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Disentangling 3D Prototypical Networks for Few-Shot Concept Learning	1 INTRODUCTION . Humans can learn new concepts from just one or a few samples . Consider the example in Figure 1 . Assuming there is a person who has no prior knowledge about blue and carrot , by showing this person an image of a blue carrot and telling him “ this is an carrot with blue color ” , the person can easily generalize from this example to ( 1 ) recognizing carrots of varying colors , 3D poses and viewing conditions and under novel background scenes , ( 2 ) recognizing the color blue on different objects , ( 3 ) combine these two concepts with other concepts to form a novel object coloring he/she has never seen before , e.g. , red carrot or blue tomato and ( 4 ) using the newly learned concepts to answer questions regarding the visual scene . Motivated by this , we explore computational models that can achieve these four types of generalization for visual concept learning . We propose disentangling 3D prototypical networks ( D3DP-Nets ) , a model that learns to disentangle RGB-D images into objects , their 3D locations , sizes , 3D shapes and styles , and the background scene , as shown in Figure 2 . Our model can learn to detect objects from a few 3D object bounding box annotations and can further disentangle objects into different attributes through a self-supervised view prediction task . Specifically , D3DP-Nets uses differentiable unprojection and rendering operations to go back and forth between the input RGB-D ( 2.5D ) image and a 3D scene feature map . From the scene feature map , our model learns to detect objects and disentangles each object into a 3D shape code and an 1D style code through a shape/style disentangling antoencoder . We use adaptive instance normalization layers ( Huang & Belongie , 2017 ) to encourage shape/style disentanglement within each object . Our key intuition is to represent objects and their shapes in terms of 3D feature representations disentangled from style variability so that the model can correspond objects with similar shape by explicitly rotating and scaling their 3D shape representations during matching . Project page : https : //mihirp1998.github.io/project_pages/d3dp/ ∗Equal contribution †Work done while at Carnegie Mellon University With the disentangled representations , D3DP-Nets can recognize new concepts regarding object shapes , styles and spatial arrangements from a few human-supplied labels by training concept classifiers only on the relevant feature subspace . Our model learns object shapes on shape codes , object colors and textures on style codes , and object spatial arrangements on object 3D locations . We show in the supplementary how the features relevant for each linguistic concept can be inferred from a few contrasting examples . Thus the classifiers attend only to the essential property of the concept and ignore irrelevant visual features . This allows them to generalize with far fewer examples and can recognize novel attribute compositions not present in the training data . We test D3DP-Nets in few-shot concept learning , visual question answering ( VQA ) and scene generation . We train concept classifiers for object shapes , object colors/materials , and spatial relationships on our inferred disentangled feature spaces , and show they outperform current stateof-the-art ( Mao et al. , 2019 ; Hu et al. , 2016 ) , which use 2D representations . We show that a VQA modular network that incorporates our concept classifiers shows improved generalization over the state-of-the-art ( Mao et al. , 2019 ) with dramatically fewer examples . Last , we empirically show that D3DP-Nets generalize their view predictions to scenes with novel number , category and styles of objects , and compare against state-of-the-art view predictive architectures of Eslami et al . ( 2018 ) . The main contribution of this paper is to identify the importance of using disentangled 3D feature representations for few-shot concept learning . We show the disentangled 3D feature representations can be learned using self-supervised view prediction , and they are useful for detecting and classifying language concepts by training them over the relevant only feature subsets . The proposed model outperforms the current state-of-the-art in VQA in the low data regime and the proposed 3D disentangled representation outperforms similar 2D or 2.5D ones in few-shot concept classification . 2 RELATION TO PREVIOUS WORKS . Few-shot concept learning Few-shot learning methods attempt to learn a new concept from one or a few annotated examples at test time , yet , at training time , these models still require labelled datasets which annotate a group of images as “ belonging to the same category ” ( Koch et al. , 2015 ; Vinyals et al. , 2016b ) . Metric-based few-shot learning approaches ( Snell et al. , 2017 ; Qi et al. , 2018 ; Schwartz et al. , 2018 ; Vinyals et al. , 2016a ) aim at learning an embedding space in which objects of the same category are closer in the latent space than objects that belong to different categories . These models needs to be trained with several ( annotated ) image collections , where each collection contains images of the same object category . Works of Misra et al . ( 2017 ) ; Purushwalkam et al . ( 2019 ) ; Nikolaus et al . ( 2019 ) ; Tokmakov et al . ( 2019 ) compose attribute and nouns to detect novel attribute-noun combinations , but their feature extractors need to be pretrained on large annotated image collections , such as Imagenet , or require annotated data with various attribute compositions . The proposed model is pretrained by predicting views , without the need for annotations regarding object classes or attributes . Our concept classifiers are related to methods that classify concepts by computing distances to prototypes produced by averaging the ( 2D CNN ) features of few labelled examples ( Snell et al. , 2017 ; Qi et al. , 2018 ; Schwartz et al. , 2018 ) . The work of Prabhudesai et al . ( 2020 ) learns 3D prototypes in a self-supervised manner , but they do not disentangle their representation into style and shape codes . We compare against 2D and 3D few shot learning methods and outperform them by a significant margin . The novel feature of our work is that we learn concept prototypes over disentangled 3D shape and 1D style codes as opposed to entangled 3D or 2D CNN features . Learning neural scene representation Our work builds upon recent view-predictive scene representation learning literature ( Tung et al. , 2019 ; Sitzmann et al. , 2019 ; Eslami et al. , 2016 ) . Our scene encoders and decoders , the view prediction objective , and the 3D neural bottleneck and egostabilization of the 3D feature maps is similar to those proposed in geometry-aware neural networks of Tung et al . ( 2019 ) . Sitzmann et al . ( 2019 ) and Eslami et al . ( 2016 ) both encode multiview images of a scene and camera poses into a scene representation , in the form of 2D scene feature maps or an implicit function . Sitzmann et al . ( 2019 ) only considers single-object scenes and needs to train a separate model for each object class . We compare generalization of our view predictions against Eslami et al . ( 2016 ) and show we have dramatically better generalization across number , type and spatial arrangements of objects . Furthermore , the above approaches do not explicitly disentangle style/shape representations of objects . Zhu et al . ( 2018 ) focuses on synthesizing natural images of objects with a disentangled 3D representation , but it remains unclear how to use the learnt embeddings to detect object concepts . Different from most inverse graphics networks ( Tung et al. , 2017 ; Kanazawa et al. , 2018 ) that aim to reconstruct detailed 3D occupancy of the objects , our model aims to learn feature representations that can detect an object across pose and scale variability , and use them for concept learning . Our shape-style disentanglement uses adaptive instance normalization layers ( Huang et al. , 2018 ; Huang & Belongie , 2017 ) that have been valuable for disentangling shape and style in 2D images . Here , we use them in a 3D latent feature space . 3 DISENTANGLING 3D PROTOTYPICAL NETWORKS ( D3DP-NETS ) . The architecture of D3DP-Nets is illustrated in Figure 2 . D3DP-Nets consists of two main components : ( a ) an image-to-scene encoder-decoder , and ( b ) an object shape/style disentanglement encoder-decoder . Next , we describe these components in detail . 3.1 IMAGE-TO-SCENE ENCODER-DECODER . A 2D-to-3D scene differentiable encoder Esc maps an input RGB-D image to a 3D feature map M ∈ Rw×h×d×c of the scene , where w , h , d , c denote width , height , depth and number of channels , respectively . Every ( x , y , z ) grid location in the 3D feature map M holds a c-channel feature vector that describes the semantic and geometric properties of a corresponding 3D physical location in the 3D world scene . We output a binary 3D occupancy map Mocc ∈ { 0 , 1 } w×h×d from M using an occupancy decoder Docc . A differentiable neural renderer Dsc neurally renders a 3D feature map M to a 2D image and a depth map from a specific viewpoint . When the input to D3DP-Nets is a sequence of images as opposed to a single image , each image It in the sequence is encoded to a corresponding 3D per frame map Mt , the 3D rotation and translation of the camera with respect to the frame map of the initial frame I0 is computed and the scene map M is computed by first rotating and translating Mt to bring it to the same coordinate frame as M0 and then averaging with the map built thus far . We will assume camera motion is known and given for this cross frame fusion operation . D3DP-Nets are self-supervised by view prediction , predicting RGB images and occupancy grids for query viewpoints . We assume there is an agent that can move around in static scenes and observes them from multiple viewpoints . The agent is equipped with a depth sensor and knowledge of its egomotion ( proprioception ) provided by the simulator in simulated environments.We train the scene encoders and decoders jointly for RGB view prediction and occupancy prediction and errors are backpropagated end-to-end to the parameters of the network : Lview−pred =‖Dsc ( rotate ( M , vq ) ) − Iq‖1 + log ( 1 + exp ( −Oq ·Docc ( ( rotate ( M , vq ) ) , vq ) ) ) , ( 1 ) where Iq and Oq are the ground truth RGB image and occupancy map respectively , vq is the query view , and rotate ( M , vq ) is a trilinear resampling operation that rotates the content of a 3D feature map M to viewpoint vq . The RGB output is trained with a regression loss , and the occupancy is trained with a logistic classification loss . Occupancy labels are computed through raycasting , similar to Harley et al . ( 2020 ) . We provide more details on the architecture of our model in the supplementary material . We train a 3D object detector that takes as input the output of the scene feature map M and predicts 3D axis-aligned bounding boxes , similar to Harley et al . ( 2020 ) . This is supervised from ground-truth 3D bounding boxes without class labels . 3.2 OBJECT SHAPE/STYLE DISENTANGLEMENT . As the style of an image can be understood as a property which is shared across its spatial dimensions , previous works ( Huang et al. , 2018 ; Karras et al. , 2019 ) use adaptive instance normalization ( Huang & Belongie , 2017 ) as an inductive bias to do style transfer between a pair of images . D3DP-Nets uses this same inductive bias in its decoder to disentangle the style and 3D shape of an object . We believe that 3D shape is not analogous to 3D occupancy , but it is a blend of 3D occupancy and texture ( spatial arrangement of color intensities ) . Given a set of 3D object boxes { bo\|o = 1 · · · \|O\| } where O is the set of objects in the scene , D3DP-Nets obtain corresponding object feature maps Mo = crop ( M , bo ) by cropping the scene feature map M using the 3D bounding box coordinates bo . We use ground-truth 3D boxes at training time and detected boxes at test time . Each object feature map is resized to a fixed resolution of 16 × 16 × 16 , and fed to an object-centric autoencoder whose encoding modules predict a 4D shape code zoshp = Eshp ( M o ) ∈ Rw×h×d×c and a 1D style code zosty = Esty ( M o ) ∈ Rc . A decoder D composes the two using adaptive instance normalization ( AIN ) layers ( Huang & Belongie , 2017 ) by adjusting the mean and variance of the 4D shape code based on the 1D style code : AIN ( z , γ , β ) = γ ( z−µ ( z ) σ ( z ) ) + β , where z is obtained by a 3D convolution on zshp , µ and σ are the channel-wise mean and standard deviation of z , and β and γ are extracted using single-layer perceptrons from zsty . The object encoders and decoders are trained with an autoencoding objective and a cycle-consistency objective which ensure that the shape and style code remain consistent after composing , decoding and encoding again ( see Figure 2 ( b ) ) : Ldis = 1 \|O\| \|O\|∑ o=1 ‖Mo −D ( Eshp ( Mo ) , Esty ( Mo ) ) ‖2︸︷︷︸ autoencoding loss + ∑ i∈O\o Lc−shp ( Mo , Mi ) + Lc−sty ( Mo , Mi ) ︸︷︷︸ cycle-consistency loss , ( 2 ) where Lc−shp ( Mo , Mi ) = ‖Eshp ( Mo ) −Eshp ( D ( Eshp ( Mo ) , Esty ( Mi ) ) ) ‖2 is the shape consistency loss and Lc−sty ( Mo , Mi ) = ‖Esty ( Mo ) −Esty ( D ( Eshp ( Mi ) , Esty ( Mo ) ) ) ‖2 is the style consistency loss . We further include a view prediction loss on the synthesized scene feature map M̄ , which is composed by replacing each object feature map Mo with its re-synthesized version D ( zoshp , zosty ) , resized to the original object size , as shown in Figure 2 ( c ) . The view prediction reads : Lview−pred−synth = ‖Dsc ( rotate ( M̄ , vt+1 ) ) − It+1‖1 . The total unsupervised optimization loss for D3DP-Nets reads : Luns = Lview−pred + Lview−pred−synth + Ldis . ( 3 )	This paper presents a modular network architecture for few-shot concept learning. The architecture consists of image-to-scene module that maps input RGBD images to 3D scene features and an object-centric disentangling auto-encoder that crops object features to generate shape and style codes, and finally a neural rendering module that put back the reconstructed object and background features to image domain. The proposed network is verified in few-short recognition task and VQA task with comparisons to state-of-the-art methods.	SP:8f46cbd5fff3557fe870cebf1ba231309fceab14
IEPT: Instance-Level and Episode-Level Pretext Tasks for Few-Shot Learning	1 INTRODUCTION . Deep convolutional neural networks ( CNNs ) ( Krizhevsky et al. , 2012 ; He et al. , 2016b ; Huang et al. , 2017 ) have seen tremendous successes in a wide range of application fields , especially in visual recognition . However , the powerful learning ability of CNNs depends on a large amount of manually labeled training data . In practice , for many visual recognition tasks , sufficient manual annotation is either too costly to collect or not feasible ( e.g. , for rare object classes ) . This has severely limited the usefulness of CNNs for real-world application scenarios . Attempts have been made recently to mitigate such a limitation from two distinct perspectives , resulting in two popular research lines , both of which aim to transfer knowledge learned from the data of a set of source tasks to a new target one : few-shot learning ( FSL ) and self-supervised learning ( SSL ) . FSL ( Fei-Fei et al. , 2006 ; Vinyals et al. , 2016 ; Finn et al. , 2017 ; Snell et al. , 2017 ; Sung et al. , 2018 ) typically takes a ‘ learning to learn ’ or meta-learning paradigm . That is , it aims to learn an algorithm for learning from few labeled samples , which generalizes well across any tasks . To that end , it adopts ∗Corresponding author . an episodic training strategy – the source tasks are arranged into learning episodes , each of which contains n classes and k labeled samples per class to simulate the setting for the target task . Part of the CNN model ( e.g. , feature extraction subnet , classification layers , or parameter initialization ) is then meta-learned for rapid adaptation to new tasks . In contrast , SSL ( Doersch et al. , 2015 ; Noroozi & Favaro , 2016 ; Iizuka et al. , 2016 ; Doersch & Zisserman , 2017 ; Noroozi et al. , 2018 ) does not require the source data to be annotated . Instead , it exploits an annotation-free pretext task on the source task data in the hope that a task-generalizable feature representation can be learned from the source tasks for easy adoption or adaptation in a target task . Such a pretext task gets its self-supervised signal at the per-instance level . Examples include rotation and context prediction ( Gidaris et al. , 2018 ; Doersch et al. , 2015 ) , jigsaw solving ( Noroozi & Favaro , 2016 ) , and colorization ( Iizuka et al. , 2016 ; Larsson et al. , 2016 ) . Since these pretext tasks are class-agnostic , solving them leads to the learning of transferable knowledge . Since both FSL and SSL aim to reduce the need of collecting a large amount of labeled training data for a target task by transferring knowledge from a set of source tasks , it is natural to consider combining them in a single framework . Indeed , two recent works ( Gidaris et al. , 2019 ; Su et al. , 2020 ) proposed to integrate SSL into FSL by adding an auxiliary SSL pretext task in an FSL model . It showed that the SSL learning objective is complementary to that of FSL and combining them leads to improved FSL performance . However , in ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , SSL is combined with FSL in a superficial way : it is only taken as a separate auxiliary task for each single training instance and has no effect on the episodic training pipeline of the FSL model . Importantly , by ignoring the class labels of samples , the instance-level SSL learning objective is weak on its own . Since meta-learning across episodes is the essence of most contemporary FSL models , we argue that adding instance-level SSL pretext tasks alone fails to exploit fully the complementarity of the aforementioned FSL and SSL , for which a closer and deeper integration is needed . To that end , in this paper we propose a novel Instance-level and Episode-level Pretext Task ( IEPT ) framework for few-shot recognition . Apart from adding an instance-level pretext SSL task as in ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , we introduce two episode-level SSL-FSL hybrid learning objectives for seamless SSL-FSL integration . Concretely , as illustrated in Figure 1 , our full model has three additional learning objectives ( besides the standard FSL one ) : ( 1 ) Different rotation transformations are applied to each original few-shot episode to generate a set of extended episodes , where each image has a rotation label for the instance-level pretext task ( i.e. , to predict the rotation label ) . ( 2 ) The consistency across the predictions of an FSL classifier from different extended episodes is maximized as an episode-level pretext task . For each training image , the rotation transformation does not change its semantic content and hence its class label ; the FSL classifier predictions across different extended episodes thus should be consistent , hence the consistency regularization objective . ( 3 ) The correlation of features across instances from these extended episodes is modeled by a transformer-based attention module , optimizing the fusion of the features of each instance/image and its various rotation-transformed versions mainly for task adaptation during meta-testing . Importantly , with these three new learning objectives introduced in IEPT , any meta-learning based FSL model can now benefit more from SSL by fully exploiting their complementarity . Our main contributions are : ( 1 ) For the first time , we propose both instance-level and episode-level pretext tasks ( IEPT ) for integrating SSL into FSL . The episode-level pretext task enables episodic training of SSL and hence closer integration of SSL with FSL . ( 2 ) In addition to these pretext tasks , FSL further benefits from SSL by integrating features extracted from various rotation-transformed versions of the original training instances . The optimal way of feature integration is learned by a transformer-based attention module , which is mainly designed for task adaptation during meta-testing . ( 3 ) Extensive experiments show that the proposed model achieves the new state-of-the-art . 