (i) is equivalent to:
+
+$\mathbb { E } _ { X \sim \mathbb { P } _ { n } } [ f ( X ) ]  \mathbb { E } _ { X \sim \mathbb { P } } [ f ( X ) ]$ for all bounded and continuous function $f \colon  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ , where $d$ is the dimension of the random variable.
+
+The $k$ -th moment of a distribution is given by
+
+$$
+\mathbb { E } \Big [ \prod _ { i = 1 } ^ { d } X _ { i } ^ { p _ { i } } \Big ] \mathrm { ~ s u c h ~ t h a t ~ } \sum _ { i = 1 } ^ { d } p _ { i } = k
+$$
+
+Our encoded code is bounded as we normalize the encoder output to lie on the unit sphere, and our generated code is also bounded to lie in $( - 1 , 1 ) ^ { n }$ by the tanh function. Hence $\begin{array} { r } { f ( X ) = \prod _ { i = 1 } ^ { d } X _ { i } ^ { q _ { i } } } \end{array}$ is a bounded continuous function for all $q _ { i } > 0$ . Therefore,
+
+$$
+\mathbb { E } _ { X \sim \mathbb { P } _ { n } } \Big [ \prod _ { i = 1 } ^ { d } X _ { i } ^ { p _ { i } } \Big ] \to \mathbb { E } _ { X \sim \mathbb { P } } \Big [ \prod _ { i = 1 } ^ { d } X _ { i } ^ { p _ { i } } \Big ]
+$$
+
+where $\textstyle \sum _ { i = 1 } ^ { d } p _ { i } = k$
+
+# 9 APPENDIX: SHEET OF STYLE-TRANSFER SAMPLES
+
+# YELP TRANSFER
+
+Table 6: Full sheet of sentiment transfer result   
+
+<table><tr><td></td><td>Positive to Negative</td><td></td><td>Negative to Positive</td></tr><tr><td>Original</td><td>great indoor mall .</td><td>Original</td><td>hell no ！</td></tr><tr><td>ARAE</td><td>no smoking mall.</td><td>ARAE</td><td>hell great！</td></tr><tr><td>Cross-AE</td><td>terrible outdoor urine .</td><td>Cross-AE</td><td>incredible pork ！</td></tr><tr><td>Original</td><td>great blooming onion.</td><td>Original</td><td>highly disappointed ！</td></tr><tr><td>ARAE</td><td>no receipt onion .</td><td>ARAE</td><td>highly recommended ！</td></tr><tr><td>Cross-AE</td><td>terrible of pie .</td><td>Cross-AE</td><td>highly clean !</td></tr><tr><td>Original</td><td>i really enjoyed getting my nails done by peter .</td><td>Original</td><td>bad products .</td></tr><tr><td>ARAE</td><td>i really needed geting my nails done by now .</td><td>ARAE</td><td>good products .</td></tr><tr><td>Cross-AE</td><td>i really really told my nails done with these things .</td><td>Cross-AE</td><td>good prices .</td></tr><tr><td>Original</td><td>definitely a great choice for sushi in las vegas ！</td><td>Original</td><td>i was so very disappointed today at lunch .</td></tr><tr><td>ARAE</td><td>definitely a_num_star rating for_num_sushi in las vegas .</td><td>ARAE</td><td>i highly recommend this place today .</td></tr><tr><td>Cross-AE</td><td>not a great choice for breakfast in las vegas vegas !</td><td>Cross-AE</td><td>i was so very pleased to this</td></tr><tr><td>Original</td><td>the best piece of meat i have ever had !</td><td>Original</td><td>i have n&#x27;t received any response to anything.</td></tr><tr><td>ARAE</td><td>the worst piece of meat i have ever been to ！</td><td>ARAE</td><td>i have n&#x27;t received any problems to please.</td></tr><tr><td>Cross-AE</td><td>the worst part of that i have ever had had !</td><td>Cross-AE</td><td>i have always the desert vet.</td></tr><tr><td>Original</td><td>really good food,super casual and really friendly</td><td>Original</td><td>all the fixes were minor and the bill ?</td></tr><tr><td>ARAE</td><td>really bad food,really generally really low and decent food.</td><td>ARAE</td><td>all the barbers were entertaining and the bill did n&#x27;t disappoint .</td></tr><tr><td>Cross-AE</td><td>really good food,super horrible and not the price .</td><td>Cross-AE</td><td>all the flavors were especially and one ！</td></tr><tr><td>Original</td><td>it has a great atmosphere,with wonderful service .</td><td>Original</td><td>small,smokey,dark and rude management .</td></tr><tr><td>ARAE</td><td>it has no taste ,with a complete jerk .</td><td>ARAE</td><td>small,intimate ,and cozy friendly staff .</td></tr><tr><td>Cross-AE</td><td>it has a great horrible food and run out service .</td><td>Cross-AE</td><td>great,,,chips and wine.</td></tr><tr><td>Original</td><td>their menu is extensive ,even have italian food .</td><td>Original</td><td>the restaurant did n&#x27;t meet our standard though .</td></tr><tr><td>ARAE</td><td>their menu is limited,even if i have an option .</td><td>ARAE</td><td>the restaurant did n&#x27;t disappoint our expectations though .</td></tr><tr><td>Cross-AE</td><td>their menu is decent ,i have gotten italian food</td><td>Cross-AE</td><td>the restaurant is always happy and knowledge .</td></tr><tr><td>Original</td><td>everyone who works there is incredibly friendly as well</td><td>Original</td><td>you could not see the stage at all ！</td></tr><tr><td>ARAE</td><td>everyone who works there is incredibly rude as well</td><td>ARAE</td><td>you could see the difference at the counter !</td></tr><tr><td>Cross-AE</td><td>everyone who works there is extremely clean and as well .</td><td>Cross-AE</td><td>you could definitely get the fuss !</td></tr><tr><td>Original</td><td>there are a couple decent places to drink and eat in here as well .</td><td>Original</td><td>room is void of all personality,no pictures or any sort of decorations .</td></tr><tr><td>ARAE</td><td>there are a couple slices of options and _num_wings in the place .</td><td>ARAE</td><td>room is eclectic,lots of flavor and all of the best .</td></tr><tr><td>Cross-AE</td><td>there are a few night places to eat the car here are a crowd .</td><td>Cross-AE</td><td>it&#x27;s a nice that amazing,that one &#x27;s some of flavor .</td></tr><tr><td>Original</td><td>if you &#x27;re in the mood to be adventurous ,this is your place ！</td><td>Original</td><td>waited in line to see how long a wait would be for three people .</td></tr><tr><td>ARAE</td><td>if you &#x27;re in the mood to be disappointed,this is not the place . if you &#x27;re in the drive to the work,this is my place !</td><td>ARAE</td><td>waited in line for a long wait and totally worth it.</td></tr><tr><td>Cross-AE</td><td></td><td>Cross-AE</td><td>another great job to see and a lot going to be from dinner .</td></tr><tr><td>Original</td><td>we came on the recommendation of a bell boy and the food was amazing .</td><td>Original</td><td>the people who ordered off the menu did n&#x27;t seem to do much better .</td></tr><tr><td>Cross-AE</td><td>we came on the recommendation and the food was a joke.</td><td>ARAE</td><td>the people who work there are super friendly and the menu is good .</td></tr><tr><td>Cross-AE</td><td>we went on the car of the time and the chicken was awful .</td><td>Cross-AE</td><td>the place,one of the office is always worth you do a business .</td></tr><tr><td>Original</td><td>service is good but not quick,just enjoy the wine and your company .</td><td>Original</td><td>they told us in the beginning to make sure they do n&#x27;teat anything .</td></tr><tr><td>ARAE</td><td>service is good but not quick,but the service is horrible .</td><td>ARAE</td><td>they told us in the mood to make sure they do great food</td></tr><tr><td>Cross-AE</td><td>service is good,and horrible,is the same and worst time ever .</td><td>Cross-AE</td><td>they &#x27;re us in the next for us as you do n&#x27;t eat.</td></tr><tr><td>Original</td><td>the steak was really juicy with my side of salsa to balance the flavor .</td><td>Original</td><td>the person who was teaching me how to control my horse was pretty rude .</td></tr><tr><td>ARAE</td><td>the steak was really bland with the sauce and mashed potatoes .</td><td>ARAE</td><td>the person who was able to give me a pretty good price .</td></tr><tr><td>Cross-AE</td><td>the fish was so much,the most of sauce had got the flavor .</td><td>Cross-AE</td><td>the owner &#x27;s was gorgeous when i had a table and was friendly .</td></tr><tr><td>Original</td><td>other than that one hell hole of a star bucks they ‘re all great !</td><td>Original</td><td>he was cleaning the table next to us with gloves on and a rag .</td></tr><tr><td>ARAE</td><td>other than that one star rating the toilet they ‘re not allowed .</td><td>ARAE</td><td>he was prompt and patient with us and the staff is awesome .</td></tr><tr><td>Cross-AE</td><td>a wonder our one came in a_num_months,you &#x27;re so better !</td><td>Cross-AE</td><td>he was like the only thing to get some with with my hair .</td></tr></table>
+
+# YAHOO TRANSFER
+
+Table 7: Full sheet of Yahoo titles transfer result   
+
+<table><tr><td></td><td>from Science</td><td></td><td>from Music</td><td></td><td>from Politics</td></tr><tr><td>Original</td><td>what is an event horizon with regards to black holes ？</td><td>Original</td><td>do you know a website that you can find people who want to join bands ?</td><td>Original</td><td>republicans :would you vote for a cheney /satan ticket in 2008 ?</td></tr><tr><td>Music</td><td>what is your favorite sitcom with adam sandler ?</td><td>Science</td><td>do you know a website that can help me with sci- ence？</td><td>Science</td><td>guys : how would you solve this question ?</td></tr><tr><td>Politics</td><td>what is an event with black people ?</td><td>Politics</td><td>do you think that you can find a person who is in prison？</td><td>Music</td><td>guys : would you rather be a good movie ?</td></tr><tr><td>Original</td><td>what did john paul jones do in the american revo- lution ？</td><td>Original</td><td>do people who quote entire poems or song lyrics ever actually get chosen best answer ?</td><td>Original</td><td>if i move to the usa do ilose my pension in canada ？</td></tr><tr><td>Music</td><td>what did john lennon do in the new york family ?</td><td>Science</td><td>do you think that scientists learn about human anatomy and physiology of life ?</td><td>Science</td><td>if i move the &lt;unk&gt; in the airi have to do my math homework ?</td></tr><tr><td>Politics</td><td>what did john mccain do in the next election ?</td><td>Politics</td><td>do people who knows anything about the recent issue of &lt;unk&gt; leadership ?</td><td>Music</td><td>if i move to the music do you thinkifeel better ?</td></tr><tr><td>Original</td><td>can anybody suggest a good topic for a statistical survey？</td><td>Original</td><td>from big brother,what is the girls name who had &lt;unk&gt; in her apt ?</td><td>Original</td><td>what is your reflection on what will be our organi- zations in the future ?</td></tr><tr><td>Music</td><td>can anybody suggest a good site fora techno ?</td><td>Science</td><td>in big bang what is the&lt;unk&gt;of &lt;unk&gt;,what is the difference between &lt;unk&gt; and &lt;unk&gt; ?</td><td>Science</td><td>what is your opinion on what will be the future in our future ？</td></tr><tr><td>Politics</td><td>can anybody suggest a good topic for a student visa ��</td><td>Politics</td><td>is big brother in the &lt;unk&gt; what do you think of her ？</td><td>Music</td><td>what is your favorite music videos on the may i find ？</td></tr><tr><td>Original</td><td>can a kidney infection effect a woman &amp;apos;s &lt;unk&gt;cycle ？</td><td>Original</td><td>where is the tickets for the filming of the suite life of zack and cody ?</td><td>Original</td><td>wouldn &amp;apos;t it be fun if we the people veto or passed bills ？</td></tr><tr><td>Music</td><td>can anyone give me a good film &lt;unk&gt; ?</td><td>Science</td><td>where is the best place of the blood stream for the production of the cell ?</td><td>Science</td><td>isnt it possible to be cloned if we put the moon or it？</td></tr><tr><td>Politics</td><td>can a landlord officer have a &lt;unk&gt; &lt;unk&gt; ?</td><td>Politics</td><td>where is the best place of the navy and the senate of the union ?</td><td>Music</td><td>isnt it possible or if we &amp;apos;re getting married ？</td></tr><tr><td>Original</td><td>where does the term &amp;quot;sweating &lt;unk&gt; &amp;quot; come from ?</td><td>Original</td><td>the &lt;unk&gt; singers was a band in 1963 who had a hit called &lt;unk&gt; man ？</td><td>Original</td><td>can anyone tell me how icould go about interview- ing north vietnamese soldiers ?</td></tr><tr><td>Music</td><td>where does the term &amp;quot; &lt;unk&gt; &amp;quot; come from ？</td><td>Science</td><td>the &lt;unk&gt;river in a &lt;unk&gt; was created by a &lt;unk&gt; who was born in the last century ?</td><td>Science</td><td>can anyone tell me how i could find how to build a robot ？</td></tr><tr><td>Politics</td><td>where does the term &amp;quot; &lt;unk&gt; &amp;quot; come from？</td><td>Politics</td><td>the &lt;unk&gt; are &lt;unk&gt; in a &lt;unk&gt; who was shot an &lt;unk&gt;？</td><td>Music</td><td>can anyone tell me how i could find out about my parents ？</td></tr><tr><td>Original</td><td>what other &lt;unk&gt; sources are there than burning fossil fuels.</td><td>Original</td><td>what is the first metal band in the early 6O &amp;apos;s ...?？？?？</td><td>Original</td><td>if the us did not exist would the world be a better place？</td></tr><tr><td>Music</td><td>what other &lt;unk&gt; are /who are the greatest gui- tarist currently on tv today ?</td><td>Science</td><td>what is the first country in the universe ?</td><td>Science</td><td>if the world did not exist,would it be possible ?</td></tr><tr><td>Politics</td><td>what other &lt;unk&gt; are there for veterans who lives ？</td><td>Politics</td><td>who is the first president in the usa ?????？？? ？？？？？？？？？？？？？？？？</td><td>Music</td><td>if you could not have a thing who would it be ?</td></tr></table>
+
+Table 8: Full sheet of Yahoo answers transfer result   
+
+<table><tr><td></td><td>from Science</td><td></td><td>from Music</td><td></td><td>from Politics</td></tr><tr><td>Original</td><td>take lml of hcl（concentrated ）and dilute it to 50ml.</td><td>Original</td><td>all three are fabulous artists,with just incredible talent！！</td><td>Original</td><td>4 years of an idiot in office +electing the idiot again=？</td></tr><tr><td>Music</td><td>take em to you and shout it to me</td><td>Science</td><td>all three are genetically bonded with water ,but just as many substances,are capable of producing</td><td>Science</td><td>4 years of an idiot in the office of science ?</td></tr><tr><td>Politics</td><td>take bribes to islam and it will be punished .</td><td>Politics</td><td>a special case. all three are competing with the government,just as far as i can.</td><td>Music</td><td>4 )&lt;unk&gt; in an idiot ,the idiot is the best of the two points ever ！</td></tr><tr><td>Original</td><td>oils do not do this,they do not &amp;quot; set &amp;quot;</td><td>Original</td><td>she ,too ,wondered about the underwear outside</td><td>Original</td><td>send me $10o and i &amp;apos;ll send you a copy</td></tr><tr><td>Music</td><td>cucumbers do not do this ,they do not &amp;quot; do</td><td>Science</td><td>the clothes . she,too,i know,the clothes outside the clothes .</td><td>Science</td><td>honest . send me an email andi &amp;apos;ll send you a copy.</td></tr><tr><td>Politics</td><td>&amp;quot;. corporations do not do this,but they do not.</td><td>Politics</td><td>she,too,ithink that the cops are theonly thing about the outside of the u.s..</td><td>Music</td><td>send me $10o andi &amp;apos;ll send you a copy.</td></tr><tr><td>Original</td><td>the average high tempsin jan and feb are about 48</td><td>Original</td><td>i like rammstein and idon &amp;apos;t speak or under-</td><td>Original</td><td>wills can be &lt;unk&gt;,or typed and signed without</td></tr><tr><td>Music</td><td>deg. the average high school in seattle and is about 15</td><td>Science</td><td>stand german. i like googling and i don &amp;apos;t understand or</td><td>Science</td><td>needing an attorney . euler can be &lt;unk&gt;,and without any type of op-</td></tr><tr><td>Politics</td><td>minutes . the average high infantry division is in afghanistan and alaska.</td><td>Politics</td><td>speak. i like mccain and idon &amp;apos;t care about it .</td><td>Music</td><td>erations,or &lt;unk&gt;. madonna can be &lt;unk&gt;,and signed without open- ing or &lt;unk&gt;.</td></tr><tr><td>Original</td><td>the light from you lamps would move away from</td><td>Original</td><td>mark is great,but the guest hosts were cool too !</td><td>Original</td><td>hungary:20 january 1945,（formerly a member</td></tr><tr><td>Music</td><td>you at light speed the light from you tube would move away from</td><td>Science</td><td>mark is great,but the water will be too busy for</td><td>Science</td><td>of the axis） nh3 :20 january,78（a）</td></tr><tr><td>Politics</td><td>you the light from you could go away from your state</td><td>Politics</td><td>the same reason. mark twain,but the great lakes ,the united states of america is too busy.</td><td>Music</td><td>1966 - 20 january 1961(a）1983 song</td></tr><tr><td>Original</td><td>van &lt;unk&gt;,on the other hand,had some serious</td><td>Original</td><td>they all offer terrific information about the cast</td><td>Original</td><td>bulgaria: 8 september 1944,(formerly a member</td></tr><tr><td>Music</td><td>issues ... van &lt;unk&gt; on the other hand,had some serious</td><td>Science</td><td>and characters,. they all offer insight about the characteristics of</td><td>Science</td><td>of the axis） moreover,83+(x+7)(x2)=(a2)</td></tr><tr><td>Politics</td><td>issues. van &lt;unk&gt;,on the other hand ,had some serious issues.</td><td>Politics</td><td>the earth,and are composed of many stars. they all offer legitimate information about the in- vasion of iraq and the u.s.,and all aspects of his-</td><td>Music</td><td>harrison :8 september 1961(a)(1995)</td></tr><tr><td>Original</td><td> just multiply the numerator of one fraction by that</td><td>Original</td><td>tory. but there are so many more i can &amp;apos;t think of</td><td>Original</td><td>anyone who doesnt have a billion dollars for all</td></tr><tr><td>Music</td><td>of the other . just multiply the fraction of the other one that</td><td>Science</td><td>！ but there are so many more of the number of ques-</td><td>Science</td><td>the publicity cant win . anyone who doesnt have a decent chance is the</td></tr><tr><td>Politics</td><td>&amp;apos;s just like it . just multiply the same fraction of other countries .</td><td>Politics</td><td>tions. but there are so many more of the can i think of</td><td>Music</td><td>same for all the other. anyone who doesnt have a lot of the show for the</td></tr><tr><td>Original</td><td>civil engineering is still an umbrella field com-</td><td>Original</td><td>today. i love zach he is sooo sweet in his own way !</td><td>Original</td><td>publicity. the theory is that cats don &amp;apos;t take to being</td></tr><tr><td>Music</td><td>prised of many related specialties . civil rights is still an art union .</td><td>Science</td><td>the answer is he &amp;apos;s definitely in his own way</td><td>Science</td><td>tied up but thats &lt;unk&gt;. the theory is that cats don &amp;apos;t grow up to</td></tr><tr><td>Politics</td><td>civil law is still an issue .</td><td>Politics</td><td>i love letting he is sooo smart in his own way </td><td>Music</td><td>&lt;unk&gt;. the theory is that dumb but don &amp;apos;t play</td></tr><tr><td>Original</td><td>h2o2（hydrogen peroxide ） naturally decomposes</td><td>Original</td><td>remember the industry is very shady so keep your</td><td>Original</td><td>&lt;unk&gt; to &lt;unk&gt;. the fear they are trying to instill in the common</td></tr><tr><td>Music</td><td>to form o2 and water . jackieand brad pittboth great albums and they are</td><td>Science</td><td>eyes open ！ remember the amount of water is so very impor-</td><td>Science</td><td>man is based on what ? the fear they are trying to find the common ances-</td></tr><tr><td>Politics</td><td>my fav. kennedy and blair hate america to invade them.</td><td>Politics</td><td>tant. remember the amount of time the politicians are</td><td>Music</td><td>tor in the world . the fear theyare trying to find out what is wrong</td></tr><tr><td>Original</td><td>the quieter it gets ,the more white noise you can</td><td>Original</td><td>open your mind . but can you fake it,for just one more show ?</td><td>Original</td><td>in the song. think about how much planning and people would</td></tr><tr><td>Music</td><td>here. the fray it gets,the more you can hear.</td><td>Science</td><td>but can you fake it,just for more than one ?</td><td>Science</td><td>have to be involved in what happened. think about how much time would you have to do</td></tr><tr><td>Politics</td><td>the gop gets it,the more you can here .</td><td>Politics</td><td>but can you fake it for more than one ?</td><td>Music</td><td>think about how much money and what would be &lt;unk&gt; about in the world ?</td></tr><tr><td>Original</td><td>h2co3（carbonic acid ） naturally decomposes to</td><td>Original</td><td>i am going to introduce you to the internet movie</td><td>Original</td><td>this restricts the availability of cash to them and</td></tr><tr><td>Music</td><td>form water and co2. phoebe and jack ,he &amp;apos;s gorgeous and she</td><td>Science</td><td>database. i am going to investigate the internet to google.</td><td>Science</td><td>other countries too start banning them . this reduces the intake of the other molecules to</td></tr><tr><td>Politics</td><td>loves to get him ！ nixon（captured）he lied and voted for bush to cause his country .</td><td>Politics</td><td>i am going to skip the internet to get you checked</td><td>Music</td><td>produce them and thus are too large . this is the cheapest package of them too.</td></tr></table>
+
+# 10 APPENDIX: SAMPLE GENERATIONS
+
+# ARAE-GAN Samples
+
+# AE Samples
+
+# LM Samples
+
+A woman preparing three fish .   
+A woman is seeing a man in the river .   
+There passes a woman near birds in the air .   
+Some ten people is sitting through their office .   
+The man got stolen with young dinner bag .   
+Monks are running in court .   
+The Two boys in glasses are all girl .   
+The man is small sitting in two men that tell a children .   
+The two children are eating the balloon animal .   
+A woman is trying on a microscope .   
+The dogs are sleeping in bed . Two Three woman in a cart tearing over of a tree A man is hugging and art .   
+The fancy skier is starting under the drag cup in . A dog are <unk> a   
+A man is not standing .   
+The Boys in their swimming .   
+A surfer and a couple waiting for a show .   
+A couple is a kids at a barbecue .   
+The motorcycles is in the ocean loading   
+I ’s bike is on empty   
+The actor was walking in a a small dog area . no dog is young their mother a man walking outside on a dirt road , sitting on the dock .   
+A large group of people is taking a photo for Christmas and at night .   
+Someone is avoiding a soccer game .   
+The man and woman are dressed for a movie .   
+Person in an empty stadium pointing at a mountain . Two children and a little boy are <unk> a man in a blue shirt .   
+A boy rides a bicycle .   
+A girl is running another in the forest .   
+the man is an indian women .
+
+# 11 APPENDIX: VECTOR ARITHMETIC
+
+We generate 1 million sentences from the ARAE-GAN and parse the sentences to obtain the main verb, subject, and modifier. Then for a given sentence, to change the main verb we subtract the mean latent vector (t) for all other sentences with the same main verb (in the first example in Figure 4 this would correspond to all sentences that had “sleeping” as the main verb) and add the mean latent vector for all sentences that have the desired transformation (with the running example this would be all sentences whose main verb was “walking”). We do the same to transform the subject and the modifier. We decode back into sentence space with the transformed latent vector via sampling from $p _ { \psi } ( g ( \mathbf { z + t } ) )$ . Some examples of successful transformations are shown in Figure 4 (right). Quantitative evaluation of the success of the vector transformations is given in Figure 4 (left). For each original vector $\mathbf { z }$ we sample 100 sentences from $p _ { \psi } ( g ( \mathbf { z } + \mathbf { t } ) )$ over the transformed new latent vector and consider it a match if any of the sentences demonstrate the desired transformation. Match $\%$ is proportion of original vectors that yield a match post transformation. As we ideally want the generated samples to only differ in the specified transformation, we also calculate the average word precision against the original sentence (Prec) for any match.
+
+# 12 APPENDIX: EXPERIMENTAL DETAILS
+
+MNIST EXPERIMENTS
+
+• The encoder is a three-layer MLP, $7 8 4 - 8 0 0 - 4 0 0 - 1 0 0$ .   
+• Additive Gaussian noise is added into c which is then fed into the decoder. The standard deviation of that noise is initialized to be 0.4, and then exponentially decayed to 0.   
+• The decoder is a four-layer MLP, $1 0 0 - 4 0 0 - 8 0 0 - 1 0 0 0 - 7 8 4$   
+• The autoencoder is optimized by Adam, with learning rate $5 \mathrm { e } { - } 0 4$ .   
+• An MLP generator $3 2 - 6 4 - 1 0 0 - 1 5 0 - 1 0 0$ , using batch normalization, and ReLU nonlinearity.   
+An MLP critic $1 0 0 { - } 1 0 0 { - } 6 0 { - } 2 0 { - } 1$ with weight clipping $\epsilon = 0 . 0 5$ . The critic is trained by 10 iterations within each GAN loop.   
+• Both components of GAN is optimized by Adam, with learning rate $5 \mathrm { e } ^ { - 0 4 }$ on the generator, and $5 \mathrm { e } \mathrm { - } 0 5$ on the critic.   
+• Weighing factor $\lambda ^ { ( 1 ) } = 0 . 2$ .
+
+# TEXT EXPERIMENTS
+
+• The encoder is an one-layer LSTM with 300 hidden units.   
+• Gaussian noise into c before feeding it into the decoder. The standard deviation of that noise is initialized to be 0.2, and then exponentially decayed every 100 iterations by a factor of 0.995.   
+• The decoder is a one-layer LSTM with 300 hidden units.   
+• The decoding process at each time step takes the top layer LSTM hidden state and concatenates it with the hidden codes c, before feeding them into the output (i.e. vocabulary projection) and the softmax layer. The word embedding is of size 300.   
+• We adopt a grad clipping on the encoder/decoder, with max grad_norm $= 1$ .   
+• The encoder/decoder is optimized by vanilla SGD with learning rate 1.   
+• An MLP generator $1 0 0 { - } 3 0 0 { - } 3 0 0$ , using batch normalization, and ReLU non-linearity.   
+• An MLP critic $3 0 0 { - } 3 0 0 { - } 1$ with weight clipping $\epsilon = 0 . 0 1$ . The critic is trained by 5 iterations within each GAN loop.   
+• Both components of GAN are optimized by Adam, with learning rate $5 \mathrm { e } \mathrm { - } 0 5$ on the generator, and $\mathtt { 1 e - 0 5 }$ on the critic.   
+• We increment the number of GAN training $\log ^ { 3 }$ by 1 (it initially is set to 1) , respectively at the beginning of epoch #2, epoch $\# 4$ and epoch #6.
+
+# SEMI-SUPERVISED EXPERIMENTS
+
+Similar to the SNLI generation experiment setup, with the following changes:
+
+• We employ larger network to GAN components: MLP generator $1 0 0 - 1 5 0 - 3 0 0 - 5 0 0$ and MLP critic $5 0 0 - 5 0 0 - 1 5 0 - 8 0 - 2 0 - 1$ with weight clipping factor $\epsilon = 0 . 0 2$ . The critic is trained by 10 iterations within each GAN loop.
+
+# YELP/YAHOO TRANSFER
+
+Similar to the SNLI setup, with the following changes
+
+• The encoder and decoder size are both increased to 500 hidden units.   
+The style adversarial classifier is an MLP with structure $3 0 0 { - } 2 0 0 { - } 1 0 0$ , with learning rate 0.1 trained with SGD.   
+• We employ both larger generator and discriminator architectures in GAN: generator $2 0 0 { - } 4 0 0 { - } 8 0 0$ with $z$ dim being set to 64; discriminator $3 0 0 { - } 1 6 0 { - } 8 0 { - } 2 0$ .   
+• Weighing factor for critic gradient $\lambda _ { a } ^ { ( 1 ) } = 1$ , $\lambda _ { b } ^ { ( 1 ) } = 1 0$ .   
+• No GAN loop scheduling is employed here.

parse/train/Owggnutk6lE/Owggnutk6lE.md

@@ -0,0 +1,245 @@
+# Alias-Free Generative Adversarial Networks
+
+Tero Karras NVIDIA tkarras@nvidia.com
+
+Miika Aittala NVIDIA maittala@nvidia.com
+
+Samuli Laine NVIDIA slaine@nvidia.com
+
+Erik Härkönen∗ Aalto University and NVIDIA erik.harkonen@aalto.fi
+
+Janne Hellsten NVIDIA jhellsten@nvidia.com
+
+Jaakko Lehtinen NVIDIA and Aalto University jlehtinen@nvidia.com
+
+Timo Aila NVIDIA taila@nvidia.com
+
+# Abstract
+
+We observe that despite their hierarchical convolutional nature, the synthesis process of typical generative adversarial networks depends on absolute pixel coordinates in an unhealthy manner. This manifests itself as, e.g., detail appearing to be glued to image coordinates instead of the surfaces of depicted objects. We trace the root cause to careless signal processing that causes aliasing in the generator network. Interpreting all signals in the network as continuous, we derive generally applicable, small architectural changes that guarantee that unwanted information cannot leak into the hierarchical synthesis process. The resulting networks match the FID of StyleGAN2 but differ dramatically in their internal representations, and they are fully equivariant to translation and rotation even at subpixel scales. Our results pave the way for generative models better suited for video and animation.
+
+# 1 Introduction
+
+The resolution and quality of images produced by generative adversarial networks (GAN) [19] have seen rapid improvement recently [27, 11, 29, 30]. They have been used for a variety of applications, including image editing [42, 47, 37, 20, 34, 3], domain translation [62, 32, 53, 36], and video generation [49, 15, 21]. While several ways of controlling the generative process have been found [8, 26, 10, 36, 22, 2, 7, 41, 6], the foundations of the synthesis process remain only partially understood.
+
+In the real world, details of different scale tend to transform hierarchically. For instance, moving a head causes the nose to move, which in turn moves the skin pores on it. The structure of a typical GAN generator is analogous: coarse, low-resolution features are hierarchically refined by upsampling layers, locally mixed by convolutions, and new detail is introduced through nonlinearities. We observe that despite this superficial similarity, current GAN architectures do not synthesize images in a natural hierarchical manner: the coarse features mainly control the presence of finer features, but not their precise positions. Instead, much of the fine detail appears to be fixed in pixel coordinates. This disturbing “texture sticking” is clearly visible in latent interpolations (see Figure 1 and our accompanying videos on the project page https://nvlabs.github.io/stylegan3), breaking the illusion of a solid and coherent object moving in space. Our goal is an architecture that exhibits a more natural transformation hierarchy, where the exact sub-pixel position of each feature is exclusively inherited from the underlying coarse features.
+
+![](images/fcc935c7c154ec662c31cc9f6bf386dee1e70d65e649b453116c3006c116b6ae.jpg)  
+Figure 1: Examples of “texture sticking”. Left: The average of images generated from a small neighborhood around a central latent (top row). The intended result is uniformly blurry because all details should move together. However, with StyleGAN2 many details (e.g., fur) stick to the same pixel coordinates, showing unwanted sharpness. Right: From a latent space interpolation (top row), we extract a short vertical segment of pixels from each generated image and stack them horizontally (bottom). The desired result is hairs moving in animation, creating a time-varying field. With StyleGAN2 the hairs mostly stick to the same coordinates, creating horizontal streaks instead.
+
+It turns out that current networks can partially bypass the ideal hierarchical construction by drawing on unintentional positional references available to the intermediate layers through image borders [25, 31, 58], per-pixel noise inputs [29] and positional encodings, and aliasing [5, 61]. Aliasing, despite being a subtle and critical issue [38], has received little attention in the GAN literature. We identify two sources for it: 1) faint after-images of the pixel grid resulting from non-ideal upsampling filters2 such as nearest, bilinear, or strided convolutions, and 2) the pointwise application of nonlinearities such as ReLU [52] or swish [40]. We find that the network has the means and motivation to amplify even the slightest amount of aliasing and combining it over multiple scales allows it to build a basis for texture motifs that are fixed in screen coordinates. This holds for all filters commonly used in deep learning [61, 51], and even high-quality filters used in image processing.
+
+How, then, do we eliminate the unwanted side information and thereby stop the network from using it? While borders can be solved by simply operating on slightly larger images, aliasing is much harder. We begin by noting that aliasing is most naturally treated in the classical Shannon-Nyquist signal processing framework, and switch focus to bandlimited functions on a continuous domain that are merely represented by discrete sample grids. Now, successful elimination of all sources of positional references means that details can be generated equally well regardless of pixel coordinates, which in turn is equivalent to enforcing continuous equivariance to sub-pixel translation (and optionally rotation) in all layers. To achieve this, we describe a comprehensive overhaul of all signal processing aspects of the StyleGAN2 generator [30]. Our contributions include the surprising finding that current upsampling filters are simply not aggressive enough in suppressing aliasing, and that extremely high-quality filters with over 100dB attenuation are required. Further, we present a principled solution to aliasing caused by pointwise nonlinearities [5] by considering their effect in the continuous domain and appropriately low-pass filtering the results. We also show that after the overhaul, a model based on $1 \times 1$ convolutions yields a strong, rotation equivariant generator.
+
+Once aliasing is adequately suppressed to force the model to implement more natural hierarchical refinement, its mode of operation changes drastically: the emergent internal representations now include coordinate systems that allow details to be correctly attached to the underlying surfaces. This promises significant improvements to models that generate video and animation. The new StyleGAN3 generator matches StyleGAN2 in terms of FID [23], while being slightly heavier computationally. Our implementation and pre-trained models are available at https://github.com/NVlabs/stylegan3
+
+Several recent works have studied the lack of translation equivariance in CNNs, mainly in the context of classification [25, 31, 58, 5, 33, 61, 12, 63, 51]. We significantly expand upon the antialiasing measures in this literature and show that doing so induces a fundamentally altered image generation behavior. Group-equivariant CNNs aim to generalize the efficiency benefits of translational weight sharing to, e.g., rotation [16, 57, 55, 54] and scale [56]. Our $1 \times 1$ convolutions can be seen an instance of a continuously E(2)-equivariant model [54] that remains compatible with, e.g., channel-wise ReLU nonlinearities and modulation. Dey et al. [17] apply $9 0 °$ rotation-and-flip equivariant CNNs [16] to GANs and show improved data efficiency. Our work is complementary, and not motivated by efficiency. Recent implicit network [45, 48, 13] based GANs [4, 46] generate each pixel independently via similar $1 \times 1$ convolutions. While equivariant, these models do not help with texture sticking, as they do not use an upsampling hierarchy or implement a shallow non-antialiased one.
+
+![](images/259d5689aa2cd9a26ad3654d37605318c1b158551e66cb76ae5d2f6f6b0e113e.jpg)  
+Figure 2: Left: Discrete representation $Z$ and continuous representation $z$ are related to each other via convolution with ideal interpolation filter $\phi _ { s }$ and pointwise multiplication with Dirac comb $\mathrm { I I I } _ { s }$ . Right: Nonlinearity $\sigma$ , ReLU in this example, may produce arbitrarily high frequencies in the continuous-domain $\sigma ( z )$ . Low-pass filtering via $\phi _ { s }$ is necessary to ensure that $Z ^ { \prime }$ captures the result.
+
+# 2 Equivariance via continuous signal interpretation
+
+To begin our analysis of equivariance in CNNs, we shall first rethink our view of what exactly is the signal that flows through a network. Even though data may be stored as values in a pixel grid, we cannot naïvely hold these values to directly represent the signal. Doing so would prevent us from considering operations as trivial as translating the contents of a feature map by half a pixel.
+
+According to the Nyquist–Shannon sampling theorem [44], a regularly sampled signal can represent any continuous signal containing frequencies between zero and half of the sampling rate. Let us consider a two-dimensional, discretely sampled feature map $Z [ { \pmb x } ]$ that consists of a regular grid of Dirac impulses of varying magnitudes, spaced $1 / s$ units apart where $s$ is the sampling rate. This is analogous to an infinite two-dimensional grid of values.