2 RELATED WORK . Few-Shot Learning . The recent FSL studies are dominated by meta-learning based methods . They can be divided into three groups : ( 1 ) Metric-based methods ( Vinyals et al. , 2016 ; Snell et al. , 2017 ; Sung et al. , 2018 ; Allen et al. , 2019 ; Xing et al. , 2019 ; Li et al. , 2019a ; b ; Wu et al. , 2019 ; Ye et al. , 2020 ; Afrasiyabi et al. , 2020 ; Liu et al. , 2020 ; Zhang et al. , 2020 ) aim to learn the distance metric between feature embeddings . The focus of these methods is often on meta-learning of a feature-extraction CNN , whilst the classifiers used are of simple form such as a nearest-neighbor classifier . ( 2 ) Optimization-based methods ( Finn et al. , 2017 ; Ravi & Larochelle , 2017 ; Rusu et al. , 2019 ; Lee et al. , 2019 ) learn to optimize the model rapidly given a few labeled samples per class in the new task . ( 3 ) Model-based methods ( Santoro et al. , 2016 ; Munkhdalai & Yu , 2017 ; Mishra et al. , 2018 ) focus on designing either specific model structures or parameters capable of rapid updating . Apart from these three groups of methods , other FSL methods have attempted feature hallucination ( Schwartz et al. , 2018 ; Hariharan & Girshick , 2017 ; Gao et al. , 2018 ; Wang et al. , 2018 ; Zhang et al. , 2019 ; Tsutsui et al. , 2019 ) which generates additional samples from the given few shots for network finetuning , and parameter predicting ( Qiao et al. , 2018 ; Qi et al. , 2018 ; Gidaris & Komodakis , 2019 ; 2018 ) which learns to predict part of the parameters of a network given few samples of new classes for quick adaptation . In this work , we adopt the metric-based Prototypical Network ( ProtoNet ) ( Snell et al. , 2017 ) as the basic FSL classifier for the main instantiation of our IEPT framework due to its simplicity and popularity . However , we show that any meta-learning based FSL method can be combined with our IEPT ( see results in Figure 2 ( c ) ) . Self-Supervised Learning . In SSL , it is assumed that the source task data is label-free and a pretext task is designed to provide self-supervision signals at the instance-level . Existing SSL approaches differ mainly in the pretext task design . These include predicting the rotation angle ( Gidaris et al. , 2018 ) and the context of image patch ( Doersch et al. , 2015 ; Nathan Mundhenk et al. , 2018 ) , jigsaw solving ( Noroozi & Favaro , 2016 ; Noroozi et al. , 2018 ) ( i.e . shuffling and then reordering image patch ) , and performing images reversion ( Iizuka et al. , 2016 ; Pathak et al. , 2016 ; Larsson et al. , 2016 ) . SSL has been shown to be beneficial to various down-steam tasks such as semantic object matching ( Novotny et al. , 2018 ) , object segmentation ( Ji et al. , 2019 ) and object detection ( Doersch & Zisserman , 2017 ) by learning transferable feature presentations for these tasks . Integrating Self-Supervised Learning into Few-Shot Learning . To the best of our knowledge , only two recent works ( Gidaris et al. , 2019 ; Su et al. , 2020 ) have attempted combining SSL with FSL . However , the integration of SSL into FSL is often shallow : the original FSL training pipeline is intact ; in the meantime , an additional loss on each image w.r.t . a self-supervised signal like the rotation angle or relative patch location is introduced . With pretext tasks solely at the instance level , combining the two approaches ( i.e. , SSL and FSL ) can only be superficial without fully exploiting the episodic training pipeline unique to FSL . Different from ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , we introduce an episode-level pretext task to integrate SSL into the episodic training in FSL fully . Specifically , the consistency across the predictions of an FSL classifier from different extended episodes is maximized to reflect the fact that various rotation transformations should not alter the class-label prediction . Moreover , features of each instance and its various rotation-transformed versions are now fused for FSL classification , to integrate SSL with FSL for the supervised classification task . Our experimental results show that thanks to the closer integration of SSL and FSL , our IEPT clearly outperforms ( Gidaris et al. , 2019 ; Su et al. , 2020 ) ( see Table 1 ) . 3 METHODOLOGY . 3.1 PRELIMINARY . Problem Setting . Given an n-way k-shot FSL task sampled from a test set Dt , to imitate the test setting , an FSL model is typically trained in an episodic way . That is , n-way k-shot episodes are randomly sampled from a training set Ds , where the class label space of Ds has no overlap with that of Dt . Each episode Ee contains a support set Se and a query set Qe . Concretely , we first randomly sample a set of n classes Ce from the training set , and then generate Se andQe by sampling k support samples and q query samples from each class in Ce , respectively . Formally , we have Se = { ( xi , yi ) \|yi ∈ Ce , i = 1 , ... , n × k } and Qe = { ( xi , yi ) \|yi ∈ Ce , i = 1 , ... , n × q } , where Se ⋂ Qe = ∅ . For simplicity , we denote lk = n × k and lq = n × q . In the meta-training stage , the training process has an inner and an outer loop in each episode : in the inner loop , the model is updated using Se ; its performance is then evaluated on the query set Qe in the outer loop to update the model parameters or algorithm that one wants to meta-learn . Basic FSL Classifier . We employ ProtoNet ( Snell et al. , 2017 ) as the basic FSL model . This model has a feature-extraction CNN and a simple non-parametric classifier . The parameter of the feature extractor is to be meta-learned . Concretely , in the inner loop of an episode , ProtoNet fixes the feature extractor and computes the mean feature embedding for each class as follows : hc = 1 k · ∑ ( xi , yi ) ∈Se fφ ( xi ) · I ( yi = c ) , ( 1 ) where class c ∈ Ce , fφ is a feature extractor with learnable parameters φ , and I is the indicator function . By computing the distance between the feature embedding of each query sample and that of the corresponding class , the loss function used to meta-learn φ in the outer loop is defined as : Lfsl ( Se , Qe ) = 1 \|Qe\| ∑ ( xi , yi ) ∈Qe − log exp ( −d ( fφ ( xi ) , hyi ) ) ∑ c∈Ce exp ( −d ( fφ ( xi ) , hc ) ) , ( 2 ) where d ( · , · ) denotes a distance function ( e.g. , the l2 distance ) .	This paper addresses the problem of few-shot classification by incorporating self-supervised learning into the standard episode-based meta learning. Specifically, it adopts the pretext task of rotation prediction into the episode design. For each sampled episode, additional episodes are constructed by using rotated examples in the original support set and query set. Two self-supervised losses are designed based on the augmented episode sets – (1) recognizing different rotation transformations as an instance-level pretext task; (2) ensuring consistent predictions of class labels across different episodes as an episode-level pretext task. Two few-shot losses are also designed – (1) predicting class labels for each individual episode; (2) predicting class labels for the fused episode set based on attention. Experimental evaluation is conducted on two standard few-shot classification benchmarks, namely miniImageNet and tieredImageNet, and shows improved performance.	SP:c812cd7dae5afa5e60ba309053fec184cdc4cad2
IEPT: Instance-Level and Episode-Level Pretext Tasks for Few-Shot Learning	1 INTRODUCTION . Deep convolutional neural networks ( CNNs ) ( Krizhevsky et al. , 2012 ; He et al. , 2016b ; Huang et al. , 2017 ) have seen tremendous successes in a wide range of application fields , especially in visual recognition . However , the powerful learning ability of CNNs depends on a large amount of manually labeled training data . In practice , for many visual recognition tasks , sufficient manual annotation is either too costly to collect or not feasible ( e.g. , for rare object classes ) . This has severely limited the usefulness of CNNs for real-world application scenarios . Attempts have been made recently to mitigate such a limitation from two distinct perspectives , resulting in two popular research lines , both of which aim to transfer knowledge learned from the data of a set of source tasks to a new target one : few-shot learning ( FSL ) and self-supervised learning ( SSL ) . FSL ( Fei-Fei et al. , 2006 ; Vinyals et al. , 2016 ; Finn et al. , 2017 ; Snell et al. , 2017 ; Sung et al. , 2018 ) typically takes a ‘ learning to learn ’ or meta-learning paradigm . That is , it aims to learn an algorithm for learning from few labeled samples , which generalizes well across any tasks . To that end , it adopts ∗Corresponding author . an episodic training strategy – the source tasks are arranged into learning episodes , each of which contains n classes and k labeled samples per class to simulate the setting for the target task . Part of the CNN model ( e.g. , feature extraction subnet , classification layers , or parameter initialization ) is then meta-learned for rapid adaptation to new tasks . In contrast , SSL ( Doersch et al. , 2015 ; Noroozi & Favaro , 2016 ; Iizuka et al. , 2016 ; Doersch & Zisserman , 2017 ; Noroozi et al. , 2018 ) does not require the source data to be annotated . Instead , it exploits an annotation-free pretext task on the source task data in the hope that a task-generalizable feature representation can be learned from the source tasks for easy adoption or adaptation in a target task . Such a pretext task gets its self-supervised signal at the per-instance level . Examples include rotation and context prediction ( Gidaris et al. , 2018 ; Doersch et al. , 2015 ) , jigsaw solving ( Noroozi & Favaro , 2016 ) , and colorization ( Iizuka et al. , 2016 ; Larsson et al. , 2016 ) . Since these pretext tasks are class-agnostic , solving them leads to the learning of transferable knowledge . Since both FSL and SSL aim to reduce the need of collecting a large amount of labeled training data for a target task by transferring knowledge from a set of source tasks , it is natural to consider combining them in a single framework . Indeed , two recent works ( Gidaris et al. , 2019 ; Su et al. , 2020 ) proposed to integrate SSL into FSL by adding an auxiliary SSL pretext task in an FSL model . It showed that the SSL learning objective is complementary to that of FSL and combining them leads to improved FSL performance . However , in ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , SSL is combined with FSL in a superficial way : it is only taken as a separate auxiliary task for each single training instance and has no effect on the episodic training pipeline of the FSL model . Importantly , by ignoring the class labels of samples , the instance-level SSL learning objective is weak on its own . Since meta-learning across episodes is the essence of most contemporary FSL models , we argue that adding instance-level SSL pretext tasks alone fails to exploit fully the complementarity of the aforementioned FSL and SSL , for which a closer and deeper integration is needed . To that end , in this paper we propose a novel Instance-level and Episode-level Pretext Task ( IEPT ) framework for few-shot recognition . Apart from adding an instance-level pretext SSL task as in ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , we introduce two episode-level SSL-FSL hybrid learning objectives for seamless SSL-FSL integration . Concretely , as illustrated in Figure 1 , our full model has three additional learning objectives ( besides the standard FSL one ) : ( 1 ) Different rotation transformations are applied to each original few-shot episode to generate a set of extended episodes , where each image has a rotation label for the instance-level pretext task ( i.e. , to predict the rotation label ) . ( 2 ) The consistency across the predictions of an FSL classifier from different extended episodes is maximized as an episode-level pretext task . For each training image , the rotation transformation does not change its semantic content and hence its class label ; the FSL classifier predictions across different extended episodes thus should be consistent , hence the consistency regularization objective . ( 3 ) The correlation of features across instances from these extended episodes is modeled by a transformer-based attention module , optimizing the fusion of the features of each instance/image and its various rotation-transformed versions mainly for task adaptation during meta-testing . Importantly , with these three new learning objectives introduced in IEPT , any meta-learning based FSL model can now benefit more from SSL by fully exploiting their complementarity . Our main contributions are : ( 1 ) For the first time , we propose both instance-level and episode-level pretext tasks ( IEPT ) for integrating SSL into FSL . The episode-level pretext task enables episodic training of SSL and hence closer integration of SSL with FSL . ( 2 ) In addition to these pretext tasks , FSL further benefits from SSL by integrating features extracted from various rotation-transformed versions of the original training instances . The optimal way of feature integration is learned by a transformer-based attention module , which is mainly designed for task adaptation during meta-testing . ( 3 ) Extensive experiments show that the proposed model achieves the new state-of-the-art . 2 RELATED WORK . Few-Shot Learning . The recent FSL studies are dominated by meta-learning based methods . They can be divided into three groups : ( 1 ) Metric-based methods ( Vinyals et al. , 2016 ; Snell et al. , 2017 ; Sung et al. , 2018 ; Allen et al. , 2019 ; Xing et al. , 2019 ; Li et al. , 2019a ; b ; Wu et al. , 2019 ; Ye et al. , 2020 ; Afrasiyabi et al. , 2020 ; Liu et al. , 2020 ; Zhang et al. , 2020 ) aim to learn the distance metric between feature embeddings . The focus of these methods is often on meta-learning of a feature-extraction CNN , whilst the classifiers used are of simple form such as a nearest-neighbor classifier . ( 2 ) Optimization-based methods ( Finn et al. , 2017 ; Ravi & Larochelle , 2017 ; Rusu et al. , 2019 ; Lee et al. , 2019 ) learn to optimize the model rapidly given a few labeled samples per class in the new task . ( 3 ) Model-based methods ( Santoro et al. , 2016 ; Munkhdalai & Yu , 2017 ; Mishra et al. , 2018 ) focus on designing either specific model structures or parameters capable of rapid updating . Apart from these three groups of methods , other FSL methods have attempted feature hallucination ( Schwartz et al. , 2018 ; Hariharan & Girshick , 2017 ; Gao et al. , 2018 ; Wang et al. , 2018 ; Zhang et al. , 2019 ; Tsutsui et al. , 2019 ) which generates additional samples from the given few shots for network finetuning , and parameter predicting ( Qiao et al. , 2018 ; Qi et al. , 2018 ; Gidaris & Komodakis , 2019 ; 2018 ) which learns to predict part of the parameters of a network given few samples of new classes for quick adaptation . In this work , we adopt the metric-based Prototypical Network ( ProtoNet ) ( Snell et al. , 2017 ) as the basic FSL classifier for the main instantiation of our IEPT framework due to its simplicity and popularity . However , we show that any meta-learning based FSL method can be combined with our IEPT ( see results in Figure 2 ( c ) ) . Self-Supervised Learning . In SSL , it is assumed that the source task data is label-free and a pretext task is designed to provide self-supervision signals at the instance-level . Existing SSL approaches differ mainly in the pretext task design . These include predicting the rotation angle ( Gidaris et al. , 2018 ) and the context of image patch ( Doersch et al. , 2015 ; Nathan Mundhenk et al. , 2018 ) , jigsaw solving ( Noroozi & Favaro , 2016 ; Noroozi et al. , 2018 ) ( i.e . shuffling and then reordering image patch ) , and performing images reversion ( Iizuka et al. , 2016 ; Pathak et al. , 2016 ; Larsson et al. , 2016 ) . SSL has been shown to be beneficial to various down-steam tasks such as semantic object matching ( Novotny et al. , 2018 ) , object segmentation ( Ji et al. , 2019 ) and object detection ( Doersch & Zisserman , 2017 ) by learning transferable feature presentations for these tasks . Integrating Self-Supervised Learning into Few-Shot Learning . To the best of our knowledge , only two recent works ( Gidaris et al. , 2019 ; Su et al. , 2020 ) have attempted combining SSL with FSL . However , the integration of SSL into FSL is often shallow : the original FSL training pipeline is intact ; in the meantime , an additional loss on each image w.r.t . a self-supervised signal like the rotation angle or relative patch location is introduced . With pretext tasks solely at the instance level , combining the two approaches ( i.e. , SSL and FSL ) can only be superficial without fully exploiting the episodic training pipeline unique to FSL . Different from ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , we introduce an episode-level pretext task to integrate SSL into the episodic training in FSL fully . Specifically , the consistency across the predictions of an FSL classifier from different extended episodes is maximized to reflect the fact that various rotation transformations should not alter the class-label prediction . Moreover , features of each instance and its various rotation-transformed versions are now fused for FSL classification , to integrate SSL with FSL for the supervised classification task . Our experimental results show that thanks to the closer integration of SSL and FSL , our IEPT clearly outperforms ( Gidaris et al. , 2019 ; Su et al. , 2020 ) ( see Table 1 ) . 3 METHODOLOGY . 3.1 PRELIMINARY . Problem Setting . Given an n-way k-shot FSL task sampled from a test set Dt , to imitate the test setting , an FSL model is typically trained in an episodic way . That is , n-way k-shot episodes are randomly sampled from a training set Ds , where the class label space of Ds has no overlap with that of Dt . Each episode Ee contains a support set Se and a query set Qe . Concretely , we first randomly sample a set of n classes Ce from the training set , and then generate Se andQe by sampling k support samples and q query samples from each class in Ce , respectively . Formally , we have Se = { ( xi , yi ) \|yi ∈ Ce , i = 1 , ... , n × k } and Qe = { ( xi , yi ) \|yi ∈ Ce , i = 1 , ... , n × q } , where Se ⋂ Qe = ∅ . For simplicity , we denote lk = n × k and lq = n × q . In the meta-training stage , the training process has an inner and an outer loop in each episode : in the inner loop , the model is updated using Se ; its performance is then evaluated on the query set Qe in the outer loop to update the model parameters or algorithm that one wants to meta-learn . Basic FSL Classifier . We employ ProtoNet ( Snell et al. , 2017 ) as the basic FSL model . This model has a feature-extraction CNN and a simple non-parametric classifier . The parameter of the feature extractor is to be meta-learned . Concretely , in the inner loop of an episode , ProtoNet fixes the feature extractor and computes the mean feature embedding for each class as follows : hc = 1 k · ∑ ( xi , yi ) ∈Se fφ ( xi ) · I ( yi = c ) , ( 1 ) where class c ∈ Ce , fφ is a feature extractor with learnable parameters φ , and I is the indicator function . By computing the distance between the feature embedding of each query sample and that of the corresponding class , the loss function used to meta-learn φ in the outer loop is defined as : Lfsl ( Se , Qe ) = 1 \|Qe\| ∑ ( xi , yi ) ∈Qe − log exp ( −d ( fφ ( xi ) , hyi ) ) ∑ c∈Ce exp ( −d ( fφ ( xi ) , hc ) ) , ( 2 ) where d ( · , · ) denotes a distance function ( e.g. , the l2 distance ) .	This paper presents a method for combining self-supervised learning (SSL) (in the form of predicting the rotation applied to an image) with few-shot learning (FSL) in the domain of image classification. Compared to prior work, this paper introduces -- (i) a consistency loss which ensures FSL episodes with different rotations agree in their class predictions; and (ii) an integration method which derives the label of an image from a fused representation of all its rotations. These lead to an improvement over 3 FSL benchmarks.	SP:c812cd7dae5afa5e60ba309053fec184cdc4cad2
IEPT: Instance-Level and Episode-Level Pretext Tasks for Few-Shot Learning	1 INTRODUCTION . Deep convolutional neural networks ( CNNs ) ( Krizhevsky et al. , 2012 ; He et al. , 2016b ; Huang et al. , 2017 ) have seen tremendous successes in a wide range of application fields , especially in visual recognition . However , the powerful learning ability of CNNs depends on a large amount of manually labeled training data . In practice , for many visual recognition tasks , sufficient manual annotation is either too costly to collect or not feasible ( e.g. , for rare object classes ) . This has severely limited the usefulness of CNNs for real-world application scenarios . Attempts have been made recently to mitigate such a limitation from two distinct perspectives , resulting in two popular research lines , both of which aim to transfer knowledge learned from the data of a set of source tasks to a new target one : few-shot learning ( FSL ) and self-supervised learning ( SSL ) . FSL ( Fei-Fei et al. , 2006 ; Vinyals et al. , 2016 ; Finn et al. , 2017 ; Snell et al. , 2017 ; Sung et al. , 2018 ) typically takes a ‘ learning to learn ’ or meta-learning paradigm . That is , it aims to learn an algorithm for learning from few labeled samples , which generalizes well across any tasks . To that end , it adopts ∗Corresponding author . an episodic training strategy – the source tasks are arranged into learning episodes , each of which contains n classes and k labeled samples per class to simulate the setting for the target task . Part of the CNN model ( e.g. , feature extraction subnet , classification layers , or parameter initialization ) is then meta-learned for rapid adaptation to new tasks . In contrast , SSL ( Doersch et al. , 2015 ; Noroozi & Favaro , 2016 ; Iizuka et al. , 2016 ; Doersch & Zisserman , 2017 ; Noroozi et al. , 2018 ) does not require the source data to be annotated . Instead , it exploits an annotation-free pretext task on the source task data in the hope that a task-generalizable feature representation can be learned from the source tasks for easy adoption or adaptation in a target task . Such a pretext task gets its self-supervised signal at the per-instance level . Examples include rotation and context prediction ( Gidaris et al. , 2018 ; Doersch et al. , 2015 ) , jigsaw solving ( Noroozi & Favaro , 2016 ) , and colorization ( Iizuka et al. , 2016 ; Larsson et al. , 2016 ) . Since these pretext tasks are class-agnostic , solving them leads to the learning of transferable knowledge . Since both FSL and SSL aim to reduce the need of collecting a large amount of labeled training data for a target task by transferring knowledge from a set of source tasks , it is natural to consider combining them in a single framework . Indeed , two recent works ( Gidaris et al. , 2019 ; Su et al. , 2020 ) proposed to integrate SSL into FSL by adding an auxiliary SSL pretext task in an FSL model . It showed that the SSL learning objective is complementary to that of FSL and combining them leads to improved FSL performance . However , in ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , SSL is combined with FSL in a superficial way : it is only taken as a separate auxiliary task for each single training instance and has no effect on the episodic training pipeline of the FSL model . Importantly , by ignoring the class labels of samples , the instance-level SSL learning objective is weak on its own . Since meta-learning across episodes is the essence of most contemporary FSL models , we argue that adding instance-level SSL pretext tasks alone fails to exploit fully the complementarity of the aforementioned FSL and SSL , for which a closer and deeper integration is needed . To that end , in this paper we propose a novel Instance-level and Episode-level Pretext Task ( IEPT ) framework for few-shot recognition . Apart from adding an instance-level pretext SSL task as in ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , we introduce two episode-level SSL-FSL hybrid learning objectives for seamless SSL-FSL integration . Concretely , as illustrated in Figure 1 , our full model has three additional learning objectives ( besides the standard FSL one ) : ( 1 ) Different rotation transformations are applied to each original few-shot episode to generate a set of extended episodes , where each image has a rotation label for the instance-level pretext task ( i.e. , to predict the rotation label ) . ( 2 ) The consistency across the predictions of an FSL classifier from different extended episodes is maximized as an episode-level pretext task . For each training image , the rotation transformation does not change its semantic content and hence its class label ; the FSL classifier predictions across different extended episodes thus should be consistent , hence the consistency regularization objective . ( 3 ) The correlation of features across instances from these extended episodes is modeled by a transformer-based attention module , optimizing the fusion of the features of each instance/image and its various rotation-transformed versions mainly for task adaptation during meta-testing . Importantly , with these three new learning objectives introduced in IEPT , any meta-learning based FSL model can now benefit more from SSL by fully exploiting their complementarity . Our main contributions are : ( 1 ) For the first time , we propose both instance-level and episode-level pretext tasks ( IEPT ) for integrating SSL into FSL . The episode-level pretext task enables episodic training of SSL and hence closer integration of SSL with FSL . ( 2 ) In addition to these pretext tasks , FSL further benefits from SSL by integrating features extracted from various rotation-transformed versions of the original training instances . The optimal way of feature integration is learned by a transformer-based attention module , which is mainly designed for task adaptation during meta-testing . ( 3 ) Extensive experiments show that the proposed model achieves the new state-of-the-art . 2 RELATED WORK . Few-Shot Learning . The recent FSL studies are dominated by meta-learning based methods . They can be divided into three groups : ( 1 ) Metric-based methods ( Vinyals et al. , 2016 ; Snell et al. , 2017 ; Sung et al. , 2018 ; Allen et al. , 2019 ; Xing et al. , 2019 ; Li et al. , 2019a ; b ; Wu et al. , 2019 ; Ye et al. , 2020 ; Afrasiyabi et al. , 2020 ; Liu et al. , 2020 ; Zhang et al. , 2020 ) aim to learn the distance metric between feature embeddings . The focus of these methods is often on meta-learning of a feature-extraction CNN , whilst the classifiers used are of simple form such as a nearest-neighbor classifier . ( 2 ) Optimization-based methods ( Finn et al. , 2017 ; Ravi & Larochelle , 2017 ; Rusu et al. , 2019 ; Lee et al. , 2019 ) learn to optimize the model rapidly given a few labeled samples per class in the new task . ( 3 ) Model-based methods ( Santoro et al. , 2016 ; Munkhdalai & Yu , 2017 ; Mishra et al. , 2018 ) focus on designing either specific model structures or parameters capable of rapid updating . Apart from these three groups of methods , other FSL methods have attempted feature hallucination ( Schwartz et al. , 2018 ; Hariharan & Girshick , 2017 ; Gao et al. , 2018 ; Wang et al. , 2018 ; Zhang et al. , 2019 ; Tsutsui et al. , 2019 ) which generates additional samples from the given few shots for network finetuning , and parameter predicting ( Qiao et al. , 2018 ; Qi et al. , 2018 ; Gidaris & Komodakis , 2019 ; 2018 ) which learns to predict part of the parameters of a network given few samples of new classes for quick adaptation . In this work , we adopt the metric-based Prototypical Network ( ProtoNet ) ( Snell et al. , 2017 ) as the basic FSL classifier for the main instantiation of our IEPT framework due to its simplicity and popularity . However , we show that any meta-learning based FSL method can be combined with our IEPT ( see results in Figure 2 ( c ) ) . Self-Supervised Learning . In SSL , it is assumed that the source task data is label-free and a pretext task is designed to provide self-supervision signals at the instance-level . Existing SSL approaches differ mainly in the pretext task design . These include predicting the rotation angle ( Gidaris et al. , 2018 ) and the context of image patch ( Doersch et al. , 2015 ; Nathan Mundhenk et al. , 2018 ) , jigsaw solving ( Noroozi & Favaro , 2016 ; Noroozi et al. , 2018 ) ( i.e . shuffling and then reordering image patch ) , and performing images reversion ( Iizuka et al. , 2016 ; Pathak et al. , 2016 ; Larsson et al. , 2016 ) . SSL has been shown to be beneficial to various down-steam tasks such as semantic object matching ( Novotny et al. , 2018 ) , object segmentation ( Ji et al. , 2019 ) and object detection ( Doersch & Zisserman , 2017 ) by learning transferable feature presentations for these tasks . Integrating Self-Supervised Learning into Few-Shot Learning . To the best of our knowledge , only two recent works ( Gidaris et al. , 2019 ; Su et al. , 2020 ) have attempted combining SSL with FSL . However , the integration of SSL into FSL is often shallow : the original FSL training pipeline is intact ; in the meantime , an additional loss on each image w.r.t . a self-supervised signal like the rotation angle or relative patch location is introduced . With pretext tasks solely at the instance level , combining the two approaches ( i.e. , SSL and FSL ) can only be superficial without fully exploiting the episodic training pipeline unique to FSL . Different from ( Gidaris et al. , 2019 ; Su et al. , 2020 ) , we introduce an episode-level pretext task to integrate SSL into the episodic training in FSL fully . Specifically , the consistency across the predictions of an FSL classifier from different extended episodes is maximized to reflect the fact that various rotation transformations should not alter the class-label prediction . Moreover , features of each instance and its various rotation-transformed versions are now fused for FSL classification , to integrate SSL with FSL for the supervised classification task . Our experimental results show that thanks to the closer integration of SSL and FSL , our IEPT clearly outperforms ( Gidaris et al. , 2019 ; Su et al. , 2020 ) ( see Table 1 ) . 3 METHODOLOGY . 3.1 PRELIMINARY . Problem Setting . Given an n-way k-shot FSL task sampled from a test set Dt , to imitate the test setting , an FSL model is typically trained in an episodic way . That is , n-way k-shot episodes are randomly sampled from a training set Ds , where the class label space of Ds has no overlap with that of Dt . Each episode Ee contains a support set Se and a query set Qe . Concretely , we first randomly sample a set of n classes Ce from the training set , and then generate Se andQe by sampling k support samples and q query samples from each class in Ce , respectively . Formally , we have Se = { ( xi , yi ) \|yi ∈ Ce , i = 1 , ... , n × k } and Qe = { ( xi , yi ) \|yi ∈ Ce , i = 1 , ... , n × q } , where Se ⋂ Qe = ∅ . For simplicity , we denote lk = n × k and lq = n × q . In the meta-training stage , the training process has an inner and an outer loop in each episode : in the inner loop , the model is updated using Se ; its performance is then evaluated on the query set Qe in the outer loop to update the model parameters or algorithm that one wants to meta-learn . Basic FSL Classifier . We employ ProtoNet ( Snell et al. , 2017 ) as the basic FSL model . This model has a feature-extraction CNN and a simple non-parametric classifier . The parameter of the feature extractor is to be meta-learned . Concretely , in the inner loop of an episode , ProtoNet fixes the feature extractor and computes the mean feature embedding for each class as follows : hc = 1 k · ∑ ( xi , yi ) ∈Se fφ ( xi ) · I ( yi = c ) , ( 1 ) where class c ∈ Ce , fφ is a feature extractor with learnable parameters φ , and I is the indicator function . By computing the distance between the feature embedding of each query sample and that of the corresponding class , the loss function used to meta-learn φ in the outer loop is defined as : Lfsl ( Se , Qe ) = 1 \|Qe\| ∑ ( xi , yi ) ∈Qe − log exp ( −d ( fφ ( xi ) , hyi ) ) ∑ c∈Ce exp ( −d ( fφ ( xi ) , hc ) ) , ( 2 ) where d ( · , · ) denotes a distance function ( e.g. , the l2 distance ) .	This paper solves the problem of few-shot learning. The recent success of SSL and FSL proves that they can handle situations that few label data are provided. Motivated by this, the author proposed a novel framework IEPT that seamlessly integrates self-supervised learning methods to few-shot learning. Unlike other trivial combination of SSL and FSL methods, this paper proposed instance-level and episode-level pretext tasks to bring on closer integration. Further, this paper proposed to use transformer to integrate features from different images and augmentations. Experiments show the model achieves new SOTA.	SP:c812cd7dae5afa5e60ba309053fec184cdc4cad2
Self-supervised and Supervised Joint Training for Resource-rich Machine Translation	1 INTRODUCTION . Self-supervised pre-training of text representations ( Peters et al. , 2018 ; Radford et al. , 2018 ) has achieved tremendous success in natural language processing applications . Inspired by BERT ( Devlin et al. , 2019 ) , recent works attempt to leverage sequence-to-sequence model pre-training for Neural Machine Translation ( NMT ) . Generally , these methods comprise two stages : pre-training and finetuning . During the pre-training stage , a proxy task , e.g . the Cloze task ( Devlin et al. , 2019 ) , is used to learn the model parameters on abundant unlabeled monolingual data . In the second stage , the full or partial model is finetuned on a downstream translation task of labeled parallel sentences . When the amount of labeled data is limited , studies have demonstrated the benefit of pre-training for low-resource translation tasks ( Lewis et al. , 2019 ; Song et al. , 2019 ) . In many NMT applications , we are confronted with resource-rich languages which are characterized by millions of labeled parallel sentences . However , for these resource-rich tasks , pre-training representations rarely endows the NMT model with superior quality and , even worse , it sometimes can undermine the model ’ s performance if improperly utilized ( Zhu et al. , 2020 ) . This is partly due to catastrophic forgetting ( French , 1999 ) where prolonged finetuning on large corpora causes the learning to overwhelm the knowledge learned during pre-training . Several mitigation methods have been proposed for resource-rich machine translation ( Edunov et al. , 2019 ; Yang et al. , 2019 ; Zhu et al. , 2020 ) , such as freezing the pre-trained representations during the finetuning stage . In this paper , we study resource-rich machine translation through a different perspective of joint training where in contrast to the conventional two-stage approaches , we train NMT models in a single stage using the self-supervised objective ( on unlabeled monolingual sentences ) in addition to the supervised objective ( on labeled parallel sentences ) . The biggest challenge for this single-stage training paradigm is that self-supervised learning is less useful in joint training because it provides a much weaker learning signal that is easily dominated by the signal obtained through supervised learning . As a result , plausible approaches such as simply combining self-supervised and supervised learning objectives perform not much better than the supervised learning objective by itself . To this end , we introduce an approach to exploit complementary self-supervised learning signals to facilitate supervised learning in a joint training framework . Inspired by chromosomal crossovers ( Rieger et al. , 2012 ) , we propose a new task called crossover encoder-decoder ( or XEnDec ) which takes two training examples as inputs ( called parents ) , shuffles their source sentences , and produces a “ virtual ” sentence ( called offspring ) by a mixture decoder model . The key to our approach is to “ interbreed ” monolingual ( unlabeled ) and parallel ( labeled ) sentences through second filial generation with a crossover encoder-decoder , which we call F2-XEnDec , and train NMT models on the F2 offspring . As the F2 offspring exhibits combinations of traits that differ from those found in either parent , it turns out to be a meaningful objective to learn NMT models from both labeled and unlabeled sentences in a joint training framework . To the best of our knowledge , the proposed F2-XEnDec is among the first joint training approaches that substantially improve resource-rich machine translation . Closest to our work are two-stage approaches by Zhu et al . ( 2020 ) and Yang et al . ( 2019 ) who designed special finetuning objectives . Compared to their approaches , our focus lies on addressing a different challenge which is making self-supervised learning complementary to joint training of supervised NMT models on large labeled parallel corpora . Our experimental results substantiate the competitiveness of the proposed joint training approach . Furthermore , our results suggest that the approach improves the robustness of NMT models ( Belinkov & Bisk , 2018 ; Cheng et al. , 2019 ) . Contemporary NMT systems often lack robustness and therefore suffer from dramatic performance drops when they are exposed to input perturbations , even though these perturbations may not be strong enough to alter the meaning of the input sentence . Our improvement in robustness is interesting as none of the two-stage training approaches have ever reported this behavior . We empirically validate our approach on the WMT ’ 14 English-German and WMT ’ 14 EnglishFrench translation benchmarks which yields an improvement of 2.13 and 1.78 BLEU points over a vanilla Transformer model baseline . We also achieve a new state of the art of 46 BLEU on the WMT ’ 14 English-French translation task when further incorporating the back translation technique into our approach . In summary , our contributions are as follows : 1 . We propose a crossover encoder-decoder ( XEnDec ) that generates ” virtual ” examples over pairs of training examples . We discuss its relation to the standard self-supervised learning objective that can be recovered by XEnDec . 