+
+Given $Z [ { \pmb x } ]$ and $s$ , the Whittaker–Shannon interpolation formula [44] states that the corresponding continuous representation $z ( \pmb { x } )$ is obtained by convolving the discretely sampled Dirac grid $Z [ { \pmb x } ]$ with an ideal interpolation filter $\phi _ { s }$ , i.e., $z ( \pmb { x } ) = \big ( \phi _ { s } * Z \big ) ( \pmb { x } )$ , where $^ *$ denotes continuous convolution and $\phi _ { s } ( \pmb { x } ) = \mathrm { s i n c } ( s x _ { 0 } ) \cdot \mathrm { s i n c } ( s x _ { 1 } )$ using the signal processing convention of defining $\operatorname { s i n c } ( x ) =$ $\sin ( \pi x ) / ( \pi x )$ . $\phi _ { s }$ has a bandlimit of $s / 2$ along the horizontal and vertical dimensions, ensuring that the resulting continuous signal captures all frequencies that can be represented with sampling rate $s$
+
+Conversion from the continuous to the discrete domain corresponds to sampling the continuous signal $z ( \pmb { x } )$ at the sampling points of $Z [ { \pmb x } ]$ that we define to be offset by half the sample spacing to lie at the “pixel centers”, see Figure 2, left. This can be expressed as a pointwise multiplication with a two-dimensional Dirac comb $\begin{array} { r } { \operatorname { I I I } _ { s } ( \pmb { x } ) = \sum _ { X \in \mathbb { Z } ^ { 2 } } \delta \big ( \pmb { x } - ( X + \frac { 1 } { 2 } ) / s \big ) } \end{array}$ .
+
+We earmark the unit square $\pmb { x } \in [ 0 , 1 ] ^ { 2 }$ in $z ( \pmb { x } )$ as our canvas for the signal of interest. In $Z [ { \pmb x } ]$ there are $s ^ { 2 }$ discrete samples in this region, but the above convolution with $\phi _ { s }$ means that values of $Z [ \pmb { x } ]$ outside the unit square also influence $z ( \pmb { x } )$ inside it. Thus storing an $s \times s$ -pixel feature map is not sufficient; in theory, we would need to store the entire infinite $Z [ { \pmb x } ]$ . As a practical solution, we store $Z [ { \pmb x } ]$ as a two-dimensional array that covers a region slightly larger than the unit square (Section 3.2).
+
+Having established correspondence between bandlimited, continuous feature maps $z ( \pmb { x } )$ and discretely sampled feature maps $Z [ { \pmb x } ]$ , we can shift our focus away from the usual pixel-centric view of the signal. In the remainder of this paper, we shall interpret $z ( \pmb { x } )$ as being the actual signal being operated on, and the discretely sampled feature map $Z [ { \pmb x } ]$ as merely a convenient encoding for it.
+
+Discrete and continuous representation of network layers Practical neural networks operate on the discretely sampled feature maps. Consider operation $\mathbf { F }$ (convolution, nonlinearity, etc.) operating on a discrete feature map: $Z ^ { \prime } = \mathbf { F } ( Z )$ . The feature map has a corresponding continuous counterpart, so we also have a corresponding mapping in the continuous domain: $z ^ { \prime } = \mathbf { f } ( z )$ . Now, an operation specified in one domain can be seen to perform a corresponding operation in the other domain:
+
+$$
+\mathbf { f } ( z ) = \phi _ { s ^ { \prime } } * \mathbf { F } ( \operatorname { I I I } _ { s } \odot z ) , \qquad \mathbf { F } ( Z ) = \operatorname { I I I } _ { s ^ { \prime } } \odot \mathbf { f } ( \phi _ { s } * Z ) ,
+$$
+
+where $\odot$ denotes pointwise multiplication and $s$ and $s ^ { \prime }$ are the input and output sampling rates. Note that in the latter case f must not introduce frequency content beyond the output bandlimit $s ^ { \prime } / 2$ .
+
+# 2.1 Equivariant network layers
+
+Operation f is equivariant with respect to a spatial transformation t of the 2D plane if it commutes with it in the continuous domain: $\mathbf { t } \circ \mathbf { f } = \mathbf { f } \circ \mathbf { t }$ . We note that when inputs are bandlimited to $s / 2$ , an equivariant operation must not generate frequency content above the output bandlimit of $s ^ { \prime } / 2$ , as otherwise no faithful discrete output representation exists.
+
+We focus on two types of equivariance in this paper: translation and rotation. In the case of rotation the spectral constraint is somewhat stricter — rotating an image corresponds to rotating the spectrum, and in order to guarantee the bandlimit in both horizontal and vertical direction, the spectrum must be limited to a disc with radius $s / 2$ . This applies to both the initial network input as well as the bandlimiting filters used for downsampling, as will be described later.
+
+We now consider the primitive operations in a typical generator network: convolution, upsampling, downsampling, and nonlinearity. Without loss of generality, we discuss the operations acting on a single feature map: pointwise linear combination of features has no effect on the analysis.
+
+Convolution Consider a standard convolution with a discrete kernel $K$ . We can interpret $K$ as living in the same grid as the input feature map, with sampling rate $s$ . The discrete-domain operation is simply ${ \bf F } _ { \mathrm { c o n v } } ( Z ) = K * Z$ , and we obtain the corresponding continuous operation from Eq. 1:
+
+$$
+\mathbf { f } _ { \mathrm { c o n v } } ( z ) = \phi _ { s } * \bigl ( K * ( \operatorname { I I I } _ { s } \odot z ) \bigr ) = K * \bigl ( \phi _ { s } * ( \operatorname { I I I } _ { s } \odot z ) \bigr ) = K * z
+$$
+
+due to commutativity of convolution and the fact that discretization followed by convolution with ideal low-pass filter, both with same sampling rate $s$ , is an identity operation, i.e., $\phi _ { s } * ( \mathrm { I I I } _ { s } \odot z ) = z$ In other words, the convolution operates by continuously sliding the discretized kernel over the continuous representation of the feature map. This convolution introduces no new frequencies, so the bandlimit requirements for both translation and rotation equivariance are trivially fulfilled.
+
+Convolution also commutes with translation in the continuous domain, and thus the operation is equivariant to translation. For rotation equivariance, the discrete kernel $K$ needs to be radially symmetric. We later show in Section 3.2 that trivially symmetric $1 \times 1$ convolution kernels are, despite their simplicity, a viable choice for rotation equivariant generative networks.
+
+Upsampling and downsampling Ideal upsampling does not modify the continuous representation. Its only purpose is to increase the output sampling rate $( s ^ { \prime } > s )$ to add headroom in the spectrum where subsequent layers may introduce additional content. Translation and rotation equivariance follow directly from upsampling being an identity operation in the continuous domain. With $\mathbf { f } _ { \mathrm { u p } } ( z ) = z$ , the discrete operation according to Eq. 1 is $\mathbf { F } _ { \mathrm { u p } } ( \bar { Z } ) = \mathrm { I I I } _ { s ^ { \prime } } \odot ( \phi _ { s } * Z )$ . If we choose $s ^ { \prime } = n \dot { s }$ with integer $n$ , this operation can be implemented by first interleaving $Z$ with zeros to increase its sampling rate and then convolving it with a discretized filter $\coprod _ { s ^ { \prime } } \odot \phi _ { s }$ .
+
+In downsampling, we must low-pass filter $z$ to remove frequencies above the output bandlimit, so that the signal can be represented faithfully in the coarser discretization. The operation in continuous domain is $\mathbf { f } _ { \mathrm { d o w n } } ( z ) = \psi _ { s ^ { \prime } } * z$ , where an ideal low-pass filter $\psi _ { s } : = s ^ { 2 } \cdot \phi _ { s }$ is simply the corresponding interpolation filter normalized to unit mass. The discrete counterpart is $\mathbf { F } _ { \mathrm { d o w n } } ( \bar { Z } ) =$ $\Pi \Pi _ { s ^ { \prime } } \odot \left( \psi _ { s ^ { \prime } } * \left( \phi _ { s } * Z \right) \right) = 1 / s ^ { 2 } \cdot \Pi \Pi _ { s ^ { \prime } } \odot \left( \psi _ { s ^ { \prime } } * \psi _ { s } * Z \right) = ( s ^ { \prime } / s ) ^ { 2 } \cdot \Pi \Pi _ { s ^ { \prime } } \odot \left( \phi _ { s ^ { \prime } } * Z \right)$ . The latter equality follows from $\psi _ { s } * \psi _ { s ^ { \prime } } = \psi _ { \mathrm { m i n } ( s , s ^ { \prime } ) }$ . Similar to upsampling, downsampling by an integer fraction can be implemented with a discrete convolution followed by dropping sample points. Translation equivariance follows automatically from the commutativity of $\mathbf { f } _ { \mathrm { d o w n } } ( z )$ with translation, but for rotation equivariance we must replace $\phi _ { s ^ { \prime } }$ with a radially symmetric filter with disc-shaped frequency response. The ideal such filter [9] is given by $\phi _ { s } ^ { \circ } ( \pmb { x } ) = \mathrm { j i n c } ( s \| \pmb { x } \| ) = 2 J _ { 1 } ( \pi s \| \pmb { x } \| ) / ( \pi s \| \pmb { x } \| )$ , where $J _ { 1 }$ is the first order Bessel function of the first kind.
+
+Figure 3: Results for FFHQ-U (unaligned FFHQ) at $2 5 6 ^ { 2 }$ . Left: Training configurations. FID is computed between 50k generated images and all training images [23, 28]; lower is better. EQ-T and EQ-R are our equivariance metrics in decibels (dB); higher is better. Right: Parameter ablations using our final configuration (R) for the filter’s support, magnification around nonlinearities, and the minimum stopband frequency at the first layer. \* indicates our default choices.   
+
+<table><tr><td></td><td>Configuration</td><td>FID↓ EQ-T↑</td><td>EQ-R↑</td><td>Parameter</td><td>FID↓</td><td>EQ-T↑</td><td>EQ-R↑</td><td>Time</td><td>Mem.</td></tr><tr><td></td><td>AStyleGAN2</td><td>5.14 1</td><td>1</td><td>Filter size n = 4</td><td>4.72</td><td>57.49</td><td>39.70</td><td>0.84×</td><td>0.99×</td></tr><tr><td></td><td>B+Fourier features</td><td>4.79</td><td>16.23 10.81</td><td>* Filter size n = 6</td><td>4.50</td><td>66.65</td><td>40.48</td><td>1.00×</td><td>1.00×</td></tr><tr><td></td><td>C+No noise inputs</td><td>4.54</td><td>15.81 10.84</td><td>Filter size n =8</td><td>4.66</td><td>65.57</td><td>42.09</td><td>1.18×</td><td>1.01×</td></tr><tr><td></td><td>D + Simplified generator</td><td>5.21</td><td>19.47 10.41</td><td>Upsampling m=1</td><td>4.38</td><td>39.96</td><td>36.42</td><td>0.65×</td><td>0.87×</td></tr><tr><td></td><td>E+Boundaries &amp; upsampling</td><td>6.02</td><td>24.62 10.97</td><td>* Upsampling m = 2</td><td>4.50</td><td>66.65</td><td>40.48</td><td>1.00×</td><td>1.00×</td></tr><tr><td></td><td>F+ Filtered nonlinearities</td><td>6.35</td><td>30.60 10.81</td><td>Upsampling m = 4</td><td>4.57</td><td>74.21</td><td>40.97</td><td>2.31×</td><td>1.62×</td></tr><tr><td></td><td>G+ Non-critical sampling</td><td>4.78</td><td>43.90 10.84</td><td>Stopband ft,0 = 21.5</td><td></td><td>51.10</td><td>29.14</td><td>0.86×</td><td></td></tr><tr><td></td><td>H + Transformed Fourier features</td><td>4.64</td><td>45.20 10.61</td><td></td><td>4.62</td><td></td><td></td><td></td><td>0.90×</td></tr><tr><td></td><td>T+Flexible layers (StyleGAN3-T)</td><td>4.62</td><td>63.01 13.12</td><td>* Stopband ft,0 = 22.1</td><td>4.50</td><td>66.65</td><td>40.48</td><td>1.00×</td><td>1.00×</td></tr><tr><td></td><td>R + Rotation equiv. (StyleGAN3-R)</td><td>4.50</td><td>66.65 40.48</td><td>Stopband ft,0 = 23.1</td><td>4.68</td><td>73.13</td><td>41.63</td><td>1.36×</td><td>1.25×</td></tr></table>
+
+Nonlinearity Applying a pointwise nonlinearity $\sigma$ in the discrete domain does not commute with fractional translation or rotation. However, in the continuous domain, any pointwise function commutes trivially with geometric transformations and is thus equivariant to translation and rotation. Fulfilling the bandlimit constraint is another question — applying, e.g., ReLU in the continuous domain may introduce arbitrarily high frequencies that cannot be represented in the output.
+
+A natural solution is to eliminate the offending high-frequency content by convolving the continuous result with the ideal low-pass filter $\psi _ { s }$ . Then, the continuous representation of the nonlinearity becomes $\mathbf { f } _ { \sigma } ( z ) = \psi _ { s } * \sigma ( \bar { z } ) = s ^ { 2 } \cdot \phi _ { s } * \sigma ( z )$ and the discrete counterpart is $\mathbf { F } _ { \sigma } ( Z ) = s ^ { 2 } \cdot \operatorname { I I I } _ { s } \dot { \odot }$ $( \phi _ { s } * \sigma ( \phi _ { s } * Z ) )$ (see Figure 2, right). This discrete operation cannot be realized without temporarily entering the continuous representation. We approximate this by upsampling the signal, applying the nonlinearity in the higher resolution, and downsampling it afterwards. Even though the nonlinearity is still performed in the discrete domain, we have found that only a $2 \times$ temporary resolution increase is sufficient for high-quality equivariance. For rotation equivariance, we must use the radially symmetric interpolation filter $\phi _ { s } ^ { \circ }$ in the downsampling step, as discussed above.
+
+Note that nonlinearity is the only operation capable of generating novel frequencies in our formulation, and that we can limit the range of these novel frequencies by applying a reconstruction filter with a lower cutoff than $s / 2$ before the final discretization operation. This gives us precise control over how much new information is introduced by each layer of a generator network (Section 3.2).
+
+# 3 Practical application to generator network
+
+We will now apply the theoretical ideas from the previous section in practice, by converting the well-established StyleGAN2 [30] generator to be fully equivariant to translation and rotation. We will introduce the necessary changes step-by-step, evaluating their impact in Figure 3. The discriminator remains unchanged in our experiments.
+
+The StyleGAN2 generator consists of two parts. First, a mapping network transforms an initial, normally distributed latent to an intermediate latent code w $\sim \ w \ w \ w$ . Then, a synthesis network $\mathbf { G }$ starts from a learned $4 \times 4 \times 5 1 2$ constant $Z _ { 0 }$ and applies a sequence of $N$ layers — consisting of convolutions, nonlinearities, upsampling, and per-pixel noise — to produce an output image $Z _ { N } = \mathbf { G } ( Z _ { 0 } ; \mathbf { w } )$ . The intermediate latent code w controls the modulation of the convolution kernels in G. The layers follow a rigid $2 \times$ upsampling schedule, where two layers are executed at each resolution and the number of feature maps is halved after each upsampling. Additionally, StyleGAN2 employs skip connections, mixing regularization [29], and path length regularization.
+
+Our goal is to make every layer of $\mathbf { G }$ equivariant w.r.t. the continuous signal, so that all finer details transform together with the coarser features of a local neighborhood. If this succeeds, the entire network becomes similarly equivariant. In other words, we aim to make the continuous operation g of the synthesis network equivariant w.r.t. transformations $\mathbf { t }$ (translations and rotations) applied on the continuous input $z _ { 0 }$ : $\mathbf { g } ( \mathbf { \bar { t } } [ z _ { 0 } ] ; \mathbf { w } ) = \mathbf { t } [ \mathbf { g } ( z _ { 0 } ; \mathbf { w } ) ]$ . To evaluate the impact of various architectural changes and practical approximations, we need a way to measure how well the network implements the equivariances. For translation equivariance, we report the peak signal-to-noise ratio (PSNR) in decibels (dB) between two sets of images, obtained by translating the input and output of the synthesis network by a random amount, resembling the definition by Zhang [61]:
+
+$$
+\begin{array} { r } { \mathrm { E Q - T } = 1 0 \cdot \log _ { 1 0 } \left( I _ { m a x } ^ { 2 } \big / \mathbb { E } _ { \mathbf { w } \sim \mathcal { W } , x \sim \mathcal { X } ^ { 2 } , p \sim \mathcal { V } , c \sim \mathcal { L } } \left[ \big ( \mathbf { g } ( \mathbf { t } _ { x } [ z _ { 0 } ] ; \mathbf { w } ) _ { c } ( p ) - \mathbf { t } _ { x } [ \mathbf { g } ( z _ { 0 } ; \mathbf { w } ) ] _ { c } ( p ) \big ) ^ { 2 } \right] \right) } \end{array}
+$$
+
+Each pair of images, corresponding to a different random choice of $\mathbf { w }$ , is sampled at integer pixel locations $p$ within their mutually valid region $\nu$ . Color channels $c$ are processed independently, and the intended dynamic range of generated images $- 1 \ldots + 1$ gives $I _ { m a x } = 2$ . Operator $\mathbf { t } _ { x }$ implements spatial translation with 2D offset $x$ , here drawn from distribution $\mathcal { X } ^ { 2 }$ of integer offsets. We define an analogous metric EQ-R for rotations, with the rotation angles drawn from $\mathcal { U } ( 0 ^ { \circ } , 3 6 0 ^ { \circ } )$ . Appendix E in the Supplement gives implementation details and our accompanying videos highlight the practical relevance of different dB values.
+
+# 3.1 Fourier features and baseline simplifications (configs B–D)
+
+To facilitate exact continuous translation and rotation of the input $z _ { \mathrm { 0 } }$ , we replace the learned input constant in StyleGAN2 with Fourier features [48, 58], which also has the advantage of naturally defining a spatially infinite map. We sample the frequencies uniformly within the circular frequency band $f _ { c } = 2$ , matching the original $4 \times 4$ input resolution, and keep them fixed over the course of training. This change (configs A and B in Figure 3, left) slightly improves FID and, crucially, allows us to compute the equivariance metrics without having to approximate the operator t. This baseline architecture is far from being equivariant; our accompanying videos show that the output images deteriorate drastically when the input features are translated or rotated from their original position.
+
+Next, we remove the per-pixel noise inputs because they are strongly at odds with our goal of a natural transformation hierarchy, i.e., that the exact sub-pixel position of each feature is exclusively inherited from the underlying coarse features. While this change (config C) is approximately FID-neutral, it fails to improve the equivariance metrics when considered in isolation.
+
+To further simplify the setup, we decrease the mapping network depth as recommended by Karras et al. [28] and disable mixing regularization and path length regularization [30]. Finally, we also eliminate the output skip connections. We hypothesize that their benefit is mostly related to gradient magnitude dynamics during training and address the underlying issue more directly using a simple normalization before each convolution. We track the exponential moving average √ $\sigma ^ { 2 } = \mathbb { E } [ x ^ { 2 } ]$ over all pixels and feature maps during training, and divide the feature maps by $\scriptstyle { \sqrt { \sigma ^ { 2 } } }$ . In practice, we bake the division into the convolution weights to improve efficiency. These changes (config D) bring FID back to the level of original StyleGAN2, while leading to a slight improvement in translation equivariance.
+
+# 3.2 Step-by-step redesign motivated by continuous interpretation
+
+Boundaries and upsampling (config E) Our theory assumes an infinite spatial extent for the feature maps, which we approximate by maintaining a fixed-size margin around the target canvas, cropping to this extended canvas after each layer. This explicit extension is necessary as border padding is known to leak absolute image coordinates into the internal representations [25, 31, 58]. In practice, we have found a 10-pixel margin to be enough; further increase has no noticeable effect on the results.
+
+Motivated by our theoretical model, we replace the bilinear $2 \times$ upsampling filter with a better approximation of the ideal low-pass filter. We use a windowed sinc filter with a relatively large Kaiser window [35] of size $n = 6$ , meaning that each output pixel is affected by 6 input pixels in upsampling and each input pixel affects 6 output pixels in downsampling. Kaiser window is a particularly good choice for our purposes, because it offers explicit control over the transition band and attenuation (Figure 4a). In the remainder of this section, we specify the transition band explicitly and compute the remaining parameters using Kaiser’s original formulas (Appendix C). For now, we choose to employ critical sampling and set the filter cutoff √ $f _ { c } = s / 2$ , i.e., exactly at the bandlimit, and transition band half-width $\dot { f } _ { h } = ( \sqrt { 2 } - 1 ) ( s / 2 )$ . Recall that sampling rate $s$ equals the width of the canvas in pixels, given our definitions in Section 2.
+
+The improved handling of boundaries and upsampling (config E) leads to better translation equivariance. However, FID is compromised by $16 \%$ , probably because we started to constrain what the feature maps can contain. In a further ablation (Figure 3, right), smaller resampling filters ${ ( n = 4 ) }$ ) hurt translation equivariance, while larger filters $( n = 8$ ) mainly increase training time.
+
+Filtered nonlinearities (config F) Our theoretical treatment of nonlinearities calls for wrapping each leaky ReLU (or any other commonly used non-linearity) between $m \times$ upsampling and $m \times$ downsampling, for some magnification factor $m$ . We further note that the order of upsampling and convolution can be switched by virtue of the signal being bandlimited, allowing us to fuse the regular $2 \times$ upsampling and a subsequent $m \times$ upsampling related to the nonlinearity into a single $2 m \times$ upsampling. In practice, we find $m = 2$ to be sufficient (Figure 3, right), again improving EQ-T (config F). Implementing the upsample-LReLU-downsample sequence is not efficient using the primitives available in current deep learning frameworks [1, 39], and thus we implement a custom CUDA kernel (Appendix D) that combines these operations (Figure 4b), leading to $1 0 \times$ faster training and considerable memory savings.
+
+![](images/f46886d986017e336472da1f6aa3acce828eb836308c48f939f2f895b31041eb.jpg)  
+Figure 4: (a) 1D example of a $2 \times$ upsampling filter with $n = 6$ , $s = 2$ , $f _ { c } = 1$ , and $f _ { h } = 0 . 4$ (blue). Setting $f _ { h } = 0 . 6$ makes the transition band wider (green), which reduces the unwanted stopband ripple and thus leads to stronger attenuation. (b) Our alias-free generator, corresponding to configs T and R in Figure 3. The main datapath consists of Fourier features and normalization (Section 3.1), modulated convolutions [30], and filtered nonlinearities (Section 3.2). (c) Flexible layer specifications (config T) with $N = 1 4$ and $s _ { N } = 1 0 2 4$ . Cutoff $f _ { c }$ (blue) and minimum acceptable stopband frequency $f _ { t }$ (orange) obey geometric progression over the layers; sampling rate $s$ (red) and actual stopband $f _ { c } + f _ { h }$ (green) are computed according to our design constraints.
+
+Non-critical sampling (config G) The critical sampling scheme — where filter cutoff is set exactly at the bandlimit — is ideal for many image processing applications as it strikes a good balance between antialiasing and the retention of high-frequency detail [50]. However, our goals are markedly different because aliasing is highly detrimental for the equivariance of the generator. While highfrequency detail is important in the output image and thus in the highest-resolution layers, it is less important in the earlier ones given that their exact resolutions are somewhat arbitrary to begin with.
+
+To suppress aliasing, we can simply lower the cutoff frequency to $f _ { c } = s / 2 - f _ { h }$ , which ensures that all alias frequencies (above $s / 2$ ) are in the stopband.3 For example, lowering the cutoff of the blue filter in Figure 4a would move its frequency response left so that the the worst-case attenuation of alias frequencies improves from $6 \mathrm { d B }$ to $4 0 \mathrm { d B }$ . This oversampling can be seen as a computational cost of better antialiasing, as we now use the same number of samples to express a slower-varying signal than before. In practice, we choose to lower $f _ { c }$ on all layers except the highest-resolution ones, because in the end the generator must be able to produce crisp images to match the training data. As the signals now contain less spatial information, we modify the heuristic used for determining the number of feature maps to be inversely proportional to $f _ { c }$ instead of the sampling rate $s$ . These changes (config G) further improve translation equivariance and push FID below the original StyleGAN2.
+
+Transformed Fourier features (config H) Equivariant generator layers are well suited for modeling unaligned and arbitrarily oriented datasets, because any geometric transformation introduced to the intermediate features $z _ { i }$ will directly carry over to the final image $z _ { N }$ . Due to the limited capability of the layers themselves to introduce global transformations, however, the input features $z _ { \mathrm { 0 } }$ play a crucial role in defining the global orientation of $z _ { N }$ . To let the orientation vary on a per-image basis, the generator should have the ability to transform $z _ { 0 }$ based on w. This motivates us to introduce a learned affine layer that outputs global translation and rotation parameters for the input Fourier features (Figure 4b and Appendix F). The layer is initialized to perform an identity transformation, but learns to use the mechanism over time when beneficial; in config H this improves the FID slightly.
+
+Flexible layer specifications (config T) Our changes have improved the equivariance quality considerably, but some visible artifacts still remain as our accompanying videos demonstrate. On closer inspection, it turns out that the attenuation of our filters (as defined for config G) is still insufficient for the lowest-resolution layers. These layers tend to have rich frequency content near their bandlimit, which calls for extremely strong attenuation to completely eliminate aliasing.
+
+So far, we have used the rigid sampling rate progression from StyleGAN2, coupled with simplistic choices for filter cutoff $f _ { c }$ and half-width $f _ { h }$ , but this need not be the case; we are free to specialize these parameters on a per-layer basis. In particular, we would like $f _ { h }$ to be high in the lowestresolution layers to maximize attenuation in the stopband, but low in the highest-resolution layers to allow matching high-frequency details of the training data.
+
+Figure 4c illustrates an example progression of filter parameters in a 14-layer generator with two critically sampled full-resolution layers at the end. The cutoff frequency grows geometrically from $f _ { c } = 2$ in the first layer to $f _ { c } = s _ { N } / 2$ in the first critically sampled layer. We choose the minimum acceptable stopband frequency to start at $f _ { t , 0 } = 2 ^ { 2 . 1 }$ , and it grows geometrically but slower than the cutoff frequency. In our tests, the stopband target at the last layer is $\overline { { f } } _ { t } = f _ { c } \cdot 2 ^ { 0 . 3 }$ , but the progression is halted at the first critically sampled layer. Next, we set the sampling rate $s$ for each layer so that it accommodates frequencies up to $f _ { t }$ , rounding up to the next power of two without exceeding the output resolution. Finally, to maximize the attenuation of aliasing frequencies, we set the transition band half-width to $f _ { h } = \operatorname* { m a x } ( s / 2 , f _ { t } ) - f _ { c }$ , i.e., making it as wide as possible within the limits of the sampling rate, but at least wide enough to reach $f _ { t }$ . The resulting improvement depends on how much slack is left between $f _ { t }$ and $s / 2$ ; as an extreme example, the first layer stopband attenuation improves from $4 2 \mathrm { d B }$ to $4 8 0 \mathrm { d B }$ using this scheme.
+
+The new layer specifications again improve translation equivariance (config T), eliminating the remaining artifacts. A further ablation (Figure 3, right) shows that $f _ { t , 0 }$ provides an effective way to trade training speed for equivariance quality. Note that the number of layers is now a free parameter that does not directly depend on the output resolution. In fact, we have found that a fixed choice of $N$ works consistently across multiple output resolutions and makes other hyperparameters such as learning rate behave more predictably. We use $N = 1 4$ in the remainder of this paper.
+
+Rotation equivariance (config R) We obtain a rotation equivariant version of the network with two changes. First, we replace the $3 \times 3$ convolutions with $1 \times 1$ on all layers and compensate for the reduced capacity by doubling the number of feature maps. Only the upsampling and downsampling operations spread information between pixels in this config. Second, we replace the sinc-based downsampling filter with a radially symmetric jinc-based one that we construct using the same Kaiser scheme (Appendix C). We do this for all layers except the two critically sampled ones, where it is important to match the potentially non-radial spectrum of the training data. These changes (config R) improve EQ-R without harming FID, even though each layer has $56 \%$ fewer trainable parameters.
+
+We also employ an additional stabilization trick in this configuration. Early on in the training, we blur all images the discriminator sees using a Gaussian filter. We start with $\sigma = 1 0$ pixels, which we ramp to zero over the first $2 0 0 \mathrm { k }$ images. This prevents the discriminator from focusing too heavily on high frequencies early on. Without this trick, config R is prone to early collapses because the generator sometimes learns to produce high frequencies with a small delay, trivializing the discriminator’s task.
+
+# 4 Results
+
+Figure 5 gives results for six datasets using StyleGAN2 [30] as well as our alias-free StyleGAN3-T and StyleGAN3-R generators. In addition to the standard FFHQ [29] and METFACES [28], we created unaligned versions of them. We also created a properly resampled version of AFHQ [14] and collected a new BEACHES dataset. Appendix B describes the datasets in detail. The results show that our FID remains competitive with StyleGAN2. StyleGAN3-T and StyleGAN3-R perform equally well in terms of FID, and both show a very high level of translation equivariance. As expected, only the latter provides rotation equivariance. In FFHQ $( 1 0 2 4 \times 1 0 2 4 )$ the three generators had 30.0M, 22.3M and $1 5 . 8 \mathbf { M }$ parameters, while the training times were 1106, 1576 $( + 4 2 \% )$ and 2248 $( + 1 0 3 \% )$ GPU hours. Our accompanying videos show side-by-side comparisons with StyleGAN2, demonstrating visually that the texture sticking problem has been solved. The resulting motion is much more natural, better sustaining an illusion that there is a coherent 3D scene being imaged.
+
+Figure 5: Left: Results for six datasets. We use adaptive discriminator augmentation (ADA) [28] for the smaller datasets. “StyleGAN2” corresponds to our baseline config B with Fourier features. Right: Ablations and comparisons for FFHQ-U (unaligned FFHQ) at $2 5 6 ^ { 2 }$ . \* indicates our default choices.   
+
+<table><tr><td>Dataset</td><td>Config</td><td>FID↓</td><td>EQ-T↑ EQ-R↑</td><td>Ablation</td><td colspan="2">Translation eq.</td><td colspan="3">+ Rotation eq.</td></tr><tr><td>FFHQ-U</td><td>StyleGAN2</td><td>3.79</td><td>15.89 10.79</td><td></td><td>FID↓</td><td>EQ-T↑</td><td>FID↓</td><td>EQ-T↑</td><td>EQ-R↑</td></tr><tr><td>70000img,10242</td><td>StyleGAN3-T(ours)</td><td>3.67</td><td>61.69 13.95</td><td>*Main configuration</td><td>4.62</td><td>63.01</td><td>4.50</td><td>66.65</td><td>40.48</td></tr><tr><td>Train from scratch</td><td>StyleGAN3-R (ours)</td><td>3.66</td><td>64.78 47.64</td><td>With mixing reg.</td><td>4.60</td><td>63.48</td><td>4.67</td><td>63.59</td><td>40.90</td></tr><tr><td>FFHQ</td><td>StyleGAN2</td><td>2.70</td><td>13.58 10.22</td><td>With noise inputs</td><td>4.96</td><td>24.46</td><td>5.79</td><td>26.71</td><td>26.80</td></tr><tr><td>70000 img,10242</td><td>StyleGAN3-T (ours)</td><td>2.79</td><td>61.21 13.82</td><td>Without flexible layers</td><td>4.64</td><td>45.20</td><td>4.65</td><td>44.74</td><td>22.52</td></tr><tr><td>Train from scratch</td><td>StyleGAN3-R (ours)</td><td>3.07</td><td>64.76 46.62</td><td>Fixed Fourier features</td><td>5.93</td><td>64.57</td><td>6.48</td><td>66.20</td><td>41.77</td></tr><tr><td>METFACES-U</td><td>StyleGAN2</td><td>18.98</td><td>18.77 13.19</td><td>With path length reg.</td><td>5.00</td><td>68.36</td><td>5.98</td><td>71.64</td><td>42.18</td></tr><tr><td>1336 img,10242</td><td>StyleGAN3-T (ours)</td><td>18.75</td><td>64.11 16.63</td><td>0.5×capacity</td><td>7.43</td><td>63.14</td><td>6.52</td><td>63.08</td><td>39.89</td></tr><tr><td>ADA, from FFHQ-U</td><td>StyleGAN3-R (ours)</td><td>18.75</td><td>66.34 48.57</td><td>* 1.0× capacity</td><td>4.62</td><td>63.01</td><td>4.50</td><td>66.65</td><td>40.48</td></tr><tr><td>METFACES</td><td>StyleGAN2</td><td>15.22</td><td>16.39 12.89</td><td>2.0× capacity</td><td>3.80</td><td>66.61</td><td>4.18</td><td>70.06</td><td>42.51</td></tr><tr><td>1336 img,10242</td><td>StyleGAN3-T(ours)</td><td>15.11</td><td>65.23 16.82</td><td>*Kaiser filter,n =6</td><td>4.62</td><td>63.01</td><td>4.50</td><td>66.65</td><td>40.48</td></tr><tr><td>ADA, from FFHQ</td><td>StyleGAN3-R (ours)</td><td>15.33</td><td>64.86 46.81</td><td>Lanczos filter, α = 2</td><td>4.69</td><td>51.93</td><td>4.44</td><td>57.70</td><td>25.25</td></tr><tr><td>AFHQv2</td><td>StyleGAN2</td><td>4.62</td><td>13.83 11.50</td><td>Gaussian filter,σ = 0.4</td><td>5.91</td><td>56.89</td><td>5.73</td><td>59.53</td><td>39.43</td></tr><tr><td>15803 img,5122</td><td>StyleGAN3-T(ours)</td><td>4.04</td><td>60.15 13.51</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ADA, from scratch</td><td>StyleGAN3-R (ours)</td><td>4.40</td><td>64.89 40.34</td><td>G-CNN comparison</td><td>FID↓</td><td>EQ-T个</td><td>EQ-R↑</td><td>Params</td><td>Time</td></tr><tr><td>BEACHES</td><td>StyleGAN2</td><td>5.03</td><td>15.73 12.69</td><td>* StyleGAN3-T(ours)</td><td>4.62</td><td>63.01</td><td>13.12</td><td>23.3M</td><td>1.00×</td></tr><tr><td>20155img,5122</td><td>StyleGAN3-T (ours)</td><td>4.32</td><td>59.33 15.88</td><td>+ p4 symmetry [16]</td><td>4.69</td><td>61.90</td><td>17.07</td><td>21.8M</td><td>2.48×</td></tr><tr><td>ADA, from scratch</td><td>StyleGAN3-R (ours)</td><td>4.57</td><td>63.66 37.42</td><td>* StyleGAN3-R (ours)</td><td>4.50</td><td>66.65</td><td>40.48</td><td>15.8M</td><td>1.37×</td></tr></table>
+
+Ablations and comparisons In Section 3.1 we disabled a number of StyleGAN2 features. We can now turn them on one by one to gauge their effect on our generators (Figure 5, right). While mixing regularization can be re-enabled without any ill effects, we also find that styles can be mixed quite reliably even without this explicit regularization (Appendix A). Re-enabling noise inputs or relying on StyleGAN2’s original layer specifications compromises equivariances significantly, and using fixed Fourier features or re-enabling path length regularization harms FID. Path length regularization is in principle at odds with translation equivariance, as it penalizes image changes upon latent space walk and thus encourages texture sticking. We suspect that the counterintuitive improvement in equivariance may come from slightly blurrier generated images, at a cost of poor FID.
+
+In a scaling test we tried changing the number of feature maps, observing that equivariances remain at a high level, but FID suffers considerably when the capacity is halved. Doubling the capacity improves result quality in terms of FID, at the cost of almost $4 \times$ training time. Finally, we consider alternatives for our windowed Kaiser filter. Lanczos is competitive in terms of FID, but as a separable filter it compromises rotation equivariance in particular. Gaussian leads to clearly worse FIDs.
+
+We compare StyleGAN3-R to an alternative where the rotation part is implemented using $p 4$ symmetric G-CNN [16, 17] on top of our StyleGAN3-T. This approach provides only modest rotation equivariance while being slower to train. Steerable filters [55] could theoretically provide competitive EQ-R, but the memory and training time requirements proved infeasible with generator networks of this size.
+
+Appendix A demonstrates that the spectral properties of generated images closely match training data, comparing favorably to several earlier architectures.
+
+Internal representations Figure 6 visualizes typical internal representations from the networks. While in StyleGAN2 all feature maps seem to encode signal magnitudes, in our networks some of the maps take a different role and encode phase information instead. Clearly this is something that is needed when the network synthesizes detail on the surfaces; it needs to invent a coordinate system. In StyleGAN3-R, the emergent positional encoding patterns appear to be somewhat more well-defined. We believe that the existence of a coordinate system that allows precise localization on the surfaces of objects will prove useful in various applications, including advanced image and video editing.
+
+# 5 Limitations, discussion, and future work
+
+In this work we modified only the generator, but it seems likely that further benefits would be available by making the discriminator equivariant as well. For example, in our FFHQ results the teeth do not move correctly when the head turns, and we suspect that this is caused by the discriminator accidentally preferring to see the front teeth at certain pixel locations. Concurrent work has identified that aliasing is detrimental for such generalization [51].
+
+![](images/b617e9fc75233f4ba35ddc487bcc878ceb52d54ee9e105bdd5d8128e41219bc4.jpg)  
+Figure 6: Example internal representations (3 feature maps as RGB) in StyleGAN2 and our generators.
+
+Our alias-free generator architecture contains implicit assumptions about the nature of the training data, and violating these may cause training difficulties. Let us consider an example. Suppose we have black-and-white cartoons as training data that we (incorrectly) pre-process using point sampling [38], leading to training images where almost all pixels are either black or white and the edges are jagged. This kind of badly aliased training data is difficult for GANs in general, but it is especially at odds with equivariance: on the one hand, we are asking the generator to be able to translate the output smoothly by subpixel amounts, but on the other hand, edges must still remain jagged and pixels only black/white, to remain faithful to the training data. The same issue can also arise with letterboxing of training images, low-quality JPEGs, or retro pixel graphics, where the jagged stair-step edges are a defining feature of the aesthetic. In such cases it may be beneficial for the generator to be aware of the pixel grid.