2 . We combine self-supervised and supervised losses in a joint training framework using our proposed F2-XEnDec and show that self-supervised learning is complementary to supervised learning for resource-rich NMT . 3 . Our approach achieves significant improvements on resource-rich translation tasks and exhibits higher robustness against input perturbations , particularly to code-switching noise . 2 BACKGROUND . 2.1 NEURAL MACHINE TRANSLATION . Under the encoder-decoder paradigm ( Bahdanau et al. , 2015 ; Gehring et al. , 2017 ; Vaswani et al. , 2017 ) , the conditional probability P ( y\|x ; θ ) of a target-language sentence y = y1 , · · · , yJ given a source-language sentence x = x1 , · · · , xI is modeled as follows : The encoder maps the source sentence x onto a sequence of I word embeddings e ( x ) = e ( x1 ) , ... , e ( xI ) . Then the word embeddings are encoded into their corresponding continuous hidden representations . The decoder acts as a conditional language model that reads embeddings e ( y ) for a shifted copy of y along with the aggregated contextual representations c. For clarity , we denote the input and output in the decoder as z and y , i.e . z = 〈s〉 , y1 , · · · , yJ−1 , where 〈s〉 is a start symbol . Conditioned on the aggregated contextual representation cj and its partial target input z≤j , the decoder generates y as P ( y\|x ; θ ) = ∏J j=1 P ( yj \|z≤j , cj ; θ ) . The aggregated contextual representation c is often calculated by summarizing the sentence x with an attention mechanism ( Bahdanau et al. , 2015 ) . A byproduct of the attention computation is a noisy alignment matrix A ∈ RJ×I which roughly captures the translation correspondence between target and source words ( Garg et al. , 2019 ) . Generally , NMT optimizes the model parameters θ by minimizing the empirical risk over a parallel training set ( x , y ) ∈ S : LS ( θ ) = E ( x , y ) ∈S [ ` ( f ( x , y ; θ ) , h ( y ) ) ] , ( 1 ) where ` is the cross entropy loss between the model prediction f ( x , y ; θ ) and h ( y ) , and h ( y ) denotes the sequence of one-hot label vectors for y with label smoothing in the Transformer ( Vaswani et al. , 2017 ) . 2.2 PRE-TRAINING FOR NEURAL MACHINE TRANSLATION . Pre-training sequence-to-sequence models for language generation is receiving increasing attention in the machine translation community ( Song et al. , 2019 ; Lewis et al. , 2019 ) . These methods generally comprise two stages : pre-training and finetuning . The pre-training takes advantage of the abundant monolingual corpus U = { y } to learn representations through a self-supervised objective called denoising autoencoder ( Vincent et al. , 2008 ) . The denoising autoencoder aims at reconstructing the original sentence y from one of its corrupted counterparts . Let y be obtained by corrupting y with a noise function n ( · ) and masking words . Then the pseudo parallel data ( y , y ) is fed into the NMT model to compute the reconstruction loss . The self-supervised loss over the monolingual corpus U is defined as : LU ( θ ) = E y∈U [ ` ( f ( y , y ; θ ) , h ( y ) ) ] , ( 2 ) The optimal model parameters θ ? are learned via a self-supervised loss LU ( θ ) and used to initialize downstream models during the finetuning on the parallel training set S. 3 CROSS-BREEDING : F2-XEnDec For resource-rich translation tasks in which a large parallel corpus S and ( virtually ) unlimited monolingual corpora U are available , our goal is to improve translation performance by exploiting selfsupervised signals to complement the supervised learning . In F2-XEnDec , we jointly train NMT models with supervised and self-supervised learning objectives in a single stage . We design a new objective LF2 and construct virtual data ( x̃ , ỹ ) to bridge the parallel data ( in supervised learning ) and the pseudo parallel data ( in self-supervised learning ) . The training loss over the virtual data ( x̃ , ỹ ) is computed as : LF2 ( θ ) = E y∈U E ( xp , yp ) ∈S [ ` ( f ( x̃ , ỹ ; θ ) , h ( ỹ ) ) ] , ( 3 ) where generating ( x̃ , ỹ ) depends on the parallel data ( xp , yp ) and the pseudo parallel data ( y , y ) . We propose a method called crossover encoder-decoder ( XEnDec ) that operates on two sentence pairs . As illustrated in Fig . 1 , the first generation ( Fig . 1 ( 1 ) ) uses XEnDec to combine monolingual sentences , thereby incurring a self-supervised proxy loss LF1 which is equivalent to LU . The second generation ( Fig . 1 ( 2 ) ) applies XEnDec between the offspring of the first generation ( y , y ) and the parallel sentence ( xp , yp ) to introduce LF2 . The final NMT models are optimized jointly on the original translation loss and the above two auxiliary losses . L ( θ ) = LS ( θ ) + LF1 ( θ ) + LF2 ( θ ) , ( 4 ) LF2 in Equation 4 is used to deeply fuse self-supervised and supervised training at instance level , rather than mixing them across instances mechanically . In the remainder of this section , we first detail XEnDec . We then discuss the relation between our approach , pre-training , and adversarial training objectives . Finally , we summarize some important techniques used in our approach and present the overall algorithm . 3.1 CROSSOVER ENCODER-DECODER This section introduces the crossover encoder-decoder ( XEnDec ) – an essential subtask for the proposed method . Different from a conventional encoder-decoder , XEnDec takes two training examples as inputs ( called parents ) , shuffles the parents ’ source sentences and produces a virtual example ( called offspring ) through a mixture decoder model . Fig . 2 illustrates this process . Formally , let ( x , y ) denote a training example where x = x1 , · · · , xI represents a source sentence of I words and y = y1 , · · · , yJ is the corresponding target sentence . In supervised training , x and y are parallel sentences . As we shall see in Section 3.4 , XEnDec can be carried out with and without supervision , although we do not distinguish both cases for now . Given a pair of examples ( x , y ) and ( x′ , y′ ) called parents , the crossover encoder shuffles the two source sequences into a new source sentence x̃ calculated from : x̃i = mixi + ( 1−mi ) x′i , ( 5 ) where m = m1 , · · · , mI ∈ { 0 , 1 } I stands for a series of Bernoulli random variables with each taking the value 1 with probability p. If mi = 1 , then the i-th word in x will be substituted with the word in x′ at the same position . For convenience , the lengths of the two sequences are aligned by padding tokens to the end of the shorter sentence . The crossover decoder employs a mixture model to generate the virtual target sentence . The embedding of the decoder ’ s input z̃ is computed as : e ( z̃j ) = 1 Z [ e ( yj−1 ) I∑ i=1 A ( j−1 ) imi + e ( y ′ j−1 ) I∑ i=1 A′ ( j−1 ) i ( 1−mi ) ] , ( 6 ) where e ( · ) is the embedding function . Z = ∑I i=1A ( j−1 ) imi+A ′ ( j−1 ) i ( 1−mi ) is the normalization term where A and A′ are the alignment matrices for the source sequences x and x′ , respectively . Equation 6 averages embeddings of y and y′ through the latent weights computed by m , A and A′ . The alignment matrix measures the contribution of the source words for generating a specific target word ( Och & Ney , 2004 ; Bahdanau et al. , 2015 ) . For example , Aj : indicates the contribution scores of each source word for the j-th word in the target sentence . For simplicity , this paper uses the attention matrix learned in the NMT model as a noisy alignment matrix ( Garg et al. , 2019 ) . Likewise , we compute label vectors for the crossover decoder by : h ( ỹj ) = 1 Z [ h ( yj ) I∑ i=1 Ajimi + h ( y ′ j ) I∑ i=1 A′ji ( 1−mi ) ] , ( 7 ) The h ( · ) function projects a word onto its label vector , e.g . a one-hot vector . The loss of XEnDec over ( x , y , x′ , y′ ) is computed as the negative log-likelihood of generating the virtual sentence ỹ : ` ( f ( x̃ , ỹ ; θ ) , h ( ỹ ) ) = − logP ( ỹ\|x̃ ; θ ) = ∑ j −Ey∈h ( ỹj ) logP ( y\|z̃≤j , cj ; θ ) , ( 8 ) Notice that even though we do not directly observe the “ virtual sentences ” z̃ and ỹ , we are still able to compute the loss using their embeddings . In practice , the length of x̃ is \|x\| whereas lengths of both ỹ and z̃ are max ( \|y\| , \|y′\| ) .	This submission proposes a joint training of self-supervised training and supervised training for neural machine translation (NMT), especially for the rich-resource language datasets. The method proposed, F$_2$-XEnDec, exploits the "crossover" operation of the monolingual sentences and bilingual data pairs in the encoder-decoder framework, and train the self-supervised (on monolingual data) objective and supervised (on bilingual data) objective, and the mixed version (crossover) together, without a clear per-train and fine-tuning stage like previous BERT related works. Experiments are conducted on two rich-resource language pairs: WMT14 English-German and WMT14 English-French translation tasks. The results set a new state-of-the-art record for English-French, and superior performances are achieved on English-German language pairs. The authors also analyze the robustness and ablation studies of their method.	SP:7ed79cb7d0bb12d3d782b5a79f07137563531484
Self-supervised and Supervised Joint Training for Resource-rich Machine Translation	1 INTRODUCTION . Self-supervised pre-training of text representations ( Peters et al. , 2018 ; Radford et al. , 2018 ) has achieved tremendous success in natural language processing applications . Inspired by BERT ( Devlin et al. , 2019 ) , recent works attempt to leverage sequence-to-sequence model pre-training for Neural Machine Translation ( NMT ) . Generally , these methods comprise two stages : pre-training and finetuning . During the pre-training stage , a proxy task , e.g . the Cloze task ( Devlin et al. , 2019 ) , is used to learn the model parameters on abundant unlabeled monolingual data . In the second stage , the full or partial model is finetuned on a downstream translation task of labeled parallel sentences . When the amount of labeled data is limited , studies have demonstrated the benefit of pre-training for low-resource translation tasks ( Lewis et al. , 2019 ; Song et al. , 2019 ) . In many NMT applications , we are confronted with resource-rich languages which are characterized by millions of labeled parallel sentences . However , for these resource-rich tasks , pre-training representations rarely endows the NMT model with superior quality and , even worse , it sometimes can undermine the model ’ s performance if improperly utilized ( Zhu et al. , 2020 ) . This is partly due to catastrophic forgetting ( French , 1999 ) where prolonged finetuning on large corpora causes the learning to overwhelm the knowledge learned during pre-training . Several mitigation methods have been proposed for resource-rich machine translation ( Edunov et al. , 2019 ; Yang et al. , 2019 ; Zhu et al. , 2020 ) , such as freezing the pre-trained representations during the finetuning stage . In this paper , we study resource-rich machine translation through a different perspective of joint training where in contrast to the conventional two-stage approaches , we train NMT models in a single stage using the self-supervised objective ( on unlabeled monolingual sentences ) in addition to the supervised objective ( on labeled parallel sentences ) . The biggest challenge for this single-stage training paradigm is that self-supervised learning is less useful in joint training because it provides a much weaker learning signal that is easily dominated by the signal obtained through supervised learning . As a result , plausible approaches such as simply combining self-supervised and supervised learning objectives perform not much better than the supervised learning objective by itself . To this end , we introduce an approach to exploit complementary self-supervised learning signals to facilitate supervised learning in a joint training framework . Inspired by chromosomal crossovers ( Rieger et al. , 2012 ) , we propose a new task called crossover encoder-decoder ( or XEnDec ) which takes two training examples as inputs ( called parents ) , shuffles their source sentences , and produces a “ virtual ” sentence ( called offspring ) by a mixture decoder model . The key to our approach is to “ interbreed ” monolingual ( unlabeled ) and parallel ( labeled ) sentences through second filial generation with a crossover encoder-decoder , which we call F2-XEnDec , and train NMT models on the F2 offspring . As the F2 offspring exhibits combinations of traits that differ from those found in either parent , it turns out to be a meaningful objective to learn NMT models from both labeled and unlabeled sentences in a joint training framework . To the best of our knowledge , the proposed F2-XEnDec is among the first joint training approaches that substantially improve resource-rich machine translation . Closest to our work are two-stage approaches by Zhu et al . ( 2020 ) and Yang et al . ( 2019 ) who designed special finetuning objectives . Compared to their approaches , our focus lies on addressing a different challenge which is making self-supervised learning complementary to joint training of supervised NMT models on large labeled parallel corpora . Our experimental results substantiate the competitiveness of the proposed joint training approach . Furthermore , our results suggest that the approach improves the robustness of NMT models ( Belinkov & Bisk , 2018 ; Cheng et al. , 2019 ) . Contemporary NMT systems often lack robustness and therefore suffer from dramatic performance drops when they are exposed to input perturbations , even though these perturbations may not be strong enough to alter the meaning of the input sentence . Our improvement in robustness is interesting as none of the two-stage training approaches have ever reported this behavior . We empirically validate our approach on the WMT ’ 14 English-German and WMT ’ 14 EnglishFrench translation benchmarks which yields an improvement of 2.13 and 1.78 BLEU points over a vanilla Transformer model baseline . We also achieve a new state of the art of 46 BLEU on the WMT ’ 14 English-French translation task when further incorporating the back translation technique into our approach . In summary , our contributions are as follows : 1 . We propose a crossover encoder-decoder ( XEnDec ) that generates ” virtual ” examples over pairs of training examples . We discuss its relation to the standard self-supervised learning objective that can be recovered by XEnDec . 2 . We combine self-supervised and supervised losses in a joint training framework using our proposed F2-XEnDec and show that self-supervised learning is complementary to supervised learning for resource-rich NMT . 3 . Our approach achieves significant improvements on resource-rich translation tasks and exhibits higher robustness against input perturbations , particularly to code-switching noise . 2 BACKGROUND . 2.1 NEURAL MACHINE TRANSLATION . Under the encoder-decoder paradigm ( Bahdanau et al. , 2015 ; Gehring et al. , 2017 ; Vaswani et al. , 2017 ) , the conditional probability P ( y\|x ; θ ) of a target-language sentence y = y1 , · · · , yJ given a source-language sentence x = x1 , · · · , xI is modeled as follows : The encoder maps the source sentence x onto a sequence of I word embeddings e ( x ) = e ( x1 ) , ... , e ( xI ) . Then the word embeddings are encoded into their corresponding continuous hidden representations . The decoder acts as a conditional language model that reads embeddings e ( y ) for a shifted copy of y along with the aggregated contextual representations c. For clarity , we denote the input and output in the decoder as z and y , i.e . z = 〈s〉 , y1 , · · · , yJ−1 , where 〈s〉 is a start symbol . Conditioned on the aggregated contextual representation cj and its partial target input z≤j , the decoder generates y as P ( y\|x ; θ ) = ∏J j=1 P ( yj \|z≤j , cj ; θ ) . The aggregated contextual representation c is often calculated by summarizing the sentence x with an attention mechanism ( Bahdanau et al. , 2015 ) . A byproduct of the attention computation is a noisy alignment matrix A ∈ RJ×I which roughly captures the translation correspondence between target and source words ( Garg et al. , 2019 ) . Generally , NMT optimizes the model parameters θ by minimizing the empirical risk over a parallel training set ( x , y ) ∈ S : LS ( θ ) = E ( x , y ) ∈S [ ` ( f ( x , y ; θ ) , h ( y ) ) ] , ( 1 ) where ` is the cross entropy loss between the model prediction f ( x , y ; θ ) and h ( y ) , and h ( y ) denotes the sequence of one-hot label vectors for y with label smoothing in the Transformer ( Vaswani et al. , 2017 ) . 2.2 PRE-TRAINING FOR NEURAL MACHINE TRANSLATION . Pre-training sequence-to-sequence models for language generation is receiving increasing attention in the machine translation community ( Song et al. , 2019 ; Lewis et al. , 2019 ) . These methods generally comprise two stages : pre-training and finetuning . The pre-training takes advantage of the abundant monolingual corpus U = { y } to learn representations through a self-supervised objective called denoising autoencoder ( Vincent et al. , 2008 ) . The denoising autoencoder aims at reconstructing the original sentence y from one of its corrupted counterparts . Let y be obtained by corrupting y with a noise function n ( · ) and masking words . Then the pseudo parallel data ( y , y ) is fed into the NMT model to compute the reconstruction loss . The self-supervised loss over the monolingual corpus U is defined as : LU ( θ ) = E y∈U [ ` ( f ( y , y ; θ ) , h ( y ) ) ] , ( 2 ) The optimal model parameters θ ? are learned via a self-supervised loss LU ( θ ) and used to initialize downstream models during the finetuning on the parallel training set S. 3 CROSS-BREEDING : F2-XEnDec For resource-rich translation tasks in which a large parallel corpus S and ( virtually ) unlimited monolingual corpora U are available , our goal is to improve translation performance by exploiting selfsupervised signals to complement the supervised learning . In F2-XEnDec , we jointly train NMT models with supervised and self-supervised learning objectives in a single stage . We design a new objective LF2 and construct virtual data ( x̃ , ỹ ) to bridge the parallel data ( in supervised learning ) and the pseudo parallel data ( in self-supervised learning ) . The training loss over the virtual data ( x̃ , ỹ ) is computed as : LF2 ( θ ) = E y∈U E ( xp , yp ) ∈S [ ` ( f ( x̃ , ỹ ; θ ) , h ( ỹ ) ) ] , ( 3 ) where generating ( x̃ , ỹ ) depends on the parallel data ( xp , yp ) and the pseudo parallel data ( y , y ) . We propose a method called crossover encoder-decoder ( XEnDec ) that operates on two sentence pairs . As illustrated in Fig . 1 , the first generation ( Fig . 1 ( 1 ) ) uses XEnDec to combine monolingual sentences , thereby incurring a self-supervised proxy loss LF1 which is equivalent to LU . The second generation ( Fig . 1 ( 2 ) ) applies XEnDec between the offspring of the first generation ( y , y ) and the parallel sentence ( xp , yp ) to introduce LF2 . The final NMT models are optimized jointly on the original translation loss and the above two auxiliary losses . L ( θ ) = LS ( θ ) + LF1 ( θ ) + LF2 ( θ ) , ( 4 ) LF2 in Equation 4 is used to deeply fuse self-supervised and supervised training at instance level , rather than mixing them across instances mechanically . In the remainder of this section , we first detail XEnDec . We then discuss the relation between our approach , pre-training , and adversarial training objectives . Finally , we summarize some important techniques used in our approach and present the overall algorithm . 3.1 CROSSOVER ENCODER-DECODER This section introduces the crossover encoder-decoder ( XEnDec ) – an essential subtask for the proposed method . Different from a conventional encoder-decoder , XEnDec takes two training examples as inputs ( called parents ) , shuffles the parents ’ source sentences and produces a virtual example ( called offspring ) through a mixture decoder model . Fig . 2 illustrates this process . Formally , let ( x , y ) denote a training example where x = x1 , · · · , xI represents a source sentence of I words and y = y1 , · · · , yJ is the corresponding target sentence . In supervised training , x and y are parallel sentences . As we shall see in Section 3.4 , XEnDec can be carried out with and without supervision , although we do not distinguish both cases for now . Given a pair of examples ( x , y ) and ( x′ , y′ ) called parents , the crossover encoder shuffles the two source sequences into a new source sentence x̃ calculated from : x̃i = mixi + ( 1−mi ) x′i , ( 5 ) where m = m1 , · · · , mI ∈ { 0 , 1 } I stands for a series of Bernoulli random variables with each taking the value 1 with probability p. If mi = 1 , then the i-th word in x will be substituted with the word in x′ at the same position . For convenience , the lengths of the two sequences are aligned by padding tokens to the end of the shorter sentence . The crossover decoder employs a mixture model to generate the virtual target sentence . The embedding of the decoder ’ s input z̃ is computed as : e ( z̃j ) = 1 Z [ e ( yj−1 ) I∑ i=1 A ( j−1 ) imi + e ( y ′ j−1 ) I∑ i=1 A′ ( j−1 ) i ( 1−mi ) ] , ( 6 ) where e ( · ) is the embedding function . Z = ∑I i=1A ( j−1 ) imi+A ′ ( j−1 ) i ( 1−mi ) is the normalization term where A and A′ are the alignment matrices for the source sequences x and x′ , respectively . Equation 6 averages embeddings of y and y′ through the latent weights computed by m , A and A′ . The alignment matrix measures the contribution of the source words for generating a specific target word ( Och & Ney , 2004 ; Bahdanau et al. , 2015 ) . For example , Aj : indicates the contribution scores of each source word for the j-th word in the target sentence . For simplicity , this paper uses the attention matrix learned in the NMT model as a noisy alignment matrix ( Garg et al. , 2019 ) . Likewise , we compute label vectors for the crossover decoder by : h ( ỹj ) = 1 Z [ h ( yj ) I∑ i=1 Ajimi + h ( y ′ j ) I∑ i=1 A′ji ( 1−mi ) ] , ( 7 ) The h ( · ) function projects a word onto its label vector , e.g . a one-hot vector . The loss of XEnDec over ( x , y , x′ , y′ ) is computed as the negative log-likelihood of generating the virtual sentence ỹ : ` ( f ( x̃ , ỹ ; θ ) , h ( ỹ ) ) = − logP ( ỹ\|x̃ ; θ ) = ∑ j −Ey∈h ( ỹj ) logP ( y\|z̃≤j , cj ; θ ) , ( 8 ) Notice that even though we do not directly observe the “ virtual sentences ” z̃ and ỹ , we are still able to compute the loss using their embeddings . In practice , the length of x̃ is \|x\| whereas lengths of both ỹ and z̃ are max ( \|y\| , \|y′\| ) .	This paper introduces a new approach to semi-supervised training of neural machine translation. During training, the traditional supervised loss is complemented with two auxiliary losses: a denoising autoencoder and what the authors call cross-breeding: shuffling together the source side of a sentence pair with an unrelated, noised monolingual sentence, and predicting a virtual target sentence whose embedings and labels are a linear interpolation of the paired target sentence and the monolingual sentence, with interpolation weights being based on an attention matrix of the input words selected during shuffling. Authors report an improvement of around 2 BLEU on WMT14 EN-DE and EN-FR over a purely supervised Transformer, and improvements of 1 BLEU (EN-DE) and 0.4 BLEU (EN-FR) over a system with back-translation. Additionally, the system shows robustness against a type of "code-switching" noise.	SP:7ed79cb7d0bb12d3d782b5a79f07137563531484
Self-supervised and Supervised Joint Training for Resource-rich Machine Translation	1 INTRODUCTION . Self-supervised pre-training of text representations ( Peters et al. , 2018 ; Radford et al. , 2018 ) has achieved tremendous success in natural language processing applications . Inspired by BERT ( Devlin et al. , 2019 ) , recent works attempt to leverage sequence-to-sequence model pre-training for Neural Machine Translation ( NMT ) . Generally , these methods comprise two stages : pre-training and finetuning . During the pre-training stage , a proxy task , e.g . the Cloze task ( Devlin et al. , 2019 ) , is used to learn the model parameters on abundant unlabeled monolingual data . In the second stage , the full or partial model is finetuned on a downstream translation task of labeled parallel sentences . When the amount of labeled data is limited , studies have demonstrated the benefit of pre-training for low-resource translation tasks ( Lewis et al. , 2019 ; Song et al. , 2019 ) . In many NMT applications , we are confronted with resource-rich languages which are characterized by millions of labeled parallel sentences . However , for these resource-rich tasks , pre-training representations rarely endows the NMT model with superior quality and , even worse , it sometimes can undermine the model ’ s performance if improperly utilized ( Zhu et al. , 2020 ) . This is partly due to catastrophic forgetting ( French , 1999 ) where prolonged finetuning on large corpora causes the learning to overwhelm the knowledge learned during pre-training . Several mitigation methods have been proposed for resource-rich machine translation ( Edunov et al. , 2019 ; Yang et al. , 2019 ; Zhu et al. , 2020 ) , such as freezing the pre-trained representations during the finetuning stage . In this paper , we study resource-rich machine translation through a different perspective of joint training where in contrast to the conventional two-stage approaches , we train NMT models in a single stage using the self-supervised objective ( on unlabeled monolingual sentences ) in addition to the supervised objective ( on labeled parallel sentences ) . The biggest challenge for this single-stage training paradigm is that self-supervised learning is less useful in joint training because it provides a much weaker learning signal that is easily dominated by the signal obtained through supervised learning . As a result , plausible approaches such as simply combining self-supervised and supervised learning objectives perform not much better than the supervised learning objective by itself . To this end , we introduce an approach to exploit complementary self-supervised learning signals to facilitate supervised learning in a joint training framework . Inspired by chromosomal crossovers ( Rieger et al. , 2012 ) , we propose a new task called crossover encoder-decoder ( or XEnDec ) which takes two training examples as inputs ( called parents ) , shuffles their source sentences , and produces a “ virtual ” sentence ( called offspring ) by a mixture decoder model . The key to our approach is to “ interbreed ” monolingual ( unlabeled ) and parallel ( labeled ) sentences through second filial generation with a crossover encoder-decoder , which we call F2-XEnDec , and train NMT models on the F2 offspring . As the F2 offspring exhibits combinations of traits that differ from those found in either parent , it turns out to be a meaningful objective to learn NMT models from both labeled and unlabeled sentences in a joint training framework . To the best of our knowledge , the proposed F2-XEnDec is among the first joint training approaches that substantially improve resource-rich machine translation . Closest to our work are two-stage approaches by Zhu et al . ( 2020 ) and Yang et al . ( 2019 ) who designed special finetuning objectives . Compared to their approaches , our focus lies on addressing a different challenge which is making self-supervised learning complementary to joint training of supervised NMT models on large labeled parallel corpora . Our experimental results substantiate the competitiveness of the proposed joint training approach . Furthermore , our results suggest that the approach improves the robustness of NMT models ( Belinkov & Bisk , 2018 ; Cheng et al. , 2019 ) . Contemporary NMT systems often lack robustness and therefore suffer from dramatic performance drops when they are exposed to input perturbations , even though these perturbations may not be strong enough to alter the meaning of the input sentence . Our improvement in robustness is interesting as none of the two-stage training approaches have ever reported this behavior . We empirically validate our approach on the WMT ’ 14 English-German and WMT ’ 14 EnglishFrench translation benchmarks which yields an improvement of 2.13 and 1.78 BLEU points over a vanilla Transformer model baseline . We also achieve a new state of the art of 46 BLEU on the WMT ’ 14 English-French translation task when further incorporating the back translation technique into our approach . In summary , our contributions are as follows : 1 . We propose a crossover encoder-decoder ( XEnDec ) that generates ” virtual ” examples over pairs of training examples . We discuss its relation to the standard self-supervised learning objective that can be recovered by XEnDec . 2 . We combine self-supervised and supervised losses in a joint training framework using our proposed F2-XEnDec and show that self-supervised learning is complementary to supervised learning for resource-rich NMT . 3 . Our approach achieves significant improvements on resource-rich translation tasks and exhibits higher robustness against input perturbations , particularly to code-switching noise . 2 BACKGROUND . 2.1 NEURAL MACHINE TRANSLATION . Under the encoder-decoder paradigm ( Bahdanau et al. , 2015 ; Gehring et al. , 2017 ; Vaswani et al. , 2017 ) , the conditional probability P ( y\|x ; θ ) of a target-language sentence y = y1 , · · · , yJ given a source-language sentence x = x1 , · · · , xI is modeled as follows : The encoder maps the source sentence x onto a sequence of I word embeddings e ( x ) = e ( x1 ) , ... , e ( xI ) . Then the word embeddings are encoded into their corresponding continuous hidden representations . The decoder acts as a conditional language model that reads embeddings e ( y ) for a shifted copy of y along with the aggregated contextual representations c. For clarity , we denote the input and output in the decoder as z and y , i.e . z = 〈s〉 , y1 , · · · , yJ−1 , where 〈s〉 is a start symbol . Conditioned on the aggregated contextual representation cj and its partial target input z≤j , the decoder generates y as P ( y\|x ; θ ) = ∏J j=1 P ( yj \|z≤j , cj ; θ ) . The aggregated contextual representation c is often calculated by summarizing the sentence x with an attention mechanism ( Bahdanau et al. , 2015 ) . A byproduct of the attention computation is a noisy alignment matrix A ∈ RJ×I which roughly captures the translation correspondence between target and source words ( Garg et al. , 2019 ) . Generally , NMT optimizes the model parameters θ by minimizing the empirical risk over a parallel training set ( x , y ) ∈ S : LS ( θ ) = E ( x , y ) ∈S [ ` ( f ( x , y ; θ ) , h ( y ) ) ] , ( 1 ) where ` is the cross entropy loss between the model prediction f ( x , y ; θ ) and h ( y ) , and h ( y ) denotes the sequence of one-hot label vectors for y with label smoothing in the Transformer ( Vaswani et al. , 2017 ) . 2.2 PRE-TRAINING FOR NEURAL MACHINE TRANSLATION . Pre-training sequence-to-sequence models for language generation is receiving increasing attention in the machine translation community ( Song et al. , 2019 ; Lewis et al. , 2019 ) . These methods generally comprise two stages : pre-training and finetuning . The pre-training takes advantage of the abundant monolingual corpus U = { y } to learn representations through a self-supervised objective called denoising autoencoder ( Vincent et al. , 2008 ) . The denoising autoencoder aims at reconstructing the original sentence y from one of its corrupted counterparts . Let y be obtained by corrupting y with a noise function n ( · ) and masking words . Then the pseudo parallel data ( y , y ) is fed into the NMT model to compute the reconstruction loss . The self-supervised loss over the monolingual corpus U is defined as : LU ( θ ) = E y∈U [ ` ( f ( y , y ; θ ) , h ( y ) ) ] , ( 2 ) The optimal model parameters θ ? are learned via a self-supervised loss LU ( θ ) and used to initialize downstream models during the finetuning on the parallel training set S. 3 CROSS-BREEDING : F2-XEnDec For resource-rich translation tasks in which a large parallel corpus S and ( virtually ) unlimited monolingual corpora U are available , our goal is to improve translation performance by exploiting selfsupervised signals to complement the supervised learning . In F2-XEnDec , we jointly train NMT models with supervised and self-supervised learning objectives in a single stage . We design a new objective LF2 and construct virtual data ( x̃ , ỹ ) to bridge the parallel data ( in supervised learning ) and the pseudo parallel data ( in self-supervised learning ) . The training loss over the virtual data ( x̃ , ỹ ) is computed as : LF2 ( θ ) = E y∈U E ( xp , yp ) ∈S [ ` ( f ( x̃ , ỹ ; θ ) , h ( ỹ ) ) ] , ( 3 ) where generating ( x̃ , ỹ ) depends on the parallel data ( xp , yp ) and the pseudo parallel data ( y , y ) . We propose a method called crossover encoder-decoder ( XEnDec ) that operates on two sentence pairs . As illustrated in Fig . 1 , the first generation ( Fig . 1 ( 1 ) ) uses XEnDec to combine monolingual sentences , thereby incurring a self-supervised proxy loss LF1 which is equivalent to LU . The second generation ( Fig . 1 ( 2 ) ) applies XEnDec between the offspring of the first generation ( y , y ) and the parallel sentence ( xp , yp ) to introduce LF2 . The final NMT models are optimized jointly on the original translation loss and the above two auxiliary losses . L ( θ ) = LS ( θ ) + LF1 ( θ ) + LF2 ( θ ) , ( 4 ) LF2 in Equation 4 is used to deeply fuse self-supervised and supervised training at instance level , rather than mixing them across instances mechanically . In the remainder of this section , we first detail XEnDec . We then discuss the relation between our approach , pre-training , and adversarial training objectives . Finally , we summarize some important techniques used in our approach and present the overall algorithm . 3.1 CROSSOVER ENCODER-DECODER This section introduces the crossover encoder-decoder ( XEnDec ) – an essential subtask for the proposed method . Different from a conventional encoder-decoder , XEnDec takes two training examples as inputs ( called parents ) , shuffles the parents ’ source sentences and produces a virtual example ( called offspring ) through a mixture decoder model . Fig . 2 illustrates this process . Formally , let ( x , y ) denote a training example where x = x1 , · · · , xI represents a source sentence of I words and y = y1 , · · · , yJ is the corresponding target sentence . In supervised training , x and y are parallel sentences . As we shall see in Section 3.4 , XEnDec can be carried out with and without supervision , although we do not distinguish both cases for now . Given a pair of examples ( x , y ) and ( x′ , y′ ) called parents , the crossover encoder shuffles the two source sequences into a new source sentence x̃ calculated from : x̃i = mixi + ( 1−mi ) x′i , ( 5 ) where m = m1 , · · · , mI ∈ { 0 , 1 } I stands for a series of Bernoulli random variables with each taking the value 1 with probability p. If mi = 1 , then the i-th word in x will be substituted with the word in x′ at the same position . For convenience , the lengths of the two sequences are aligned by padding tokens to the end of the shorter sentence . The crossover decoder employs a mixture model to generate the virtual target sentence . The embedding of the decoder ’ s input z̃ is computed as : e ( z̃j ) = 1 Z [ e ( yj−1 ) I∑ i=1 A ( j−1 ) imi + e ( y ′ j−1 ) I∑ i=1 A′ ( j−1 ) i ( 1−mi ) ] , ( 6 ) where e ( · ) is the embedding function . Z = ∑I i=1A ( j−1 ) imi+A ′ ( j−1 ) i ( 1−mi ) is the normalization term where A and A′ are the alignment matrices for the source sequences x and x′ , respectively . Equation 6 averages embeddings of y and y′ through the latent weights computed by m , A and A′ . The alignment matrix measures the contribution of the source words for generating a specific target word ( Och & Ney , 2004 ; Bahdanau et al. , 2015 ) . For example , Aj : indicates the contribution scores of each source word for the j-th word in the target sentence . For simplicity , this paper uses the attention matrix learned in the NMT model as a noisy alignment matrix ( Garg et al. , 2019 ) . Likewise , we compute label vectors for the crossover decoder by : h ( ỹj ) = 1 Z [ h ( yj ) I∑ i=1 Ajimi + h ( y ′ j ) I∑ i=1 A′ji ( 1−mi ) ] , ( 7 ) The h ( · ) function projects a word onto its label vector , e.g . a one-hot vector . The loss of XEnDec over ( x , y , x′ , y′ ) is computed as the negative log-likelihood of generating the virtual sentence ỹ : ` ( f ( x̃ , ỹ ; θ ) , h ( ỹ ) ) = − logP ( ỹ\|x̃ ; θ ) = ∑ j −Ey∈h ( ỹj ) logP ( y\|z̃≤j , cj ; θ ) , ( 8 ) Notice that even though we do not directly observe the “ virtual sentences ” z̃ and ỹ , we are still able to compute the loss using their embeddings . In practice , the length of x̃ is \|x\| whereas lengths of both ỹ and z̃ are max ( \|y\| , \|y′\| ) .	This paper proposes a joint training strategy that combines supervised learning on parallel data and self-supervised learning on monolingual data for NMT. The monolingual sentences are corrupted with word order shuffling and masking. And a cross encoder-decoder is introduced to fuse the parallel source-side sentence and the corrupted monolingual sentence. The NMT models are optimized jointly with cross-entropy loss of parallel data, reconstruction loss for monolingual data, and cross-entropy of the virtual fused data.	SP:7ed79cb7d0bb12d3d782b5a79f07137563531484