+
+In future, it might be interesting to re-introduce noise inputs (stochastic variation) in a way that is consistent with hierarchical synthesis. A better path length regularization would encourage neighboring features to move together, not discourage them from moving at all. It might be beneficial to try to extend our approach to equivariance w.r.t. scaling, anisotropic scaling, or even arbitrary homeomorphisms. Finally, it is well known that antialiasing should be done before tone mapping. So far, all GANs — including ours — have operated in the sRGB color space (after tone mapping).
+
+Attention layers in the middle of a generator [60] could likely be dealt with similarly to non-linearities by temporarily switching to higher resolution – although the time complexity of attention layers may make this somewhat challenging in practice. Recent attention-based GANs that start with a tokenizing transformer (e.g., VQGAN [18]) may be at odds with equivariance. Whether it is possible to make them equivariant is an important open question.
+
+Potential negative societal impacts of (image-producing) GANs include many forms of disinformation, from fake portraits in social media [24] to propaganda videos of world leaders [43]. Our contribution eliminates certain characteristic artifacts from videos, potentially making them more convincing or deceiving, depending on the application. Viable solutions include model watermarking [59] along with large-scale authenticity assessment in major social media sites. This entire project consumed 92 GPU years and 225 MWh of electricity on an in-house cluster of NVIDIA V100s. The new StyleGAN3 generator is only marginally costlier to train or use than that of StyleGAN2.
+
+# 6 Acknowledgments
+
+We thank David Luebke, Ming-Yu Liu, Koki Nagano, Tuomas Kynkäänniemi, and Timo Viitanen for reviewing early drafts and helpful suggestions. Frédo Durand for early discussions. Tero Kuosmanen for maintaining our compute infrastructure. AFHQ authors for an updated version of their dataset. Getty Images for the training images in the BEACHES dataset. We did not receive external funding or additional revenues for this project.
+
+# References
+
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+        "text": "Erik Härkönen∗ Aalto University and NVIDIA erik.harkonen@aalto.fi ",
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+        "type": "text",
+        "text": "Jaakko Lehtinen NVIDIA and Aalto University jlehtinen@nvidia.com ",
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+        "text": "Timo Aila NVIDIA taila@nvidia.com ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
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+        "text": "We observe that despite their hierarchical convolutional nature, the synthesis process of typical generative adversarial networks depends on absolute pixel coordinates in an unhealthy manner. This manifests itself as, e.g., detail appearing to be glued to image coordinates instead of the surfaces of depicted objects. We trace the root cause to careless signal processing that causes aliasing in the generator network. Interpreting all signals in the network as continuous, we derive generally applicable, small architectural changes that guarantee that unwanted information cannot leak into the hierarchical synthesis process. The resulting networks match the FID of StyleGAN2 but differ dramatically in their internal representations, and they are fully equivariant to translation and rotation even at subpixel scales. Our results pave the way for generative models better suited for video and animation. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
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+        "text": "The resolution and quality of images produced by generative adversarial networks (GAN) [19] have seen rapid improvement recently [27, 11, 29, 30]. They have been used for a variety of applications, including image editing [42, 47, 37, 20, 34, 3], domain translation [62, 32, 53, 36], and video generation [49, 15, 21]. While several ways of controlling the generative process have been found [8, 26, 10, 36, 22, 2, 7, 41, 6], the foundations of the synthesis process remain only partially understood. ",
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+        "text": "In the real world, details of different scale tend to transform hierarchically. For instance, moving a head causes the nose to move, which in turn moves the skin pores on it. The structure of a typical GAN generator is analogous: coarse, low-resolution features are hierarchically refined by upsampling layers, locally mixed by convolutions, and new detail is introduced through nonlinearities. We observe that despite this superficial similarity, current GAN architectures do not synthesize images in a natural hierarchical manner: the coarse features mainly control the presence of finer features, but not their precise positions. Instead, much of the fine detail appears to be fixed in pixel coordinates. This disturbing “texture sticking” is clearly visible in latent interpolations (see Figure 1 and our accompanying videos on the project page https://nvlabs.github.io/stylegan3), breaking the illusion of a solid and coherent object moving in space. Our goal is an architecture that exhibits a more natural transformation hierarchy, where the exact sub-pixel position of each feature is exclusively inherited from the underlying coarse features. ",
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+        "img_path": "images/fcc935c7c154ec662c31cc9f6bf386dee1e70d65e649b453116c3006c116b6ae.jpg",
+        "image_caption": [
+            "Figure 1: Examples of “texture sticking”. Left: The average of images generated from a small neighborhood around a central latent (top row). The intended result is uniformly blurry because all details should move together. However, with StyleGAN2 many details (e.g., fur) stick to the same pixel coordinates, showing unwanted sharpness. Right: From a latent space interpolation (top row), we extract a short vertical segment of pixels from each generated image and stack them horizontally (bottom). The desired result is hairs moving in animation, creating a time-varying field. With StyleGAN2 the hairs mostly stick to the same coordinates, creating horizontal streaks instead. "
+        ],
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+        "text": "It turns out that current networks can partially bypass the ideal hierarchical construction by drawing on unintentional positional references available to the intermediate layers through image borders [25, 31, 58], per-pixel noise inputs [29] and positional encodings, and aliasing [5, 61]. Aliasing, despite being a subtle and critical issue [38], has received little attention in the GAN literature. We identify two sources for it: 1) faint after-images of the pixel grid resulting from non-ideal upsampling filters2 such as nearest, bilinear, or strided convolutions, and 2) the pointwise application of nonlinearities such as ReLU [52] or swish [40]. We find that the network has the means and motivation to amplify even the slightest amount of aliasing and combining it over multiple scales allows it to build a basis for texture motifs that are fixed in screen coordinates. This holds for all filters commonly used in deep learning [61, 51], and even high-quality filters used in image processing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "How, then, do we eliminate the unwanted side information and thereby stop the network from using it? While borders can be solved by simply operating on slightly larger images, aliasing is much harder. We begin by noting that aliasing is most naturally treated in the classical Shannon-Nyquist signal processing framework, and switch focus to bandlimited functions on a continuous domain that are merely represented by discrete sample grids. Now, successful elimination of all sources of positional references means that details can be generated equally well regardless of pixel coordinates, which in turn is equivalent to enforcing continuous equivariance to sub-pixel translation (and optionally rotation) in all layers. To achieve this, we describe a comprehensive overhaul of all signal processing aspects of the StyleGAN2 generator [30]. Our contributions include the surprising finding that current upsampling filters are simply not aggressive enough in suppressing aliasing, and that extremely high-quality filters with over 100dB attenuation are required. Further, we present a principled solution to aliasing caused by pointwise nonlinearities [5] by considering their effect in the continuous domain and appropriately low-pass filtering the results. We also show that after the overhaul, a model based on $1 \\times 1$ convolutions yields a strong, rotation equivariant generator. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Once aliasing is adequately suppressed to force the model to implement more natural hierarchical refinement, its mode of operation changes drastically: the emergent internal representations now include coordinate systems that allow details to be correctly attached to the underlying surfaces. This promises significant improvements to models that generate video and animation. The new StyleGAN3 generator matches StyleGAN2 in terms of FID [23], while being slightly heavier computationally. Our implementation and pre-trained models are available at https://github.com/NVlabs/stylegan3 ",
+        "bbox": [
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+        "type": "text",
+        "text": "Several recent works have studied the lack of translation equivariance in CNNs, mainly in the context of classification [25, 31, 58, 5, 33, 61, 12, 63, 51]. We significantly expand upon the antialiasing measures in this literature and show that doing so induces a fundamentally altered image generation behavior. Group-equivariant CNNs aim to generalize the efficiency benefits of translational weight sharing to, e.g., rotation [16, 57, 55, 54] and scale [56]. Our $1 \\times 1$ convolutions can be seen an instance of a continuously E(2)-equivariant model [54] that remains compatible with, e.g., channel-wise ReLU nonlinearities and modulation. Dey et al. [17] apply $9 0 °$ rotation-and-flip equivariant CNNs [16] to GANs and show improved data efficiency. Our work is complementary, and not motivated by efficiency. Recent implicit network [45, 48, 13] based GANs [4, 46] generate each pixel independently via similar $1 \\times 1$ convolutions. While equivariant, these models do not help with texture sticking, as they do not use an upsampling hierarchy or implement a shallow non-antialiased one. ",
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+        "img_path": "images/259d5689aa2cd9a26ad3654d37605318c1b158551e66cb76ae5d2f6f6b0e113e.jpg",
+        "image_caption": [
+            "Figure 2: Left: Discrete representation $Z$ and continuous representation $z$ are related to each other via convolution with ideal interpolation filter $\\phi _ { s }$ and pointwise multiplication with Dirac comb $\\mathrm { I I I } _ { s }$ . Right: Nonlinearity $\\sigma$ , ReLU in this example, may produce arbitrarily high frequencies in the continuous-domain $\\sigma ( z )$ . Low-pass filtering via $\\phi _ { s }$ is necessary to ensure that $Z ^ { \\prime }$ captures the result. "
+        ],
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+    {
+        "type": "text",
+        "text": "2 Equivariance via continuous signal interpretation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To begin our analysis of equivariance in CNNs, we shall first rethink our view of what exactly is the signal that flows through a network. Even though data may be stored as values in a pixel grid, we cannot naïvely hold these values to directly represent the signal. Doing so would prevent us from considering operations as trivial as translating the contents of a feature map by half a pixel. ",
+        "bbox": [
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+        "type": "text",
+        "text": "According to the Nyquist–Shannon sampling theorem [44], a regularly sampled signal can represent any continuous signal containing frequencies between zero and half of the sampling rate. Let us consider a two-dimensional, discretely sampled feature map $Z [ { \\pmb x } ]$ that consists of a regular grid of Dirac impulses of varying magnitudes, spaced $1 / s$ units apart where $s$ is the sampling rate. This is analogous to an infinite two-dimensional grid of values. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given $Z [ { \\pmb x } ]$ and $s$ , the Whittaker–Shannon interpolation formula [44] states that the corresponding continuous representation $z ( \\pmb { x } )$ is obtained by convolving the discretely sampled Dirac grid $Z [ { \\pmb x } ]$ with an ideal interpolation filter $\\phi _ { s }$ , i.e., $z ( \\pmb { x } ) = \\big ( \\phi _ { s } * Z \\big ) ( \\pmb { x } )$ , where $^ *$ denotes continuous convolution and $\\phi _ { s } ( \\pmb { x } ) = \\mathrm { s i n c } ( s x _ { 0 } ) \\cdot \\mathrm { s i n c } ( s x _ { 1 } )$ using the signal processing convention of defining $\\operatorname { s i n c } ( x ) =$ $\\sin ( \\pi x ) / ( \\pi x )$ . $\\phi _ { s }$ has a bandlimit of $s / 2$ along the horizontal and vertical dimensions, ensuring that the resulting continuous signal captures all frequencies that can be represented with sampling rate $s$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Conversion from the continuous to the discrete domain corresponds to sampling the continuous signal $z ( \\pmb { x } )$ at the sampling points of $Z [ { \\pmb x } ]$ that we define to be offset by half the sample spacing to lie at the “pixel centers”, see Figure 2, left. This can be expressed as a pointwise multiplication with a two-dimensional Dirac comb $\\begin{array} { r } { \\operatorname { I I I } _ { s } ( \\pmb { x } ) = \\sum _ { X \\in \\mathbb { Z } ^ { 2 } } \\delta \\big ( \\pmb { x } - ( X + \\frac { 1 } { 2 } ) / s \\big ) } \\end{array}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We earmark the unit square $\\pmb { x } \\in [ 0 , 1 ] ^ { 2 }$ in $z ( \\pmb { x } )$ as our canvas for the signal of interest. In $Z [ { \\pmb x } ]$ there are $s ^ { 2 }$ discrete samples in this region, but the above convolution with $\\phi _ { s }$ means that values of $Z [ \\pmb { x } ]$ outside the unit square also influence $z ( \\pmb { x } )$ inside it. Thus storing an $s \\times s$ -pixel feature map is not sufficient; in theory, we would need to store the entire infinite $Z [ { \\pmb x } ]$ . As a practical solution, we store $Z [ { \\pmb x } ]$ as a two-dimensional array that covers a region slightly larger than the unit square (Section 3.2). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having established correspondence between bandlimited, continuous feature maps $z ( \\pmb { x } )$ and discretely sampled feature maps $Z [ { \\pmb x } ]$ , we can shift our focus away from the usual pixel-centric view of the signal. In the remainder of this paper, we shall interpret $z ( \\pmb { x } )$ as being the actual signal being operated on, and the discretely sampled feature map $Z [ { \\pmb x } ]$ as merely a convenient encoding for it. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Discrete and continuous representation of network layers Practical neural networks operate on the discretely sampled feature maps. Consider operation $\\mathbf { F }$ (convolution, nonlinearity, etc.) operating on a discrete feature map: $Z ^ { \\prime } = \\mathbf { F } ( Z )$ . The feature map has a corresponding continuous counterpart, so we also have a corresponding mapping in the continuous domain: $z ^ { \\prime } = \\mathbf { f } ( z )$ . Now, an operation specified in one domain can be seen to perform a corresponding operation in the other domain: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddc397a731f52eb33965303cae60ff95d0a5556bbd1eedf0b16fe143385e2d64.jpg",
+        "text": "$$\n\\mathbf { f } ( z ) = \\phi _ { s ^ { \\prime } } * \\mathbf { F } ( \\operatorname { I I I } _ { s } \\odot z ) , \\qquad \\mathbf { F } ( Z ) = \\operatorname { I I I } _ { s ^ { \\prime } } \\odot \\mathbf { f } ( \\phi _ { s } * Z ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ denotes pointwise multiplication and $s$ and $s ^ { \\prime }$ are the input and output sampling rates. Note that in the latter case f must not introduce frequency content beyond the output bandlimit $s ^ { \\prime } / 2$ . ",
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+    {
+        "type": "text",
+        "text": "2.1 Equivariant network layers ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Operation f is equivariant with respect to a spatial transformation t of the 2D plane if it commutes with it in the continuous domain: $\\mathbf { t } \\circ \\mathbf { f } = \\mathbf { f } \\circ \\mathbf { t }$ . We note that when inputs are bandlimited to $s / 2$ , an equivariant operation must not generate frequency content above the output bandlimit of $s ^ { \\prime } / 2$ , as otherwise no faithful discrete output representation exists. ",
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+    {
+        "type": "text",
+        "text": "We focus on two types of equivariance in this paper: translation and rotation. In the case of rotation the spectral constraint is somewhat stricter — rotating an image corresponds to rotating the spectrum, and in order to guarantee the bandlimit in both horizontal and vertical direction, the spectrum must be limited to a disc with radius $s / 2$ . This applies to both the initial network input as well as the bandlimiting filters used for downsampling, as will be described later. ",
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+        "type": "text",
+        "text": "We now consider the primitive operations in a typical generator network: convolution, upsampling, downsampling, and nonlinearity. Without loss of generality, we discuss the operations acting on a single feature map: pointwise linear combination of features has no effect on the analysis. ",
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+        "type": "text",
+        "text": "Convolution Consider a standard convolution with a discrete kernel $K$ . We can interpret $K$ as living in the same grid as the input feature map, with sampling rate $s$ . The discrete-domain operation is simply ${ \\bf F } _ { \\mathrm { c o n v } } ( Z ) = K * Z$ , and we obtain the corresponding continuous operation from Eq. 1: ",
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+    {
+        "type": "equation",
+        "img_path": "images/a35786017d799d8b681cf9cfb0bb4f8f5128944e0ff9001947e8bd2b885a9676.jpg",
+        "text": "$$\n\\mathbf { f } _ { \\mathrm { c o n v } } ( z ) = \\phi _ { s } * \\bigl ( K * ( \\operatorname { I I I } _ { s } \\odot z ) \\bigr ) = K * \\bigl ( \\phi _ { s } * ( \\operatorname { I I I } _ { s } \\odot z ) \\bigr ) = K * z\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "due to commutativity of convolution and the fact that discretization followed by convolution with ideal low-pass filter, both with same sampling rate $s$ , is an identity operation, i.e., $\\phi _ { s } * ( \\mathrm { I I I } _ { s } \\odot z ) = z$ In other words, the convolution operates by continuously sliding the discretized kernel over the continuous representation of the feature map. This convolution introduces no new frequencies, so the bandlimit requirements for both translation and rotation equivariance are trivially fulfilled. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Convolution also commutes with translation in the continuous domain, and thus the operation is equivariant to translation. For rotation equivariance, the discrete kernel $K$ needs to be radially symmetric. We later show in Section 3.2 that trivially symmetric $1 \\times 1$ convolution kernels are, despite their simplicity, a viable choice for rotation equivariant generative networks. ",
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+    {
+        "type": "text",
+        "text": "Upsampling and downsampling Ideal upsampling does not modify the continuous representation. Its only purpose is to increase the output sampling rate $( s ^ { \\prime } > s )$ to add headroom in the spectrum where subsequent layers may introduce additional content. Translation and rotation equivariance follow directly from upsampling being an identity operation in the continuous domain. With $\\mathbf { f } _ { \\mathrm { u p } } ( z ) = z$ , the discrete operation according to Eq. 1 is $\\mathbf { F } _ { \\mathrm { u p } } ( \\bar { Z } ) = \\mathrm { I I I } _ { s ^ { \\prime } } \\odot ( \\phi _ { s } * Z )$ . If we choose $s ^ { \\prime } = n \\dot { s }$ with integer $n$ , this operation can be implemented by first interleaving $Z$ with zeros to increase its sampling rate and then convolving it with a discretized filter $\\coprod _ { s ^ { \\prime } } \\odot \\phi _ { s }$ . ",
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+    {
+        "type": "text",
+        "text": "In downsampling, we must low-pass filter $z$ to remove frequencies above the output bandlimit, so that the signal can be represented faithfully in the coarser discretization. The operation in continuous domain is $\\mathbf { f } _ { \\mathrm { d o w n } } ( z ) = \\psi _ { s ^ { \\prime } } * z$ , where an ideal low-pass filter $\\psi _ { s } : = s ^ { 2 } \\cdot \\phi _ { s }$ is simply the corresponding interpolation filter normalized to unit mass. The discrete counterpart is $\\mathbf { F } _ { \\mathrm { d o w n } } ( \\bar { Z } ) =$ $\\Pi \\Pi _ { s ^ { \\prime } } \\odot \\left( \\psi _ { s ^ { \\prime } } * \\left( \\phi _ { s } * Z \\right) \\right) = 1 / s ^ { 2 } \\cdot \\Pi \\Pi _ { s ^ { \\prime } } \\odot \\left( \\psi _ { s ^ { \\prime } } * \\psi _ { s } * Z \\right) = ( s ^ { \\prime } / s ) ^ { 2 } \\cdot \\Pi \\Pi _ { s ^ { \\prime } } \\odot \\left( \\phi _ { s ^ { \\prime } } * Z \\right)$ . The latter equality follows from $\\psi _ { s } * \\psi _ { s ^ { \\prime } } = \\psi _ { \\mathrm { m i n } ( s , s ^ { \\prime } ) }$ . Similar to upsampling, downsampling by an integer fraction can be implemented with a discrete convolution followed by dropping sample points. Translation equivariance follows automatically from the commutativity of $\\mathbf { f } _ { \\mathrm { d o w n } } ( z )$ with translation, but for rotation equivariance we must replace $\\phi _ { s ^ { \\prime } }$ with a radially symmetric filter with disc-shaped frequency response. The ideal such filter [9] is given by $\\phi _ { s } ^ { \\circ } ( \\pmb { x } ) = \\mathrm { j i n c } ( s \\| \\pmb { x } \\| ) = 2 J _ { 1 } ( \\pi s \\| \\pmb { x } \\| ) / ( \\pi s \\| \\pmb { x } \\| )$ , where $J _ { 1 }$ is the first order Bessel function of the first kind. ",
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+    {
+        "type": "table",
+        "img_path": "images/15a51c5c58d34fa8353fbf0d77beda1236504742a945e1c44d3d2de8e883887c.jpg",
+        "table_caption": [
+            "Figure 3: Results for FFHQ-U (unaligned FFHQ) at $2 5 6 ^ { 2 }$ . Left: Training configurations. FID is computed between 50k generated images and all training images [23, 28]; lower is better. EQ-T and EQ-R are our equivariance metrics in decibels (dB); higher is better. Right: Parameter ablations using our final configuration (R) for the filter’s support, magnification around nonlinearities, and the minimum stopband frequency at the first layer. \\* indicates our default choices. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Configuration</td><td>FID↓ EQ-T↑</td><td>EQ-R↑</td><td>Parameter</td><td>FID↓</td><td>EQ-T↑</td><td>EQ-R↑</td><td>Time</td><td>Mem.</td></tr><tr><td></td><td>AStyleGAN2</td><td>5.14 1</td><td>1</td><td>Filter size n = 4</td><td>4.72</td><td>57.49</td><td>39.70</td><td>0.84×</td><td>0.99×</td></tr><tr><td></td><td>B+Fourier features</td><td>4.79</td><td>16.23 10.81</td><td>* Filter size n = 6</td><td>4.50</td><td>66.65</td><td>40.48</td><td>1.00×</td><td>1.00×</td></tr><tr><td></td><td>C+No noise inputs</td><td>4.54</td><td>15.81 10.84</td><td>Filter size n =8</td><td>4.66</td><td>65.57</td><td>42.09</td><td>1.18×</td><td>1.01×</td></tr><tr><td></td><td>D + Simplified generator</td><td>5.21</td><td>19.47 10.41</td><td>Upsampling m=1</td><td>4.38</td><td>39.96</td><td>36.42</td><td>0.65×</td><td>0.87×</td></tr><tr><td></td><td>E+Boundaries &amp; upsampling</td><td>6.02</td><td>24.62 10.97</td><td>* Upsampling m = 2</td><td>4.50</td><td>66.65</td><td>40.48</td><td>1.00×</td><td>1.00×</td></tr><tr><td></td><td>F+ Filtered nonlinearities</td><td>6.35</td><td>30.60 10.81</td><td>Upsampling m = 4</td><td>4.57</td><td>74.21</td><td>40.97</td><td>2.31×</td><td>1.62×</td></tr><tr><td></td><td>G+ Non-critical sampling</td><td>4.78</td><td>43.90 10.84</td><td>Stopband ft,0 = 21.5</td><td></td><td>51.10</td><td>29.14</td><td>0.86×</td><td></td></tr><tr><td></td><td>H + Transformed Fourier features</td><td>4.64</td><td>45.20 10.61</td><td></td><td>4.62</td><td></td><td></td><td></td><td>0.90×</td></tr><tr><td></td><td>T+Flexible layers (StyleGAN3-T)</td><td>4.62</td><td>63.01 13.12</td><td>* Stopband ft,0 = 22.1</td><td>4.50</td><td>66.65</td><td>40.48</td><td>1.00×</td><td>1.00×</td></tr><tr><td></td><td>R + Rotation equiv. (StyleGAN3-R)</td><td>4.50</td><td>66.65 40.48</td><td>Stopband ft,0 = 23.1</td><td>4.68</td><td>73.13</td><td>41.63</td><td>1.36×</td><td>1.25×</td></tr></table>",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Nonlinearity Applying a pointwise nonlinearity $\\sigma$ in the discrete domain does not commute with fractional translation or rotation. However, in the continuous domain, any pointwise function commutes trivially with geometric transformations and is thus equivariant to translation and rotation. Fulfilling the bandlimit constraint is another question — applying, e.g., ReLU in the continuous domain may introduce arbitrarily high frequencies that cannot be represented in the output. ",
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+    {
+        "type": "text",
+        "text": "A natural solution is to eliminate the offending high-frequency content by convolving the continuous result with the ideal low-pass filter $\\psi _ { s }$ . Then, the continuous representation of the nonlinearity becomes $\\mathbf { f } _ { \\sigma } ( z ) = \\psi _ { s } * \\sigma ( \\bar { z } ) = s ^ { 2 } \\cdot \\phi _ { s } * \\sigma ( z )$ and the discrete counterpart is $\\mathbf { F } _ { \\sigma } ( Z ) = s ^ { 2 } \\cdot \\operatorname { I I I } _ { s } \\dot { \\odot }$ $( \\phi _ { s } * \\sigma ( \\phi _ { s } * Z ) )$ (see Figure 2, right). This discrete operation cannot be realized without temporarily entering the continuous representation. We approximate this by upsampling the signal, applying the nonlinearity in the higher resolution, and downsampling it afterwards. Even though the nonlinearity is still performed in the discrete domain, we have found that only a $2 \\times$ temporary resolution increase is sufficient for high-quality equivariance. For rotation equivariance, we must use the radially symmetric interpolation filter $\\phi _ { s } ^ { \\circ }$ in the downsampling step, as discussed above. ",
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+        "text": "Note that nonlinearity is the only operation capable of generating novel frequencies in our formulation, and that we can limit the range of these novel frequencies by applying a reconstruction filter with a lower cutoff than $s / 2$ before the final discretization operation. This gives us precise control over how much new information is introduced by each layer of a generator network (Section 3.2). ",
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+        "text": "3 Practical application to generator network ",
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+        "text": "We will now apply the theoretical ideas from the previous section in practice, by converting the well-established StyleGAN2 [30] generator to be fully equivariant to translation and rotation. We will introduce the necessary changes step-by-step, evaluating their impact in Figure 3. The discriminator remains unchanged in our experiments. ",
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+        "text": "The StyleGAN2 generator consists of two parts. First, a mapping network transforms an initial, normally distributed latent to an intermediate latent code w $\\sim \\ w \\ w \\ w$ . Then, a synthesis network $\\mathbf { G }$ starts from a learned $4 \\times 4 \\times 5 1 2$ constant $Z _ { 0 }$ and applies a sequence of $N$ layers — consisting of convolutions, nonlinearities, upsampling, and per-pixel noise — to produce an output image $Z _ { N } = \\mathbf { G } ( Z _ { 0 } ; \\mathbf { w } )$ . The intermediate latent code w controls the modulation of the convolution kernels in G. The layers follow a rigid $2 \\times$ upsampling schedule, where two layers are executed at each resolution and the number of feature maps is halved after each upsampling. Additionally, StyleGAN2 employs skip connections, mixing regularization [29], and path length regularization. ",
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+        "text": "Our goal is to make every layer of $\\mathbf { G }$ equivariant w.r.t. the continuous signal, so that all finer details transform together with the coarser features of a local neighborhood. If this succeeds, the entire network becomes similarly equivariant. In other words, we aim to make the continuous operation g of the synthesis network equivariant w.r.t. transformations $\\mathbf { t }$ (translations and rotations) applied on the continuous input $z _ { 0 }$ : $\\mathbf { g } ( \\mathbf { \\bar { t } } [ z _ { 0 } ] ; \\mathbf { w } ) = \\mathbf { t } [ \\mathbf { g } ( z _ { 0 } ; \\mathbf { w } ) ]$ . To evaluate the impact of various architectural changes and practical approximations, we need a way to measure how well the network implements the equivariances. For translation equivariance, we report the peak signal-to-noise ratio (PSNR) in decibels (dB) between two sets of images, obtained by translating the input and output of the synthesis network by a random amount, resembling the definition by Zhang [61]: ",
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+        "img_path": "images/8ba0b4a00620a9e806632f729a474c8328b186cf89105d8bf8b77976695ceb38.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { E Q - T } = 1 0 \\cdot \\log _ { 1 0 } \\left( I _ { m a x } ^ { 2 } \\big / \\mathbb { E } _ { \\mathbf { w } \\sim \\mathcal { W } , x \\sim \\mathcal { X } ^ { 2 } , p \\sim \\mathcal { V } , c \\sim \\mathcal { L } } \\left[ \\big ( \\mathbf { g } ( \\mathbf { t } _ { x } [ z _ { 0 } ] ; \\mathbf { w } ) _ { c } ( p ) - \\mathbf { t } _ { x } [ \\mathbf { g } ( z _ { 0 } ; \\mathbf { w } ) ] _ { c } ( p ) \\big ) ^ { 2 } \\right] \\right) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Each pair of images, corresponding to a different random choice of $\\mathbf { w }$ , is sampled at integer pixel locations $p$ within their mutually valid region $\\nu$ . Color channels $c$ are processed independently, and the intended dynamic range of generated images $- 1 \\ldots + 1$ gives $I _ { m a x } = 2$ . Operator $\\mathbf { t } _ { x }$ implements spatial translation with 2D offset $x$ , here drawn from distribution $\\mathcal { X } ^ { 2 }$ of integer offsets. We define an analogous metric EQ-R for rotations, with the rotation angles drawn from $\\mathcal { U } ( 0 ^ { \\circ } , 3 6 0 ^ { \\circ } )$ . Appendix E in the Supplement gives implementation details and our accompanying videos highlight the practical relevance of different dB values. ",
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+    {
+        "type": "text",
+        "text": "3.1 Fourier features and baseline simplifications (configs B–D) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To facilitate exact continuous translation and rotation of the input $z _ { \\mathrm { 0 } }$ , we replace the learned input constant in StyleGAN2 with Fourier features [48, 58], which also has the advantage of naturally defining a spatially infinite map. We sample the frequencies uniformly within the circular frequency band $f _ { c } = 2$ , matching the original $4 \\times 4$ input resolution, and keep them fixed over the course of training. This change (configs A and B in Figure 3, left) slightly improves FID and, crucially, allows us to compute the equivariance metrics without having to approximate the operator t. This baseline architecture is far from being equivariant; our accompanying videos show that the output images deteriorate drastically when the input features are translated or rotated from their original position. ",
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+        "text": "Next, we remove the per-pixel noise inputs because they are strongly at odds with our goal of a natural transformation hierarchy, i.e., that the exact sub-pixel position of each feature is exclusively inherited from the underlying coarse features. While this change (config C) is approximately FID-neutral, it fails to improve the equivariance metrics when considered in isolation. ",
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+        "type": "text",
+        "text": "To further simplify the setup, we decrease the mapping network depth as recommended by Karras et al. [28] and disable mixing regularization and path length regularization [30]. Finally, we also eliminate the output skip connections. We hypothesize that their benefit is mostly related to gradient magnitude dynamics during training and address the underlying issue more directly using a simple normalization before each convolution. We track the exponential moving average √ $\\sigma ^ { 2 } = \\mathbb { E } [ x ^ { 2 } ]$ over all pixels and feature maps during training, and divide the feature maps by $\\scriptstyle { \\sqrt { \\sigma ^ { 2 } } }$ . In practice, we bake the division into the convolution weights to improve efficiency. These changes (config D) bring FID back to the level of original StyleGAN2, while leading to a slight improvement in translation equivariance. ",
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+        "type": "text",
+        "text": "3.2 Step-by-step redesign motivated by continuous interpretation ",
+        "text_level": 1,
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+        "text": "Boundaries and upsampling (config E) Our theory assumes an infinite spatial extent for the feature maps, which we approximate by maintaining a fixed-size margin around the target canvas, cropping to this extended canvas after each layer. This explicit extension is necessary as border padding is known to leak absolute image coordinates into the internal representations [25, 31, 58]. In practice, we have found a 10-pixel margin to be enough; further increase has no noticeable effect on the results. ",
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+        "type": "text",
+        "text": "Motivated by our theoretical model, we replace the bilinear $2 \\times$ upsampling filter with a better approximation of the ideal low-pass filter. We use a windowed sinc filter with a relatively large Kaiser window [35] of size $n = 6$ , meaning that each output pixel is affected by 6 input pixels in upsampling and each input pixel affects 6 output pixels in downsampling. Kaiser window is a particularly good choice for our purposes, because it offers explicit control over the transition band and attenuation (Figure 4a). In the remainder of this section, we specify the transition band explicitly and compute the remaining parameters using Kaiser’s original formulas (Appendix C). For now, we choose to employ critical sampling and set the filter cutoff √ $f _ { c } = s / 2$ , i.e., exactly at the bandlimit, and transition band half-width $\\dot { f } _ { h } = ( \\sqrt { 2 } - 1 ) ( s / 2 )$ . Recall that sampling rate $s$ equals the width of the canvas in pixels, given our definitions in Section 2. ",
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+        "type": "text",
+        "text": "The improved handling of boundaries and upsampling (config E) leads to better translation equivariance. However, FID is compromised by $16 \\%$ , probably because we started to constrain what the feature maps can contain. In a further ablation (Figure 3, right), smaller resampling filters ${ ( n = 4 ) }$ ) hurt translation equivariance, while larger filters $( n = 8$ ) mainly increase training time. ",
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+        "text": "Filtered nonlinearities (config F) Our theoretical treatment of nonlinearities calls for wrapping each leaky ReLU (or any other commonly used non-linearity) between $m \\times$ upsampling and $m \\times$ downsampling, for some magnification factor $m$ . We further note that the order of upsampling and convolution can be switched by virtue of the signal being bandlimited, allowing us to fuse the regular $2 \\times$ upsampling and a subsequent $m \\times$ upsampling related to the nonlinearity into a single $2 m \\times$ upsampling. In practice, we find $m = 2$ to be sufficient (Figure 3, right), again improving EQ-T (config F). Implementing the upsample-LReLU-downsample sequence is not efficient using the primitives available in current deep learning frameworks [1, 39], and thus we implement a custom CUDA kernel (Appendix D) that combines these operations (Figure 4b), leading to $1 0 \\times$ faster training and considerable memory savings. ",
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+    {
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+        "img_path": "images/f46886d986017e336472da1f6aa3acce828eb836308c48f939f2f895b31041eb.jpg",
+        "image_caption": [
+            "Figure 4: (a) 1D example of a $2 \\times$ upsampling filter with $n = 6$ , $s = 2$ , $f _ { c } = 1$ , and $f _ { h } = 0 . 4$ (blue). Setting $f _ { h } = 0 . 6$ makes the transition band wider (green), which reduces the unwanted stopband ripple and thus leads to stronger attenuation. (b) Our alias-free generator, corresponding to configs T and R in Figure 3. The main datapath consists of Fourier features and normalization (Section 3.1), modulated convolutions [30], and filtered nonlinearities (Section 3.2). (c) Flexible layer specifications (config T) with $N = 1 4$ and $s _ { N } = 1 0 2 4$ . Cutoff $f _ { c }$ (blue) and minimum acceptable stopband frequency $f _ { t }$ (orange) obey geometric progression over the layers; sampling rate $s$ (red) and actual stopband $f _ { c } + f _ { h }$ (green) are computed according to our design constraints. "
+        ],
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+        "type": "text",
+        "text": "Non-critical sampling (config G) The critical sampling scheme — where filter cutoff is set exactly at the bandlimit — is ideal for many image processing applications as it strikes a good balance between antialiasing and the retention of high-frequency detail [50]. However, our goals are markedly different because aliasing is highly detrimental for the equivariance of the generator. While highfrequency detail is important in the output image and thus in the highest-resolution layers, it is less important in the earlier ones given that their exact resolutions are somewhat arbitrary to begin with. ",
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+    {
+        "type": "text",
+        "text": "To suppress aliasing, we can simply lower the cutoff frequency to $f _ { c } = s / 2 - f _ { h }$ , which ensures that all alias frequencies (above $s / 2$ ) are in the stopband.3 For example, lowering the cutoff of the blue filter in Figure 4a would move its frequency response left so that the the worst-case attenuation of alias frequencies improves from $6 \\mathrm { d B }$ to $4 0 \\mathrm { d B }$ . This oversampling can be seen as a computational cost of better antialiasing, as we now use the same number of samples to express a slower-varying signal than before. In practice, we choose to lower $f _ { c }$ on all layers except the highest-resolution ones, because in the end the generator must be able to produce crisp images to match the training data. As the signals now contain less spatial information, we modify the heuristic used for determining the number of feature maps to be inversely proportional to $f _ { c }$ instead of the sampling rate $s$ . These changes (config G) further improve translation equivariance and push FID below the original StyleGAN2. ",
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+        "text": "Transformed Fourier features (config H) Equivariant generator layers are well suited for modeling unaligned and arbitrarily oriented datasets, because any geometric transformation introduced to the intermediate features $z _ { i }$ will directly carry over to the final image $z _ { N }$ . Due to the limited capability of the layers themselves to introduce global transformations, however, the input features $z _ { \\mathrm { 0 } }$ play a crucial role in defining the global orientation of $z _ { N }$ . To let the orientation vary on a per-image basis, the generator should have the ability to transform $z _ { 0 }$ based on w. This motivates us to introduce a learned affine layer that outputs global translation and rotation parameters for the input Fourier features (Figure 4b and Appendix F). The layer is initialized to perform an identity transformation, but learns to use the mechanism over time when beneficial; in config H this improves the FID slightly. ",
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+        "type": "text",
+        "text": "Flexible layer specifications (config T) Our changes have improved the equivariance quality considerably, but some visible artifacts still remain as our accompanying videos demonstrate. On closer inspection, it turns out that the attenuation of our filters (as defined for config G) is still insufficient for the lowest-resolution layers. These layers tend to have rich frequency content near their bandlimit, which calls for extremely strong attenuation to completely eliminate aliasing. ",
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+        "type": "text",
+        "text": "So far, we have used the rigid sampling rate progression from StyleGAN2, coupled with simplistic choices for filter cutoff $f _ { c }$ and half-width $f _ { h }$ , but this need not be the case; we are free to specialize these parameters on a per-layer basis. In particular, we would like $f _ { h }$ to be high in the lowestresolution layers to maximize attenuation in the stopband, but low in the highest-resolution layers to allow matching high-frequency details of the training data. ",
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+    {
+        "type": "text",
+        "text": "Figure 4c illustrates an example progression of filter parameters in a 14-layer generator with two critically sampled full-resolution layers at the end. The cutoff frequency grows geometrically from $f _ { c } = 2$ in the first layer to $f _ { c } = s _ { N } / 2$ in the first critically sampled layer. We choose the minimum acceptable stopband frequency to start at $f _ { t , 0 } = 2 ^ { 2 . 1 }$ , and it grows geometrically but slower than the cutoff frequency. In our tests, the stopband target at the last layer is $\\overline { { f } } _ { t } = f _ { c } \\cdot 2 ^ { 0 . 3 }$ , but the progression is halted at the first critically sampled layer. Next, we set the sampling rate $s$ for each layer so that it accommodates frequencies up to $f _ { t }$ , rounding up to the next power of two without exceeding the output resolution. Finally, to maximize the attenuation of aliasing frequencies, we set the transition band half-width to $f _ { h } = \\operatorname* { m a x } ( s / 2 , f _ { t } ) - f _ { c }$ , i.e., making it as wide as possible within the limits of the sampling rate, but at least wide enough to reach $f _ { t }$ . The resulting improvement depends on how much slack is left between $f _ { t }$ and $s / 2$ ; as an extreme example, the first layer stopband attenuation improves from $4 2 \\mathrm { d B }$ to $4 8 0 \\mathrm { d B }$ using this scheme. ",
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+    {
+        "type": "text",
+        "text": "The new layer specifications again improve translation equivariance (config T), eliminating the remaining artifacts. A further ablation (Figure 3, right) shows that $f _ { t , 0 }$ provides an effective way to trade training speed for equivariance quality. Note that the number of layers is now a free parameter that does not directly depend on the output resolution. In fact, we have found that a fixed choice of $N$ works consistently across multiple output resolutions and makes other hyperparameters such as learning rate behave more predictably. We use $N = 1 4$ in the remainder of this paper. ",
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+    {
+        "type": "text",
+        "text": "Rotation equivariance (config R) We obtain a rotation equivariant version of the network with two changes. First, we replace the $3 \\times 3$ convolutions with $1 \\times 1$ on all layers and compensate for the reduced capacity by doubling the number of feature maps. Only the upsampling and downsampling operations spread information between pixels in this config. Second, we replace the sinc-based downsampling filter with a radially symmetric jinc-based one that we construct using the same Kaiser scheme (Appendix C). We do this for all layers except the two critically sampled ones, where it is important to match the potentially non-radial spectrum of the training data. These changes (config R) improve EQ-R without harming FID, even though each layer has $56 \\%$ fewer trainable parameters. ",
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+    {
+        "type": "text",
+        "text": "We also employ an additional stabilization trick in this configuration. Early on in the training, we blur all images the discriminator sees using a Gaussian filter. We start with $\\sigma = 1 0$ pixels, which we ramp to zero over the first $2 0 0 \\mathrm { k }$ images. This prevents the discriminator from focusing too heavily on high frequencies early on. Without this trick, config R is prone to early collapses because the generator sometimes learns to produce high frequencies with a small delay, trivializing the discriminator’s task. ",
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+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 5 gives results for six datasets using StyleGAN2 [30] as well as our alias-free StyleGAN3-T and StyleGAN3-R generators. In addition to the standard FFHQ [29] and METFACES [28], we created unaligned versions of them. We also created a properly resampled version of AFHQ [14] and collected a new BEACHES dataset. Appendix B describes the datasets in detail. The results show that our FID remains competitive with StyleGAN2. StyleGAN3-T and StyleGAN3-R perform equally well in terms of FID, and both show a very high level of translation equivariance. As expected, only the latter provides rotation equivariance. In FFHQ $( 1 0 2 4 \\times 1 0 2 4 )$ the three generators had 30.0M, 22.3M and $1 5 . 8 \\mathbf { M }$ parameters, while the training times were 1106, 1576 $( + 4 2 \\% )$ and 2248 $( + 1 0 3 \\% )$ GPU hours. Our accompanying videos show side-by-side comparisons with StyleGAN2, demonstrating visually that the texture sticking problem has been solved. The resulting motion is much more natural, better sustaining an illusion that there is a coherent 3D scene being imaged. ",
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+    {
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+        "img_path": "images/65b5eec04b29a15d818b3836e3ba7c88d6be29eb648f9d4d606d906c90a50a62.jpg",
+        "table_caption": [
+            "Figure 5: Left: Results for six datasets. We use adaptive discriminator augmentation (ADA) [28] for the smaller datasets. “StyleGAN2” corresponds to our baseline config B with Fourier features. Right: Ablations and comparisons for FFHQ-U (unaligned FFHQ) at $2 5 6 ^ { 2 }$ . \\* indicates our default choices. "
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+        "table_body": "<table><tr><td>Dataset</td><td>Config</td><td>FID↓</td><td>EQ-T↑ EQ-R↑</td><td>Ablation</td><td colspan=\"2\">Translation eq.</td><td colspan=\"3\">+ Rotation eq.</td></tr><tr><td>FFHQ-U</td><td>StyleGAN2</td><td>3.79</td><td>15.89 10.79</td><td></td><td>FID↓</td><td>EQ-T↑</td><td>FID↓</td><td>EQ-T↑</td><td>EQ-R↑</td></tr><tr><td>70000img,10242</td><td>StyleGAN3-T(ours)</td><td>3.67</td><td>61.69 13.95</td><td>*Main configuration</td><td>4.62</td><td>63.01</td><td>4.50</td><td>66.65</td><td>40.48</td></tr><tr><td>Train from scratch</td><td>StyleGAN3-R (ours)</td><td>3.66</td><td>64.78 47.64</td><td>With mixing reg.</td><td>4.60</td><td>63.48</td><td>4.67</td><td>63.59</td><td>40.90</td></tr><tr><td>FFHQ</td><td>StyleGAN2</td><td>2.70</td><td>13.58 10.22</td><td>With noise inputs</td><td>4.96</td><td>24.46</td><td>5.79</td><td>26.71</td><td>26.80</td></tr><tr><td>70000 img,10242</td><td>StyleGAN3-T (ours)</td><td>2.79</td><td>61.21 13.82</td><td>Without flexible layers</td><td>4.64</td><td>45.20</td><td>4.65</td><td>44.74</td><td>22.52</td></tr><tr><td>Train from scratch</td><td>StyleGAN3-R (ours)</td><td>3.07</td><td>64.76 46.62</td><td>Fixed Fourier features</td><td>5.93</td><td>64.57</td><td>6.48</td><td>66.20</td><td>41.77</td></tr><tr><td>METFACES-U</td><td>StyleGAN2</td><td>18.98</td><td>18.77 13.19</td><td>With path length reg.</td><td>5.00</td><td>68.36</td><td>5.98</td><td>71.64</td><td>42.18</td></tr><tr><td>1336 img,10242</td><td>StyleGAN3-T (ours)</td><td>18.75</td><td>64.11 16.63</td><td>0.5×capacity</td><td>7.43</td><td>63.14</td><td>6.52</td><td>63.08</td><td>39.89</td></tr><tr><td>ADA, from FFHQ-U</td><td>StyleGAN3-R (ours)</td><td>18.75</td><td>66.34 48.57</td><td>* 1.0× capacity</td><td>4.62</td><td>63.01</td><td>4.50</td><td>66.65</td><td>40.48</td></tr><tr><td>METFACES</td><td>StyleGAN2</td><td>15.22</td><td>16.39 12.89</td><td>2.0× capacity</td><td>3.80</td><td>66.61</td><td>4.18</td><td>70.06</td><td>42.51</td></tr><tr><td>1336 img,10242</td><td>StyleGAN3-T(ours)</td><td>15.11</td><td>65.23 16.82</td><td>*Kaiser filter,n =6</td><td>4.62</td><td>63.01</td><td>4.50</td><td>66.65</td><td>40.48</td></tr><tr><td>ADA, from FFHQ</td><td>StyleGAN3-R (ours)</td><td>15.33</td><td>64.86 46.81</td><td>Lanczos filter, α = 2</td><td>4.69</td><td>51.93</td><td>4.44</td><td>57.70</td><td>25.25</td></tr><tr><td>AFHQv2</td><td>StyleGAN2</td><td>4.62</td><td>13.83 11.50</td><td>Gaussian filter,σ = 0.4</td><td>5.91</td><td>56.89</td><td>5.73</td><td>59.53</td><td>39.43</td></tr><tr><td>15803 img,5122</td><td>StyleGAN3-T(ours)</td><td>4.04</td><td>60.15 13.51</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ADA, from scratch</td><td>StyleGAN3-R (ours)</td><td>4.40</td><td>64.89 40.34</td><td>G-CNN comparison</td><td>FID↓</td><td>EQ-T个</td><td>EQ-R↑</td><td>Params</td><td>Time</td></tr><tr><td>BEACHES</td><td>StyleGAN2</td><td>5.03</td><td>15.73 12.69</td><td>* StyleGAN3-T(ours)</td><td>4.62</td><td>63.01</td><td>13.12</td><td>23.3M</td><td>1.00×</td></tr><tr><td>20155img,5122</td><td>StyleGAN3-T (ours)</td><td>4.32</td><td>59.33 15.88</td><td>+ p4 symmetry [16]</td><td>4.69</td><td>61.90</td><td>17.07</td><td>21.8M</td><td>2.48×</td></tr><tr><td>ADA, from scratch</td><td>StyleGAN3-R (ours)</td><td>4.57</td><td>63.66 37.42</td><td>* StyleGAN3-R (ours)</td><td>4.50</td><td>66.65</td><td>40.48</td><td>15.8M</td><td>1.37×</td></tr></table>",
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+        "text": "Ablations and comparisons In Section 3.1 we disabled a number of StyleGAN2 features. We can now turn them on one by one to gauge their effect on our generators (Figure 5, right). While mixing regularization can be re-enabled without any ill effects, we also find that styles can be mixed quite reliably even without this explicit regularization (Appendix A). Re-enabling noise inputs or relying on StyleGAN2’s original layer specifications compromises equivariances significantly, and using fixed Fourier features or re-enabling path length regularization harms FID. Path length regularization is in principle at odds with translation equivariance, as it penalizes image changes upon latent space walk and thus encourages texture sticking. We suspect that the counterintuitive improvement in equivariance may come from slightly blurrier generated images, at a cost of poor FID. ",
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+        "text": "In a scaling test we tried changing the number of feature maps, observing that equivariances remain at a high level, but FID suffers considerably when the capacity is halved. Doubling the capacity improves result quality in terms of FID, at the cost of almost $4 \\times$ training time. Finally, we consider alternatives for our windowed Kaiser filter. Lanczos is competitive in terms of FID, but as a separable filter it compromises rotation equivariance in particular. Gaussian leads to clearly worse FIDs. ",
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+        "text": "We compare StyleGAN3-R to an alternative where the rotation part is implemented using $p 4$ symmetric G-CNN [16, 17] on top of our StyleGAN3-T. This approach provides only modest rotation equivariance while being slower to train. Steerable filters [55] could theoretically provide competitive EQ-R, but the memory and training time requirements proved infeasible with generator networks of this size. ",
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+        "text": "Appendix A demonstrates that the spectral properties of generated images closely match training data, comparing favorably to several earlier architectures. ",
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+        "text": "Internal representations Figure 6 visualizes typical internal representations from the networks. While in StyleGAN2 all feature maps seem to encode signal magnitudes, in our networks some of the maps take a different role and encode phase information instead. Clearly this is something that is needed when the network synthesizes detail on the surfaces; it needs to invent a coordinate system. In StyleGAN3-R, the emergent positional encoding patterns appear to be somewhat more well-defined. We believe that the existence of a coordinate system that allows precise localization on the surfaces of objects will prove useful in various applications, including advanced image and video editing. ",
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+        "type": "text",
+        "text": "5 Limitations, discussion, and future work ",
+        "text_level": 1,
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+        "text": "In this work we modified only the generator, but it seems likely that further benefits would be available by making the discriminator equivariant as well. For example, in our FFHQ results the teeth do not move correctly when the head turns, and we suspect that this is caused by the discriminator accidentally preferring to see the front teeth at certain pixel locations. Concurrent work has identified that aliasing is detrimental for such generalization [51]. ",
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+    {
+        "type": "image",
+        "img_path": "images/b617e9fc75233f4ba35ddc487bcc878ceb52d54ee9e105bdd5d8128e41219bc4.jpg",
+        "image_caption": [
+            "Figure 6: Example internal representations (3 feature maps as RGB) in StyleGAN2 and our generators. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Our alias-free generator architecture contains implicit assumptions about the nature of the training data, and violating these may cause training difficulties. Let us consider an example. Suppose we have black-and-white cartoons as training data that we (incorrectly) pre-process using point sampling [38], leading to training images where almost all pixels are either black or white and the edges are jagged. This kind of badly aliased training data is difficult for GANs in general, but it is especially at odds with equivariance: on the one hand, we are asking the generator to be able to translate the output smoothly by subpixel amounts, but on the other hand, edges must still remain jagged and pixels only black/white, to remain faithful to the training data. The same issue can also arise with letterboxing of training images, low-quality JPEGs, or retro pixel graphics, where the jagged stair-step edges are a defining feature of the aesthetic. In such cases it may be beneficial for the generator to be aware of the pixel grid. ",
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+    {
+        "type": "text",
+        "text": "In future, it might be interesting to re-introduce noise inputs (stochastic variation) in a way that is consistent with hierarchical synthesis. A better path length regularization would encourage neighboring features to move together, not discourage them from moving at all. It might be beneficial to try to extend our approach to equivariance w.r.t. scaling, anisotropic scaling, or even arbitrary homeomorphisms. Finally, it is well known that antialiasing should be done before tone mapping. So far, all GANs — including ours — have operated in the sRGB color space (after tone mapping). ",
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+    {
+        "type": "text",
+        "text": "Attention layers in the middle of a generator [60] could likely be dealt with similarly to non-linearities by temporarily switching to higher resolution – although the time complexity of attention layers may make this somewhat challenging in practice. Recent attention-based GANs that start with a tokenizing transformer (e.g., VQGAN [18]) may be at odds with equivariance. Whether it is possible to make them equivariant is an important open question. ",
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+    {
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+        "text": "Potential negative societal impacts of (image-producing) GANs include many forms of disinformation, from fake portraits in social media [24] to propaganda videos of world leaders [43]. Our contribution eliminates certain characteristic artifacts from videos, potentially making them more convincing or deceiving, depending on the application. Viable solutions include model watermarking [59] along with large-scale authenticity assessment in major social media sites. This entire project consumed 92 GPU years and 225 MWh of electricity on an in-house cluster of NVIDIA V100s. The new StyleGAN3 generator is only marginally costlier to train or use than that of StyleGAN2. ",
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+    {
+        "type": "text",
+        "text": "6 Acknowledgments ",
+        "text_level": 1,
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+        "text": "We thank David Luebke, Ming-Yu Liu, Koki Nagano, Tuomas Kynkäänniemi, and Timo Viitanen for reviewing early drafts and helpful suggestions. Frédo Durand for early discussions. Tero Kuosmanen for maintaining our compute infrastructure. AFHQ authors for an updated version of their dataset. Getty Images for the training images in the BEACHES dataset. We did not receive external funding or additional revenues for this project. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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parse/train/S1VaB4cex/S1VaB4cex.md

@@ -0,0 +1,259 @@
+# FRACTALNET: ULTRA-DEEP NEURAL NETWORKS WITHOUT RESIDUALS
+
+Gustav Larsson University of Chicago larsson@cs.uchicago.edu
+
+Michael Maire TTI Chicago mmaire@ttic.edu
+
+Gregory Shakhnarovich TTI Chicago greg@ttic.edu
+
+# ABSTRACT
+
+We introduce a design strategy for neural network macro-architecture based on selfsimilarity. Repeated application of a simple expansion rule generates deep networks whose structural layouts are precisely truncated fractals. These networks contain interacting subpaths of different lengths, but do not include any pass-through or residual connections; every internal signal is transformed by a filter and nonlinearity before being seen by subsequent layers. In experiments, fractal networks match the excellent performance of standard residual networks on both CIFAR and ImageNet classification tasks, thereby demonstrating that residual representations may not be fundamental to the success of extremely deep convolutional neural networks. Rather, the key may be the ability to transition, during training, from effectively shallow to deep. We note similarities with student-teacher behavior and develop drop-path, a natural extension of dropout, to regularize co-adaptation of subpaths in fractal architectures. Such regularization allows extraction of highperformance fixed-depth subnetworks. Additionally, fractal networks exhibit an anytime property: shallow subnetworks provide a quick answer, while deeper subnetworks, with higher latency, provide a more accurate answer.
+
+# 1 INTRODUCTION
+
+Residual networks (He et al., 2016a), or ResNets, lead a recent and dramatic increase in both depth and accuracy of convolutional neural networks, facilitated by constraining the network to learn residuals. ResNet variants (He et al., 2016a;b; Huang et al., 2016b) and related architectures (Srivastava et al., 2015) employ the common technique of initializing and anchoring, via a pass-through channel, a network to the identity function. Training now differs in two respects. First, the objective changes to learning residual outputs, rather than unreferenced absolute mappings. Second, these networks exhibit a type of deep supervision (Lee et al., 2014), as near-identity layers effectively reduce distance to the loss. He et al. (2016a) speculate that the former, the residual formulation itself, is crucial.
+
+We show otherwise, by constructing a competitive extremely deep architecture that does not rely on residuals. Our design principle is pure enough to communicate in a single word, fractal, and a simple diagram (Figure 1). Yet, fractal networks implicitly recapitulate many properties hard-wired into previous successful architectures. Deep supervision not only arises automatically, but also drives a type of student-teacher learning (Ba & Caruana, 2014; Urban et al., 2017) internal to the network. Modular building blocks of other designs (Szegedy et al., 2015; Liao & Carneiro, 2015) resemble special cases of a fractal network’s nested substructure.
+
+For fractal networks, simplicity of training mirrors simplicity of design. A single loss, attached to the final layer, suffices to drive internal behavior mimicking deep supervision. Parameters are randomly initialized. As they contain subnetworks of many depths, fractal networks are robust to choice of overall depth; make them deep enough and training will carve out a useful assembly of subnetworks.
+
+The entirety of emergent behavior resulting from a fractal design may erode the need for recent engineering tricks intended to achieve similar effects. These tricks include residual functional forms with identity initialization, manual deep supervision, hand-crafted architectural modules, and studentteacher training regimes. Section 2 reviews this large body of related techniques. Hybrid designs could certainly integrate any of them with a fractal architecture; we leave open the question of the degree to which such hybrids are synergistic.
+
+![](images/389e48bd72030062303ac6da089f1bd92ce9277af84c158fee8a805381e555a4.jpg)  
+Figure 1: Fractal architecture. Left: A simple expansion rule generates a fractal architecture with $C$ intertwined columns. The base case, $f _ { 1 } ( z )$ , has a single layer of the chosen type (e.g. convolutional) between input and output. Join layers compute element-wise mean. Right: Deep convolutional networks periodically reduce spatial resolution via pooling. A fractal version uses $f _ { C }$ as a building block between pooling layers. Stacking $B$ such blocks yields a network whose total depth, measured in terms of convolution layers, is $B \cdot \bar { 2 } ^ { C - 1 }$ . This example has depth 40 $B = 5$ , $C = 4$ ).
+
+Our main contribution is twofold:
+
+• We introduce FractalNet, the first simple alternative to ResNet. FractalNet shows that explicit residual learning is not a requirement for building ultra-deep neural networks. • Through analysis and experiments, we elucidate connections between FractalNet and an array of phenomena engineered into previous deep network designs.
+
+As an additional contribution, we develop drop-path, a novel regularization protocol for ultradeep fractal networks. Without data augmentation, fractal networks, trained with drop-path and dropout (Hinton et al., 2012), exceed the performance of residual networks regularized via stochastic depth (Huang et al., 2016b). Though, like stochastic depth, it randomly removes macro-scale components, drop-path further exploits our fractal structure in choosing which components to disable.
+
+Drop-path constitutes not only a regularization strategy, but also provides means of optionally imparting fractal networks with anytime behavior. A particular schedule of dropped paths during learning prevents subnetworks of different depths from co-adapting. As a consequence, both shallow and deep subnetworks must individually produce correct output. Querying a shallow subnetwork thus yields a quick and moderately accurate result in advance of completion of the full network.
+
+Section 3 elaborates the technical details of fractal networks and drop-path. Section 4 provides experimental comparisons to residual networks across the CIFAR-10, CIFAR-100 (Krizhevsky, 2009), SVHN (Netzer et al., 2011), and ImageNet (Deng et al., 2009) datasets. We also evaluate regularization and data augmentation strategies, investigate subnetwork student-teacher behavior during training, and benchmark anytime networks obtained using drop-path. Section 5 provides synthesis. By virtue of encapsulating many known, yet seemingly distinct, design principles, selfsimilar structure may materialize as a fundamental component of neural architectures.
+
+# 2 RELATED WORK
+
+Deepening feed-forward neural networks has generally returned dividends in performance. A striking example within the computer vision community is the improvement on the ImageNet (Deng et al., 2009) classification task when transitioning from AlexNet (Krizhevsky et al., 2012) to VGG (Simonyan & Zisserman, 2015) to GoogLeNet (Szegedy et al., 2015) to ResNet (He et al., 2016a). Unfortunately, greater depth also makes training more challenging, at least when employing a firstorder optimization method with randomly initialized layers. As the network grows deeper and more non-linear, the linear approximation of a gradient step becomes increasingly inappropriate. Desire to overcome these difficulties drives research on both optimization techniques and network architectures.
+
+On the optimization side, much recent work yields improvements. To prevent vanishing gradients, ReLU activation functions now widely replace sigmoid and tanh units (Nair & Hinton, 2010). This subject remains an area of active inquiry, with various tweaks on ReLUs, e.g. PReLUs (He et al., 2015), and ELUs (Clevert et al., 2016). Even with ReLUs, employing batch normalization (Ioffe & Szegedy, 2015) speeds training by reducing internal covariate shift. Good initialization can also ameliorate this problem (Glorot & Bengio, 2010; Mishkin & Matas, 2016). Path-SGD (Neyshabur et al., 2015) offers an alternative normalization scheme. Progress in optimization is somewhat orthogonal to our architectural focus, with the expectation that advances in either are ripe for combination.
+
+Notable ideas in architecture reach back to skip connections, the earliest example of a nontrivial routing pattern within a neural network. Recent work further elaborates upon them (Maire et al., 2014; Hariharan et al., 2015). Highway networks (Srivastava et al., 2015) and ResNet (He et al., 2016a;b) offer additional twists in the form of parameterized pass-through and gating. In work subsequent to our own, Huang et al. (2016a) investigate a ResNet variant with explicit skip connections. These methods share distinction as the only other designs demonstrated to scale to hundreds of layers and beyond. ResNet’s building block uses the identity map as an anchor point and explicitly parameterizes an additive correction term (the residual). Identity initialization also appears in the context of recurrent networks (Le et al., 2015). A tendency of ResNet and highway networks to fall-back to the identity map may make their effective depth much smaller than their nominal depth.
+
+Some prior results hint at what we experimentally demonstrate in Section 4. Namely, reduction of effective depth is key to training extremely deep networks; residuals are incidental. Huang et al. (2016b) provide one clue in their work on stochastic depth: randomly dropping layers from ResNet during training, thereby shrinking network depth by a constant factor, provides additional performance benefit. We build upon this intuition through drop-path, which shrinks depth much more drastically.
+
+The success of deep supervision (Lee et al., 2014) provides another clue that effective depth is crucial. Here, an auxiliary loss, forked off mid-level layers, introduces a shorter path during backpropagation. The layer at the fork receives two gradients, originating from the main loss and the auxiliary loss, that are added together. Deep supervision is now common, being adopted, for example, by GoogLeNet (Szegedy et al., 2015). However, irrelevance of the auxiliary loss at test time introduces the drawback of having a discrepancy between the actual objective and that used for training.
+
+Exploration of the student-teacher paradigm (Ba & Caruana, 2014) illuminates the potential for interplay between networks of different depth. In the model compression scenario, a deeper network (previously trained) guides and improves the learning of a shallower and faster student network (Ba & Caruana, 2014; Urban et al., 2017). This is accomplished by feeding unlabeled data through the teacher and having the student mimic the teacher’s soft output predictions. FitNets (Romero et al., 2015) explicitly couple students and teachers, forcing mimic behavior across several intermediate points in the network. Our fractal networks capture yet another alternative, in the form of implicit coupling, with the potential for bidirectional information flow between shallow and deep subnetworks.
+
+Widening networks, by using larger modules in place of individual layers, has also produced performance gains. For example, an Inception module (Szegedy et al., 2015) concatenates results of convolutional layers of different receptive field size. Stacking these modules forms the GoogLeNet architecture. Liao & Carneiro (2015) employ a variant with maxout in place of concatenation. Figure 1 makes apparent our connection with such work. As a fractal network deepens, it also widens. Moreover, note that stacking two 2D convolutional layers with the same spatial receptive field (e.g. $3 \times 3 ,$ ) achieves a larger $( 5 \times 5 )$ receptive field. A horizontal cross-section of a fractal network is reminiscent of an Inception module, except with additional joins due to recursive structure.
+
+# 3 FRACTAL NETWORKS
+
+We begin with a more formal presentation of the ideas sketched in Figure 1. Convolutional neural networks serve as our running example and, in the subsequent section, our experimental platform. However, it is worth emphasizing that our framework is more general. In principle, convolutional layers in Figure 1 could be replaced by a different layer type, or even a custom-designed module or subnetwork, in order to generate other fractal architectures.
+
+Let $C$ denote the index of the truncated fractal $f _ { C } ( \cdot )$ . Our network’s structure, connections and layer types, is defined by $f _ { C } ( \cdot )$ . A network consisting of a single convolutional layer is the base case:
+
+$$
+f _ { 1 } ( z ) = \mathrm { c o n v } ( z )
+$$
+
+We define successive fractals recursively:
+
+$$
+f _ { C + 1 } ( z ) = \left[ ( f _ { C } \circ f _ { C } ) ( z ) \right] \oplus [ \mathrm { c o n v } ( z ) ]
+$$
+
+where $\bigcirc$ denotes composition and $\textcircled{+}$ a join operation. When drawn in the style of Figure 1, $C$ corresponds to the number of columns, or width, of network $f _ { C } ( \cdot )$ . Depth, defined to be the number of conv layers on the longest path between input and output, scales as $2 ^ { \overbrace { C } - 1 }$ . Convolutional networks for classification typically intersperse pooling layers. We achieve the same by using $f _ { C } ( \cdot )$ as a building block and stacking it with subsequent pooling layers $B$ times, yielding total depth $B \cdot 2 ^ { C - 1 }$
+
+The join operation $\textcircled{+}$ merges two feature blobs into one. Here, a blob is the result of a conv layer: a tensor holding activations for a fixed number of channels over a spatial domain. The channel count corresponds to the size of the filter set in the preceding conv layer. As the fractal is expanded, we collapse neighboring joins into a single join layer which spans multiple columns, as shown on the right side of Figure 1. The join layer merges all of its input feature blobs into a single output blob.
+
+Several choices seem reasonable for the action of a join layer, including concatenation and addition. We instantiate each join to compute the element-wise mean of its inputs. This is appropriate for convolutional networks in which channel count is set the same for all conv layers within a fractal block. Averaging might appear similar to ResNet’s addition operation, but there are critical differences:
+
+• ResNet makes clear distinction between pass-through and residual signals. In FractalNet, no signal is privileged. Every input to a join layer is the output of an immediately preceding conv layer. The network structure alone cannot identify any as being primary. Drop-path regularization, as described next in Section 3.1, forces each input to a join to be individually reliable. This reduces the reward for even implicitly learning to allocate part of one signal to act as a residual for another. Experiments show that we can extract high-performance subnetworks consisting of a single column (Section 4.2). Such a subnetwork is effectively devoid of joins, as only a single path is active throughout. They produce no signal to which a residual could be added.
+
+Together, these properties ensure that join layers are not an alternative method of residual learning.
+
+# 3.1 REGULARIZATION VIA DROP-PATH
+
+Dropout (Hinton et al., 2012) and drop-connect (Wan et al., 2013) modify interactions between sequential network layers in order to discourage co-adaptation. Since fractal networks contain additional macro-scale structure, we propose to complement these techniques with an analogous coarse-scale regularization scheme.
+
+Figure 2 illustrates drop-path. Just as dropout prevents co-adaptation of activations, drop-path prevents co-adaptation of parallel paths by randomly dropping operands of the join layers. This discourages the network from using one input path as an anchor and another as a corrective term (a configuration that, if not prevented, is prone to overfitting). We consider two sampling strategies:
+
+• Local: a join drops each input with fixed probability, but we make sure at least one survives. • Global: a single path is selected for the entire network. We restrict this path to be a single column, thereby promoting individual columns as independently strong predictors.
+
+![](images/90121a668d72375db114b3617bbdcce2116550e84a2571d28c89b49996affdc3.jpg)  
+Figure 2: Drop-path. A fractal network block functions with some connections between layers disabled, provided some path from input to output is still available. Drop-path guarantees at least one such path, while sampling a subnetwork with many other paths disabled. During training, presenting a different active subnetwork to each mini-batch prevents co-adaptation of parallel paths. A global sampling strategy returns a single column as a subnetwork. Alternating it with local sampling encourages the development of individual columns as performant stand-alone subnetworks.
+
+As with dropout, signals may need appropriate rescaling. With element-wise means, this is trivial;   
+each join computes the mean of only its active inputs.
+
+In experiments, we train with dropout and a mixture model of $5 0 \%$ local and $5 0 \%$ global sampling for drop-path. We sample a new subnetwork each mini-batch. With sufficient memory, we can simultaneously evaluate one local sample and all global samples for each mini-batch by keeping separate networks and tying them together via weight sharing.
+
+While fractal connectivity permits the use of paths of any length, global drop-path forces the use of many paths whose lengths differ by orders of magnitude (powers of 2). The subnetworks sampled by drop-path thus exhibit large structural diversity. This property stands in contrast to stochastic depth regularization of ResNet, which, by virtue of using a fixed drop probability for each layer in a chain, samples subnetworks with a concentrated depth distribution (Huang et al., 2016b).
+
+Global drop-path serves not only as a regularizer, but also as a diagnostic tool. Monitoring performance of individual columns provides insight into both the network and training mechanisms, as Section 4.3 discusses in more detail. Individually strong columns of various depths also give users choices in the trade-off between speed (shallow) and accuracy (deep).
+
+# 3.2 DATA AUGMENTATION
+
+Data augmentation can reduce the need for regularization. ResNet demonstrates this, achieving $2 7 . 2 2 \%$ error rate on CIFAR-100 with augmentation compared to $4 4 . 7 6 \%$ without (Huang et al., 2016b). While augmentation benefits fractal networks, we show that drop-path provides highly effective regularization, allowing them to achieve competitive results even without data augmentation.
+
+# 3.3 IMPLEMENTATION DETAILS
+
+We implement FractalNet using Caffe (Jia et al., 2014). Purely for convenience, we flip the order of pool and join layers at the end of a block in Figure 1. We pool individual columns immediately before the joins spanning all columns, rather than pooling once immediately after them.
+
+We train fractal networks using stochastic gradient descent with momentum. As now standard, we employ batch normalization together with each conv layer (convolution, batch norm, then ReLU).
+
+<table><tr><td>Method</td><td>C100</td><td>C100+</td><td>C100++</td><td>C10</td><td>C10+</td><td>C10++</td><td>SVHN</td></tr><tr><td>Network in Network (Lin et al.,2013)</td><td>35.68</td><td>1</td><td></td><td>10.41 1</td><td>8.81</td><td></td><td>2.35</td></tr><tr><td>Generalized Pooling (Lee et al., 2016)</td><td>32.37</td><td>1</td><td></td><td>7.62 1</td><td>6.05</td><td></td><td>1.69</td></tr><tr><td>Recurrent CNN (Liang &amp; Hu, 2015)</td><td>31.75</td><td>=</td><td></td><td>8.69</td><td>7.09</td><td>=</td><td>1.77</td></tr><tr><td>Multi-scale (Liao &amp; Carneiro, 2015)</td><td>27.56</td><td>一</td><td></td><td>6.87 一</td><td></td><td>=</td><td>1.76</td></tr><tr><td>FitNet Romero et al. (2015)</td><td>1</td><td>一 35.04</td><td>=</td><td>- 一</td><td>8.39</td><td>=</td><td>2.42</td></tr><tr><td>Deeply Supervised (Lee et al., 2014)</td><td>=</td><td>一 34.57 一</td><td></td><td>9.69 一</td><td>7.97</td><td>1</td><td>1.92</td></tr><tr><td>All-CNN (Springenberg et al., 2014)</td><td></td><td>33.71 一</td><td>=</td><td>9.08</td><td>7.25</td><td>4.41</td><td>1</td></tr><tr><td>Highway Net (Srivastava et al., 2015)</td><td></td><td>32.39 一</td><td></td><td>=</td><td>7.72</td><td>、</td><td>1</td></tr><tr><td>ELU (Clevert et al., 2016)</td><td></td><td>一 24.28</td><td></td><td>一 =</td><td>6.55</td><td>-</td><td>1</td></tr><tr><td>Scalable BO (Snoek et al.,2015)</td><td></td><td>一</td><td>27.04</td><td>一 = 一</td><td>=</td><td>6.37</td><td>1.77</td></tr><tr><td>Fractional Max-Pool (Graham,2014)</td><td>=</td><td>一 1 1</td><td>26.32</td><td>= 一</td><td>1</td><td>3.47</td><td>1</td></tr><tr><td>FitResNet (Mishkin &amp; Matas, 2016)</td><td>=</td><td>一 27.66</td><td></td><td>一 =</td><td>5.84</td><td>=</td><td>1</td></tr><tr><td>ResNet (He et al., 2016a)</td><td>1</td><td>一 -</td><td>=</td><td>=</td><td>1 6.61</td><td>=</td><td>=</td></tr><tr><td>ResNet by (Huang et al., 2016b)</td><td>44.76</td><td>一 27.22</td><td></td><td>13.63</td><td>一 6.41</td><td>=</td><td>2.01</td></tr><tr><td>Stochastic Depth (Huang et al., 2016b)</td><td>37.80</td><td>一 24.58</td><td></td><td>11.66</td><td>一 5.23</td><td></td><td>1.75</td></tr><tr><td>Identity Mapping (He et al.,2016b)</td><td>=</td><td>一 22.68</td><td></td><td></td><td>4.69</td><td></td><td>-</td></tr><tr><td>ResNet in ResNet (Targ et al., 2016)</td><td></td><td>一 22.90</td><td></td><td>=</td><td>5.01</td><td></td><td></td></tr><tr><td>Wide (Zagoruyko &amp; Komodakis, 2016)</td><td>1</td><td>一 20.50 一</td><td></td><td>- 1</td><td>4.17</td><td></td><td>= 1</td></tr><tr><td>DenseNet-BC (Huang et al., 2016a)1</td><td>19.64</td><td>一 17.60</td><td>=</td><td>5.19</td><td>一 3.62</td><td></td><td>1.74</td></tr><tr><td>FractalNet (20 layers, 38.6M params)</td><td>35.34</td><td>一 23.30</td><td>22.85</td><td>10.18 一</td><td></td><td>-</td><td></td></tr><tr><td>+ drop-path + dropout</td><td>28.20</td><td>一 23.73</td><td>23.36</td><td>7.33 一</td><td>5.22 4.60</td><td>5.11</td><td>2.01</td></tr><tr><td>Ldeepest column alone</td><td>29.05</td><td>24.32</td><td>23.60</td><td>7.27</td><td>4.68</td><td>4.59 4.63</td><td>1.87</td></tr><tr><td>FractalNet (40layers,2.9params)</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td>1.89</td></tr><tr><td></td><td></td><td>一 22.49</td><td>21.49</td><td>1</td><td>一 5.24</td><td>5.21</td><td>1</td></tr></table>
+
+Table 1: CIFAR-100/CIFAR-10/SVHN. We compare test error $( \% )$ with other leading methods, trained with either no data augmentation, translation/mirroring $( + )$ , or more substantial augmentation $( + + )$ . Our main point of comparison is ResNet. We closely match its benchmark results using data augmentation, and outperform it by large margins without data augmentation. Training with drop-path, we can extract from FractalNet single-column (plain) networks that are highly competitive.