Differentiable Weighted Finite-State Transducers	1 INTRODUCTION . Weighted finite-state transducers ( WFSTs ) are a commonly used tool in speech and language processing ( Knight & May , 2009 ; Mohri et al. , 2002 ) . They are most frequently used to combine predictions from multiple already trained models . In speech recognition , for example , WFSTs are used to combine constraints from an acoustic-to-phoneme model , a lexicon mapping words to pronunciations , and a word-level language model . However , combining separately learned models using WFSTs only at inference time has several drawbacks , including the well-known problems of exposure bias ( Ranzato et al. , 2015 ) and label bias ( Bottou , 1991 ; Lafferty et al. , 2001 ) . Given that gradients may be computed for most WFST operations , using them only at the inference stage of a learning system is not a hard limitation . We speculate that this limitation is primarily due to practical considerations . Historically , hardware has not been sufficiently performant to make training with WFSTs tractable . Also , no implementation exists with the required operations which supports automatic differentiation in a high-level yet efficient manner . We develop a framework for automatic differentiation through operations on WFSTs . We show the utility of this framework by leveraging it to design and experiment with existing and novel learning algorithms . Automata are a more convenient structure than tensors to encode prior knowledge into a learning algorithm . However , not training with them limits the extent to which this prior knowledge can be incorporated in a useful manner . A framework for differentiable WFSTs allows the model to jointly learn from training data as well as prior knowledge encoded in WFSTs . This enables the learning algorithm to incorporate such knowledge in the best possible way . Use of WFSTs conveniently decomposes operations from data ( i.e . graphs ) . For example , rather than hand-coding sequence-level loss functions such as Connectionist Temporal Classification ( CTC ) ( Graves et al. , 2006 ) or the Automatic Segmentation Criterion ( ASG ) ( Collobert et al. , 2016 ) , we may specify the core assumptions of the criteria in graphs and compute the resulting loss with graph operations . This facilitates exploration in the space of such structured loss functions . We show the utility of the differentiable WFST framework by designing and testing several algorithms . For example , bi-gram transitions may be added to CTC with a transition WFST . We scale transitions to large token set sizes by encoding pruning and back-off in the transition graph . Word pieces are commonly used as the output of speech recognition and machine translation models ( Chiu et al. , 2018 ; Sennrich et al. , 2016 ) . The word piece decomposition for a word is learned with a task-independent model . Instead , we use WFSTs to marginalize over the latent word piece decomposition at training time . This lets the model learn decompositions salient to the task at hand . Finally , we show that WFSTs may be used as layers themselves intermixed with tensor-based layers . We propose a convolutional WFST layer which maps lower-level representations to higher-level representations . The WFST convolution can be trained with the rest of the model and results in improved accuracy with fewer parameters and operations as compared to a traditional convolution . In summary , our contributions are : • A framework for automatic differentiation with WFSTs . The framework supports both C++ and Python front-ends and is available at https : //www.anonymized.com . • We show that the framework may be used to express both existing sequence-level loss functions and to design novel sequence-level loss functions . • We propose a convolutional WFST layer which can be used in the interior of a deep neural network to map lower-level representations to higher-level representations . • We demonstrate the effectiveness of using WFSTs in the manners described above with experiments in automatic speech and handwriting recognition . 2 RELATED WORK . A wealth of prior work exists using weighted finite-state automata in speech recognition , natural language processing , optical character recognition , and other applications ( Breuel , 2008 ; Knight & May , 2009 ; Mohri , 1997 ; Mohri et al. , 2008 ; Pereira et al. , 1994 ) . However , the use of WFSTs is limited mostly to the inference stage of a predictive system . For example , Kaldi , a commonly used toolkit for automatic speech recognition , uses WFSTs extensively , but in most cases for inference or to estimate the parameters of shallow models ( Povey et al. , 2011 ) . In some cases , WFSTs are used statically to incorporate fixed lattices in discriminative sequence criteria ( Kingsbury , 2009 ; Kingsbury et al. , 2012 ; Su et al. , 2013 ; Veselỳ et al. , 2013 ) . Implementations of sequence criteria in end-to-end style training are typically hand-crafted with careful consideration for speed ( Amodei et al. , 2016 ; Collobert et al. , 2019 ; Povey et al. , 2016 ) . The use of hand-crafted implementations reduces flexibility which limits research . In some cases , such as the fully differentiable beam search of Collobert et al . ( 2019 ) , achieving the necessary computational efficiency with a WFST-based implementation may not yet be tractable . However , as a first step , we show that in many common cases we can have the expressiveness afforded by the differentiable WFST framework without paying an unacceptable penalty in execution time . The ideas of learning with WFSTs ( Eisner , 2002 ) and automatic differentiation through operations on graphs ( Bottou et al. , 1997 ) are not new . However , no simple and efficient frameworks exist . Especially related to this work , and inspiring the name of our framework , are the graph transformer networks of Bottou et al . ( 1997 ) . Generalized graph transducers ( Bottou et al. , 1996 ) are more expressive than WFSTs , allowing arbitrary data as edge labels . When composing these graphs , one defines an edge matching function and a “ transformer ” to construct the resulting structure . While not this general , differentiable WFSTs nevertheless allow for a vast design space of interesting algorithms , and perhaps make a more pragmatic trade-off between flexibility and efficiency . Highly efficient libraries for operations on WFSTs exist , notably OpenFST and its predecessor FSM ( Allauzen et al. , 2007 ; Mohri et al. , 2000 ) . We take inspiration from OpenFst in the interface and implementation of many of our functions . However , the design implications of operating on WFSTs with automatic differentiation are quite different than those of the use cases OpenFST has been optimized for . We also draw inspiration from libraries for automatic differentiation and deep learning ( Collobert et al. , 2011 ; Paszke et al. , 2019 ; Pratap et al. , 2019 ; Tokui et al. , 2015 ) . Some of the algorithms we propose , with the goal of demonstrating the utility of the differentiable WFST library , are inspired by prior work . Prior work has explored pruning the set of allowed alignments with CTC , and in particular limiting the spacing between output tokens ( Liu et al. , 2018 ) . Learning n-gram word decompositions with both differentiable ( Liu et al. , 2017 ) and nondifferentiable ( Chan et al. , 2017 ) loss functions has also been explored . 3 DIFFERENTIABLE WEIGHTED FINITE-STATE TRANSDUCERS . A weighted finite-state acceptor A is a 6-tuple consisting of an alphabet Σ , a set of states Q , start states Qs , accepting states Qa , a transition function π ( q , p ) which maps elements of Q × Σ to elements of Q , and a weight function ω ( q , p ) which maps elements of Q × Σ to R. A weighted finite-state transducer T is a 7-tuple which augments an acceptor with an output alphabet ∆ . The transition function π ( q , p , r ) and weight function ω ( q , p , r ) map elements of Q×Σ×∆ to elements of Q and R respectively . In other words , each edge of a transducer connects two states and has an input label p , an output label r and a weight w ∈ R. We denote an input by p = [ p1 , . . . , pT ] where each pi ∈ Σ . An acceptor A accepts the input p if there exists a sequence of states qi+1 = π ( qi , pi ) ∈ Q such that q1 ∈ Qs and qT+1 ∈ Qa . With a slight abuse of notation , we let p ∈ A denote that A accepts p. The score of p is given by s ( p ) =∑T i=1 ω ( qi , pi ) . Let r = [ r1 , . . . , rT ] be a path with ri ∈ ∆ . A transducer T transduces the input p to the output r ( i.e . ( p , r ) ∈ T ) if there exists a sequence of states qi+1 = π ( qi , pi , ri ) ∈ Q such that q1 ∈ Qs and qT+1 ∈ Qa . The score of the pair ( p , r ) is given by s ( p , r ) = ∑T i=1 ω ( qi , pi , ri ) . We restrict operations to the log and tropical semirings . In both cases the accumulation of weights along a path is with addition . In the log semiring the accumulation over path scores is with logsum-exp which we denote by logaddi si = log ∑ i e si . In the tropical semiring , the accumulation over path scores is with the max , maxi si . We allow transitions in both acceptors and transducers . An input ( or output ) on an edge means the edge can be traversed without consuming an input ( or emitting an output ) . This allows inputs to map to outputs of differing length . 3.1 OPERATIONS . We briefly describe a subset of the most important operations implemented in our differentiable WFST framework . For a more detailed discussion of operations on WFSTs see e.g . Mohri ( 2009 ) . Intersection We denote by B = A1 ◦ A2 the intersection of two acceptors A1 and A2 . The intersected graph B contains all paths which are accepted by both inputs . The score of any path in B is the sum of the scores of the two corresponding paths in A1 and A2 . Composition The same symbol denotes the composition of two transducers U = T1 ◦ T2 . If T1 transduces p to u and T2 transduces u to r then U transduces p to r. As in intersection , the score of the path in the composed graph is the sum of the path scores from the input graphs . Forward and Viterbi Score The forward score of T is logadd ( p , r ) ∈T s ( p , r ) . Similarly , the Viterbi score of T is max ( p , r ) ∈T s ( p , r ) . 0 1a : a/0 a : ε/0 ( a ) Token graph Ta 0 1a : ε/0 2b : ab/0 ( b ) Label graph Y 0 1a : ε/0 a : ε/0 2b : ab/0 b : ε/0 ( c ) Alignment graph T ◦ Y Viterbi Path The Viterbi path of T is given by arg max ( p , r ) ∈T s ( p , r ) . The forward score , Viterbi score , and the Viterbi path for an acceptorA are defined in the same way using just p. In a directed acyclic graph these operations can be computed in time linear in the size of the graph with the well-known forward and Viterbi algorithms ( Jurafsky , 2000 ; Rabiner , 1989 ) . We support standard rational operations including the union ( A + B ) , concatenation ( AB ) , and the Kleene closure ( A∗ ) of WFSTs . To enable automatic differentiation through complete computations , we support slightly non-standard operations . We allow for the negation of all the arcs weights in a graph and the addition or subtraction of the arc weights of two identically structured graphs .	This is a well-written paper describing the incorporation of WFSTs into an "auto-diff" framework for Deep Learning. The central point is that by providing partial derivatives, in addition to the standard forward scores, to the major WFST operations, one can embed those WFST operations into a gradient-based deep learning framework as just another differentiable module. Given the continued demonstrated utility of WFSTs in the speech and natural language community, many researchers will welcome the motivation. The paper details how some specific WFST models can help implement models such as CTC or ASG, presents a less-clear application to convolutional models, and presents results evaluating the use of differentiable WFSTs for tasks in ASR and handwriting recognition.	SP:f09077a0cb9dfb47788db6c18a0ec92c18d608b9
Differentiable Weighted Finite-State Transducers	1 INTRODUCTION . Weighted finite-state transducers ( WFSTs ) are a commonly used tool in speech and language processing ( Knight & May , 2009 ; Mohri et al. , 2002 ) . They are most frequently used to combine predictions from multiple already trained models . In speech recognition , for example , WFSTs are used to combine constraints from an acoustic-to-phoneme model , a lexicon mapping words to pronunciations , and a word-level language model . However , combining separately learned models using WFSTs only at inference time has several drawbacks , including the well-known problems of exposure bias ( Ranzato et al. , 2015 ) and label bias ( Bottou , 1991 ; Lafferty et al. , 2001 ) . Given that gradients may be computed for most WFST operations , using them only at the inference stage of a learning system is not a hard limitation . We speculate that this limitation is primarily due to practical considerations . Historically , hardware has not been sufficiently performant to make training with WFSTs tractable . Also , no implementation exists with the required operations which supports automatic differentiation in a high-level yet efficient manner . We develop a framework for automatic differentiation through operations on WFSTs . We show the utility of this framework by leveraging it to design and experiment with existing and novel learning algorithms . Automata are a more convenient structure than tensors to encode prior knowledge into a learning algorithm . However , not training with them limits the extent to which this prior knowledge can be incorporated in a useful manner . A framework for differentiable WFSTs allows the model to jointly learn from training data as well as prior knowledge encoded in WFSTs . This enables the learning algorithm to incorporate such knowledge in the best possible way . Use of WFSTs conveniently decomposes operations from data ( i.e . graphs ) . For example , rather than hand-coding sequence-level loss functions such as Connectionist Temporal Classification ( CTC ) ( Graves et al. , 2006 ) or the Automatic Segmentation Criterion ( ASG ) ( Collobert et al. , 2016 ) , we may specify the core assumptions of the criteria in graphs and compute the resulting loss with graph operations . This facilitates exploration in the space of such structured loss functions . We show the utility of the differentiable WFST framework by designing and testing several algorithms . For example , bi-gram transitions may be added to CTC with a transition WFST . We scale transitions to large token set sizes by encoding pruning and back-off in the transition graph . Word pieces are commonly used as the output of speech recognition and machine translation models ( Chiu et al. , 2018 ; Sennrich et al. , 2016 ) . The word piece decomposition for a word is learned with a task-independent model . Instead , we use WFSTs to marginalize over the latent word piece decomposition at training time . This lets the model learn decompositions salient to the task at hand . Finally , we show that WFSTs may be used as layers themselves intermixed with tensor-based layers . We propose a convolutional WFST layer which maps lower-level representations to higher-level representations . The WFST convolution can be trained with the rest of the model and results in improved accuracy with fewer parameters and operations as compared to a traditional convolution . In summary , our contributions are : • A framework for automatic differentiation with WFSTs . The framework supports both C++ and Python front-ends and is available at https : //www.anonymized.com . • We show that the framework may be used to express both existing sequence-level loss functions and to design novel sequence-level loss functions . • We propose a convolutional WFST layer which can be used in the interior of a deep neural network to map lower-level representations to higher-level representations . • We demonstrate the effectiveness of using WFSTs in the manners described above with experiments in automatic speech and handwriting recognition . 2 RELATED WORK . A wealth of prior work exists using weighted finite-state automata in speech recognition , natural language processing , optical character recognition , and other applications ( Breuel , 2008 ; Knight & May , 2009 ; Mohri , 1997 ; Mohri et al. , 2008 ; Pereira et al. , 1994 ) . However , the use of WFSTs is limited mostly to the inference stage of a predictive system . For example , Kaldi , a commonly used toolkit for automatic speech recognition , uses WFSTs extensively , but in most cases for inference or to estimate the parameters of shallow models ( Povey et al. , 2011 ) . In some cases , WFSTs are used statically to incorporate fixed lattices in discriminative sequence criteria ( Kingsbury , 2009 ; Kingsbury et al. , 2012 ; Su et al. , 2013 ; Veselỳ et al. , 2013 ) . Implementations of sequence criteria in end-to-end style training are typically hand-crafted with careful consideration for speed ( Amodei et al. , 2016 ; Collobert et al. , 2019 ; Povey et al. , 2016 ) . The use of hand-crafted implementations reduces flexibility which limits research . In some cases , such as the fully differentiable beam search of Collobert et al . ( 2019 ) , achieving the necessary computational efficiency with a WFST-based implementation may not yet be tractable . However , as a first step , we show that in many common cases we can have the expressiveness afforded by the differentiable WFST framework without paying an unacceptable penalty in execution time . The ideas of learning with WFSTs ( Eisner , 2002 ) and automatic differentiation through operations on graphs ( Bottou et al. , 1997 ) are not new . However , no simple and efficient frameworks exist . Especially related to this work , and inspiring the name of our framework , are the graph transformer networks of Bottou et al . ( 1997 ) . Generalized graph transducers ( Bottou et al. , 1996 ) are more expressive than WFSTs , allowing arbitrary data as edge labels . When composing these graphs , one defines an edge matching function and a “ transformer ” to construct the resulting structure . While not this general , differentiable WFSTs nevertheless allow for a vast design space of interesting algorithms , and perhaps make a more pragmatic trade-off between flexibility and efficiency . Highly efficient libraries for operations on WFSTs exist , notably OpenFST and its predecessor FSM ( Allauzen et al. , 2007 ; Mohri et al. , 2000 ) . We take inspiration from OpenFst in the interface and implementation of many of our functions . However , the design implications of operating on WFSTs with automatic differentiation are quite different than those of the use cases OpenFST has been optimized for . We also draw inspiration from libraries for automatic differentiation and deep learning ( Collobert et al. , 2011 ; Paszke et al. , 2019 ; Pratap et al. , 2019 ; Tokui et al. , 2015 ) . Some of the algorithms we propose , with the goal of demonstrating the utility of the differentiable WFST library , are inspired by prior work . Prior work has explored pruning the set of allowed alignments with CTC , and in particular limiting the spacing between output tokens ( Liu et al. , 2018 ) . Learning n-gram word decompositions with both differentiable ( Liu et al. , 2017 ) and nondifferentiable ( Chan et al. , 2017 ) loss functions has also been explored . 3 DIFFERENTIABLE WEIGHTED FINITE-STATE TRANSDUCERS . A weighted finite-state acceptor A is a 6-tuple consisting of an alphabet Σ , a set of states Q , start states Qs , accepting states Qa , a transition function π ( q , p ) which maps elements of Q × Σ to elements of Q , and a weight function ω ( q , p ) which maps elements of Q × Σ to R. A weighted finite-state transducer T is a 7-tuple which augments an acceptor with an output alphabet ∆ . The transition function π ( q , p , r ) and weight function ω ( q , p , r ) map elements of Q×Σ×∆ to elements of Q and R respectively . In other words , each edge of a transducer connects two states and has an input label p , an output label r and a weight w ∈ R. We denote an input by p = [ p1 , . . . , pT ] where each pi ∈ Σ . An acceptor A accepts the input p if there exists a sequence of states qi+1 = π ( qi , pi ) ∈ Q such that q1 ∈ Qs and qT+1 ∈ Qa . With a slight abuse of notation , we let p ∈ A denote that A accepts p. The score of p is given by s ( p ) =∑T i=1 ω ( qi , pi ) . Let r = [ r1 , . . . , rT ] be a path with ri ∈ ∆ . A transducer T transduces the input p to the output r ( i.e . ( p , r ) ∈ T ) if there exists a sequence of states qi+1 = π ( qi , pi , ri ) ∈ Q such that q1 ∈ Qs and qT+1 ∈ Qa . The score of the pair ( p , r ) is given by s ( p , r ) = ∑T i=1 ω ( qi , pi , ri ) . We restrict operations to the log and tropical semirings . In both cases the accumulation of weights along a path is with addition . In the log semiring the accumulation over path scores is with logsum-exp which we denote by logaddi si = log ∑ i e si . In the tropical semiring , the accumulation over path scores is with the max , maxi si . We allow transitions in both acceptors and transducers . An input ( or output ) on an edge means the edge can be traversed without consuming an input ( or emitting an output ) . This allows inputs to map to outputs of differing length . 3.1 OPERATIONS . We briefly describe a subset of the most important operations implemented in our differentiable WFST framework . For a more detailed discussion of operations on WFSTs see e.g . Mohri ( 2009 ) . Intersection We denote by B = A1 ◦ A2 the intersection of two acceptors A1 and A2 . The intersected graph B contains all paths which are accepted by both inputs . The score of any path in B is the sum of the scores of the two corresponding paths in A1 and A2 . Composition The same symbol denotes the composition of two transducers U = T1 ◦ T2 . If T1 transduces p to u and T2 transduces u to r then U transduces p to r. As in intersection , the score of the path in the composed graph is the sum of the path scores from the input graphs . Forward and Viterbi Score The forward score of T is logadd ( p , r ) ∈T s ( p , r ) . Similarly , the Viterbi score of T is max ( p , r ) ∈T s ( p , r ) . 0 1a : a/0 a : ε/0 ( a ) Token graph Ta 0 1a : ε/0 2b : ab/0 ( b ) Label graph Y 0 1a : ε/0 a : ε/0 2b : ab/0 b : ε/0 ( c ) Alignment graph T ◦ Y Viterbi Path The Viterbi path of T is given by arg max ( p , r ) ∈T s ( p , r ) . The forward score , Viterbi score , and the Viterbi path for an acceptorA are defined in the same way using just p. In a directed acyclic graph these operations can be computed in time linear in the size of the graph with the well-known forward and Viterbi algorithms ( Jurafsky , 2000 ; Rabiner , 1989 ) . We support standard rational operations including the union ( A + B ) , concatenation ( AB ) , and the Kleene closure ( A∗ ) of WFSTs . To enable automatic differentiation through complete computations , we support slightly non-standard operations . We allow for the negation of all the arcs weights in a graph and the addition or subtraction of the arc weights of two identically structured graphs .	This paper presents how weighted finite-state transducers (WFST) and a few common operations performed on them can be integrated in a differentiable model, and therefore contribute to the training of complete systems. The authors propose a few case studies, mainly in language applications, where the WFSTs are used to compute a sequence-level loss function, to keep ambiguity and let the model decide word-peices decomposition, or gracefully replace convolution layers. The code associated with the presented methods will be available.	SP:f09077a0cb9dfb47788db6c18a0ec92c18d608b9
Differentiable Weighted Finite-State Transducers	1 INTRODUCTION . Weighted finite-state transducers ( WFSTs ) are a commonly used tool in speech and language processing ( Knight & May , 2009 ; Mohri et al. , 2002 ) . They are most frequently used to combine predictions from multiple already trained models . In speech recognition , for example , WFSTs are used to combine constraints from an acoustic-to-phoneme model , a lexicon mapping words to pronunciations , and a word-level language model . However , combining separately learned models using WFSTs only at inference time has several drawbacks , including the well-known problems of exposure bias ( Ranzato et al. , 2015 ) and label bias ( Bottou , 1991 ; Lafferty et al. , 2001 ) . Given that gradients may be computed for most WFST operations , using them only at the inference stage of a learning system is not a hard limitation . We speculate that this limitation is primarily due to practical considerations . Historically , hardware has not been sufficiently performant to make training with WFSTs tractable . Also , no implementation exists with the required operations which supports automatic differentiation in a high-level yet efficient manner . We develop a framework for automatic differentiation through operations on WFSTs . We show the utility of this framework by leveraging it to design and experiment with existing and novel learning algorithms . Automata are a more convenient structure than tensors to encode prior knowledge into a learning algorithm . However , not training with them limits the extent to which this prior knowledge can be incorporated in a useful manner . A framework for differentiable WFSTs allows the model to jointly learn from training data as well as prior knowledge encoded in WFSTs . This enables the learning algorithm to incorporate such knowledge in the best possible way . Use of WFSTs conveniently decomposes operations from data ( i.e . graphs ) . For example , rather than hand-coding sequence-level loss functions such as Connectionist Temporal Classification ( CTC ) ( Graves et al. , 2006 ) or the Automatic Segmentation Criterion ( ASG ) ( Collobert et al. , 2016 ) , we may specify the core assumptions of the criteria in graphs and compute the resulting loss with graph operations . This facilitates exploration in the space of such structured loss functions . We show the utility of the differentiable WFST framework by designing and testing several algorithms . For example , bi-gram transitions may be added to CTC with a transition WFST . We scale transitions to large token set sizes by encoding pruning and back-off in the transition graph . Word pieces are commonly used as the output of speech recognition and machine translation models ( Chiu et al. , 2018 ; Sennrich et al. , 2016 ) . The word piece decomposition for a word is learned with a task-independent model . Instead , we use WFSTs to marginalize over the latent word piece decomposition at training time . This lets the model learn decompositions salient to the task at hand . Finally , we show that WFSTs may be used as layers themselves intermixed with tensor-based layers . We propose a convolutional WFST layer which maps lower-level representations to higher-level representations . The WFST convolution can be trained with the rest of the model and results in improved accuracy with fewer parameters and operations as compared to a traditional convolution . In summary , our contributions are : • A framework for automatic differentiation with WFSTs . The framework supports both C++ and Python front-ends and is available at https : //www.anonymized.com . • We show that the framework may be used to express both existing sequence-level loss functions and to design novel sequence-level loss functions . • We propose a convolutional WFST layer which can be used in the interior of a deep neural network to map lower-level representations to higher-level representations . • We demonstrate the effectiveness of using WFSTs in the manners described above with experiments in automatic speech and handwriting recognition . 2 RELATED WORK . A wealth of prior work exists using weighted finite-state automata in speech recognition , natural language processing , optical character recognition , and other applications ( Breuel , 2008 ; Knight & May , 2009 ; Mohri , 1997 ; Mohri et al. , 2008 ; Pereira et al. , 1994 ) . However , the use of WFSTs is limited mostly to the inference stage of a predictive system . For example , Kaldi , a commonly used toolkit for automatic speech recognition , uses WFSTs extensively , but in most cases for inference or to estimate the parameters of shallow models ( Povey et al. , 2011 ) . In some cases , WFSTs are used statically to incorporate fixed lattices in discriminative sequence criteria ( Kingsbury , 2009 ; Kingsbury et al. , 2012 ; Su et al. , 2013 ; Veselỳ et al. , 2013 ) . Implementations of sequence criteria in end-to-end style training are typically hand-crafted with careful consideration for speed ( Amodei et al. , 2016 ; Collobert et al. , 2019 ; Povey et al. , 2016 ) . The use of hand-crafted implementations reduces flexibility which limits research . In some cases , such as the fully differentiable beam search of Collobert et al . ( 2019 ) , achieving the necessary computational efficiency with a WFST-based implementation may not yet be tractable . However , as a first step , we show that in many common cases we can have the expressiveness afforded by the differentiable WFST framework without paying an unacceptable penalty in execution time . The ideas of learning with WFSTs ( Eisner , 2002 ) and automatic differentiation through operations on graphs ( Bottou et al. , 1997 ) are not new . However , no simple and efficient frameworks exist . Especially related to this work , and inspiring the name of our framework , are the graph transformer networks of Bottou et al . ( 1997 ) . Generalized graph transducers ( Bottou et al. , 1996 ) are more expressive than WFSTs , allowing arbitrary data as edge labels . When composing these graphs , one defines an edge matching function and a “ transformer ” to construct the resulting structure . While not this general , differentiable WFSTs nevertheless allow for a vast design space of interesting algorithms , and perhaps make a more pragmatic trade-off between flexibility and efficiency . Highly efficient libraries for operations on WFSTs exist , notably OpenFST and its predecessor FSM ( Allauzen et al. , 2007 ; Mohri et al. , 2000 ) . We take inspiration from OpenFst in the interface and implementation of many of our functions . However , the design implications of operating on WFSTs with automatic differentiation are quite different than those of the use cases OpenFST has been optimized for . We also draw inspiration from libraries for automatic differentiation and deep learning ( Collobert et al. , 2011 ; Paszke et al. , 2019 ; Pratap et al. , 2019 ; Tokui et al. , 2015 ) . Some of the algorithms we propose , with the goal of demonstrating the utility of the differentiable WFST library , are inspired by prior work . Prior work has explored pruning the set of allowed alignments with CTC , and in particular limiting the spacing between output tokens ( Liu et al. , 2018 ) . Learning n-gram word decompositions with both differentiable ( Liu et al. , 2017 ) and nondifferentiable ( Chan et al. , 2017 ) loss functions has also been explored . 3 DIFFERENTIABLE WEIGHTED FINITE-STATE TRANSDUCERS . A weighted finite-state acceptor A is a 6-tuple consisting of an alphabet Σ , a set of states Q , start states Qs , accepting states Qa , a transition function π ( q , p ) which maps elements of Q × Σ to elements of Q , and a weight function ω ( q , p ) which maps elements of Q × Σ to R. A weighted finite-state transducer T is a 7-tuple which augments an acceptor with an output alphabet ∆ . The transition function π ( q , p , r ) and weight function ω ( q , p , r ) map elements of Q×Σ×∆ to elements of Q and R respectively . In other words , each edge of a transducer connects two states and has an input label p , an output label r and a weight w ∈ R. We denote an input by p = [ p1 , . . . , pT ] where each pi ∈ Σ . An acceptor A accepts the input p if there exists a sequence of states qi+1 = π ( qi , pi ) ∈ Q such that q1 ∈ Qs and qT+1 ∈ Qa . With a slight abuse of notation , we let p ∈ A denote that A accepts p. The score of p is given by s ( p ) =∑T i=1 ω ( qi , pi ) . Let r = [ r1 , . . . , rT ] be a path with ri ∈ ∆ . A transducer T transduces the input p to the output r ( i.e . ( p , r ) ∈ T ) if there exists a sequence of states qi+1 = π ( qi , pi , ri ) ∈ Q such that q1 ∈ Qs and qT+1 ∈ Qa . The score of the pair ( p , r ) is given by s ( p , r ) = ∑T i=1 ω ( qi , pi , ri ) . We restrict operations to the log and tropical semirings . In both cases the accumulation of weights along a path is with addition . In the log semiring the accumulation over path scores is with logsum-exp which we denote by logaddi si = log ∑ i e si . In the tropical semiring , the accumulation over path scores is with the max , maxi si . We allow transitions in both acceptors and transducers . An input ( or output ) on an edge means the edge can be traversed without consuming an input ( or emitting an output ) . This allows inputs to map to outputs of differing length . 3.1 OPERATIONS . We briefly describe a subset of the most important operations implemented in our differentiable WFST framework . For a more detailed discussion of operations on WFSTs see e.g . Mohri ( 2009 ) . Intersection We denote by B = A1 ◦ A2 the intersection of two acceptors A1 and A2 . The intersected graph B contains all paths which are accepted by both inputs . The score of any path in B is the sum of the scores of the two corresponding paths in A1 and A2 . Composition The same symbol denotes the composition of two transducers U = T1 ◦ T2 . If T1 transduces p to u and T2 transduces u to r then U transduces p to r. As in intersection , the score of the path in the composed graph is the sum of the path scores from the input graphs . Forward and Viterbi Score The forward score of T is logadd ( p , r ) ∈T s ( p , r ) . Similarly , the Viterbi score of T is max ( p , r ) ∈T s ( p , r ) . 0 1a : a/0 a : ε/0 ( a ) Token graph Ta 0 1a : ε/0 2b : ab/0 ( b ) Label graph Y 0 1a : ε/0 a : ε/0 2b : ab/0 b : ε/0 ( c ) Alignment graph T ◦ Y Viterbi Path The Viterbi path of T is given by arg max ( p , r ) ∈T s ( p , r ) . The forward score , Viterbi score , and the Viterbi path for an acceptorA are defined in the same way using just p. In a directed acyclic graph these operations can be computed in time linear in the size of the graph with the well-known forward and Viterbi algorithms ( Jurafsky , 2000 ; Rabiner , 1989 ) . We support standard rational operations including the union ( A + B ) , concatenation ( AB ) , and the Kleene closure ( A∗ ) of WFSTs . To enable automatic differentiation through complete computations , we support slightly non-standard operations . We allow for the negation of all the arcs weights in a graph and the addition or subtraction of the arc weights of two identically structured graphs .	The authors introduce a library for differential weighted finite-state transducers. WFST are commonly used in speech or handwriting recognition systems but are generally not trained jointly with the deep neural networks components such as ConvNN. This is not due to theoretical limitation of WFST but rather to a lack of available implementation and the need of important computational power to train them. The authors show that this new library can be used to encode the ASG criterion, by combining the emission graph (coming from a NN for example), the token graph (base recognition units) and the label graph (the sequence annotation) on one hand and the emission graph and a language model graph on the other hand. The authors show how word pieces decomposition can be learnt through marginalisation. Finally, convolution Wfst are rapidly presented. Preliminary experiments are reported on wSj data base for speech recognition and IAM database for handwriting recognition.	SP:f09077a0cb9dfb47788db6c18a0ec92c18d608b9
BeBold: Exploration Beyond the Boundary of Explored Regions	1 INTRODUCTION . Deep reinforcement learning ( RL ) has experienced significant progress over the last several years , with impressive performance in games like Atari ( Mnih et al. , 2015 ; Badia et al. , 2020a ) , StarCraft ( Vinyals et al. , 2019 ) and Chess ( Silver et al. , 2016 ; 2017 ; 2018 ) . However , most work requires either a manually-designed dense reward ( Brockman et al. , 2016 ) or a perfect environment model ( Silver et al. , 2017 ; Moravčı́k et al. , 2017 ) . This is impractical for real-world settings , where the reward is sparse ; in fact , the proper reward function for a task is often even unknown due to lack of domain knowledge . Random exploration ( e.g. , -greedy ) in these environments is often insufficient and leads to poor performance ( Bellemare et al. , 2016 ) . Recent approaches have proposed to use intrinsic rewards ( IR ) ( Schmidhuber , 2010 ) to motivate agents for exploration before any extrinsic rewards are obtained . Various criteria have been proposed , including curiosity/surprise-driven ( Pathak et al. , 2017 ) , count-based ( Bellemare et al. , 2016 ; Burda et al. , 2018a ; b ; Ostrovski et al. , 2017 ; Badia et al. , 2020b ) , and state-diff approaches ( Zhang et al. , 2019 ; Marino et al. , 2019 ) . Each approach has its upsides and downsides : Curiosity-driven approaches look for prediction errors in the learned dynamics model and may be misled by the noisy TV ( Burda et al. , 2018b ) problem , where environment dynamics are inherently stochastic . Count-based approaches favor novel states in the environment but suffer from detachment and derailment ( Ecoffet et al. , 2019 ) , in which the agent gets trapped into one ( long ) corridor and fails to try other choices . Count-based approaches are also short-sighted : the agent often settles in local minima , sometimes oscillating between two states that alternately feature lower visitation counts ( Burda et al. , 2018b ) . Finally , state-diff approaches offer rewards if , for each trajectory , representations of consecutive states differ significantly . While these approaches consider the entire trajectory of the agent rather than a local state , it is asymptotically inconsistent : the intrinsic reward remains positive when the visitation counts approach infinity . As a result , the final policy does not necessarily maximize the cumulative extrinsic reward . In this paper , we propose a novel exploration criterion that combines count-based and state-diff approaches : instead of using the difference of state representations , we use the regulated difference of inverse visitation counts of consecutive states in a trajectory . The inverse visitation count is approximated by Random Network Distillation ( Burda et al. , 2018b ) . Our IR provides two benefits : ( 1 ) This addresses asymptotic inconsistency in the state-diff , since the inverse visitation count vanishes with sufficient explorations . ( 2 ) Our IR is large at the end of a trajectory and at the boundary between the explored and the unexplored regions ( Fig . 