+
+# 4 EXPERIMENTS
+
+The CIFAR, SVHN, and ImageNet datasets serve as testbeds for comparison to prior work and analysis of FractalNet’s internal behavior. We evaluate performance on the standard classification task associated with each dataset. For CIFAR and SVHN, which consist of $3 2 \times 3 2$ images, we set our fractal network to have 5 blocks $\mathrm { \Delta B = 5 }$ ) with $2 \times 2$ non-overlapping max-pooling and subsampling applied after each. This reduces the input $3 2 \times 3 2$ spatial resolution to $1 \times 1$ over the course of the entire network. A softmax prediction layer attaches at the end of the network. Unless otherwise noted, we set the number of filter channels within blocks 1 through 5 as p64, 128, 256, 512, 512q, mostly matching the convention of doubling the number of channels after halving spatial resolution.
+
+For ImageNet, we choose a fractal architecture to facilitate direct comparison with the 34-layer ResNet of He et al. (2016a). We use the same first and last layer as ResNet-34, but change the middle of the network to consist of 4 blocks $B = 4$ ), each of 8 layers $C = 4$ columns). We use a filter channel progression of p128, 256, 512, 1024q in blocks 1 through 4.
+
+# 4.1 TRAINING
+
+For experiments using dropout, we fix drop rate per block at $0 \%$ , $1 0 \%$ , $2 0 \%$ , $3 0 \%$ , $4 0 \%$ q, similar to Clevert et al. (2016). Local drop-path uses $1 5 \%$ drop rate across the entire network.
+
+Table 2: ImageNet (validation set, 10-crop).   
+
+<table><tr><td>Method</td><td>Top-1 (%)</td><td>Top-5 (%)</td></tr><tr><td>VGG-16</td><td>28.07</td><td>9.33</td></tr><tr><td>ResNet-34 C</td><td>24.19</td><td>7.40</td></tr><tr><td>FractalNet-34</td><td>24.12</td><td>7.39</td></tr></table>
+
+<table><tr><td>Model</td><td>Depth</td><td>Train Loss</td><td>Error (%)</td></tr><tr><td>Plain</td><td>5</td><td>0.786</td><td>36.62</td></tr><tr><td>Plain</td><td>10</td><td>0.159</td><td>32.47</td></tr><tr><td>Plain</td><td>20</td><td>0.037</td><td>31.31</td></tr><tr><td>Plain</td><td>40</td><td>0.580</td><td>38.84</td></tr><tr><td>Fractal Col #1</td><td>5</td><td>0.677</td><td>37.23</td></tr><tr><td>Fractal Col #2</td><td>10</td><td>0.141</td><td>32.85</td></tr><tr><td>Fractal Col #3</td><td>20</td><td>0.029</td><td>31.31</td></tr><tr><td>Fractal Col #4</td><td>40</td><td>0.016</td><td>31.75</td></tr><tr><td>Fractal Full</td><td>40</td><td>0.015</td><td>27.40</td></tr></table>
+
+Table 3: Ultra-deep fractal networks (CIFAR- $1 0 0 { + + }$ ). Increasing depth greatly improves accuracy until eventual diminishing returns. Contrast with plain networks, which are not trainable if made too deep (Table 4).   
+
+<table><tr><td>Cols.</td><td>Depth</td><td>Params.</td><td>Error (%)</td></tr><tr><td>1</td><td>5</td><td>0.3M</td><td>37.32</td></tr><tr><td>2</td><td>10</td><td>0.8M</td><td>30.71</td></tr><tr><td>3</td><td>20</td><td>2.1M</td><td>27.69</td></tr><tr><td>4</td><td>40</td><td>4.8M</td><td>27.38</td></tr><tr><td>5</td><td>80</td><td>10.2M</td><td>26.46</td></tr><tr><td>6</td><td>160</td><td>21.1M</td><td>27.38</td></tr></table>
+
+Table 4: Fractal structure as a training apparatus (CIFAR- $1 0 0 + +$ ). Plain networks perform well if moderately deep, but exhibit worse convergence during training if instantiated with great depth. However, as a column trained within, and then extracted from, a fractal network with mixed drop-path, we recover a plain network that overcomes such depth limitation (possibly due to a student-teacher effect).
+
+We run for 400 epochs on CIFAR, 20 epochs on SVHN, and 70 epochs on ImageNet. Our learning rate starts at 0.02 (for ImageNet, 0.001) and we train using stochastic gradient descent with batch size 100 (for ImageNet, 32) and momentum 0.9. For CIFAR/SVHN, we drop the learning rate by a factor of 10 whenever the number of remaining epochs halves. For ImageNet, we drop by a factor of 10 at epochs 50 and 65. We use Xavier initialization (Glorot & Bengio, 2010).
+
+A widely employed (Lin et al., 2013; Clevert et al., 2016; Srivastava et al., 2015; He et al., 2016a;b; Huang et al., 2016b; Targ et al., 2016) scheme for data augmentation on CIFAR consists of only horizontal mirroring and translation (uniform offsets in $[ - 4 , 4 ] )$ , with images zero-padded where needed after mean subtraction. We denote results achieved using no more than this degree of augmentation by appending a $" + "$ to the dataset name (e.g. CIFAR- $1 0 0 +$ ). A $" + + "$ marks results reliant on more data augmentation; here exact schemes may vary. Our entry in this category is modest and simply changes the zero-padding to reflect-padding.
+
+# 4.2 RESULTS
+
+Table 1 compares performance of FractalNet on CIFAR and SVHN with competing methods. FractalNet (depth 20) outperforms the original ResNet across the board. With data augmentation, our CIFAR-100 accuracy is close to that of the best ResNet variants. With neither augmentation nor regularization, FractalNet’s performance on CIFAR is superior to both ResNet and ResNet with stochastic depth, suggesting that FractalNet may be less prone to overfitting. Most methods perform similarly on SVHN. Increasing depth to 40, while borrowing some parameter reduction tricks (Iandola et al., 2016), reveals FractalNet’s performance to be consistent across a range of configuration choices.
+
+Experiments without data augmentation highlight the power of drop-path regularization. On CIFAR100, drop-path reduces FractalNet’s error rate from $3 5 . 3 4 \%$ to $2 8 . 2 0 \%$ . Unregularized ResNet is far behind $( 4 4 . 7 6 \% )$ and ResNet with stochastic depth $( 3 7 . 8 0 \% )$ ) does not catch up to our unregularized starting point of $3 5 . 3 4 \%$ . CIFAR-10 mirrors this story. With data augmentation, drop-path provides a boost (CIFAR-10), or does not significantly influence FractalNet’s performance (CIFAR-100).
+
+Note that the performance of the deepest column of the fractal network is close to that of the full network (statistically equivalent on CIFAR-10). This suggests that the fractal structure may be more important as a learning framework than as a final model architecture.
+
+Table 2 shows that FractalNet scales to ImageNet, matching ResNet (He et al., 2016a) at equal depth. Note that, concurrent with our work, refinements to the residual network paradigm further improve the state-of-the-art on ImageNet. Wide residual networks (Zagoruyko & Komodakis, 2016) of 34-layers reduce single-crop Top-1 and Top-5 validation error by approximately $2 \%$ and $1 \%$ , respectively, over
+
+![](images/a6248c9240c55983fd50617f4bd57bfcc51f4b9ffadcbc4d3106e909a5485df5.jpg)  
+Figure 3: Implicit deep supervision. Left: Evolution of loss for plain networks of depth 5, 10, 20 and 40 trained on CIFAR-100. Training becomes increasingly difficult for deeper networks. At 40 layers, we are unable to train the network satisfactorily. Right: We train a 4 column fractal network with mixed drop-path, monitoring its loss as well as the losses of its four subnetworks corresponding to individual columns of the same depth as the plain networks. As the 20-layer subnetwork starts to stabilize, drop-path puts pressure on the 40-layer column to adapt, with the rest of the network as its teacher. This explains the elbow-shaped learning curve for Col #4 that occurs around 25 epochs.
+
+ResNet-34 by doubling feature channels in each layer. DenseNets (Huang et al., 2016a) substantially improve performance by building residual blocks that concatenate rather than add feature channels.
+
+Table 3 demonstrates that FractalNet resists performance degradation as we increase $C$ to obtain extremely deep networks (160 layers for $C \ = \ 6$ ). Scores in this table are not comparable to those in Table 1. For time and memory efficiency, we reduced block-wise feature channels to p16, 32, 64, 128, 128q and the batch size to 50 for the supporting experiments in Tables 3 and 4.
+
+Table 4 provides a baseline showing that training of plain deep networks begins to degrade by the time their depth reaches 40 layers. In our experience, a plain 160-layer completely fails to converge. This table also highlights the ability to use FractalNet and drop-path as an engine for extracting trained networks (columns) with the same topology as plain networks, but much higher test performance.
+
+# 4.3 INTROSPECTION
+
+With Figure 3, we examine the evolution of a 40-layer FractalNet during training. Tracking columns individually (recording their losses when run as stand-alone networks), we observe that the 40-layer column initially improves slowly, but picks up once the loss of the rest of the network begins to stabilize. Contrast with a plain 40-layer network trained alone (dashed blue line), which never makes fast progress. The column has the same initial plateau, but subsequently improves after 25 epochs, producing a loss curve uncharacteristic of plain networks.
+
+We hypothesize that the fractal structure triggers effects akin to deep supervision and lateral studentteacher information flow. Column #4 joins with column #3 every other layer, and in every fourth layer this join involves no other columns. Once the fractal network partially relies on the signal going through column #3, drop-path puts pressure on column #4 to produce a replacement signal when column #3 is dropped. This task has constrained scope. A particular drop only requires two consecutive layers in column $\# 4$ to substitute for one in column #3 (a mini student-teacher problem).
+
+This explanation of FractalNet dynamics parallels what, in concurrent work, Greff et al. (2017) claim for ResNet. Specifically, Greff et al. (2017) suggest residual networks learn unrolled iterative estimation, with each layer performing a gradual refinement on its input representation. The deepest FractalNet column could behave in the same manner, with the remainder of the network acting as a scaffold for building smaller refinement steps by doubling layers from one column to the next.
+
+These interpretations appear not to mesh with the conclusions of Veit et al. (2016), who claim that ensemble-like behavior underlies the success of ResNet. This is certainly untrue of some very deep networks, as FractalNet provides a counterexample: we can extract a single column (plain network topology) and it alone (no ensembling) performs nearly as well as the entire network. Moreover, the gradual refinement view may offer an alternative explanation for the experiments of Veit et al. (2016). If each layer makes only a small modification, removing one may look, to the subsequent portion of the network, like injecting a small amount of input noise. Perhaps noise tolerance explains the gradual performance degradation that Veit et al. (2016) observe when removing ResNet layers.
+
+# 5 CONCLUSION
+
+Our experiments with fractal networks provide strong evidence that path length is fundamental for training ultra-deep neural networks; residuals are incidental. Key is the shared characteristic of FractalNet and ResNet: large nominal network depth, but effectively shorter paths for gradient propagation during training. Fractal architectures are arguably the simplest means of satisfying this requirement, and match residual networks in experimental performance. Fractal networks are resistant to being too deep; extra depth may slow training, but does not impair accuracy.
+
+With drop-path, regularization of extremely deep fractal networks is intuitive and effective. Drop-path doubles as a method of enforcing speed (latency) vs. accuracy tradeoffs. For applications where fast responses have utility, we can obtain fractal networks whose partial evaluation yields good answers.
+
+Our analysis connects the internal behavior of fractal networks with phenomena engineered into other networks. Their substructure resembles hand-crafted modules used as components in prior work. Their training evolution may emulate deep supervision and student-teacher learning.
+
+# ACKNOWLEDGMENTS
+
+We gratefully acknowledge the support of NVIDIA Corporation with the donation of GPUs used for this research.
+
+# REFERENCES
+
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parse/train/S1VaB4cex/S1VaB4cex_content_list.json

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+    {
+        "type": "text",
+        "text": "FRACTALNET: ULTRA-DEEP NEURAL NETWORKS WITHOUT RESIDUALS ",
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Gustav Larsson University of Chicago larsson@cs.uchicago.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Michael Maire TTI Chicago mmaire@ttic.edu ",
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+            170,
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+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Gregory Shakhnarovich TTI Chicago greg@ttic.edu ",
+        "bbox": [
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+            170,
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+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            250,
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+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a design strategy for neural network macro-architecture based on selfsimilarity. Repeated application of a simple expansion rule generates deep networks whose structural layouts are precisely truncated fractals. These networks contain interacting subpaths of different lengths, but do not include any pass-through or residual connections; every internal signal is transformed by a filter and nonlinearity before being seen by subsequent layers. In experiments, fractal networks match the excellent performance of standard residual networks on both CIFAR and ImageNet classification tasks, thereby demonstrating that residual representations may not be fundamental to the success of extremely deep convolutional neural networks. Rather, the key may be the ability to transition, during training, from effectively shallow to deep. We note similarities with student-teacher behavior and develop drop-path, a natural extension of dropout, to regularize co-adaptation of subpaths in fractal architectures. Such regularization allows extraction of highperformance fixed-depth subnetworks. Additionally, fractal networks exhibit an anytime property: shallow subnetworks provide a quick answer, while deeper subnetworks, with higher latency, provide a more accurate answer. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            540
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Residual networks (He et al., 2016a), or ResNets, lead a recent and dramatic increase in both depth and accuracy of convolutional neural networks, facilitated by constraining the network to learn residuals. ResNet variants (He et al., 2016a;b; Huang et al., 2016b) and related architectures (Srivastava et al., 2015) employ the common technique of initializing and anchoring, via a pass-through channel, a network to the identity function. Training now differs in two respects. First, the objective changes to learning residual outputs, rather than unreferenced absolute mappings. Second, these networks exhibit a type of deep supervision (Lee et al., 2014), as near-identity layers effectively reduce distance to the loss. He et al. (2016a) speculate that the former, the residual formulation itself, is crucial. ",
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+    {
+        "type": "text",
+        "text": "We show otherwise, by constructing a competitive extremely deep architecture that does not rely on residuals. Our design principle is pure enough to communicate in a single word, fractal, and a simple diagram (Figure 1). Yet, fractal networks implicitly recapitulate many properties hard-wired into previous successful architectures. Deep supervision not only arises automatically, but also drives a type of student-teacher learning (Ba & Caruana, 2014; Urban et al., 2017) internal to the network. Modular building blocks of other designs (Szegedy et al., 2015; Liao & Carneiro, 2015) resemble special cases of a fractal network’s nested substructure. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For fractal networks, simplicity of training mirrors simplicity of design. A single loss, attached to the final layer, suffices to drive internal behavior mimicking deep supervision. Parameters are randomly initialized. As they contain subnetworks of many depths, fractal networks are robust to choice of overall depth; make them deep enough and training will carve out a useful assembly of subnetworks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The entirety of emergent behavior resulting from a fractal design may erode the need for recent engineering tricks intended to achieve similar effects. These tricks include residual functional forms with identity initialization, manual deep supervision, hand-crafted architectural modules, and studentteacher training regimes. Section 2 reviews this large body of related techniques. Hybrid designs could certainly integrate any of them with a fractal architecture; we leave open the question of the degree to which such hybrids are synergistic. ",
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+    {
+        "type": "image",
+        "img_path": "images/389e48bd72030062303ac6da089f1bd92ce9277af84c158fee8a805381e555a4.jpg",
+        "image_caption": [
+            "Figure 1: Fractal architecture. Left: A simple expansion rule generates a fractal architecture with $C$ intertwined columns. The base case, $f _ { 1 } ( z )$ , has a single layer of the chosen type (e.g. convolutional) between input and output. Join layers compute element-wise mean. Right: Deep convolutional networks periodically reduce spatial resolution via pooling. A fractal version uses $f _ { C }$ as a building block between pooling layers. Stacking $B$ such blocks yields a network whose total depth, measured in terms of convolution layers, is $B \\cdot \\bar { 2 } ^ { C - 1 }$ . This example has depth 40 $B = 5$ , $C = 4$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contribution is twofold: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "• We introduce FractalNet, the first simple alternative to ResNet. FractalNet shows that explicit residual learning is not a requirement for building ultra-deep neural networks. • Through analysis and experiments, we elucidate connections between FractalNet and an array of phenomena engineered into previous deep network designs. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As an additional contribution, we develop drop-path, a novel regularization protocol for ultradeep fractal networks. Without data augmentation, fractal networks, trained with drop-path and dropout (Hinton et al., 2012), exceed the performance of residual networks regularized via stochastic depth (Huang et al., 2016b). Though, like stochastic depth, it randomly removes macro-scale components, drop-path further exploits our fractal structure in choosing which components to disable. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Drop-path constitutes not only a regularization strategy, but also provides means of optionally imparting fractal networks with anytime behavior. A particular schedule of dropped paths during learning prevents subnetworks of different depths from co-adapting. As a consequence, both shallow and deep subnetworks must individually produce correct output. Querying a shallow subnetwork thus yields a quick and moderately accurate result in advance of completion of the full network. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Section 3 elaborates the technical details of fractal networks and drop-path. Section 4 provides experimental comparisons to residual networks across the CIFAR-10, CIFAR-100 (Krizhevsky, 2009), SVHN (Netzer et al., 2011), and ImageNet (Deng et al., 2009) datasets. We also evaluate regularization and data augmentation strategies, investigate subnetwork student-teacher behavior during training, and benchmark anytime networks obtained using drop-path. Section 5 provides synthesis. By virtue of encapsulating many known, yet seemingly distinct, design principles, selfsimilar structure may materialize as a fundamental component of neural architectures. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Deepening feed-forward neural networks has generally returned dividends in performance. A striking example within the computer vision community is the improvement on the ImageNet (Deng et al., 2009) classification task when transitioning from AlexNet (Krizhevsky et al., 2012) to VGG (Simonyan & Zisserman, 2015) to GoogLeNet (Szegedy et al., 2015) to ResNet (He et al., 2016a). Unfortunately, greater depth also makes training more challenging, at least when employing a firstorder optimization method with randomly initialized layers. As the network grows deeper and more non-linear, the linear approximation of a gradient step becomes increasingly inappropriate. Desire to overcome these difficulties drives research on both optimization techniques and network architectures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On the optimization side, much recent work yields improvements. To prevent vanishing gradients, ReLU activation functions now widely replace sigmoid and tanh units (Nair & Hinton, 2010). This subject remains an area of active inquiry, with various tweaks on ReLUs, e.g. PReLUs (He et al., 2015), and ELUs (Clevert et al., 2016). Even with ReLUs, employing batch normalization (Ioffe & Szegedy, 2015) speeds training by reducing internal covariate shift. Good initialization can also ameliorate this problem (Glorot & Bengio, 2010; Mishkin & Matas, 2016). Path-SGD (Neyshabur et al., 2015) offers an alternative normalization scheme. Progress in optimization is somewhat orthogonal to our architectural focus, with the expectation that advances in either are ripe for combination. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Notable ideas in architecture reach back to skip connections, the earliest example of a nontrivial routing pattern within a neural network. Recent work further elaborates upon them (Maire et al., 2014; Hariharan et al., 2015). Highway networks (Srivastava et al., 2015) and ResNet (He et al., 2016a;b) offer additional twists in the form of parameterized pass-through and gating. In work subsequent to our own, Huang et al. (2016a) investigate a ResNet variant with explicit skip connections. These methods share distinction as the only other designs demonstrated to scale to hundreds of layers and beyond. ResNet’s building block uses the identity map as an anchor point and explicitly parameterizes an additive correction term (the residual). Identity initialization also appears in the context of recurrent networks (Le et al., 2015). A tendency of ResNet and highway networks to fall-back to the identity map may make their effective depth much smaller than their nominal depth. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Some prior results hint at what we experimentally demonstrate in Section 4. Namely, reduction of effective depth is key to training extremely deep networks; residuals are incidental. Huang et al. (2016b) provide one clue in their work on stochastic depth: randomly dropping layers from ResNet during training, thereby shrinking network depth by a constant factor, provides additional performance benefit. We build upon this intuition through drop-path, which shrinks depth much more drastically. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The success of deep supervision (Lee et al., 2014) provides another clue that effective depth is crucial. Here, an auxiliary loss, forked off mid-level layers, introduces a shorter path during backpropagation. The layer at the fork receives two gradients, originating from the main loss and the auxiliary loss, that are added together. Deep supervision is now common, being adopted, for example, by GoogLeNet (Szegedy et al., 2015). However, irrelevance of the auxiliary loss at test time introduces the drawback of having a discrepancy between the actual objective and that used for training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Exploration of the student-teacher paradigm (Ba & Caruana, 2014) illuminates the potential for interplay between networks of different depth. In the model compression scenario, a deeper network (previously trained) guides and improves the learning of a shallower and faster student network (Ba & Caruana, 2014; Urban et al., 2017). This is accomplished by feeding unlabeled data through the teacher and having the student mimic the teacher’s soft output predictions. FitNets (Romero et al., 2015) explicitly couple students and teachers, forcing mimic behavior across several intermediate points in the network. Our fractal networks capture yet another alternative, in the form of implicit coupling, with the potential for bidirectional information flow between shallow and deep subnetworks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Widening networks, by using larger modules in place of individual layers, has also produced performance gains. For example, an Inception module (Szegedy et al., 2015) concatenates results of convolutional layers of different receptive field size. Stacking these modules forms the GoogLeNet architecture. Liao & Carneiro (2015) employ a variant with maxout in place of concatenation. Figure 1 makes apparent our connection with such work. As a fractal network deepens, it also widens. Moreover, note that stacking two 2D convolutional layers with the same spatial receptive field (e.g. $3 \\times 3 ,$ ) achieves a larger $( 5 \\times 5 )$ receptive field. A horizontal cross-section of a fractal network is reminiscent of an Inception module, except with additional joins due to recursive structure. ",
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+    {
+        "type": "text",
+        "text": "3 FRACTAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            387,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We begin with a more formal presentation of the ideas sketched in Figure 1. Convolutional neural networks serve as our running example and, in the subsequent section, our experimental platform. However, it is worth emphasizing that our framework is more general. In principle, convolutional layers in Figure 1 could be replaced by a different layer type, or even a custom-designed module or subnetwork, in order to generate other fractal architectures. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $C$ denote the index of the truncated fractal $f _ { C } ( \\cdot )$ . Our network’s structure, connections and layer types, is defined by $f _ { C } ( \\cdot )$ . A network consisting of a single convolutional layer is the base case: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7eb9079055a9ee8de45b03e1fdab5a324fa791eab35cf9db66afa3e940ba46bc.jpg",
+        "text": "$$\nf _ { 1 } ( z ) = \\mathrm { c o n v } ( z )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We define successive fractals recursively: ",
+        "bbox": [
+            176,
+            270,
+            446,
+            285
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/761a0d2ca2fcdeb55c99e1ba279398080d5d4f3afcbac6540c857162ecc14c6e.jpg",
+        "text": "$$\nf _ { C + 1 } ( z ) = \\left[ ( f _ { C } \\circ f _ { C } ) ( z ) \\right] \\oplus [ \\mathrm { c o n v } ( z ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            291,
+            632,
+            310
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\bigcirc$ denotes composition and $\\textcircled{+}$ a join operation. When drawn in the style of Figure 1, $C$ corresponds to the number of columns, or width, of network $f _ { C } ( \\cdot )$ . Depth, defined to be the number of conv layers on the longest path between input and output, scales as $2 ^ { \\overbrace { C } - 1 }$ . Convolutional networks for classification typically intersperse pooling layers. We achieve the same by using $f _ { C } ( \\cdot )$ as a building block and stacking it with subsequent pooling layers $B$ times, yielding total depth $B \\cdot 2 ^ { C - 1 }$ ",
+        "bbox": [
+            174,
+            316,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The join operation $\\textcircled{+}$ merges two feature blobs into one. Here, a blob is the result of a conv layer: a tensor holding activations for a fixed number of channels over a spatial domain. The channel count corresponds to the size of the filter set in the preceding conv layer. As the fractal is expanded, we collapse neighboring joins into a single join layer which spans multiple columns, as shown on the right side of Figure 1. The join layer merges all of its input feature blobs into a single output blob. ",
+        "bbox": [
+            174,
+            393,
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+            464
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Several choices seem reasonable for the action of a join layer, including concatenation and addition. We instantiate each join to compute the element-wise mean of its inputs. This is appropriate for convolutional networks in which channel count is set the same for all conv layers within a fractal block. Averaging might appear similar to ResNet’s addition operation, but there are critical differences: ",
+        "bbox": [
+            176,
+            469,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• ResNet makes clear distinction between pass-through and residual signals. In FractalNet, no signal is privileged. Every input to a join layer is the output of an immediately preceding conv layer. The network structure alone cannot identify any as being primary. Drop-path regularization, as described next in Section 3.1, forces each input to a join to be individually reliable. This reduces the reward for even implicitly learning to allocate part of one signal to act as a residual for another. Experiments show that we can extract high-performance subnetworks consisting of a single column (Section 4.2). Such a subnetwork is effectively devoid of joins, as only a single path is active throughout. They produce no signal to which a residual could be added. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Together, these properties ensure that join layers are not an alternative method of residual learning. ",
+        "bbox": [
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+            702
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 REGULARIZATION VIA DROP-PATH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            718,
+            457,
+            733
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Dropout (Hinton et al., 2012) and drop-connect (Wan et al., 2013) modify interactions between sequential network layers in order to discourage co-adaptation. Since fractal networks contain additional macro-scale structure, we propose to complement these techniques with an analogous coarse-scale regularization scheme. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 illustrates drop-path. Just as dropout prevents co-adaptation of activations, drop-path prevents co-adaptation of parallel paths by randomly dropping operands of the join layers. This discourages the network from using one input path as an anchor and another as a corrective term (a configuration that, if not prevented, is prone to overfitting). We consider two sampling strategies: ",
+        "bbox": [
+            176,
+            808,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Local: a join drops each input with fixed probability, but we make sure at least one survives. • Global: a single path is selected for the entire network. We restrict this path to be a single column, thereby promoting individual columns as independently strong predictors. ",
+        "bbox": [
+            217,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/90121a668d72375db114b3617bbdcce2116550e84a2571d28c89b49996affdc3.jpg",
+        "image_caption": [
+            "Figure 2: Drop-path. A fractal network block functions with some connections between layers disabled, provided some path from input to output is still available. Drop-path guarantees at least one such path, while sampling a subnetwork with many other paths disabled. During training, presenting a different active subnetwork to each mini-batch prevents co-adaptation of parallel paths. A global sampling strategy returns a single column as a subnetwork. Alternating it with local sampling encourages the development of individual columns as performant stand-alone subnetworks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            108,
+            816,
+            348
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As with dropout, signals may need appropriate rescaling. With element-wise means, this is trivial;   \neach join computes the mean of only its active inputs. ",
+        "bbox": [
+            174,
+            472,
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+            501
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In experiments, we train with dropout and a mixture model of $5 0 \\%$ local and $5 0 \\%$ global sampling for drop-path. We sample a new subnetwork each mini-batch. With sufficient memory, we can simultaneously evaluate one local sample and all global samples for each mini-batch by keeping separate networks and tying them together via weight sharing. ",
+        "bbox": [
+            174,
+            507,
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+            563
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While fractal connectivity permits the use of paths of any length, global drop-path forces the use of many paths whose lengths differ by orders of magnitude (powers of 2). The subnetworks sampled by drop-path thus exhibit large structural diversity. This property stands in contrast to stochastic depth regularization of ResNet, which, by virtue of using a fixed drop probability for each layer in a chain, samples subnetworks with a concentrated depth distribution (Huang et al., 2016b). ",
+        "bbox": [
+            174,
+            570,
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+            640
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Global drop-path serves not only as a regularizer, but also as a diagnostic tool. Monitoring performance of individual columns provides insight into both the network and training mechanisms, as Section 4.3 discusses in more detail. Individually strong columns of various depths also give users choices in the trade-off between speed (shallow) and accuracy (deep). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 DATA AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            720,
+            372,
+            734
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Data augmentation can reduce the need for regularization. ResNet demonstrates this, achieving $2 7 . 2 2 \\%$ error rate on CIFAR-100 with augmentation compared to $4 4 . 7 6 \\%$ without (Huang et al., 2016b). While augmentation benefits fractal networks, we show that drop-path provides highly effective regularization, allowing them to achieve competitive results even without data augmentation. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            820,
+            405,
+            834
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We implement FractalNet using Caffe (Jia et al., 2014). Purely for convenience, we flip the order of pool and join layers at the end of a block in Figure 1. We pool individual columns immediately before the joins spanning all columns, rather than pooling once immediately after them. ",
+        "bbox": [
+            174,
+            845,
+            823,
+            888
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train fractal networks using stochastic gradient descent with momentum. As now standard, we employ batch normalization together with each conv layer (convolution, batch norm, then ReLU). ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/72279064f13aec4902cffd3b631b94a1185e8da1bae5325ba46f87db1fa38a72.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>C100</td><td>C100+</td><td>C100++</td><td>C10</td><td>C10+</td><td>C10++</td><td>SVHN</td></tr><tr><td>Network in Network (Lin et al.,2013)</td><td>35.68</td><td>1</td><td></td><td>10.41 1</td><td>8.81</td><td></td><td>2.35</td></tr><tr><td>Generalized Pooling (Lee et al., 2016)</td><td>32.37</td><td>1</td><td></td><td>7.62 1</td><td>6.05</td><td></td><td>1.69</td></tr><tr><td>Recurrent CNN (Liang &amp; Hu, 2015)</td><td>31.75</td><td>=</td><td></td><td>8.69</td><td>7.09</td><td>=</td><td>1.77</td></tr><tr><td>Multi-scale (Liao &amp; Carneiro, 2015)</td><td>27.56</td><td>一</td><td></td><td>6.87 一</td><td></td><td>=</td><td>1.76</td></tr><tr><td>FitNet Romero et al. (2015)</td><td>1</td><td>一 35.04</td><td>=</td><td>- 一</td><td>8.39</td><td>=</td><td>2.42</td></tr><tr><td>Deeply Supervised (Lee et al., 2014)</td><td>=</td><td>一 34.57 一</td><td></td><td>9.69 一</td><td>7.97</td><td>1</td><td>1.92</td></tr><tr><td>All-CNN (Springenberg et al., 2014)</td><td></td><td>33.71 一</td><td>=</td><td>9.08</td><td>7.25</td><td>4.41</td><td>1</td></tr><tr><td>Highway Net (Srivastava et al., 2015)</td><td></td><td>32.39 一</td><td></td><td>=</td><td>7.72</td><td>、</td><td>1</td></tr><tr><td>ELU (Clevert et al., 2016)</td><td></td><td>一 24.28</td><td></td><td>一 =</td><td>6.55</td><td>-</td><td>1</td></tr><tr><td>Scalable BO (Snoek et al.,2015)</td><td></td><td>一</td><td>27.04</td><td>一 = 一</td><td>=</td><td>6.37</td><td>1.77</td></tr><tr><td>Fractional Max-Pool (Graham,2014)</td><td>=</td><td>一 1 1</td><td>26.32</td><td>= 一</td><td>1</td><td>3.47</td><td>1</td></tr><tr><td>FitResNet (Mishkin &amp; Matas, 2016)</td><td>=</td><td>一 27.66</td><td></td><td>一 =</td><td>5.84</td><td>=</td><td>1</td></tr><tr><td>ResNet (He et al., 2016a)</td><td>1</td><td>一 -</td><td>=</td><td>=</td><td>1 6.61</td><td>=</td><td>=</td></tr><tr><td>ResNet by (Huang et al., 2016b)</td><td>44.76</td><td>一 27.22</td><td></td><td>13.63</td><td>一 6.41</td><td>=</td><td>2.01</td></tr><tr><td>Stochastic Depth (Huang et al., 2016b)</td><td>37.80</td><td>一 24.58</td><td></td><td>11.66</td><td>一 5.23</td><td></td><td>1.75</td></tr><tr><td>Identity Mapping (He et al.,2016b)</td><td>=</td><td>一 22.68</td><td></td><td></td><td>4.69</td><td></td><td>-</td></tr><tr><td>ResNet in ResNet (Targ et al., 2016)</td><td></td><td>一 22.90</td><td></td><td>=</td><td>5.01</td><td></td><td></td></tr><tr><td>Wide (Zagoruyko &amp; Komodakis, 2016)</td><td>1</td><td>一 20.50 一</td><td></td><td>- 1</td><td>4.17</td><td></td><td>= 1</td></tr><tr><td>DenseNet-BC (Huang et al., 2016a)1</td><td>19.64</td><td>一 17.60</td><td>=</td><td>5.19</td><td>一 3.62</td><td></td><td>1.74</td></tr><tr><td>FractalNet (20 layers, 38.6M params)</td><td>35.34</td><td>一 23.30</td><td>22.85</td><td>10.18 一</td><td></td><td>-</td><td></td></tr><tr><td>+ drop-path + dropout</td><td>28.20</td><td>一 23.73</td><td>23.36</td><td>7.33 一</td><td>5.22 4.60</td><td>5.11</td><td>2.01</td></tr><tr><td>Ldeepest column alone</td><td>29.05</td><td>24.32</td><td>23.60</td><td>7.27</td><td>4.68</td><td>4.59 4.63</td><td>1.87</td></tr><tr><td>FractalNet (40layers,2.9params)</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td>1.89</td></tr><tr><td></td><td></td><td>一 22.49</td><td>21.49</td><td>1</td><td>一 5.24</td><td>5.21</td><td>1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: CIFAR-100/CIFAR-10/SVHN. We compare test error $( \\% )$ with other leading methods, trained with either no data augmentation, translation/mirroring $( + )$ , or more substantial augmentation $( + + )$ . Our main point of comparison is ResNet. We closely match its benchmark results using data augmentation, and outperform it by large margins without data augmentation. Training with drop-path, we can extract from FractalNet single-column (plain) networks that are highly competitive. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            541,
+            326,
+            558
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The CIFAR, SVHN, and ImageNet datasets serve as testbeds for comparison to prior work and analysis of FractalNet’s internal behavior. We evaluate performance on the standard classification task associated with each dataset. For CIFAR and SVHN, which consist of $3 2 \\times 3 2$ images, we set our fractal network to have 5 blocks $\\mathrm { \\Delta B = 5 }$ ) with $2 \\times 2$ non-overlapping max-pooling and subsampling applied after each. This reduces the input $3 2 \\times 3 2$ spatial resolution to $1 \\times 1$ over the course of the entire network. A softmax prediction layer attaches at the end of the network. Unless otherwise noted, we set the number of filter channels within blocks 1 through 5 as p64, 128, 256, 512, 512q, mostly matching the convention of doubling the number of channels after halving spatial resolution. ",
+        "bbox": [
+            174,
+            575,
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+            688
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For ImageNet, we choose a fractal architecture to facilitate direct comparison with the 34-layer ResNet of He et al. (2016a). We use the same first and last layer as ResNet-34, but change the middle of the network to consist of 4 blocks $B = 4$ ), each of 8 layers $C = 4$ columns). We use a filter channel progression of p128, 256, 512, 1024q in blocks 1 through 4. ",
+        "bbox": [
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+            751
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            771,
+            285,
+            786
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For experiments using dropout, we fix drop rate per block at $0 \\%$ , $1 0 \\%$ , $2 0 \\%$ , $3 0 \\%$ , $4 0 \\%$ q, similar to Clevert et al. (2016). Local drop-path uses $1 5 \\%$ drop rate across the entire network. ",
+        "bbox": [
+            174,
+            799,
+            823,
+            828
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c6c3f84b145e0352bbf190d6364e6b5fc105355adbd8e8d1285e0806b21e152c.jpg",
+        "table_caption": [
+            "Table 2: ImageNet (validation set, 10-crop). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Top-1 (%)</td><td>Top-5 (%)</td></tr><tr><td>VGG-16</td><td>28.07</td><td>9.33</td></tr><tr><td>ResNet-34 C</td><td>24.19</td><td>7.40</td></tr><tr><td>FractalNet-34</td><td>24.12</td><td>7.39</td></tr></table>",
+        "bbox": [
+            186,
+            101,
+            459,
+            167
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0a1ac5ad36d48572ab06b8bd60850d477f189d9fe80e5620180e07399cc6b033.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Depth</td><td>Train Loss</td><td>Error (%)</td></tr><tr><td>Plain</td><td>5</td><td>0.786</td><td>36.62</td></tr><tr><td>Plain</td><td>10</td><td>0.159</td><td>32.47</td></tr><tr><td>Plain</td><td>20</td><td>0.037</td><td>31.31</td></tr><tr><td>Plain</td><td>40</td><td>0.580</td><td>38.84</td></tr><tr><td>Fractal Col #1</td><td>5</td><td>0.677</td><td>37.23</td></tr><tr><td>Fractal Col #2</td><td>10</td><td>0.141</td><td>32.85</td></tr><tr><td>Fractal Col #3</td><td>20</td><td>0.029</td><td>31.31</td></tr><tr><td>Fractal Col #4</td><td>40</td><td>0.016</td><td>31.75</td></tr><tr><td>Fractal Full</td><td>40</td><td>0.015</td><td>27.40</td></tr></table>",
+        "bbox": [
+            490,
+            101,
+            825,
+            263
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/9da0eecea8c3d7fbab1803736d44b28e59b0e645f2e8783fd453dd0c501be609.jpg",
+        "table_caption": [
+            "Table 3: Ultra-deep fractal networks (CIFAR- $1 0 0 { + + }$ ). Increasing depth greatly improves accuracy until eventual diminishing returns. Contrast with plain networks, which are not trainable if made too deep (Table 4). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Cols.</td><td>Depth</td><td>Params.</td><td>Error (%)</td></tr><tr><td>1</td><td>5</td><td>0.3M</td><td>37.32</td></tr><tr><td>2</td><td>10</td><td>0.8M</td><td>30.71</td></tr><tr><td>3</td><td>20</td><td>2.1M</td><td>27.69</td></tr><tr><td>4</td><td>40</td><td>4.8M</td><td>27.38</td></tr><tr><td>5</td><td>80</td><td>10.2M</td><td>26.46</td></tr><tr><td>6</td><td>160</td><td>21.1M</td><td>27.38</td></tr></table>",
+        "bbox": [
+            194,
+            200,
+            449,
+            308
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Fractal structure as a training apparatus (CIFAR- $1 0 0 + +$ ). Plain networks perform well if moderately deep, but exhibit worse convergence during training if instantiated with great depth. However, as a column trained within, and then extracted from, a fractal network with mixed drop-path, we recover a plain network that overcomes such depth limitation (possibly due to a student-teacher effect). ",
+        "bbox": [
+            490,
+            275,
+            825,
+            386
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We run for 400 epochs on CIFAR, 20 epochs on SVHN, and 70 epochs on ImageNet. Our learning rate starts at 0.02 (for ImageNet, 0.001) and we train using stochastic gradient descent with batch size 100 (for ImageNet, 32) and momentum 0.9. For CIFAR/SVHN, we drop the learning rate by a factor of 10 whenever the number of remaining epochs halves. For ImageNet, we drop by a factor of 10 at epochs 50 and 65. We use Xavier initialization (Glorot & Bengio, 2010). ",
+        "bbox": [
+            174,
+            415,
+            825,
+            484
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A widely employed (Lin et al., 2013; Clevert et al., 2016; Srivastava et al., 2015; He et al., 2016a;b; Huang et al., 2016b; Targ et al., 2016) scheme for data augmentation on CIFAR consists of only horizontal mirroring and translation (uniform offsets in $[ - 4 , 4 ] )$ , with images zero-padded where needed after mean subtraction. We denote results achieved using no more than this degree of augmentation by appending a $\" + \"$ to the dataset name (e.g. CIFAR- $1 0 0 +$ ). A $\" + + \"$ marks results reliant on more data augmentation; here exact schemes may vary. Our entry in this category is modest and simply changes the zero-padding to reflect-padding. ",
+        "bbox": [
+            174,
+            491,
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+            590
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
+            277,
+            625
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 compares performance of FractalNet on CIFAR and SVHN with competing methods. FractalNet (depth 20) outperforms the original ResNet across the board. With data augmentation, our CIFAR-100 accuracy is close to that of the best ResNet variants. With neither augmentation nor regularization, FractalNet’s performance on CIFAR is superior to both ResNet and ResNet with stochastic depth, suggesting that FractalNet may be less prone to overfitting. Most methods perform similarly on SVHN. Increasing depth to 40, while borrowing some parameter reduction tricks (Iandola et al., 2016), reveals FractalNet’s performance to be consistent across a range of configuration choices. ",
+        "bbox": [
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+            734
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experiments without data augmentation highlight the power of drop-path regularization. On CIFAR100, drop-path reduces FractalNet’s error rate from $3 5 . 3 4 \\%$ to $2 8 . 2 0 \\%$ . Unregularized ResNet is far behind $( 4 4 . 7 6 \\% )$ and ResNet with stochastic depth $( 3 7 . 8 0 \\% )$ ) does not catch up to our unregularized starting point of $3 5 . 3 4 \\%$ . CIFAR-10 mirrors this story. With data augmentation, drop-path provides a boost (CIFAR-10), or does not significantly influence FractalNet’s performance (CIFAR-100). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that the performance of the deepest column of the fractal network is close to that of the full network (statistically equivalent on CIFAR-10). This suggests that the fractal structure may be more important as a learning framework than as a final model architecture. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows that FractalNet scales to ImageNet, matching ResNet (He et al., 2016a) at equal depth. Note that, concurrent with our work, refinements to the residual network paradigm further improve the state-of-the-art on ImageNet. Wide residual networks (Zagoruyko & Komodakis, 2016) of 34-layers reduce single-crop Top-1 and Top-5 validation error by approximately $2 \\%$ and $1 \\%$ , respectively, over ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/a6248c9240c55983fd50617f4bd57bfcc51f4b9ffadcbc4d3106e909a5485df5.jpg",
+        "image_caption": [
+            "Figure 3: Implicit deep supervision. Left: Evolution of loss for plain networks of depth 5, 10, 20 and 40 trained on CIFAR-100. Training becomes increasingly difficult for deeper networks. At 40 layers, we are unable to train the network satisfactorily. Right: We train a 4 column fractal network with mixed drop-path, monitoring its loss as well as the losses of its four subnetworks corresponding to individual columns of the same depth as the plain networks. As the 20-layer subnetwork starts to stabilize, drop-path puts pressure on the 40-layer column to adapt, with the rest of the network as its teacher. This explains the elbow-shaped learning curve for Col #4 that occurs around 25 epochs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            104,
+            813,
+            329
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ResNet-34 by doubling feature channels in each layer. DenseNets (Huang et al., 2016a) substantially improve performance by building residual blocks that concatenate rather than add feature channels. ",
+        "bbox": [
+            173,
+            468,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3 demonstrates that FractalNet resists performance degradation as we increase $C$ to obtain extremely deep networks (160 layers for $C \\ = \\ 6$ ). Scores in this table are not comparable to those in Table 1. For time and memory efficiency, we reduced block-wise feature channels to p16, 32, 64, 128, 128q and the batch size to 50 for the supporting experiments in Tables 3 and 4. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4 provides a baseline showing that training of plain deep networks begins to degrade by the time their depth reaches 40 layers. In our experience, a plain 160-layer completely fails to converge. This table also highlights the ability to use FractalNet and drop-path as an engine for extracting trained networks (columns) with the same topology as plain networks, but much higher test performance. ",
+        "bbox": [
+            174,
+            568,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 INTROSPECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            645,
+            328,
+            659
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "With Figure 3, we examine the evolution of a 40-layer FractalNet during training. Tracking columns individually (recording their losses when run as stand-alone networks), we observe that the 40-layer column initially improves slowly, but picks up once the loss of the rest of the network begins to stabilize. Contrast with a plain 40-layer network trained alone (dashed blue line), which never makes fast progress. The column has the same initial plateau, but subsequently improves after 25 epochs, producing a loss curve uncharacteristic of plain networks. ",
+        "bbox": [
+            174,
+            672,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We hypothesize that the fractal structure triggers effects akin to deep supervision and lateral studentteacher information flow. Column #4 joins with column #3 every other layer, and in every fourth layer this join involves no other columns. Once the fractal network partially relies on the signal going through column #3, drop-path puts pressure on column #4 to produce a replacement signal when column #3 is dropped. This task has constrained scope. A particular drop only requires two consecutive layers in column $\\# 4$ to substitute for one in column #3 (a mini student-teacher problem). ",
+        "bbox": [
+            174,
+            763,
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+            847
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This explanation of FractalNet dynamics parallels what, in concurrent work, Greff et al. (2017) claim for ResNet. Specifically, Greff et al. (2017) suggest residual networks learn unrolled iterative estimation, with each layer performing a gradual refinement on its input representation. The deepest FractalNet column could behave in the same manner, with the remainder of the network acting as a scaffold for building smaller refinement steps by doubling layers from one column to the next. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "These interpretations appear not to mesh with the conclusions of Veit et al. (2016), who claim that ensemble-like behavior underlies the success of ResNet. This is certainly untrue of some very deep networks, as FractalNet provides a counterexample: we can extract a single column (plain network topology) and it alone (no ensembling) performs nearly as well as the entire network. Moreover, the gradual refinement view may offer an alternative explanation for the experiments of Veit et al. (2016). If each layer makes only a small modification, removing one may look, to the subsequent portion of the network, like injecting a small amount of input noise. Perhaps noise tolerance explains the gradual performance degradation that Veit et al. (2016) observe when removing ResNet layers. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            234,
+            318,
+            251
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our experiments with fractal networks provide strong evidence that path length is fundamental for training ultra-deep neural networks; residuals are incidental. Key is the shared characteristic of FractalNet and ResNet: large nominal network depth, but effectively shorter paths for gradient propagation during training. Fractal architectures are arguably the simplest means of satisfying this requirement, and match residual networks in experimental performance. Fractal networks are resistant to being too deep; extra depth may slow training, but does not impair accuracy. ",
+        "bbox": [
+            174,
+            267,
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+            352
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "With drop-path, regularization of extremely deep fractal networks is intuitive and effective. Drop-path doubles as a method of enforcing speed (latency) vs. accuracy tradeoffs. For applications where fast responses have utility, we can obtain fractal networks whose partial evaluation yields good answers. ",
+        "bbox": [
+            176,
+            358,
+            825,
+            400
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our analysis connects the internal behavior of fractal networks with phenomena engineered into other networks. Their substructure resembles hand-crafted modules used as components in prior work. Their training evolution may emulate deep supervision and student-teacher learning. ",
+        "bbox": [
+            176,
+            406,
+            825,
+            449
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "We gratefully acknowledge the support of NVIDIA Corporation with the donation of GPUs used for this research. ",
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+            521
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+        "page_idx": 8
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parse/train/S1zk9iRqF7/S1zk9iRqF7.md

@@ -0,0 +1,458 @@
+# PATE-GAN: GENERATING SYNTHETIC DATA WITHDIFFERENTIAL PRIVACY GUARANTEES
+
+James Jordon∗   
+Engineering Science Department   
+University of Oxford, UK   
+james.jordon@wolfson.ox.ac.uk   
+Jinsung Yoon∗   
+Department of Electrical and Computer Engineering   
+UCLA, California, USA   
+jsyoon0823@g.ucla.edu   
+Mihaela van der Schaar   
+University of Cambridge, UK   
+Department of Electrical and Computer Engineering, UCLA, California, USA   
+Alan Turing Institute, London, UK   
+mihaela@ee.ucla.edu
+
+# ABSTRACT
+
+Machine learning has the potential to assist many communities in using the large datasets that are becoming more and more available. Unfortunately, much of that potential is not being realized because it would require sharing data in a way that compromises privacy. In this paper, we investigate a method for ensuring (differential) privacy of the generator of the Generative Adversarial Nets (GAN) framework. The resulting model can be used for generating synthetic data on which algorithms can be trained and validated, and on which competitions can be conducted, without compromising the privacy of the original dataset. Our method modifies the Private Aggregation of Teacher Ensembles (PATE) framework and applies it to GANs. Our modified framework (which we call PATE-GAN) allows us to tightly bound the influence of any individual sample on the model, resulting in tight differential privacy guarantees and thus an improved performance over models with the same guarantees. We also look at measuring the quality of synthetic data from a new angle; we assert that for the synthetic data to be useful for machine learning researchers, the relative performance of two algorithms (trained and tested) on the synthetic dataset should be the same as their relative performance (when trained and tested) on the original dataset. Our experiments, on various datasets, demonstrate that PATE-GAN consistently outperforms the stateof-the-art method with respect to this and other notions of synthetic data quality.
+
+# 1 INTRODUCTION
+
+More and more large datasets are becoming available in a wide variety of communities. In the U.S. medical community, for example, the fraction of providers using electronic health records (EHR) increased from $9 . 4 \%$ in 2008 to $8 3 . 8 \%$ in 2015 [20]. The availability of large datasets presents enormous opportunities for collaboration between the data-holders and the machine learning community. However, many of these large datasets, especially EHR, include sensitive information that prevents data-holders from sharing the data.
+
+The most common way to mitigate the privacy risk of sharing sensitive records is to de-identify the records - but it is by now well-known that records that have been de-identified can be easily re-identified by linking them to other identifiable datasets [30; 13; 24; 22; 14]. (This is especially true for medical records of patients who have rare diseases.) However, if the purpose of sharing the data is to develop and validate machine learning methods for a particular task (e.g. prognostic risk scoring), real data is not necessary; it would suffice to have synthetic data that is sufficiently like the real data.
+
+Precisely what this means depends on how the synthetic data will be used. For example, the synthetic data may be used to train models that will be deployed directly on real data. In this setting it is important that these methods (which we trained entirely on synthetic data) perform as well as if they had been trained on real data. Another setting to consider is one in which data-holders wish to use the synthetic data to identify the best method(s) to be used on the real data [11]. In this setting, it is not important that training on synthetic data leads to good performance on real data, but rather that comparing two methods on the synthetic data results in conclusions similar to those that would have been drawn from comparing the two methods on the real data. We evaluate our method in both settings.
+
+Generative Adversarial Networks (GAN) [19] provide a powerful method for using real data to generate synthetic data but it does not provide any rigorous privacy guarantees. Our method modifies the GAN machinery in a way that does guarantee privacy; the synthetic data is (differentially) private [12] with respect to the original data. To do this we modify the training procedure of the discriminator to be differentially private by using a modified version of the Private Aggregation of Teacher Ensembles (PATE) [25; 26] framework. The Post-Processing Theorem [12] then guarantees that the GAN generator - which is trained only using the differentially private discriminator - will also be differentially private and thus so will the synthetic data it generates. We call our proposed framework PATE-GAN.
+
+Using two Kaggle datasets, two different real-world medical datasets and two UCI datasets, we evaluate the utility of the samples generated by PATE-GAN in various settings with various levels of differential privacy. In line with the settings outlined above, we consider two methods for evaluating the similarity of synthetic datasets with a real dataset. The first method, first proposed in [15], compares the predictive performance of models trained on the synthetic datasets and tested on the real dataset. The second method, which we propose for the first time here, compares the performance rankings of predictive models on the synthetic datasets with their performance rankings on the real dataset. We demonstrate that, for both of these methods, PATE-GAN consistently produces synthetic datasets that are ”more like” the original real dataset than the synthetic datasets produced by the state-of-the-art benchmark (DPGAN [32]).
+
+The contributions of this paper can be summarized as follows: (1) we modify the PATE framework and apply it to GANs to generate synthetic data, (2) we demonstrate in the experiments section that using PATE to enforce differential privacy results in higher quality synthetic data than DPGAN using various real-world datasets, (3) we propose a novel new metric for evaluating the generated synthetic data.
+
+# 2 RELATED WORKS
+
+The most related previous work to this paper is DPGAN [32]. Like us, DPGAN proposes a framework for modifying the GAN framework to be differentially private, also relying on the PostProcessing Theorem to change the problem of learning a differentially private generator to learning a differentially private discriminator. Their work uses a technique introduced by [1] that provides a differentially private mechanism for training deep networks. The key idea is that noise is added to the gradient of the discriminator during training to create differential privacy guarantees. These ideas are also used in [2]. Our method is similar in spirit; during training of the discriminator differentially private training data is used, which results in noisy gradients, however, we use the mechanism introduced in [25] which we believe gives tighter differential privacy guarantees (via tighter bounds on the effect of a single sample) than those provided in [1]. This means that for the same privacy guarantees, our method is capable of producing higher quality synthetic data. For a visual representation of both PATE-GAN and DPGAN, see the Appendix.
+
+The proposed model modifies the PATE framework [25; 26] for use in a generative model setting (specifically for use with GANs). The key to the GAN framework is that the discriminator is a differentiable module trained to classify samples as either real or generated. The PATE framework provides a differentially private mechanism for classification by training multiple teacher models on disjoint partitions of the data. To classify a new sample each teacher’s output is evaluated on the sample and then all outputs are noisily aggregated. This noisy aggregation, though, results in a classifier that is not differentiable with respect to the parameters of the generator. In order to overcome this problem we follow the idea of the student model, also proposed in [25], that involves taking some public unlabelled data, labelling it using the standard PATE mechanism and then training the student using the resulting labelled data. Because access to any public data is often an unreasonable assumption in synthetic data generation, we adapt this training paradigm in a way that does not require public data by training the student using only outputs from the (differentially private) generator.
+
+Some previous works generate synthetic data using summary statistics of the original data [23] or based on specific domain-knowledge [6]; however, those methods are limited to low-dimensional feature spaces, specific fields and do not provide any differential privacy guarantees. [9] generates synthetic patient records using a GAN framework. However, [9] focuses only on generating discrete variables, whereas PATE-GAN is capable of generating mixed-type (continuous, discrete, and binary) variables. Furthermore, [9] also does not provide any differential privacy guarantees and instead uses ad-hoc notions of privacy which are only validated empirically.
+
+Finally, it is worth remarking that it is known to be hard in the worst-case to generate private synthetic data [31] and so techniques such as GANs are necessary to address this challenge.
+
+# 3 BACKGROUND
+
+Let us denote the feature space by $\mathcal { X }$ , the label space by $\mathcal { V }$ and write $\mathcal { U } = \mathcal { X } \times \mathcal { Y }$ . Let the dimension of $\mathcal { U }$ be $d$ . Suppose that $\mathbf { X }$ and $Y$ are random variables over $\mathcal { X }$ and $\mathcal { V }$ . We write $\mathbf { U } = ( \mathbf { X } , Y )$ and $\mathbf { x } , y , \mathbf { u }$ to denote realizations med i.i.d. according to $\mathbf { X } , Y$ and oted $\mathbf { U }$ , $\mathcal { D }$ onsists of . $N$ samples of $\mathbf { u }$ ${ \mathcal { P } } _ { U }$ $\mathcal { D } \overset { ^ { . } } { = } \{ \mathbf { u } _ { i } \} _ { i = 1 } ^ { \check { N } } = \{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$
+
+# 3.1 DIFFERENTIAL PRIVACY
+
+We first provide some preliminaries on differential privacy [12] before describing PATE-GAN; we refer interested readers to [12] for a thorough exposition of differential privacy. We will denote an algorithm by $\mathcal { M }$ , which takes as input a dataset $\mathcal { D }$ and outputs a value from some output space, $\mathcal { O }$ .
+
+Definition 1. (Neighboring Datasets) Two datasets $\mathcal { D } , \mathcal { D } ^ { \prime }$ are said to be neighboring if
+
+$$
+\exists x \in { \mathcal { D } } s . t . { \mathcal { D } } \setminus \{ x \} = { \mathcal { D } } ^ { \prime } .
+$$
+
+Definition 2. (Differential Privacy) $A$ randomized algorithm, $\mathcal { M }$ , is $( \epsilon , \delta )$ -differentially private if for all $s \subset \mathcal { O }$ and for all neighboring datasets $\mathcal { D } , \mathcal { D } ^ { \prime }$ :
+
+$$
+\mathbb { P } ( \mathcal { M } ( \mathcal { D } ) \in \mathcal { S } ) \leq e ^ { \epsilon } \mathbb { P } ( \mathcal { M } ( \mathcal { D ^ { \prime } } ) \in \mathcal { S } ) + \delta
+$$
+
+where $\mathbb { P }$ is taken with respect to the randomness of $\mathcal { M }$
+
+Differential privacy provides an intuitively understandable notion of privacy - a particular sample’s inclusion or exclusion in the dataset does not change the probability of a particular outcome very much: it does so by a multiplicative factor of $e ^ { \epsilon }$ and an additive amount, $\delta$ .
+
+The following theorem, a proof of which can be found in [12], allows us to move the burden of differential privacy to the discriminator; the differential privacy of the generator will follow by the theorem.
+
+Theorem. (Post-processing) Let $\mathcal { M }$ be an $( \epsilon , \delta )$ -differentially private algorithm and let $f : \mathcal { O }  \mathcal { O } ^ { \prime }$ where $\mathcal { O } ^ { \prime }$ is any arbitrary space. Then $f \circ { \mathcal { M } }$ is $( \epsilon , \delta )$ -differentially private.
+
+# 3.2 PRIVATE AGGREGATION OF TEACHER ENSEMBLES (PATE)
+
+In this section we describe the PATE mechanism first defined in [25] and later improved upon by [26]. The PATE mechanism provides a differentially private method for classification, a core component of the GAN framework; the discriminator is a classifier trained to identify whether samples are real/fake.
+
+In order to build a differentially private classifier, the dataset is first divided into $k$ disjoint subsets $\mathcal { D } _ { 1 } , . . . , \mathcal { D } _ { k }$ . $k$ classifiers, $T _ { 1 } , . . . , T _ { k }$ (referred to as teachers) are then trained separately on the $k$ sub-datasets (i.e. $T _ { i }$ is only trained on $\mathcal { D } _ { i }$ ). Given a new input feature vector $\mathbf { x }$ to classify, the differentially private output is given by passing $\mathbf { x }$ to each of the $k$ teachers, and then performing a noisy aggregation of the resulting outputs.
+
+Formally, given the $k$ teachers, $m$ possible classes and an input feature vector, $\mathbf { x }$ , set
+
+$$
+n _ { j } ( \mathbf { x } ) = | \{ T _ { i } : T _ { i } ( \mathbf { x } ) = j \} | \ \mathrm { f o r } \ j = 1 , . . . , m
+$$
+
+so that $n _ { j } ( \mathbf { x } )$ is the number of teachers that output class $j$ for $\mathbf { x }$ . The output of the $\mathrm { P A T E } _ { \lambda }$ mechanism for input $\mathbf { x }$ is then defined as
+
+$$
+\mathrm { P A T E } _ { \lambda } ( \mathbf { x } ) = \underset { j \in [ m ] } { \arg \operatorname* { m a x } } ( n _ { j } ( \mathbf { x } ) + Y _ { j } )
+$$
+
+where $Y _ { 1 } , . . . , Y _ { m }$ are i.i.d. $L a p ( \lambda )$ random variables. The following result, found in [25], follows from [12].
+
+Theorem. The output of a single query to the $P A T E _ { \lambda }$ mechanism is $\textstyle { \bigl ( } { \frac { 1 } { \lambda } } , 0 { \bigr ) }$ -differentially private.
+
+In order to apply this framework in the GAN framework, however, we require that the discriminator be differentiable, which the output of this classification mechanism is not (note that accessing the internal parameters of the teachers would violate differential privacy, the only thing we have access to in this case is the output). Instead, we draw on the PATE extension (also introduced in [25]) in which a student model is trained. This student model (after being trained) is free to access, not only its outputs given inputs but also its internal parameters. The model itself is differentially private.
+
+Formally, the student, $S$ , is a classifier that is trained by taking some public, unlabelled data, $\mathbf { \mathcal { P } } = \{ \tilde { \mathbf { x } _ { i } } \} _ { i = 1 } ^ { K }$ , passing each sample, $\mathbf { x } _ { i }$ , through the (standard) PATE mechanism, to receive a differentially private label, $\hat { y } _ { i }$ , and forming a new (noisy-)teacher-labelled dataset $\hat { \mathcal { P } } = \{ ( \mathbf { x } _ { i } , \hat { y } _ { i } ) \} _ { i = 1 } ^ { K }$ on which the student is then trained.
+
+Importantly, we can make the student differentiable - it can be modelled using any classifier, such as a neural net. Moreover, querying the student is “free” - there is no privacy cost associated with passing an input to the student and receiving an output, the only privacy cost is in acquiring the data on which to train the student. We state the following result which follows from the analysis in [25].
+
+Theorem. The student, $S$ , trained on the dataset $\hat { \mathcal { P } }$ where labels were generated according to the $P A T E _ { \lambda }$ mechanism using $\begin{array} { r } { \lambda = \frac { K } { 2 \epsilon } } \end{array}$ , is $( \epsilon , 0 )$ -differentially private with respect to the original data $\mathcal { D }$ .
+
+# 4 PROPOSED METHOD: PATE-GAN
+
+The proposed method builds on GAN and PATE frameworks. We replace the GAN discriminator with a PATE mechanism so that our discriminator is differentially private, but require the (differentiable) student version to allow back-propagation to the generator. We modify the implementation of the student, noting that the training paradigm presented in [25] is not appropriate for this setting due to the lack of publicly available data. Before training, we partition the dataset into $k$ subsets, $\mathcal { D } _ { 1 } , . . . , \mathcal { D } _ { k }$ , with $\begin{array} { r } { | \mathcal { D } _ { i } | = \frac { | \mathcal { D } | } { k } } \end{array}$ for $\forall i$ .
+
+# 4.1 GENERATOR
+
+The generator, $G$ , is as in the standard GAN framework. Formally it is a function $G ( \cdot ; \theta _ { G } ) \ :$ $[ 0 , 1 ] ^ { \breve { d } }  \mathcal { U }$ , parametrized by $\theta _ { G }$ that takes random noise, $\mathbf { z } \sim \mathrm { U n i f } ( [ 0 , 1 ] ^ { d } )$ , as input and outputs a vector in $\mathcal { U } = \mathcal { X } \times \mathcal { Y }$ . The generator will be trained to minimize its loss with respect to the student-discriminator. Given $n$ i.i.d. samples of $\mathrm { U n i f } ( [ 0 , 1 ] ^ { d } )$ , $\mathbf { z } _ { 1 } , . . . , \mathbf { z } _ { n }$ , the empirical loss of $G$ at $\theta$ for fixed $S$ is defined by
+
+$$
+\mathcal { L } _ { G } ( \theta _ { G } ; S ) = \sum _ { j = 1 } ^ { n } \log ( 1 - S ( G ( \mathbf { z } _ { j } ; \theta _ { G } ) ) ) .
+$$
+
+We will denote by $\mathcal { P } _ { G }$ the distribution induced by $G$ over $\mathcal { U }$ .
+
+# 4.2 DISCRIMINATOR
+
+In the standard GAN framework, there is a single discriminator, $D$ , that is trained in a directly adversarial fashion with $G$ , where at each iteration either $G$ is trying to improve its loss with respect to $D$ or $D$ is trying to improve its loss with respect to $G$ . In our proposed model, however, we replace $D$ with the PATE mechanism. This means we introduce $k$ teacher-discriminators, $T ^ { 1 } , . . . , T ^ { k }$ , and a student discriminator, $S$ . A noticeable difference is that the adversarial training is no longer symmetrical: the teachers are now being trained to improve their loss with respect to $G$ but $G$ is being trained to improve its loss with respect to the student $S$ which in turn is being trained to improve its loss with respect to the teachers.
+
+# 4.2.1 TEACHER-DISCRIMINATORS
+
+Formally, the teacher-discriminators (which we will refer to simply as teachers) are functions $T _ { 1 } ( \cdot ; \theta _ { T } ^ { 1 } ) , . . . , T _ { k } ( \cdot ; \theta _ { T } ^ { k } ) : \mathcal { U } \to [ 0 , 1 ]$ each parametrized by $\theta _ { T } ^ { i }$ . The teachers are given either a real sample from their corresponding partition of the dataset (i.e. $T _ { i }$ may receive a sample from $\mathcal { D } _ { i }$ ) as input or a sample from the generator. The teachers are then trained to classify them.
+
+Given $n$ i.i.d. samples of $\mathrm { U n i f } ( [ 0 , 1 ] ^ { d } )$ , $\mathbf { z } _ { 1 } , . . . , \mathbf { z } _ { n }$ , we define the empirical loss of teacher $i$ with weights $\theta _ { T } ^ { i }$ for fixed $G$ by
+
+$$
+\mathcal { L } _ { T } ^ { i } ( \theta _ { T } ^ { i } ) = - \Big [ \sum _ { \mathbf { u } \in \mathcal { D } _ { i } } \log T _ { i } ( \mathbf { u } ; \theta _ { T } ^ { i } ) + \sum _ { j = 1 } ^ { n } \log ( 1 - T _ { i } ( G ( \mathbf { z } _ { j } ) ; \theta _ { T } ^ { i } ) ) \Big ] .
+$$
+
+Each teacher is trained in the same way the discriminator is trained in a standard GAN framework, except that here the teacher only ever sees its partition of the real data.
+
+# 4.2.2 STUDENT-DISCRIMINATORS
+
+The main innovation of our paper comes from our implementation of the student-discriminator (which we will refer to simply as the student) in this setting. The differential privacy guarantee provided by the standard student model is only with respect to the original data, $\mathcal { D }$ , and not the public data, $\mathcal { P }$ , used to train the student. In our setting, where the entire focus is on generating synthetic data because no data is publicly available, we must propose a novel methodology to train the student without public data.
+
+We first note, that the student training paradigm described in [25] would involve training the student using data similar to that used to train the generator - i.e. by taking an equal number of samples from each and then labelling those using the standard $\mathrm { P A T E } _ { \lambda }$ mechanism (where here “labelling” refers to assigning them a real/fake label - not the label $y$ present in the data). We consider the implications of training the student on teacher-labelled generated samples only.
+
+We first observe that during training of the generator, the discriminator is only evaluated on samples from the generator itself, and not the real data, so by training the student only on generated samples we are in fact training it on the distribution we need it to perform well on. However, we note that if the student only sees unrealistic samples from the generator (i.e. generated samples that most teachers label as fake), then the student will not contain any information that the generator can use to improve its generated samples. It is therefore important that some of the generated samples the student is trained on are realistic. We then note that if $\operatorname { S u p p } ( \mathcal { P } _ { U } ) \subset \operatorname { S u p p } ( \mathcal { P } _ { G } )$ then some of the generated samples will be realistic.
+
+In order to ensure $\operatorname { S u p p } ( { \mathcal { P } } _ { U } ) \subset \operatorname { S u p p } ( { \mathcal { P } } _ { G } )$ , we normalize the data into $[ 0 , 1 ] ^ { d }$ and then initialize the parameters of the generator randomly using Xavier initialization. It follows that $\operatorname { S u p p } ( \mathcal { P } ) \subset$ $[ 0 , \hat { 1 ] } ^ { d } \subset G ( [ 0 , 1 ] ^ { d } ) = \mathbf { \bar { \Gamma } } ^ { } G ( \operatorname { S u p p } ( \mathbf { Z } ) ) = \operatorname { S u p p } ( G ( \mathbf { Z } ) )$ when $\mathbf { Z } \sim \operatorname { U n i f } ( [ 0 , 1 ] ^ { d } )$ .
+
+We create our training data for the student by taking $n$ i.i.d. samples of $\mathrm { U n i f } ( [ 0 , 1 ] ^ { d } )$ , $\mathbf { z } _ { 1 } , . . . , \mathbf { z } _ { n }$ , generating $n$ samples using the generator, $\hat { \mathbf { u } } _ { 1 } , . . . , \hat { \mathbf { u } } _ { n }$ with $\hat { \mathbf { u } } _ { j } = G ( \mathbf { z } _ { j } )$ , and using the teachers to label these using $\mathrm { P A T E } _ { \lambda }$ , setting $r _ { j } = \mathrm { P A T E } _ { \lambda } ( \hat { \mathbf { u } } _ { j } )$ . We train the student, $S ( \cdot ; \theta _ { S } ) \overset { \cdot } { : } \mathcal { U }  [ 0 , 1 ]$ , to maximize the standard cross-entropy loss on this teacher-labelled data, i.e.
+
+$$
+\mathcal { L } _ { S } ( \boldsymbol { \theta } _ { S } ) = \sum _ { j = 1 } ^ { n } r _ { j } \log S ( \hat { \mathbf { u } } _ { j } ; \boldsymbol { \theta } _ { S } ) + ( 1 - r _ { j } ) \log ( 1 - S ( \hat { \mathbf { u } } _ { j } ; \boldsymbol { \theta } _ { S } ) ) .
+$$
+
+Although a priori the above mechanism does not appear to depend on the number of teachers, it should be noted that for fixed $\lambda$ , more teachers results in the teacher-labelled dataset being less noisy - the noise being added is smaller relative to the counts $n _ { j }$ . This introduces a trade-off - for a small number of teachers, the noise may be too large and thus render the output meaningless; with a larger number of teachers, less data can be used to train each teacher, which may also render the output meaningless, even though the noise has a smaller effect. Finding the right balance in this problem is key. In our experiments, we use $d$ real and $d$ generated samples to train each teacher where $d$ is the dimension of the input space. Although the utility of a single teacher may be low, by aggregating (even noisily) the resulting classifier actually has high utility. Moreover, by using a minimal number of samples for each teacher, the effect of any individual sample on the output is small (because there are more teachers and each sample can effect at most 1 teacher) which means that our differential privacy guarantees are tighter - if we used fewer teachers, the mechanism still assumes that, in the worst case, the presence (or absence) of a single sample can completely flip a teacher’s vote and so we still need to add the same noise.
+
+We train $G , T ^ { 1 } , . . . , T ^ { k }$ and $S$ iteratively1, with each iteration of $G$ consisting of first performing $n _ { T }$ updates on all teachers, then performing $n _ { S }$ updates of the student. We perform generator iterations until our privacy constraint, $\epsilon .$ , has been reached. A block diagram of PATE-GAN can be found in the Appendix.
+
+To calculate the privacy of our algorithm we use the moments accountant method given in [25] to derive a data-dependent privacy guarantee at run-time. Details of its definition, and key results we use can be found in the Appendix. We denote the moments accountant of PATE-GAN by $\alpha ( l )$ . The moments accountant allows us to more tightly bound the total privacy cost of our mechanism than standard composition theorems would, and moreover attributes a lower privacy cost to accessing the noisy aggregation of the teachers when the teachers have a stronger consensus with the intuition being that when the teachers have a strong consensus, a single teacher (and therefore a single sample) has a much lower influence on the output than when the votes $\cdot n _ { 0 }$ and $n _ { 1 }$ ) are close. Pseudo-code for PATE-GAN can be found in Algorithm 1.
+
+We now state the main theorem of the paper, which follows from the theory in [25].
+
+Theorem 1. Algorithm 1, which takes as input $\delta > 0$ , a dataset, $\mathcal { D }$ , and outputs $G$ and $\epsilon$ is $( \epsilon , \delta )$ differentially private.
+
+The proof relies on applying the post-processing theorem where the discriminator corresponds to the mechanism $\mathcal { M }$ which takes outputs in $\mathcal { O }$ (in our case this corresponds to the weights of the discriminator), and the generator corresponds to the function $f$ which maps from $\mathcal { O }$ to $\mathcal { O }$ (which corresponds to the weights of the generator). For full details of the proof and further details of the theory required for it, see the Appendix.
+
+# 5 EXPERIMENTS
+
+In this section, we use a real-world Kaggle dataset (Credit card fraud detection dataset [11]) to evaluate PATE-GAN against the state-of-the-art benchmark (DPGAN [32]). In addition, we provide high-level (average) results for five additional datasets (with various characteristics): MAGGIC [27], UNOS-Heart wait-list [7], Kaggle cervical cancer dataset [16], UCI ISOLET dataset and UCI Epileptic Seizure Recognition dataset. A more detailed breakdown of the results for these datasets is given in the Appendix. Details of all six datasets can be also found in the Appendix.
+
+# 5.1 EXPERIMENTAL SETTINGS
+
+To empirically validate the quality of the generated dataset we introduce three different trainingtesting settings. Setting A: train the predictive models on the real training set, test the performance of the models on the real testing set. Setting $B$ : train on the synthetic training set, test on the real testing set ([15]), Setting $C .$ : train on the synthetic training set, test on the synthetic testing set. Note that the training set and the testing set are disjoint in both the real and synthetic datasets.
+
+We are interested in two comparisons. If we see a high predictive performance on the real data for models that were trained on synthetic data (Setting B), we can infer that the synthetic data has
+
+# Algorithm 1 Pseudo-code of PATE-GAN
+
+1: Input: δ, $\mathcal { D }$ , $n _ { T }$ , $n _ { S }$ , batch size $n$ , number of teachers $k$ , noise size $\lambda$   
+2: Initialize: $\theta _ { G }$ ${ \bf \Sigma } _ { G } , \theta _ { T } ^ { 1 } , . . . , \theta _ { T } ^ { k }$ , $\theta _ { S }$ , $\alpha ( l ) = 0$ for $l = 1 , . . . , L$   
+3: Partition dataset into $k$ subsets $\mathcal { D } _ { 1 } , . . . , \mathcal { D } _ { k }$ of size $\frac { | { \mathcal { D } } | } { k }$   
+4: while $\hat { \epsilon } < \epsilon$ do   
+5: 6: for $t _ { 2 } = 1 , . . . , n _ { T }$ Sample $\mathbf { z } _ { 1 } , . . . , \mathbf { z } _ { n } \overset { \mathrm { i . i . d . } } { \sim } \mathcal { P } _ { \mathcal { Z } }$ do   
+7: for $i = 1 , . . . , k$ do   
+8: Sample $\mathbf { u } _ { 1 } , . . . , \mathbf { u } _ { n } \stackrel { \mathrm { i . i . d . } } { \sim } { \mathcal { D } } _ { i }$   
+9: Update teacher, $T _ { i }$ , using SGD   
+10: $\begin{array} { r } { \nabla _ { \theta _ { T } ^ { i } } - \left[ \sum _ { j = 1 } ^ { d } \log ( T _ { i } ( \mathbf { u } _ { j } ) ) + \log ( 1 - T _ { i } ( G ( \mathbf { z } _ { j } ) ) ) \right] } \end{array}$   
+11: 12: for Sample $t _ { 3 } = 1 , . . . , n _ { S }$ $\mathbf { z } _ { 1 } , . . . , \mathbf { z } _ { n } \overset { ^ { 1 . 1 . 0 } } { \sim } \mathcal { P } _ { \mathcal { Z } }$ do   
+13: for $j = 1 , . . . , n$ do   
+14: $\hat { \mathbf { u } } _ { j }  G ( \mathbf { z } _ { j } )$   
+15: $r _ { j } \gets \mathrm { P A T E } _ { \lambda } ( \hat { \mathbf { u } } _ { i } )$ for $j = 1 , . . . , n$   
+16: Update moments accountant   
+17: q ← 4 exp(λ|n0−n1|) 2+λ|n0−n1|   
+18: for $l = 1 , . . . , L$ do   
+19: $\alpha ( l ) \gets \alpha ( l ) + \operatorname* { m i n } \{ 2 \lambda ^ { 2 } l ( l + 1 ) , \log ( \left( 1 - q \right) \left( \frac { 1 - q } { 1 - e ^ { 2 \lambda } q } \right) ^ { l } + q e ^ { 2 \lambda l } ) \}$   
+20: Update the student, $S$ , using SGD   
+21: $\begin{array} { r } { \dot { \nabla _ { \theta _ { S } } } - \sum _ { j = 1 } ^ { n } r _ { j } \log S ( \hat { \mathbf { u } } _ { j } ) \stackrel { \smile } { + } ( 1 - r _ { j } ) \log ( 1 - S ( \hat { \mathbf { u } } _ { j } ) ) } \end{array}$   
+22: Sample z1, ..., zn i.i.d. ∼ PZ   
+23: Update the generator, $G$ , using SGD   
+24: ∇θG - Pni=1 log(1 − S(G(zi))   
+25: ˆ ← min α(l)+log( 1δ ) l l
+
+captured the relationship between features and labels well. Moreover, synthetic data that does well in this setting can be used to train models without ever seeing the real data.
+
+On the other hand, when we consider synthetic data for use in competitions such as Kaggle, we need synthetic data that allows researchers to do meaningful comparisons on the synthetic data. In this setting, the researchers will only be able to use the synthetic data as both the training and testing set, and will need to develop their algorithms using results on the synthetic data. Now it becomes important that the relative performance of two algorithms when trained and tested on the synthetic data (Setting C), is similar to their relative performance when trained and tested on the real data (Setting A). A simple requirement would be that if model 1 is better than model 2 on the real data, then model 1 is better than model 2 on the synthetic data. This allows researchers to use the synthetic data to choose the best method(s) to try on the real data (or rather to give to the data-holder to try on the real data).
+
+For both comparisons, we use 12 different predictive models, shown in Table 1. We use two performance metrics to measure the capability of each model in predicting the label: (1) area under the receiver operating characteristics curve (AUROC), (2) area under the precision recall curve (AUPRC). Throughout the experiments we fix $\delta = 1 0 ^ { - 5 }$ for use as input to PATE-GAN and DPGAN. We also report the performance of the original GAN framework (”GAN”), which serves to indicate an upper bound on performance and allows us to see how much performance is lost due to the two differential privacy mechanisms (PATE-GAN and DPGAN). The details of hyper-parameter optimization and benchmark implementations can be found in the Appendix.
+
+<table><tr><td rowspan="2"></td><td colspan="3">三 AUROC</td><td colspan="3">川I AUPRC</td></tr><tr><td>II GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>Logistic Regression</td><td>0.8950</td><td>0.8728</td><td>0.8720</td><td>0.4069</td><td>0.3907</td><td>0.3923</td></tr><tr><td>Random Forests [5]</td><td>0.9075</td><td>0.8980</td><td>0.8730</td><td>0.3219</td><td>0.3157</td><td>0.2926</td></tr><tr><td>Gaussian Naive Bayes [29]</td><td>0.8861</td><td>0.8817</td><td>0.8522</td><td>0.1963</td><td>0.1858</td><td>0.1601</td></tr><tr><td>Bermoulli Naive Bayes [29]</td><td>0.8997</td><td>0.8968</td><td>0.8891</td><td>0.2169</td><td>0.2099</td><td>0.2069</td></tr><tr><td>Linear SVM[10]</td><td>0.7611</td><td>0.7523</td><td>0.7502</td><td>0.4473</td><td>0.4466</td><td>0.4464</td></tr><tr><td>Decision Tree [28]</td><td>0.9102</td><td>0.9011</td><td>0.8647</td><td>0.4071</td><td>0.3978</td><td>0.3672</td></tr><tr><td>LDA [3]</td><td>Ⅱ 0.8710</td><td>0.8510</td><td>0.8487</td><td>0.1956</td><td>0.1852</td><td>0.1788</td></tr><tr><td>AdaBoost [17]</td><td>0.9143</td><td>0.8952</td><td>0.8809</td><td>0.4530</td><td>0.4366</td><td>0.4234</td></tr><tr><td>Bagging [4]</td><td>0.8951</td><td>0.8877</td><td>0.8657</td><td>0.3303</td><td>0.3221</td><td>0.3073</td></tr><tr><td>GBM[18]</td><td>1 0.8848</td><td>0.8709</td><td>0.8499</td><td>0.3057</td><td>0.2974</td><td>0.2773</td></tr><tr><td>Multi-layer Perceptron</td><td>0.9086</td><td>0.8925</td><td>0.8787</td><td>0.4790</td><td>0.4693</td><td>0.4600</td></tr><tr><td>XgBoost[8]</td><td>1 0.9058</td><td>0.8904</td><td>0.8637</td><td>0.3837</td><td>0.3700</td><td>0.3440</td></tr><tr><td>Average</td><td>1 0.8866</td><td>0.8737</td><td>0.8578</td><td>0.3453</td><td>0.3351</td><td>0.3219</td></tr></table>
+
+Table 1: Performance comparison of 12 different predictive models in Setting B (trained on synthetic, tested on real) in terms of AUROC and AUPRC (the generators of PATE-GAN and DPGAN are $( 1 , 1 0 ^ { - 5 } )$ -differentially private).
+
+# 5.2 RESULTS WITH SETTING B
+
+In this subsection, we evaluate PATE-GAN and DPGAN in Setting B (trained on synthetic, tested on real) to understand whether or not the models are capturing the feature-label relationships well. Intuitively, if a synthetic dataset is such that a model trained on it performs well when performance is measured on real data, then the relationship between feature and label in the synthetic data is similar to that in the real data. In Table 1 we give the results for the Kaggle Credit dataset for all 12 predictive models. In Table 2, we give the performance on each dataset averaged across the 12 methods for each of the 6 datasets. A breakdown of the performance of each predictive model for each dataset can be found in the Appendix. Across all datasets, we see that PATE-GAN is capable of generating synthetic samples that better preserve the feature-label relationship (according to AUROC and AUPRC) than DPGAN.
+
+<table><tr><td rowspan="2">Datasets</td><td colspan="3">I AUROC</td><td colspan="3">I AUPRC</td></tr><tr><td>Ⅱ GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>Kaggle Credit</td><td>Ⅱ 0.8866</td><td>0.8737</td><td>0.8578</td><td>0.3453</td><td>0.3351</td><td>0.3219</td></tr><tr><td>MAGGIC</td><td>三 0.6574</td><td>0.6446</td><td>0.6286</td><td>0.3054</td><td>0.2952</td><td>0.2820</td></tr><tr><td>UNOS</td><td>0.6277</td><td>0.5996</td><td>0.5552</td><td>0.6554</td><td>0.6282</td><td>0.5862</td></tr><tr><td>Kaggle Cervical Cancer</td><td>0.9268</td><td>0.9108</td><td>0.8699</td><td>0.5994</td><td>0.5460</td><td>0.4851</td></tr><tr><td>UCI ISOLET</td><td>1 0.8171</td><td>0.6399</td><td>0.5577</td><td>0.5561</td><td>0.2953</td><td>0.2146</td></tr><tr><td>UCI Epileptic Seizure Recognition</td><td>0.9173</td><td>0.7681</td><td>0.6718</td><td>0.8133</td><td>0.6512</td><td>0.5369</td></tr></table>
+
+Table 2: Performance comparison of 12 different predictive models in Setting B (trained on synthetic, tested on real) in terms of AUROC and AUPRC (the generators of PATE-GAN and DPGAN are $( 1 , 1 \dot { 0 } ^ { - 5 } )$ -differentially private) over 6 different datasets. GAN is $( \infty , \infty )$ -differentially private and is given to indicate an upper bound of PATE-GAN and DPGAN.
+
+We note that the performance of all models, including the original GAN model (i.e. PATE-GAN - or equivalently DPGAN - with $( \infty , \infty )$ -differential privacy) in the high dimensional UCI ISOLET and UCI Epileptic Seizure Recognition datasets is lower than in lower dimensional datasets (when compared to the baseline AUROC and AUPRC found in the Appendix). We do, however, see that both PATE-GAN and DPGAN show more significant decreases in performance than the original GAN in these high-dimensional settings. In the case of PATE-GAN, we believe this may be due to the fact that the student discriminator is trained only using data from the generator, and therefore requires that some of the generated data look somewhat realistic from the start, which is a harder requirement to satisfy as the data has more dimensions. On the other hand, in DPGAN, noise must be added to each component of the gradient (of the discriminator) and so in higher dimensions the norm of the noise added is larger. Note that in PATE-GAN, noise is added only to the teacher outputs, whose dimension (typically 1) does not depend on the dimension of the input data, and so this phenomena does not present itself in PATE-GAN. The results on both the UCI datasets would suggest that the loss from increasing noise norm (for DPGAN) is greater than from difficulty in randomly generating realistic samples (for PATE-GAN).
+
+# 5.3 VARYING THE PRIVACY CONSTRAINT ()
+
+![](images/6e325ea9f3839f5631b73dba0204ee5467047c62f3a8c41be66f013cc319f54d.jpg)  
+Figure 1: Average AUROC performance across 12 different predictive models trained on the synthetic dataset generated by PATE-GAN and DPGAN with various $\epsilon$ (with $\dot { \delta } = 1 0 ^ { - 5 }$ ) (Setting B).
+
+In Fig. 1, we investigate the trade off between privacy constraint and utility. In the table we report the average performance of AUROC over the 12 different predictive models for PATE-GAN and the benchmark for various $\epsilon$ (with $\delta = 1 0 ^ { - 5 }$ ). As can be seen in Fig. 1, PATE-GAN is consistently better than DPGAN over the entire range of tested $\epsilon$ . We believe this is because the PATE mechanism allows us to more tightly bound the influence of a single sample on the discriminator, and hence we can provide tighter differential privacy guarantees - when the differential privacy guarantee is fixed, this results in higher quality synthetic data. Of course, as we increase $\epsilon$ (i.e. decrease the required privacy) both methods converge to the performance of GAN and the increase in performance of PATE-GAN over DPGAN becomes smaller.
+
+# 5.4 SETTING A VS SETTING C: PRESERVING THE RANKING OF PREDICTIVE MODELS
+
+As discussed at the beginning of this section, it is important that a synthetic dataset respects the ranking of models (in terms of their prediction performances) [21]. To evaluate this, we now introduce a new metric, which we refer to as the Synthetic Ranking Agreement (SRA). Suppose that we have $L$ predictive models, $f _ { 1 } , f _ { 2 } , . . . , f _ { L } { } ^ { 2 }$ . Furthermore, suppose that the performance of model $i$ when trained and tested on the real data (Setting A) is $A _ { i } \in \mathbb { R }$ and that the performance of model $i$ when trained and tested on the synthetic data (Setting C) is $C _ { i } \in \mathbb { R }$ . Then we define the Synthetic Ranking Agreement by
+
+$$
+\mathbf { S R A } ( \{ A _ { i } \} _ { i = 1 } ^ { L } , \{ C _ { i } \} _ { i = 1 } ^ { L } ) = { \frac { 1 } { L ( L - 1 ) } } \sum _ { j = 1 } ^ { L } \sum _ { k \neq j } \mathbb { I } \Bigl ( ( A _ { j } - A _ { k } ) \times ( C _ { j } - C _ { k } ) > 0 \Bigr )
+$$
+
+where $\mathbb { I }$ is an indicator function. Note that the summand is 1 when the ordering of algorithms $j$ and $k$ are the same in both settings, and is 0 when the ordering in one setting differs from the ordering in the other.
+
+$$
+\begin{array}{c} \frac { \mathrm { ~ \textcircled ~ { 1 } ~ P A T E - G A N ~ } \mid \mathrm { ~ \bf ~ D P G A N ~ } \mid } { \epsilon = 0 . 0 1 ~ \parallel } ~ \mathrm { ~ \bf ~ O . B O 0 0 ~ }  \\ { \frac { \epsilon = 0 . 0 1 ~ } { \epsilon = 0 . 0 5 ~ \parallel } ~ 0 . 0 9 9 ~ \mid ~ 0 . 5 2 7 3 ~ \parallel ~ \epsilon = 1 ~ \parallel ~ 0 . 8 3 6 4 ~ \mid ~ 0 . 8 0 0 0 } \\ { \frac { \epsilon = 0 . 0 5 ~ } { \epsilon = 0 . 1 ~ \parallel } ~ 0 . 0 7 4 5 5 ~ \mid ~ 0 . 6 9 0 9 ~ \parallel ~ \epsilon = 5 ~ \parallel ~ 0 . 8 9 0 9 ~ \mid ~ 0 . 8 3 6 4 } \\ { \frac { \epsilon = 0 . 1 ~ \parallel } { \epsilon = 0 . 1 ~ \parallel } ~ 0 . 0 8 1 ~ \mid ~ 0 . 7 4 5 5 ~ \parallel ~ \epsilon = 1 0 ~ \parallel ~ 0 . 9 0 9 1 ~ \mid ~ 0 . 8 9 0 9 } \\ { \frac { \epsilon = 0 . 5 ~ } { \epsilon = 0 . 5 ~ \parallel } ~ 0 . 0 . 8 0 0 0 ~ \mid ~ 0 . 7 8 1 8 ~ \parallel ~ \epsilon = 5 0 ~ \parallel ~ 0 . 9 0 9 1 ~ \mid ~ 0 . 9 0 9 1 ~ } \end{array}
+$$
+
+Table 3: Synthetic Ranking Probability of PATE-GAN and the benchmark when comparing Setting A and Setting C for various $\epsilon$ (with $\delta = 1 0 ^ { - 5 }$ ) in terms of AUROC. The Synthetic Ranking Agreement of Original GAN is 0.9091, which is also attained by both PATE-GAN and DPGAN for $\epsilon = 5 0$ .
+
+We compare the SRA of PATE-GAN and the benchmark for various $\epsilon$ (with $\delta = 1 0 ^ { - 5 } )$ 3 . As can be seen in Table 3, PATE-GAN achieves the best SRA across all values of $\epsilon$ .
+
+In the Appendix, we perform a similar experiment in which we compare the ranking of features by their importance (determined by their absolute Pearson correlation coefficient with the label) on the original dataset and on the synthetic dataset (generated by PATE-GAN and the benchmark) and report the results using a metric that is identical to SRA, with the model performances $( \{ A _ { i } \} , \{ C _ { i } \} )$ substituted for feature importances.
+
+# 5.5 QUANTITATIVE ANALYSIS ON THE NUMBER OF TEACHERS
+
+The number of teachers is a hyper-parameter of PATE-GAN and we choose the number of teachers among $\{ N / 1 0 , N / 5 0 , N / 1 0 0 , N / 5 0 0 , N / 1 0 0 0 , N / 5 0 0 0 , N / 1 0 0 0 0 \}$ where $N$ is the total number of samples. As we described in the previous section, there is a trade-off between number of teachers and the corresponding quality of the synthetic data. Table 4 quantitatively shows the trade-off between the number of teachers and the performance (in terms of both AUROC and AUPRC).
+
+<table><tr><td># of teachers</td><td>N/10</td><td>N/50</td><td>N/100</td><td>N/500</td><td>N/1000</td><td>N/5000</td><td>N/10000</td></tr><tr><td>AUROC AUPRC</td><td>0.5425 0.1273</td><td>0.6398 0.2484</td><td>0.7638 0.2900</td><td>0.8343 0.3184</td><td>0.8737 0.3351</td><td>0.8655 0.3278</td><td>0.8282 0.3092</td></tr></table>
+
+Table 4: Trade-off between the number of teachers and the performances (AUROC, AUPRC)
+
+# 6 DISCUSSION
+
+In this paper we introduced a novel methodology for generating differentially private synthetic data. Through several experiments we demonstrated the ability of our method to produce high quality synthetic data while being able to give strict differential privacy guarantees.
+
+In order to apply PATE to the GAN setting, we needed to use the original GAN framework. Extending PATE to the regression setting so that, for example, a Wasserstein GAN can be used instead, is an open and interesting question, and a potential direction for future research.
+
+# ACKNOWLEDGEMENT
+
+The authors would like to thank the reviewers for their helpful comments. The research presented in this paper was supported by the Office of Naval Research (ONR) and the NSF (Grant number: ECCS1462245, ECCS1533983, and ECCS1407712).
+
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+
+# APPENDIX
+
+THEORY REQUIRED FOR THEOREM 1
+
+Theorem 2. Algorithm 1, which takes as input $\delta > 0$ , a dataset, $\mathcal { D }$ , and outputs $G$ and $\epsilon$ is $( \epsilon , \delta )$ - differentially private.
+
+In order to prove our theorem, we define the moments accountant [1] and state the theorems that the data-dependent privacy guarantees of the PATE mechanism rely on. For proofs of the results below, see [25] and the references therein.
+
+Definition 3. (Privacy Loss) Let $\mathcal { M }$ be a randomized algorithm taking outputs in a space $\mathcal { O }$ and $\mathcal { D } , \mathcal { D } ^ { \prime }$ be neighbouring datasets. Let aux denote an auxiliary input. For an outcome $o \in \mathcal { O }$ , the privacy loss at o is defined as:
+
+$$
+c ( o ; \mathcal { M } , a u x , \mathcal { D } , \mathcal { D ^ { \prime } } ) = \log \frac { \mathbb { P } ( \mathcal { M } ( a u x , \mathcal { D } ) = o ) } { \mathbb { P } \left( \mathcal { M } ( a u x , \mathcal { D ^ { \prime } } ) = o \right) } .
+$$
+
+The privacy loss random variable $C ( \mathcal { M } , a u x , \mathcal { D } , \mathcal { D } ^ { \prime } )$ is defined as $c ( \mathcal { M } ( \mathcal { D } ) ; \mathcal { M } , a u x , \mathcal { D } , \mathcal { D } ^ { \prime } )$ , i.e.   
+the random variable defined by evaluating the privacy loss at an outcome sampled from $\mathcal { M } ( \mathcal { D } )$ .
+
+Definition 4. (Moments accountant) Let $\mathcal { M }$ be a randomized algorithm. The moments accountant is defined as:
+
+$$
+\alpha _ { \mathcal { M } } ( l ) = \operatorname* { m a x } _ { a u x , \mathcal { D } , \mathcal { D ^ { \prime } } } \alpha _ { \mathcal { M } } ( l ; a u x , \mathcal { D } , \mathcal { D ^ { \prime } } )
+$$
+
+where $\alpha _ { \mathcal { M } } ( l ; a u x , \mathcal { D } , \mathcal { D ^ { \prime } } ) = \log \mathbb { E } ( \exp ( l C ( \mathcal { M } , a u x , \mathcal { D } , \mathcal { D ^ { \prime } } ) ) )$ ) is the moment generating function of the privacy loss random variable and the max is taken over neighbouring datasets $\mathcal { D } , \mathcal { D } ^ { \prime }$ .
+
+Theorem 3. (Composability) Suppose that an algorithm $\mathcal { M }$ consists of a sequence of adaptive algorithms $\mathcal { M } _ { 1 } , . . . , \mathcal { M } _ { k }$ where $\mathcal { M } _ { i }$ outputs in $\mathcal { O } _ { i }$ and takes inputs from $\Pi _ { j = 1 } ^ { i - 1 } { \mathcal { O } } _ { j }$ as well as the dataset $\mathcal { D }$ . Then for any output sequence $o _ { 1 } , . . . , o _ { k - 1 }$ and any $l$
+
+$$
+\alpha _ { \mathcal { M } } ( l ; \mathcal { D } , \mathcal { D } ^ { \prime } ) = \sum _ { i = 1 } ^ { k } \alpha _ { \mathcal { M } _ { i } } ( l ; o _ { 1 } , . . . , o _ { i - 1 } , \mathcal { D } , \mathcal { D } ^ { \prime } )
+$$
+
+where $\alpha _ { \mathcal { M } }$ is conditioned on each $\mathcal { M } _ { i }$ ’s output being $o _ { i }$
+
+Theorem 4. (Tail bound) Let $\mathcal { M }$ be a randomized algorithm. For any $\epsilon > 0$ , $\mathcal { M }$ is $( \epsilon , \delta )$ differentially private for
+
+$$
+\delta = \operatorname* { m i n } _ { l } \exp ( \alpha _ { \mathcal { M } } ( l ) - l \epsilon ) .
+$$
+
+The following theorem combines Theorems 2, 3 and Lemma 4 from [25].
+
+Theorem 5. (Data-dependent privacy guarantee for PATE) Let $\mathcal { M }$ be the PATE mechanism defined in Section 3 of the paper. Let n0, n1 be as defined in Equation 3 of the paper. Let q = 2+λ|n0−n1|4 exp(λ|n −n |) . Then
+
+$$
+\alpha _ { \mathcal { M } } ( l ) \leq \operatorname* { m i n } \{ 2 \lambda ^ { 2 } l ( l + 1 ) , \log ( ( 1 - q ) \left( \frac { 1 - q } { 1 - e ^ { 2 \lambda } q } \right) ^ { l } + q e ^ { 2 \lambda l } ) \} .
+$$
+
+Proof of Theorem 2. We use Theorem 5 to bound the moments accountant for each query to the PATE mechanism during the training of our algorithm (i.e. each time a generated sample is labeled by the teachers). Theorem 3 then allows us to sum the individual bounds for each query to bound the moments accountant of the entire algorithm. Theorem 4 then allows us to derive a value for $\epsilon$ given $\delta$ . □
+
+# DATA DESCRIPTIONS
+
+# KAGGLE CREDIT CARD FRAUD DETECTION DATA DESCRIPTION
+
+The Kaggle credit card fraud detection dataset [11] contains transactions made by credit cards in September 2013 by European cardholders and the label is whether or not the transaction is fraudulent. The total number of features is 29 (binary: 0, continuous: 29) and the number of samples in this dataset is 284,807. Among the 284,807 samples, 492 $( 0 . 2 \% )$ samples are fraudulent transactions.
+
+# MAGGIC DATA DESCRIPTION
+
+The Meta-analysis Global Group in Chronic Heart Failure (MAGGIC) dataset [27] is a collection of 30 different datasets from 30 different medical studies containing patients who experienced heart failure. We set the label of each patient as 1-year all-cause mortality, excluding all patients who are censored before 1-year. The total number of features is 29 (binary: 20, continuous: 9) and the number of patients in this dataset is 30,389. Among the 30,389 patients, 5,723 $( 1 8 . 8 \% )$ patients died within 1 year.
+
+# UNOS DATA DESCRIPTION
+
+The United Network for Organ Transplantation (UNOS) dataset [7] provides information about all patients in the U.S. who have received a transplantation or were on the wait-list during the period 1985-2015. In this paper, we focus on the patients who were on the heart transplant wait-list. The objective is to predict 1-year all-cause mortality. The total number of features is 20 (binary: 18, continuous: 2) and the number of patients in this dataset is 23,706. Among the 23,706 patients, 12,606 $( 5 3 . 2 \% )$ patients died within 1 year.
+
+# KAGGLE CERVICAL CANCER DATA DESCRIPTION
+
+The Kaggle cervical cancer dataset [16] was collected at ’Hospital Universitario de Caracas’ in Caracas, Venezuela. It contains demographic information, habits, and historic medical records. The total number of features is 35 (binary: 24, continuous: 11) and the number of patients in this dataset is 858. Among the 858 patients, 55 $( 6 . 4 \% )$ patients have positive biopsy.
+
+# UCI ISOLET DATA DESCRIPTION
+
+The UCI ISOLET dataset https://archive.ics.uci.edu/ml/datasets/isolet was generated by speaking the name of each letter of the alphabet. The task is to classify each spoken letter as either a vowel or a consonant (binary classification). The total number of features is 617 and the number of samples in this dataset is 7797. Among the 7797 samples, 1500 $1 9 . 2 \% )$ samples are vowels.
+
+# UCI EPILEPTIC SEIZURE RECOGNITION DATA DESCRIPTION
+
+The UCI Epileptic Seizure Recognition dataset https://archive.ics.uci.edu/ml/ datasets/Epileptic+Seizure+Recognition was generated by recording brain activity. The task is to classify activity as seizure activity (binary classification). The total number of features is 179 and the number of samples in this dataset is 11500. Among the 11500 samples, 2300 $( 2 0 . 0 \% )$ samples correspond to seizure activity.
+
+# DATA SUMMARY AND SETTING A PERFORMANCE
+
+Table 5 summarises the 6 datasets we use and provides a baseline performance for a predictive model on each dataset - recall that Setting A refers to training and testing on the real data. The AUROC and AUPRC in this setting are upper bounds on the AUROC and AUPRC we could hope to achieve in Setting B.   
+
+<table><tr><td rowspan=1 colspan=2>Datasets                No of samples</td><td rowspan=1 colspan=3> No of features AUROC AUPRC</td></tr><tr><td rowspan=1 colspan=1>Kaggle Credit</td><td rowspan=1 colspan=1>284807</td><td rowspan=1 colspan=2>29         0.9438</td><td rowspan=1 colspan=1>0.7020</td></tr><tr><td rowspan=1 colspan=1>MAGGIC</td><td rowspan=1 colspan=1>30389</td><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>0.7069</td><td rowspan=1 colspan=1>0.3638</td></tr><tr><td rowspan=1 colspan=1>UNOS</td><td rowspan=1 colspan=1>23706</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>0.6416</td><td rowspan=1 colspan=1>0.6677</td></tr><tr><td rowspan=1 colspan=1> Kaggle Cervical cancer</td><td rowspan=1 colspan=1>858</td><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>0.9354</td><td rowspan=1 colspan=1>0.6314</td></tr><tr><td rowspan=1 colspan=1>UCI ISOLET</td><td rowspan=1 colspan=1>7797</td><td rowspan=1 colspan=1>617</td><td rowspan=1 colspan=1>0.9671</td><td rowspan=1 colspan=1>0.8758</td></tr><tr><td rowspan=1 colspan=1>UCI Epileptic Seizure Recognition</td><td rowspan=1 colspan=1>11500</td><td rowspan=1 colspan=1>179</td><td rowspan=1 colspan=1>0.9809</td><td rowspan=1 colspan=1>0.9511</td></tr></table>
+
+Table 5: No of samples, No of features, Average AUROC and AUPRC performance across 12 different predictive models trained and tested on the real data (Setting A) for the 6 datasets: Kaggle Credit, MAGGIC, UNOS, Kaggle Cervical Cancer, UCI ISOLET, UCI Epileptic Seizure Recognition.
+
+# HYPER-PARAMETER OPTIMIZATION
+
+In all experiments, the depth of the generator and discriminator (student-discriminator in our case) in both PATE-GAN and the DPGAN benchmark [32] is set to 3. The depth of the teacher discriminators is set to 1. The number of hidden nodes in each layer is $d , d / 2$ and $d$ (where $d$ is the feature dimension), respectively. We use relu as the activation functions of each layer except for the output layer where we use the sigmoid activation function and the batch size is 64 for both the generator and discriminator. We set $n _ { T } = n _ { S } = 5$ . Using cross validation, we select the number of teachers, $k$ , among $\mathrm { N } / 1 0 \mathrm { N } / 5 0 \mathrm { N } / 1 0 0 \mathrm { N } / 5 0 0 \mathrm { N } / 1 0 0 0 \mathrm { N } / 5 0 0 0 \mathrm { N } / 1 0 0 0 0$ . The learning rate is $1 0 ^ { - 4 }$ and we use Adam Optimizer to minimize the loss function.
+
+We use tensorflow to implement PATE-GAN and DPGAN. For DPGAN we use the code from the following link: https://github.com/illidanlab. We use the sklearn package in python to implement the 12 predictive models: Logistic Regression (LogisticRegression), Random Forests (RandomForestClassifier), Gaussian Naive Bayes (GaussianNB), Bernoulli Naive Bayes (BernoulliNB), Linear Support Vector Machine (svm), Decision Tree (DecisionTree), Linear Discriminant Analysis Classifier (LinearDiscriminantAnalysis), Adaptive Boosting (AdaBoost) (AdaBoostClassifier), Bootstrap Aggregating (Bagging) (BaggingClassifier), Gradient Boosting Machine (GBM) (GradientBoostingClassifier), Multi-layer Perceptron (MLPClassifier), and XgBoost (XGBoostRegressor).
+
+# VARYING THE PRIVACY CONSTRAINT () IN TERMS OF AUPRC
+
+![](images/acac5f4590f7e198ea53b875e95af5e77177f1387bc837fc24538ae593ca4a08.jpg)  
+Figure 2: Average AUPRC performance across 12 different predictive models trained on the synthetic dataset generated by PATE-GAN and DPGAN with various $\epsilon$ (with $\dot { \delta } = 1 0 ^ { - 5 }$ ) (Setting B).
+
+Similar to Fig. 1 in the main manuscript, Fig. 2 shows the trade off between the privacy constraint and utility, where utility is now measured in terms of AUPRC (rather than AUROC). We report the average performance in terms of AUPRC over the 12 different predictive models for PATE-GAN and the benchmark for various $\epsilon$ (with $\delta = 1 0 ^ { - 5 }$ ). As can be seen in Fig. 2, PATE-GAN consistently outperforms DPGAN over the entire range of tested $\epsilon$ in terms of AUPRC as well.
+
+# PRESERVING THE RANKING OF VARIABLE IMPORTANCE IN KAGGLE CREDIT DATASET
+
+We compare the ranking of variables by their importance (according to absolute Pearson correlation coefficient with the label) on the original dataset and on both synthetic datasets. We report the results using agreed ranking probability. As can be seen in Table 6, PATE-GAN achieves consistently better agreed ranking probability across all values of tested $\epsilon$ (with $\delta = 1 0 ^ { - 5 }$ ).
+
+$$
+\frac { \mathrm { ~ \# ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ \phi ~ } ~ } } { \epsilon = 0 . 0 1 ~ \| \mathrm { ~ { ~ \bf ~ \phi ~ } ~ { ~ \bf ~ 0 . 8 7 8 } ~ } } 
+$$
+
+Table 6: Agreed ranking probability of PATE-GAN and the benchmark to order the features by variable importance in terms of absolute Pearson correlation coefficient
+
+HIGH-DIMENSIONAL RESULTS: UCI ISOLET AND UCI EPILEPTIC SEIZURE RECOGNITION   
+
+<table><tr><td rowspan="2">三 (e,8）</td><td colspan="3">AUROC</td><td colspan="3">AUPRC</td></tr><tr><td>I GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>三 GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>(10,10-5) (1,10-5)</td><td>0.8171</td><td>0.7688 0.6399</td><td>0.7390 0.5577</td><td>0.5561</td><td>0.4734 0.2953</td><td>0.3831 0.2146</td></tr></table>
+
+Table 7: Average AUROC and AUPRC performance of 12 different predictive models trained on the synthetic datasets for $\epsilon = 1 , 1 0$ with $\delta = 1 0 ^ { - 5 }$ - Setting B using UCI ISOLET dataset. GAN is $( \infty , \infty )$ -differentially private and is given to indicate an upper bound of PATE-GAN and DPGAN.
+
+<table><tr><td rowspan="2">川 (,）</td><td colspan="3">AUROC 1</td><td colspan="3">AUPRC</td></tr><tr><td>I GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>(10,10-5) (1,10-5)</td><td>0.9173</td><td>0.8718 0.7681</td><td>0.8189 0.6718</td><td>0.8133</td><td>0.7662 0.6512</td><td>0.7201 0.5369</td></tr></table>
+
+Table 8: Average AUROC and AUPRC performance across 12 different predictive models trained on the synthetic datasets with various $\epsilon = 1 , 1 0$ with $\delta = 1 0 ^ { - 5 }$ - Setting B using UCI Epileptic Seizure Recognition dataset. GAN is $( \infty , \infty )$ -differentially private and is given to indicate an upper bound of PATE-GAN and DPGAN.
+
+# MAGGIC DATASET RESULT
+
+<table><tr><td rowspan="2">I</td><td colspan="3">AUROC</td><td colspan="3">I AUPRC</td></tr><tr><td>川 GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>Logistic Regression</td><td>0.6645 1</td><td>0.6413</td><td>0.6415</td><td>0.3113</td><td>0.2951</td><td>0.2967</td></tr><tr><td>Random Forests</td><td>0.6492</td><td>0.6397</td><td>0.6147</td><td>0.2953</td><td>0.2891</td><td>0.2660</td></tr><tr><td>Gaussian Naive Bayes</td><td>0.6770</td><td>0.6726</td><td>0.6431</td><td>0.3258</td><td>0.3153</td><td>0.2896</td></tr><tr><td>Bernoulli Naive Bayes</td><td>0.6647</td><td>0.6618</td><td>0.6541</td><td>0.3008</td><td>0.2938</td><td>0.2908</td></tr><tr><td>Linear SVM</td><td>0.6410</td><td>0.6301</td><td>0.6322</td><td>0.2911</td><td>0.2904</td><td>0.2902</td></tr><tr><td>Decision Tree</td><td> 0.6689</td><td>0.6598</td><td>0.6234</td><td>0.3163</td><td>0.3070</td><td>0.2764</td></tr><tr><td>LDA</td><td>I 0.6656</td><td>0.6433</td><td>0.6456</td><td>0.3118</td><td>0.2950</td><td>0.3014</td></tr><tr><td>AdaBoost</td><td>Ⅱ 0.6524</td><td>0.6333</td><td>0.6190</td><td>0.3054</td><td>0.2890</td><td>0.2758</td></tr><tr><td>Bagging</td><td> 0.6454</td><td>0.6380</td><td>0.6160</td><td>0.2912</td><td>0.2830</td><td>0.2682</td></tr><tr><td>GBM</td><td>0.6609</td><td>0.6470</td><td>0.6260</td><td>0.3106</td><td>0.3023</td><td>0.2822</td></tr><tr><td>Multi-layer Perceptron</td><td>0.6390</td><td>0.6229</td><td>0.6091</td><td>0.2921</td><td>0.2824</td><td>0.2731</td></tr><tr><td>XgBoost</td><td>0.6604</td><td>0.6450</td><td>0.6183</td><td>0.3133</td><td>0.2996</td><td>0.2736</td></tr><tr><td>Average</td><td>1 0.6574</td><td>0.6446</td><td>0.6286</td><td>0.3054</td><td>0.2952</td><td>0.2820</td></tr></table>
+
+Table 9: Performance comparison of 12 different predictive models in Setting B (trained on synthetic, tested on real) in terms of AUROC and AUPRC (the generators of PATE-GAN and DPGAN are $( 1 , 1 \dot { 0 } ^ { - 5 } )$ -differentially private). GAN is $( \infty , \infty )$ -differentially private and is given to indicate an upper bound of PATE-GAN and DPGAN.
+
+UNOS HEART WAIT DATASET RESULT   
+
+<table><tr><td rowspan="2">I</td><td colspan="3">AUROC 三</td><td colspan="3">AUPRC</td></tr><tr><td>Ⅱ GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>Logistic Regression</td><td>0.6407 1</td><td>0.6155</td><td>0.5548</td><td>0.6691</td><td>0.6450</td><td>0.5901</td></tr><tr><td>Random Forests</td><td>Ⅱ 0.6159</td><td>0.5950</td><td>0.5574</td><td>0.6436</td><td>0.6129</td><td>0.5768</td></tr><tr><td>Gaussian Naive Bayes</td><td>0.6323</td><td>0.6015</td><td>0.5343</td><td>0.6648</td><td>0.6371</td><td>0.5824</td></tr><tr><td>Bernoulli Naive Bayes</td><td>0.6213</td><td>0.6045</td><td>0.5763</td><td>0.6501</td><td>0.6363</td><td>0.6077</td></tr><tr><td>Linear SVM</td><td>Ⅱ 0.6244</td><td>0.5979</td><td>0.5581</td><td>0.6486</td><td>0.6254</td><td>0.5892</td></tr><tr><td>Decision Tree</td><td>I 0.6209</td><td>0.6019</td><td>0.5590</td><td>0.6496</td><td>0.6284</td><td>0.5819</td></tr><tr><td>LDA</td><td>Ⅱ 0.6403</td><td>0.6077</td><td>0.5530</td><td>0.6682</td><td>0.6406</td><td>0.5882</td></tr><tr><td>AdaBoost</td><td>Ⅱ 0.6222</td><td>0.5928</td><td>0.5527</td><td>0.6404 1</td><td>0.6194</td><td>0.5826</td></tr><tr><td>Bagging</td><td> 0.6084</td><td>0.5858</td><td>0.5493</td><td>0.6325</td><td>0.6074</td><td>0.5693</td></tr><tr><td>GBM</td><td>0.6374</td><td>0.6040</td><td>0.5585</td><td>0.6679</td><td>0.6352</td><td>0.5920</td></tr><tr><td>Multi-layer Perceptron</td><td>0.6328</td><td>0.5927</td><td>0.5562</td><td>0.6629</td><td>0.6240</td><td>0.5856</td></tr><tr><td>XgBoost</td><td> 0.6362</td><td>0.5956</td><td>0.5533</td><td>0.6676</td><td>0.6267</td><td>0.5880</td></tr><tr><td>Average</td><td> 0.6277</td><td>0.5996</td><td>0.5552</td><td>0.6554</td><td>0.6282</td><td>0.5862</td></tr></table>
+
+Table 10: Performance comparison of 12 different predictive models in Setting B (trained on synthetic, tested on real) in terms of AUROC and AUPRC (the generators of PATE-GAN and DPGAN are $( 1 , 1 0 ^ { - 5 } )$ -differentially private). GAN is $( \infty , \infty )$ -differentially private and is given to indicate an upper bound of PATE-GAN and DPGAN.
+
+KAGGLE CERVICAL CANCER DATASET RESULT   
+
+<table><tr><td rowspan="2">I</td><td colspan="3">AUROC 三</td><td colspan="3">AUPRC</td></tr><tr><td>Ⅱ GAN</td><td>PATE-GAN</td><td>DPGAN</td><td>GAN</td><td>PATE-GAN</td><td>DPGAN</td></tr><tr><td>Logistic Regression</td><td>0.9188 1</td><td>0.9102</td><td>0.8945</td><td>0.5949</td><td>0.5605</td><td>0.4672</td></tr><tr><td>Random Forests</td><td>Ⅱ 0.9515</td><td>0.9373</td><td>0.9237</td><td>0.6366</td><td>0.6361</td><td>0.5735</td></tr><tr><td>Gaussian Naive Bayes</td><td>0.9393</td><td>0.8890</td><td>0.7973</td><td>0.5605</td><td>0.4422</td><td>0.3702</td></tr><tr><td>Bernoulli Naive Bayes</td><td>0.8421</td><td>0.8331</td><td>0.8296</td><td>0.2491</td><td>0.2211</td><td>0.2160</td></tr><tr><td>Linear SVM</td><td>1 0.9282</td><td>0.9086</td><td>0.9050</td><td>0.6031</td><td>0.5921</td><td>0.5665</td></tr><tr><td>Decision Tree</td><td>Ⅱ 0.9451</td><td>0.9434</td><td>0.9283</td><td>0.6455</td><td>0.6094</td><td>0.5734</td></tr><tr><td>LDA</td><td>1 0.9358</td><td>0.9155</td><td>0.8667</td><td>0.6518</td><td>0.6061</td><td>0.5629</td></tr><tr><td>AdaBoost</td><td>Ⅱ 0.9361</td><td>0.8898</td><td>0.7989</td><td>0.6881 1</td><td>0.5587</td><td>0.4281</td></tr><tr><td>Bagging</td><td>I 0.9425</td><td>0.9275</td><td>0.9080</td><td>0.6257</td><td>0.5871</td><td>0.5809</td></tr><tr><td>GBM</td><td>1 0.9398</td><td>0.9333</td><td>0.9017</td><td>0.6927</td><td>0.6165</td><td>0.5422</td></tr><tr><td>Multi-layer Perceptron</td><td>0.9005</td><td>0.9064</td><td>0.7933</td><td>0.5675</td><td>0.5246</td><td>0.3746</td></tr><tr><td>XgBoost</td><td>I 0.9408</td><td>0.9351</td><td>0.8919</td><td>0.6784</td><td>0.5978</td><td>0.5657</td></tr><tr><td>Average</td><td>Ⅱ 0.9268</td><td>0.9108</td><td>0.8699</td><td>0.5994</td><td>0.5460</td><td>0.4851</td></tr></table>
+
+Table 11: Performance comparison of 12 different predictive models in Setting B (trained on synthetic, tested on real) in terms of AUROC and AUPRC (the generators of PATE-GAN and DPGAN are $( 1 , 1 0 ^ { - 5 } )$ -differentially private). GAN is $( \infty , \infty )$ -differentially private and is given to indicate an upper bound of PATE-GAN and DPGAN.
+
+# BLOCK DIAGRAMS
+
+# PATE-GAN
+
+The two figures below indicate the iterative training procedure carried out by PATE-GAN; the figures correspond to a single generator update.
+
+![](images/8870fb48cfcf4ae0f51b091d74b00f1fa86db1f69acf366d2dd788762bd87de4.jpg)  
+Figure 3: Block diagram of the training procedure for the teacher-discriminator during a single generator iteration. Teacher-discriminators are trained to minimize the classification loss when classifying samples as real samples or generated samples. During this step only the parameters of the teachers are updates (and not the generator).
+
+![](images/88df77a63a3fb0339f743d2aaf36fcdbed4d779ca7dc03736fc950c508cd3642.jpg)  
+Figure 4: Block diagram of the training procedure for the student-discriminator and the generator. The studentdiscriminator is trained using noisy teacher-labelled generated samples (the noise provides the DP guarantees). The student is trained to minimize classification loss on this noisily labelled dataset, while the generator is trained to maximize the student loss. Note that the teachers are not updated during this step, only the student and the generator.
+
+# DPGAN [32]
+
+![](images/dfc4424c04898a25bbcad9e7912df95f1c35b2c3f30f91192742a2c8e1d73467.jpg)  
+Figure 5: Block diagram of the DPGAN benchmark. It uses the standard WGAN framework. To guarantee differential privacy of the generator (with Post-processing Theorem), noise is added to the gradient of the discriminator during training to create a differentially private discriminator.

parse/train/x9jS8pX3dkx/x9jS8pX3dkx.md

@@ -0,0 +1,211 @@
+# Gradient Inversion with Generative Image Prior
+
+Jinwoo $\mathbf { J e o n ^ { 1 * } }$ , Jaechang $\mathbf { K i m ^ { 2 * } }$ , Kangwook $\mathbf { L e e ^ { 3 } }$ , Sewoong $\mathbf { O h ^ { 4 } }$ , Jungseul $\mathbf { O k } ^ { 1 , 2 }$   
+1 Department of Computer Science & Engineering, Pohang University of Science and Technology 2 Graduate School of Artificial Intelligence, Pohang University of Science and Technology   
+3 Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison 4 Paul G. Allen School of Computer Science & Engineering, University of Washington
+
+# Abstract
+
+Federated Learning (FL) is a distributed learning framework, in which the local data never leaves clients’ devices to preserve privacy, and the server trains models on the data via accessing only the gradients of those local data. Without further privacy mechanisms such as differential privacy, this leaves the system vulnerable against an attacker who inverts those gradients to reveal clients’ sensitive data. However, a gradient is often insufficient to reconstruct the user data without any prior knowledge. By exploiting a generative model pretrained on the data distribution, we demonstrate that data privacy can be easily breached. Further, when such prior knowledge is unavailable, we investigate the possibility of learning the prior from a sequence of gradients seen in the process of FL training. We experimentally show that the prior in a form of generative model is learnable from iterative interactions in FL. Our findings strongly suggest that additional mechanisms are necessary to prevent privacy leakage in FL.
+
+# 1 Introduction
+
+Federated learning (FL) is an emerging framework for distributed learning, where central server aggregates model updates, rather than user data, from end users [5, 17]. The main premise of federated learning is that this particular way of distributed learning can protect users’ data privacy as there is no explicit data shared by the end users with the central server.
+
+However, a recent line of work [34, 31, 9, 29] demonstrates that one may recover the private user data used for training by observing the gradients. This process of recovering the training data from gradients, so-called gradient inversion, poses a huge threat to the federated learning community, as it may imply the fundamental flaw of its main premise.
+
+Even more worryingly, recent works suggest that such gradient inversion attacks can be made even stronger if certain side-information is available. For instance, Geiping et al. [9] show that if the attacker knows a prior that user data consists of natural images, then the gradient inversion attack can leverage such prior, achieving a more accurate recovery of the user data. Another instance is when batch norm statistics are available at the attacker in addition to gradients. This can actually happen if the end users share their local batch norm statistics as in [17]. Yin et al. [29] show that such batch normalization statistics can significantly improve the strength of the gradient inversion attack, enabling precise recovery of high-resolution images.
+
+In this paper, we systematically study how one can maximally utilize and even obtain the prior information when inverting gradients. We first consider the case that the attacker has a generative model pretrained on the exact or approximate distribution of the user data as a prior. For this, we propose an efficient gradient inversion algorithm that utilizes the generative model prior. In particular, the algorithm consists of two steps, in which the first step searches the latent space (of lower dimension) defined by the generative model instead of the ambient input space (of higher dimension), and then the second step adapts the generative model to each input given the gradient. Each step provides substantial improvement in the reconstruction. We name the algorithm as gradient inversion in alternative spaces (GIAS). Figure 1 represents reconstruction results with the proposed method and existing one.
+
+![](images/b81bb14fb3a48de6867d8a38581f8cf29cb13a06887b9c8990eaed2b11ce75bc.jpg)  
+existing method [9]   
+Figure 1: An example showing the superiority of GIAS compared to existing method. Images of the authors are reconstructed from gradients by exploiting a generative model pretrained on human face images.
+
+We then consider a realistic scenario in which the user data distribution is not known in advance, and thus the attacker needs to learn it from gradients. For this scenario, we develop a meta-learning framework, called gradient inversion to meta-learn (GIML), which learns a generative model on user data from observing and inverting multiple gradients computed on the data, e.g. across different FL epochs or participating nodes. Our experimental results demonstrate that one can learn a generative model via GIML and reconstruct data by making use of the learned generative model.
+
+This implies a great threat on privacy leakage in FL since our methods can be applied for any data type in most FL scenarios unless a specialized architecture prevents the gradient leakage explicitly, e.g., [18].
+
+Our main contributions are as follows:
+
+• We introduce GIAS that fully utilizes a pretrained generative model to invert gradient. In addition, we propose GIML which can train generative model from gradients only in FL. • We demonstrate significant privacy leakage occurring by GIAS with a pretrained generative model in various FL scenarios which are challenging to other existing methods, e.g., [9, 29]. • We experimentally show that GIML can learn a generative model on the user data from only gradients, which provides the same level of data recovery with a given pretrained model. To our best knowledge, GIML is the first capable of learning explicit prior on a set of gradient inversion tasks. We note that a gradient inversion technique defines a standard on defence mechanism in FL for privacy [28]. By substantiating that our proposed methods are able to break down defense mechanisms that were safe according to the previous standard, we give a strong warning to the FL community to use a higher standard defined by our attack methods, and raise the necessity of a more conservative choice of defense mechanisms.
+
+# 2 Related work
+
+Privacy attacks in FL. Early works [19, 24] investigate membership inference from gradients to check the possibility of privacy leakage in FL. Phong et al. [21] demonstrate that it is possible to reconstruct detailed input image when FL trains a shallow network such as single-layer perceptron. Fan et al. [7] and Zhu and Blaschko [32] consider a wider class of learning model and propose an analytical approach solving a sequence of linear systems to reveal the output of each layer recursively. To study the limit of the gradient inversion in practical scenarios of training deep networks via FL, a sequence of effort has been made formulating optimization problem to minimize discrepancy comparing gradients from true data and reconstructed data [9, 27, 29, 31, 34].
+
+Gradient inversion with prior. The optimization-based approaches are particularly useful as one can easily utilize prior knowledge by adding regularization terms, e.g., total variation [27, 9] and BN statistics [29], or changing discrepancy measure [9] . In [29], a privacy attack technique using a generative model is introduced. They however require a pretrained model, while we propose a meta learning framework training generative model from gradients only. In addition, our method of inverting gradient maximally exploit a given generative model by alternating search spaces, which are analogous to the state-of-the-art GAN inversion techniques [3, 4, 33].
+
+Generative model revealing private data. Training a generative model with transmitted gradients also demonstrates privacy leakage in FL. Hitaj et al. [11] introduce an algorithm to train a GAN regarding shared model in FL framework as a discriminator. Wang et al. [27] use reconstructed data from gradient to train a GAN. Those works require some auxiliary dataset given in advance to enable the training of GAN, while we train a generative model using transmitted gradients only. Also, we not only train a generative model but also utilize it for reconstruction, while the generative models in [11, 27] are not used for the reconstruction. Hence, in our approach, the generative model and reconstruction can be improved interactively to each other as shown in Figure 6. In addition, [27] is less sample-efficient than ours in a sense that they use gradients to reconstruct images and then train a generative model with the reconstructed images, i.e., if the reconstruction fails, then the corresponding update of the generative model fails too, whereas we train the generative model directly from gradients.
+
+# 3 Problem formulation
+
+In this section, we formally describe the gradient inversion (GI) problem. Consider a standard supervised learning for classification, which optimizes neural network model $f _ { \theta }$ parameterized by $\theta$ as follows:
+
+$$
+\operatorname* { m i n } _ { \theta } \sum _ { ( x , y ) \in \mathcal { D } } \ell ( f _ { \theta } ( x ) , y ) \ ,
+$$
+
+where $\ell$ is a point-wise loss function and $\mathcal { D }$ is a dataset of input $x \in \mathbb { R } ^ { m }$ and label $y \in \{ 0 , 1 \} ^ { L }$ (one-hot vector). In federated learning framework, each node reports the gradient of $\ell ( f _ { \theta } ( x ) , y )$ for sampled data $( x , y )$ ’s instead of directly transferring the data. The problem of inverting gradient is to reconstruct the sampled data used to compute the reported gradient. Specifically, when a node computes the gradient $g$ using a batch $\{ ( x _ { 1 } ^ { * } , y _ { 1 } ^ { * } ) , . . . , ( x _ { B } ^ { * } , y _ { B } ^ { * } ) \}$ , i.e., $\begin{array} { r } { \boldsymbol { g } = \frac { \top } { B } \sum _ { j = 1 } ^ { B } \dot { \nabla } \ell ( f _ { \theta } ( x _ { j } ^ { * } ) , y _ { j } ^ { * } ) } \end{array}$ we consider the following problem of inverting gradient:
+
+$$
+\operatorname* { m i n } _ { ( x _ { 1 } , y _ { 1 } ) , \cdots , ( x _ { B } , y _ { B } ) } d \left( \frac { 1 } { B } \sum _ { j = 1 } ^ { B } \nabla \ell ( f _ { \theta } ( x _ { j } ) , y _ { j } ) , g \right) \ ,
+$$
+
+where $d ( \cdot , \cdot )$ is a measure of the discrepancy between two gradient, e.g., $\ell _ { 2 }$ -distance [34, 29] or negative cosine similarity [9]. It is known that label $y$ can be almost accurately recovered by simple methods just observing the gradient at the last layer [31, 29], while reconstructing input $x$ remains still challenging as it is often under-determined even when the true label is given. For simplicity, we hence focus on the following minimization to reveal the inputs from the gradient given the true labels:
+
+$$
+\operatorname* { m i n } _ { \substack { x _ { 1 } , \ldots , x _ { B } \in \mathbb { R } ^ { m } } } c \left( x _ { 1 } , . . . , x _ { B } ; \theta , g \right) ,
+$$
+
+where we denote by $c \left( x _ { 1 } , . . . , x _ { B } ; \theta , g \right)$ the cost function in (2) given $y _ { j } = y _ { j } ^ { * }$ for each $j = 1 , . . . , B$
+
+# 4 Methods
+
+The key challenge of inverting gradient is that solving (2) is often under-determined, i.e., a gradient contains only insufficient information to recover data. Such an issue is observed even when the dimension of gradient is much larger than that of input data. Indeed, Zhu and Blaschko [32] show that there exist a pair of different data having the same gradient, so called twin data, even when the learning model is large. To alleviate this issue, a set of prior knowledge on the nature of data can be considered.
+
+When inverting images, Geiping et al. [9] propose to add the total variation regularization $R _ { \mathrm { T V } } ( x )$ to the cost function in (3) since neighboring pixels of natural images are likely to have similar values. More formally,
+
+$$
+R _ { \mathrm { T V } } ( x ) : = \sum _ { ( i , j ) } \sum _ { ( i ^ { \prime } , j ^ { \prime } ) \in \partial ( i , j ) } \Vert x ( i , j ) - x ( i ^ { \prime } , j ^ { \prime } ) \Vert ^ { 2 } ,
+$$
+
+where $\partial ( i , j )$ is the set of neighbors of $( i , j )$ . This method is limited to the natural image data.
+
+For general type of data, one can consider exploiting the batch normalization (BN) statistics from nodes. This is available in the case that the server wants to utilize batch normalization (BN) in $\mathrm { F L }$ , and thus collects the BN statistics (mean and variance) of batch from each node, in addition, with every gradient report [17]. To be specific, Yin et al. [29] propose to employ the regularizer $R _ { \mathrm { B N } } ( x _ { 1 } , . . . , x _ { B } ; \theta )$ which quantifies the discrepancy between the BN statistics of estimated $x _ { j }$ ’s and those of true $\boldsymbol { x } _ { j } ^ { * }$ ’s on each layer of the learning model. More formally,
+
+$$
+R _ { \mathrm { B N } } ( x _ { 1 } , . . . , x _ { B } ; \theta ) : = \sum _ { l } \| \mu _ { l } - \mu _ { l , \mathrm { e x a c t } } \| _ { 2 } + \| \sigma _ { l } ^ { 2 } - \sigma _ { l , \mathrm { e x a c t } } ^ { 2 } \| _ { 2 } ,
+$$
+
+where $\mu _ { l } ( x _ { 1 } , . . . , x _ { B } ; \theta )$ and $\sigma _ { l } ^ { 2 } ( x _ { 1 } , . . . , x _ { B } ; \theta )$ (resp. $\mu _ { l , \mathrm { { e x a c t } } } ( x _ { 1 } ^ { * } , . . . , x _ { B } ^ { * } ; \theta )$ and $\sigma _ { l , \mathrm { { e x a c t } } } ^ { 2 } ( x _ { 1 } ^ { * } , . . . , x _ { B } ^ { * } ; \theta ) )$ are the mean and variance of $l$ -th layer feature maps for the estimated batch $x _ { 1 } , . . . , x _ { B }$ (resp. the true batch $x _ { 1 } ^ { * } , . . . , x _ { B } ^ { * } )$ given $\theta$ . This is available only if clients agree to report their exact BN statistics at every round. But not every FL framework report BN statistics [15, 2]. In that case, Yin et al. [29] also propose to use the BN statistics over the entire data distribution as a proxy of the true BN statistics, and reports that the gain from the approximated BN statistics is comparable to that from the exact ones. The applicability of $R _ { \mathrm { B N } }$ with the approximated BN statistics is still limited as the proxy needs to be additionally recomputed over the entire data distribution at every change of $\theta$ . However, this demonstrates the significant impact of knowing the data distribution in the gradient inversion and motivates our methods using and learning a generative model on the user data, described in what follows.
+
+# 4.1 Gradient inversion with trained generative model
+
+Consider a decent generative model $G _ { w } : \mathbb { R } ^ { k } \mapsto \mathbb { R } ^ { m }$ trained on the approximate (possibly exact) distribution of user data $\mathcal { D }$ such that $x ^ { * } \approx G _ { w } ( z ^ { * } )$ for $( x ^ { \ast } , \cdot ) \in \bar { \mathcal { D } }$ and its latent code $z ^ { * } =$ $\mathrm { a r g m i n } _ { z } \| G _ { w } ( z ) - x ^ { * } \|$ . To fully utilize such a pretrained generative model, we propose gradient inversion in alternative spaces (GIAS), of which pseudocode is presented in Appendix A, which performs latent space search over $z$ and then parameter space search over $w$ . We also illustrate the overall procedure of GIAS in Figure 2.
+
+Latent space search. Note that the latent space is typically much smaller than the ambient input space, i.e., $k \ll m$ , for instances, DCGAN [25] of $k = 1 0 0$ and StyleGAN [12] of $k = 5 1 2 \times 1 6$ for image data of $m = ( \mathrm { w i d t h } ) \times ( \mathrm { h e i g h t } ) \times ( \mathrm { c o l o r } )$ such as $3 2 \times 3 2 \times 3$ , $2 5 6 \times 2 5 6 \times 3$ , or larger. Using such a pretrained generative model with $k \ll m$ , the under-determined issues of (3) can be directly mitigated by narrowing down the searching space from $\mathbb { R } ^ { m }$ to $\{ G _ { w } ( z ) : z \in \mathbb { R } ^ { k } \}$ . Hence, GIAS first performs the latent space search in the followings:
+
+$$
+\operatorname* { m i n } _ { \substack { z _ { 1 } , \ldots , z _ { B } \in \mathbb { R } ^ { k } } } c \left( G _ { w } ( z _ { 1 } ) , . . . , G _ { w } ( z _ { B } ) \right) .
+$$
+
+Considering a canonical class of neural network model $f _ { \theta }$ , we can show that the reconstruction of $x ^ { * }$ by latent space search in (5) aligns with that by input space search in (3) if the generative model $G _ { w }$ approximates input data with small enough error.
+
+![](images/86214e463c23aef68db3947ced6b1f76f55c86ba1e2dd66943856f0c1dfa79be.jpg)  
+Figure 2: An overview of GIAS. GIAS optimizes a latent code $z$ and generative model parameters $w$ to reconstruct the data which matches the gradient.
+
+Property 1. For an input data $x ^ { * } \in [ 0 , 1 ] ^ { m }$ consider the gradient inversion problem of minimizing cost c in (3), where a canonical form of deep learning for classification is considered and the discrepancy measure $d$ is $\ell _ { 2 }$ -distance. Suppose that it has the unique global minimizer at $x ^ { * }$ . Let $\varepsilon \geq 0$ be the approximation error bound on $x ^ { * }$ for generative model $\mathring { G } _ { w } : [ 0 , 1 ] ^ { k } \mapsto [ 0 , 1 ] ^ { m }$ Then, there exists $\delta ( \varepsilon ) \geq 0$ such that for any $z ^ { * } \in \arg \operatorname* { m i n } _ { z } c ( G _ { w } ( z ) ) ,$ ,
+
+$$
+\| G _ { w } ( z ^ { * } ) - x ^ { * } \| \leq \delta ( \varepsilon ) ,
+$$
+
+of which upper bound $\delta ( \varepsilon )  0$ as $\varepsilon \to 0$ .
+
+A rigorous statement of Property 1 and its proof are provided in Appendix B, where we prove and use that the cost function is continuous around $x ^ { * }$ under the assumptions. This property justifies solving the latent space search in (5) for $\mathrm { F L }$ scenarios training neural network model while it requires an accurate generative model.
+
+Parameter space search. Using the latent space search only, there can be inevitable reconstruction error due to the imperfection of generative model. This is mainly because we cannot perfectly prepare the generative model for every plausible data in advance. Similar difficulty of the latent space search has been reported even when inverting GAN [33, 3, 4] for plausible but new data directly, i.e., $\mathrm { m i n } _ { z } \parallel G _ { w } ( z ) - x ^ { * } \parallel$ given $x ^ { * }$ , rather than inverting gradient. Bau et al. [3] propose an instance-specific model adaptation, which slightly adjusts the model parameter $w$ to (a part of source image) $x ^ { * }$ after obtaining a latent code $z ^ { * }$ for $x ^ { * }$ . Inspired by such an instance-specific adaptation, GIAS performs the following parameter space search over $w$ preceded by the latent space search over $z$ :
+
+$$
+\operatorname * { m i n } _ { w _ { 1 } , . . . , w _ { B } } \ c \left( G _ { w _ { 1 } } ( z _ { 1 } ) , . . . , G _ { w _ { B } } ( z _ { B } ) \right) \ ,
+$$
+
+where $z _ { 1 } , \dots , z _ { B }$ are obtained from (5).
+
+Remark. We propose the optimization over $w$ followed by that over $z$ sequentially This is to maximally utilize the benefit of mitigating the under-determined issue from reducing the searching space on the pretrained model. However, the benefit would be degenerated if $z$ and $w$ are optimized jointly or $w$ is optimized first. We provide an empirical justification on the proposed searching strategy in Section 5.1.
+
+We perform each search in GIAS using a standard gradient method to the cost function directly. It is worth noting that those optimizations (5) and (7) with generative model can be tackled in a recursive manner as R-GAP [32] reconstructs each layer from output to input. We provide details and performance of the recursive procedure in Appendix C, where employing generative model improves the inversion accuracy of R-GAP substantially, while R-GAP apparently suffers from an error accumulation issue when $f _ { \theta }$ is a deep neural network.
+
+![](images/3e98af5f62fd12872d10b40932ee68ec2b13001baf19a78e4df336bdfc8cd337.jpg)  
+Figure 3: Comparison of different searching spaces. (a) Each row shows reconstructed images of different optimization domains. The first three rows share the same latent space search of 1, 500 iterations, and ${ \mathrm { G I } } { - z } / w$ is verified to be the best option to fully exploits the knowledge inside the generative model. (b) Cost function over iterations of different optimization domains.
+
+# 4.2 Gradient inversion to meta-learn generative model
+
+For the case that pretrained generative model is unavailable, we devise an algorithm to train a generative model $G _ { w }$ for a set $\mathcal { S } = \{ ( \theta _ { i } , g _ { i } ) \}$ of gradient inversion tasks. Since each inversion task can be considered as a small learning task to adapt generative model per data, we hence call it gradient inversions to meta-learn (GIML). The detailed procedure of GIML is presented in Appendix A. We start with an arbitrary initialization of $w$ , and iteratively update toward $w ^ { \prime }$ from a variant of GIAS for $N$ tasks sub-sampled from $s$ , which is different than multiple applications of GIAS for each task in two folds: (i) $\ell _ { 2 }$ -regularization in latent space search; and (ii) an integrated optimization on model parameter. The variant first finds optimal latent codes $\boldsymbol { z } _ { i } ^ { * } = ( z _ { i 1 } ^ { * } , . . . , z _ { i B } ^ { * } )$ for each task $i$ with respect to the same cost function of GIAS but additional $\ell _ { 2 }$ -regularization. Note that the latent space search with untrained generative model easily diverges. The $\ell _ { 2 }$ -regularization is added to prevent the divergence of $z _ { i } ^ { * }$ . Once we obtained $z _ { i } ^ { * }$ ’s, $w ^ { \prime }$ is computed by few steps of gradient descents for an integrated parameter search to minimize $\begin{array} { r } { \sum _ { i } c ( G _ { w ^ { \prime } } ( z _ { i 1 } ^ { * } ) , . . . , \bar { G } _ { w ^ { \prime } } ( z _ { i B } ^ { * } ) ; \theta _ { i } , g _ { i } ) } \end{array}$ . This is because in GIML, we want meta information $w$ to help GIAS for each task rather than solving individual tasks, while after performing GIML to train $w$ , we perform GIAS to invert gradient with the trained $w$ . This is analogous to the Reptile in [20].
+
+# 5 Experiments
+
+Setup. Unless stated otherwise, we consider the image classification task on the validation set of ImageNet [22] dataset scaled down to $6 4 \times 6 4$ pixels (for computational tractability) and use a randomly initialized ResNet18 [10] for training. For deep generative models in GIAS, we use StyleGAN2 [13] trained on ImageNet. We use a batch size of $B \ = \ 4$ as default and use the negative cosine to measure the gradient dissimilarity $d ( \cdot , \cdot )$ . We present detailed setup in Appendix H. Our experiment code is available at https://github.com/ml-postech/ gradient-inversion-generative-image-prior.
+
+Algorithms. We evaluate several algorithms for the gradient inversion (GI) task in (3). They differ mainly in which spaces each algorithm searches over: the input $x$ , the latent code $z$ , and/or the model parameter $w$ . Each algorithm is denoted by GI-(·), where the suffix indicates the search space(s). For instances, GI- $z / w$ is identical to the proposed method, GIAS, and GI- $x$ is the one proposed by Geiping et al. [9].
+
+![](images/05927611c6cf240c845850b9daf59ed7a0462bf6d541e6a943eb9a3ea192a47a.jpg)  
+Figure 4: Comparison of state-of-the-art models and ours. Replacing GI- $x$ with GI- $z / w$ (GIAS) regardless of using BN [29] or not [9] provides substantial improvement in the reconstruction accuracy. (a) Average PSNR and best PSNR in a batch throughout the experiments. (b) An ablation study and comparison of reconstruction results with our models and state-of-the-art models. We highlight the proposed models in bold.
+
+Table 1: Comparison of our methods with state-of-the-art methods. Adding our method makes performance improvement versus two baseline methods. PSNR, SSIM, and LPIPS[30] are used to evaluate reconstruction results. We highlight the best performances in bold.   
+
+<table><tr><td>Method</td><td>GI-x [9]</td><td>GI-z (ours)</td><td>GI-w (ours)</td><td>GI-z/w (GIAS,ours)</td><td>GI-x+BN[29]</td><td>GI-z/w+BN (ours)</td></tr><tr><td>PSNR↑</td><td>13.78</td><td>14.27</td><td>14.70</td><td>15.58</td><td>15.52</td><td>16.31</td></tr><tr><td>SSIM↑</td><td>0.2542</td><td>0.3106</td><td>0.3519</td><td>0.3895</td><td>0.3513</td><td>0.4311</td></tr><tr><td>LPIPS↓</td><td>0.4376</td><td>0.3233</td><td>0.5121</td><td>0.3023</td><td>0.3645</td><td>0.2861</td></tr></table>
+
+# 5.1 Justification of GIAS design
+
+We first provide an empirical justification of the specific order of searching spaces in GIAS (corresponding to $\mathbf { G I } { - } z / w ,$ ) to fully utilize a pretrained generative model. To do so, we provide Figure 4b comparing algorithms with different searching spaces: GI- $z / w$ , GI- $z / x$ , $\mathrm { G I } { - } z$ , and $_ { \mathrm { G I - } x }$ , of which the first three share the same latent space search over $z$ for the first 1, 500 iterations. As shown in Figure 3(a), the latent space search over $z$ quickly finds plausible image in a much shorter number of iterations than GI- $x$ , while it does not improve after a certain point due to the imperfection of pretrained generative model. Such a limitation of $\mathrm { G I } { - } z$ is also captured in Figure 3(b), where the cost function of GI- $z$ is not decreasing after a certain number of optimization steps. To further minimize the cost function, one alternative to GI- $z / w$ (GIAS) is ${ \mathrm { G I } } { - } z / x$ , which can further reduce the loss function whereas the parameter search in GI- $z / w$ seems to provide more natural reconstruction of the image than ${ \mathrm { G I } } { - } z / { \bar { x } }$ . The superiority of ${ \mathrm { G I } } { - z } / w$ over ${ \mathrm { G I } } { - } z / x$ may come from that the parameter space search exploits an implicit bias from optimizing a good architecture for expressing images, c.f., deep image prior [26]. In Appendix E and Figure 1, we also present the same comparison on FFHQ (human-face images) [12] where diversity is much smaller than that of ImageNet. On such a less diverse dataset, the distribution can be easily learned, and the gain from training a generative model is larger.
+
+# 5.2 The gain from fully exploiting pretrained generative model
+
+Comparison with state-of-the-art models. Our method can be easily added to previous methods [9, 29]. In Table 1 and Figure 4, we compare the state-of-the-art methods both with and without the proposed generative modelling. In Table 1, comparing GI- $x$ to ${ \mathrm { G I } } { - } z / w$ and GI- $x + { \tt B N }$ to ${ \mathrm { G I } } { - } z / w +$ BN, adding the proposed generative modelling provides additional gain in terms of all the measures (PSNR, SSIM, LPIPS) of reconstruction quality. GI- $z / w$ without BN has lower reconstruction error than GI- $x + { \tt B N }$ , which is the method of [29]. This implies that the gain from the generative model is comparable to that from BN statistics. However, while the generative model only requires a global (and hence coarse) knowledge on the entire dataset, BN statistics are local to the batch in hand and hence requires significantly more detailed information on the exact batch used to compute gradient.
+
+![](images/eaebaa7d2c36a894aebb95c062ddacad082b8c4df28b7ce3593656951181c37f.jpg)  
+Figure 5: Comparison of state-of-the-art models and $G I - z / w$ with varying difficulties. Larger batch size, higher sparsity, and larger gradient noise increases reconstruction difficulty. GI- $z / w$ always surpasses GI- $x$ thanks to the pretrained generative model. All subfigures share the y-axis.
+
+As shown in Figure 4, the superiority of our method compared to the others is clear in terms of the best-in-batch performance than the average one, where the former is more suitable to show actual privacy threat in the worst case than the latter. It is also interesting to note that GI- $w$ with untrained $w$ provides substantial gain compared to GI- $x$ . This may imply that there is a gain of the implicit bias, c.f., [26], from training the architecture of deep generative model.
+
+Evaluation against possible defense methods We evaluate the gain of using a generative model for various FL scenarios with varying levels of difficulty in the inversion. As batch size, gradient sparsity1 [28] and gradient noise level increase, the risk of having under-determined inversion increases and the inversion task becomes more challenging. Figure 5 shows that for all the levels of difficulty, the generative model provides significant gain in reconstruction quality. In particular, the averaged PSNR of GI- $x$ with a batch size of 4 is comparable to that of ${ \mathrm { G I } } { - z } / w$ with a batch size 32. It is also comparable to that of ${ \mathrm { G I } } { - z } / w$ with a gradient sparsity of $9 9 \%$ . To measure the impact of the noisy gradient, we experimented gradient inversion with varying gaussian noise level in aforementioned settings. Figure 5(c) shows that adding enough noise to the gradient can mitigate the privacy leakage. GI- $z / w$ with a noise level of 0.01, which is relatively large, still surpasses GI- $_ x$ without noise. A large noise of 0.1 can diminish the gain of exploiting a pretrained generative model. However, the fact that adding large noise to the gradient slows down training makes it difficult for FL practitioners to choose suitable hyperparameters. The results imply our method is more robust to defense methods against gradient inversion, but can be blocked by a high threshold. Note that our results of gradient sparsity and gradient noise implies the Differential Privacy(DP) is still a valid defense method, when applied with a more conservative threshold. For more discussion about possible defense methods in FL framework, see Appendix F.
+
+# 5.3 Learning generative model from gradients
+
+We demonstrate the possibility of training a generative model only with gradients. For computational tractability, we use DCGAN and images from FFHQ [12] resized to $3 2 \mathbf { x } 3 2$ . We generate a set of gradients from 4 rounds of gradient reports from 200 nodes, in which node computes gradient for a classification task based on the annotation provided in [6]. From the set of gradients, we perform GIML to train a DCGAN to potentially generate FFHQ data.
+
+Figure 6 shows the evolution of generative model improves the reconstruction quality when performing either $\mathrm { G I } { - } z$ and ${ \mathrm { G I } } { - z } / w$ . We can clearly see the necessity of parameter space search. Figure 6(a) shows that the quality of images from the generative model is evolving in the training process of GIML. As the step $t$ of GIML increases, the generative model $G _ { w ^ { ( t ) } } ( z )$ for arbitrary $z$ outputs more plausible image of human face. When using generative model trained on wrong dataset (CIFAR10), GI- $z$ completely fails at recovering data.
+
+![](images/11a9bf0deccf6f26645320b1714e23bf518f3cf71177c576ecd107d8a4e5a1fa.jpg)  
+Figure 6: Qualitative and quantitative result of GIML. (a) Results validating generative model trained with GIML. Images on the first row are sampled from different GIML training steps. The same latent code $z$ was used to sample images in same rows. Images on the second row and third row are results of $\mathrm { G I } { - } z$ and GI- $z / w$ using generative model trained with GIML and pretrained model which is trained with CIFAR10 images. Experiments were done with gradient sparsity 0.95 for comparison in difficult setting. Last column represents the ground truth image and result of GI- $\boldsymbol { w }$ with untrained model. (b) A comparison of GIAS with meta-learned generative model and GIAS using improper generative model. Proper generative model boosts GIAS performance.
+
+In Figure $6 ( \mathbf { b } )$ , as GIML iteration step increases, the performance of $\mathrm { G I } { - } z$ and ${ \mathrm { G I } } { - z } / w$ with GIML surpass GI- $z$ and ${ \mathrm { G I } } { - z } / w$ with wrong prior knowledge. GI- $z / w$ using generative model trained on wrong dataset and GI- $\boldsymbol { \cdot } \boldsymbol { w }$ which starts with an untrained generative model show lower averaged PSNR compared to ${ \mathrm { G I } } { - z } / w$ with GIML. GI- $z / w$ with GIML to train generative model on right data shows the best performance in terms of not only quality (Figure 6) but also convergence speed. We provide a comparison of the convergence speed in Appendix G.
+
+# 6 Conclusion
+
+We propose GIAS fully exploit the prior information on user data from a pretrained generative model when inverting gradient. We demonstrate significant privacy leakage using GIAS with pretrained generative model in various challenging scenarios, where our method provides substantial gain additionally to any other existing methods [9, 29]. In addition, we propose GIML which can train a generative model using only the gradients seen in the FL classifier training. We experimentally show that GIML can meta-learn a generative model on the user data from only gradients, which improves the quality of each individual recovered image. To our best knowledge, GIML is the first capable of learning explicit prior on a set of gradient inversion tasks.
+
+# Acknowledgments
+
+This work was partly supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2019-0-01906, Artificial Intelligence Graduate School Program (POSTECH)) and (No. 2021-0-00739, Development of Distributed/Cooperative AI based $^ { 5 \mathrm { G } + }$ Network Data Analytics Functions and Control Technology). Jinwoo Jeon and Jaechang Kim were supported by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by Korea(MSIT) (2020-0-01594, PSAI industry-academic joint research and education program). Kangwook Lee was supported by NSF/Intel Partnership on Machine Learning for Wireless Networking Program under Grant No. CNS-2003129 and NSF Award DMS-2023239. Sewoong Oh acknowledges funding from NSF IIS-1929955, NSF CCF 2019844, and Google faculty research award.
+
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