1 ) . This motivates the agent to move Beyond the Boundary of the explored regions and step into the unknown , mitigating the short-sighted issue in count-based approaches . Following this simple criterion , we propose a novel algorithm BeBold and evaluate it on two very challenging procedurally-generated ( PG ) environments : MiniGrid ( Chevalier-Boisvert et al. , 2018 ) and NetHack ( Küttler et al. , 2020 ) . MiniGrid is a popular benchmark for evaluating exploration algorithms ( Raileanu and Rocktäschel , 2020 ; Campero et al. , 2020 ; Goyal et al. , 2019 ) and NetHack is a much more realistic environment with complex goals and skills . BeBold manages to solve the 12 most challenging environments in MiniGrid within 120M environment steps , without curriculum learning . In contrast , ( Campero et al. , 2020 ) solves 50 % of the tasks , which were categorized as “ easy ” and “ medium ” , by training a separate goal-generating teacher network in 500M steps . In NetHack , a more challenging procedurally-generated environment , BeBold also outperforms all baselines with a significant margin on various tasks . In addition , we analyze BeBold extensively in MiniGrid . The quantitative results show that BeBold largely mitigates the detachment problem , with a much simpler design than Go-Explore ( Ecoffet et al. , 2020 ) which contains multiple handtune stages and hyper-parameters . Most Related Works . RIDE ( Raileanu and Rocktäschel , 2020 ) also combines multiple criteria together . RIDE learns the state representation with curiosity-driven approaches , and then uses the difference of learned representation along a trajectory as the reward , weighted by pseudo counts of the state . However , as a two-stage approach , RIDE heavily relies on the quality of generalization of the learned representation on novel states . As a result , BeBold shows substantially better performance in the same procedurally-generated environments . Go-Explore ( Ecoffet et al. , 2020 ) stores many visited states ( including boundaries ) , reaches these states without exploration , and explores from them . BeBold focuses on boundaries , perform exploration without human-designed cell representation ( e.g. , image downsampling ) and is end-to-end . Frontier-based exploration ( Yamauchi , 1997 ; 1998 ; Topiwala et al. , 2018 ) is used to help specific robots explore the map by maximizing the information gain . The “ frontier ” is defined as the 2D spatial regions out of the explored parts . No automatic policy optimization with deep models is used . In contrast , BeBold can be applied to more general partial observable MDPs with deep policies . 2 BACKGROUND . Following single agent Markov Decision Process ( MDP ) , we define a state space S , an action space A , and a ( non-deterministic ) transition function T : S × A → P ( S ) where P ( S ) is the probability of next state given the current state and action . The goal is to maximize the expected reward R = E [ ∑T k=0 γ krt+k=1 ] where rt is the reward , γ is the discount factor , and the expectation is taken w.r.t . the policy π and MDP transition P ( S ) . In this paper , the total reward received at time step t is given by rt = ret +αr i t , where r e t is the extrinsic reward given by the environment , r i t is the intrinsic reward from the exploration criterion , and α is a scaling hyperparameter . 3 EXPLORATION BEYOND THE BOUNDARY . Our new exploration criterion combines both counting-based and state-diff-based criteria . Our criterion doesn ’ t suffer from short-sightedness and is asymptomatically consistent . We ’ ll first introduce BeBold and then analyse the advantages of BeBold over existing criteria in Sec . 4 . Exploration Beyond the Boundary . BeBold gives intrinsic reward ( IR ) to the agent when it explores beyond the boundary of explored regions , i.e. , along a trajectory , the previous state st has been sufficiently explored but st+1 is new : ri ( st , at , st+1 ) = max ( 1 N ( st+1 ) − 1 N ( st ) , 0 ) , ( 1 ) Here N is the visitation counts . We clip the IR here because we don ’ t want to give a negative IR to the agent if it transits back from a novel state to a familiar state . From the equation , only crossing the frontier matters to the intrinsic reward ; if both N ( st ) and N ( st+1 ) are high or low , their difference would be small . As we will show in Sec . 4 , for each trajectory going towards the frontier/boundary , BeBold assigns an approximately equal IR , regardless of their length . As a result , the agent will continue pushing the frontier of exploration in a much more uniform manner than RND and won ’ t suffer from short-sightedness . This motivates the agent to explore different trajectories uniformly . Also Eq . 1 is asymptotically consistent as ri → 0 when N →∞ . Like RIDE ( Raileanu and Rocktäschel , 2020 ) , in our implementation , partial observation ot are used instead of the real state st , when st is not available . Episodic Restriction on Intrinsic Reward ( ERIR ) . In many environments where the state transition is reversible , simply using intrinsic reward to guide exploration would result in the agent going back and forth between novel states st+1 and their previous states st. RIDE ( Raileanu and Rocktäschel , 2020 ) avoids this by scaling the intrinsic reward r ( s ) by the inverse of the state visitation counts . BeBold puts a more aggressive restriction : the agent is only rewarded when it visits the state s for the first time in an episode . Thus , the intrinsic reward of BeBold becomes : ri ( st , at , st+1 ) = max ( 1 N ( st+1 ) − 1 N ( st ) , 0 ) ∗ 1 { Ne ( st+1 ) = 1 } ( 2 ) Ne here stands for episodic state count and is reset every episode . In contrast , the visitation count N is a life-long memory bank counting state visitation across all of training . Inverse visitation counts as prediction difference . We use the difference between a teacher φ and a student network φ′ to approximate visitation counts : N ( st+1 ) ≈ 1\|\|φ ( ot+1 ) −φ′ ( ot+1 ) \|\|2 , here ot+1 is the observation of the agent in state st+1 . This yields the following implementation of BeBold : ri ( st , at , st+1 ) =max ( \|\|φ ( ot+1 ) − φ′ ( ot+1 ) \|\|2 − \|\|φ ( ot ) − φ′ ( ot ) \|\|2 , 0 ) ∗ 1 { Ne ( ot+1 ) = 1 } ) ( 3 ) Shared visitation counts N ( st ) in the training of Procedurally-Generated ( PG ) Environments . During training , the environment changes constantly ( e.g. , blue keys becomes red ) , while the semantic links of these objects remain the same . We use a shared RND ( φ , φ′ ) across different PG environments , and treat these semantically similar states as new without using domain knowledge ( e.g. , image downsampling like in Go-Explore ( Ecoffet et al. , 2019 ) ) . Partial observability and generalization of neural network φ handles these differences and leads to count-sharing . For episodic count Ne ( ot+1 ) , since it is not shared across episodes ( and environments ) , we use a hash table . 4 CONCEPTUAL ADVANTAGES OF BEBOLD OVER EXISTING CRITERIA . Short-sightedness and Detachment . One issue in the count-based approach is its short-sightedness . Let ’ s assume in a simple environment , there are M corridors { τj } Mj=1 starting at s0 and extending to different parts of the environment . The corridor τj has a length of Tj . The agent starts at s0 . For each visited state , the agent receives the reward of 1N ( s ) where N ( · ) is the visitation count , and learns with Q-learning . Then with some calculation ( See Appendix ) , we see that the agent has a strong preference on exploring the longest corridor first ( say τ1 ) , and only after a long period does it start to explore the second longest . This is because the agent initially receives high IR in τ1 due to its length , which makes the policy π visit τ1 more often , until it depletes the IR in τ1 . This behavior of “ dedication ” could lead to serious issues . If M ≥ 3 and 2 corridors are long enough ( say τ1 and τ2 are long ) , then before the agent is able to explore other corridors , its policy π has already been trained long enough so that it only remembers how to get into τ1 and τ2 . When τ1 has depleted its IR , the agent goes to τ2 following the policy . After that , the IR in τ1 revives since the visitation counts in τ1 is now comparable or even smaller than τ2 , which lures the agent to explore τ1 again following the policy . This leaves other corridors ( e.g. , τ3 ) unexplored for a very long time . Note that using a neural-network-approximated IR ( RND ) instead of tabular IR could potentially alleviate this issue , but it is often far less than enough in complex environments . As mentioned in Go-Explore series ( Ecoffet et al. , 2019 ; 2020 ) , count-based approaches also suffer from detachment : if the agent by chance starts exploring τ2 after briefly exploring the first few states of τ1 , it would not return and explore τ1 further since τ1 is now “ shorter ” than τ2 and has lower IR than τ2 for a long period . Go-Explore tries to resolve this dilemma between “ dedication ” and “ exploration ” by using a two-stage approach with many hand-tuned parameters . In contrast , IR of BeBold depends on the difference of the visitation counts along the trajectory , and is insensitive to the length of the corridor . This leads to simultaneous exploration of multiple corridors and yields a diverse policy π ( See Sec . 5.2 for empirical evidence ) . Moreover , the IR focuses on the boundary between explored and unexplored regions , where the two goals ( dedication and exploration ) align , yielding a much cleaner , one-stage method . Asymptotic Inconsistency . Approaches that define IR as the difference between state representations ‖ψ ( s ) − ψ ( s′ ) ‖ ( ψ is a learned embedding network ) ( Zhang et al. , 2019 ; Marino et al. , 2019 ) suffer from asymptotic inconsistency . In other words , their IR does not vanish even after sufficient exploration : ri 6→ 0 whenN →∞ . This is because when the embedding network ψ converges after sufficient exploration , the agent can always obtain non-zero IR if a major change in state representation occurs ( e.g. , opening a door or picking up a key in MiniGrid ) . Therefore , the learned policy does not maximize the extrinsic reward re , deviating from the goal of RL . Automatic curriculum approaches ( Campero et al. , 2020 ) ) have similar issues due to an ever-present IR . For this , ( Zhang et al. , 2019 ) proposes to learn a separate scheduler to switch between intrinsic and extrinsic rewards , and ( Raileanu and Rocktäschel , 2020 ) divides the state representation difference by the square root of visitation counts . In comparison , BeBold does not require any extra stage and is a much simpler solution .	This paper is a presentation of BeBold, a new method using an intrinsic reward for exploration, meant for procedurally generated, episodic environments. The method includes two major components: the first being intrinsically rewarding the agent for entering states that are less visited than the current state and the second being only intrinsically rewarding the agent for the first time it enters a state during an episode. The agent's intrinsic reward is larger if the difference in visit counts is larger. The paper includes some discussion of the conceptual advantages of BeBold over prior work as well as empirical demonstrations in MiniGrid and NetHack.	SP:cb17cc8e64068c1b5294af47cc07ccc3ebcada5b
BeBold: Exploration Beyond the Boundary of Explored Regions	1 INTRODUCTION . Deep reinforcement learning ( RL ) has experienced significant progress over the last several years , with impressive performance in games like Atari ( Mnih et al. , 2015 ; Badia et al. , 2020a ) , StarCraft ( Vinyals et al. , 2019 ) and Chess ( Silver et al. , 2016 ; 2017 ; 2018 ) . However , most work requires either a manually-designed dense reward ( Brockman et al. , 2016 ) or a perfect environment model ( Silver et al. , 2017 ; Moravčı́k et al. , 2017 ) . This is impractical for real-world settings , where the reward is sparse ; in fact , the proper reward function for a task is often even unknown due to lack of domain knowledge . Random exploration ( e.g. , -greedy ) in these environments is often insufficient and leads to poor performance ( Bellemare et al. , 2016 ) . Recent approaches have proposed to use intrinsic rewards ( IR ) ( Schmidhuber , 2010 ) to motivate agents for exploration before any extrinsic rewards are obtained . Various criteria have been proposed , including curiosity/surprise-driven ( Pathak et al. , 2017 ) , count-based ( Bellemare et al. , 2016 ; Burda et al. , 2018a ; b ; Ostrovski et al. , 2017 ; Badia et al. , 2020b ) , and state-diff approaches ( Zhang et al. , 2019 ; Marino et al. , 2019 ) . Each approach has its upsides and downsides : Curiosity-driven approaches look for prediction errors in the learned dynamics model and may be misled by the noisy TV ( Burda et al. , 2018b ) problem , where environment dynamics are inherently stochastic . Count-based approaches favor novel states in the environment but suffer from detachment and derailment ( Ecoffet et al. , 2019 ) , in which the agent gets trapped into one ( long ) corridor and fails to try other choices . Count-based approaches are also short-sighted : the agent often settles in local minima , sometimes oscillating between two states that alternately feature lower visitation counts ( Burda et al. , 2018b ) . Finally , state-diff approaches offer rewards if , for each trajectory , representations of consecutive states differ significantly . While these approaches consider the entire trajectory of the agent rather than a local state , it is asymptotically inconsistent : the intrinsic reward remains positive when the visitation counts approach infinity . As a result , the final policy does not necessarily maximize the cumulative extrinsic reward . In this paper , we propose a novel exploration criterion that combines count-based and state-diff approaches : instead of using the difference of state representations , we use the regulated difference of inverse visitation counts of consecutive states in a trajectory . The inverse visitation count is approximated by Random Network Distillation ( Burda et al. , 2018b ) . Our IR provides two benefits : ( 1 ) This addresses asymptotic inconsistency in the state-diff , since the inverse visitation count vanishes with sufficient explorations . ( 2 ) Our IR is large at the end of a trajectory and at the boundary between the explored and the unexplored regions ( Fig . 1 ) . This motivates the agent to move Beyond the Boundary of the explored regions and step into the unknown , mitigating the short-sighted issue in count-based approaches . Following this simple criterion , we propose a novel algorithm BeBold and evaluate it on two very challenging procedurally-generated ( PG ) environments : MiniGrid ( Chevalier-Boisvert et al. , 2018 ) and NetHack ( Küttler et al. , 2020 ) . MiniGrid is a popular benchmark for evaluating exploration algorithms ( Raileanu and Rocktäschel , 2020 ; Campero et al. , 2020 ; Goyal et al. , 2019 ) and NetHack is a much more realistic environment with complex goals and skills . BeBold manages to solve the 12 most challenging environments in MiniGrid within 120M environment steps , without curriculum learning . In contrast , ( Campero et al. , 2020 ) solves 50 % of the tasks , which were categorized as “ easy ” and “ medium ” , by training a separate goal-generating teacher network in 500M steps . In NetHack , a more challenging procedurally-generated environment , BeBold also outperforms all baselines with a significant margin on various tasks . In addition , we analyze BeBold extensively in MiniGrid . The quantitative results show that BeBold largely mitigates the detachment problem , with a much simpler design than Go-Explore ( Ecoffet et al. , 2020 ) which contains multiple handtune stages and hyper-parameters . Most Related Works . RIDE ( Raileanu and Rocktäschel , 2020 ) also combines multiple criteria together . RIDE learns the state representation with curiosity-driven approaches , and then uses the difference of learned representation along a trajectory as the reward , weighted by pseudo counts of the state . However , as a two-stage approach , RIDE heavily relies on the quality of generalization of the learned representation on novel states . As a result , BeBold shows substantially better performance in the same procedurally-generated environments . Go-Explore ( Ecoffet et al. , 2020 ) stores many visited states ( including boundaries ) , reaches these states without exploration , and explores from them . BeBold focuses on boundaries , perform exploration without human-designed cell representation ( e.g. , image downsampling ) and is end-to-end . Frontier-based exploration ( Yamauchi , 1997 ; 1998 ; Topiwala et al. , 2018 ) is used to help specific robots explore the map by maximizing the information gain . The “ frontier ” is defined as the 2D spatial regions out of the explored parts . No automatic policy optimization with deep models is used . In contrast , BeBold can be applied to more general partial observable MDPs with deep policies . 2 BACKGROUND . Following single agent Markov Decision Process ( MDP ) , we define a state space S , an action space A , and a ( non-deterministic ) transition function T : S × A → P ( S ) where P ( S ) is the probability of next state given the current state and action . The goal is to maximize the expected reward R = E [ ∑T k=0 γ krt+k=1 ] where rt is the reward , γ is the discount factor , and the expectation is taken w.r.t . the policy π and MDP transition P ( S ) . In this paper , the total reward received at time step t is given by rt = ret +αr i t , where r e t is the extrinsic reward given by the environment , r i t is the intrinsic reward from the exploration criterion , and α is a scaling hyperparameter . 3 EXPLORATION BEYOND THE BOUNDARY . Our new exploration criterion combines both counting-based and state-diff-based criteria . Our criterion doesn ’ t suffer from short-sightedness and is asymptomatically consistent . We ’ ll first introduce BeBold and then analyse the advantages of BeBold over existing criteria in Sec . 4 . Exploration Beyond the Boundary . BeBold gives intrinsic reward ( IR ) to the agent when it explores beyond the boundary of explored regions , i.e. , along a trajectory , the previous state st has been sufficiently explored but st+1 is new : ri ( st , at , st+1 ) = max ( 1 N ( st+1 ) − 1 N ( st ) , 0 ) , ( 1 ) Here N is the visitation counts . We clip the IR here because we don ’ t want to give a negative IR to the agent if it transits back from a novel state to a familiar state . From the equation , only crossing the frontier matters to the intrinsic reward ; if both N ( st ) and N ( st+1 ) are high or low , their difference would be small . As we will show in Sec . 4 , for each trajectory going towards the frontier/boundary , BeBold assigns an approximately equal IR , regardless of their length . As a result , the agent will continue pushing the frontier of exploration in a much more uniform manner than RND and won ’ t suffer from short-sightedness . This motivates the agent to explore different trajectories uniformly . Also Eq . 1 is asymptotically consistent as ri → 0 when N →∞ . Like RIDE ( Raileanu and Rocktäschel , 2020 ) , in our implementation , partial observation ot are used instead of the real state st , when st is not available . Episodic Restriction on Intrinsic Reward ( ERIR ) . In many environments where the state transition is reversible , simply using intrinsic reward to guide exploration would result in the agent going back and forth between novel states st+1 and their previous states st. RIDE ( Raileanu and Rocktäschel , 2020 ) avoids this by scaling the intrinsic reward r ( s ) by the inverse of the state visitation counts . BeBold puts a more aggressive restriction : the agent is only rewarded when it visits the state s for the first time in an episode . Thus , the intrinsic reward of BeBold becomes : ri ( st , at , st+1 ) = max ( 1 N ( st+1 ) − 1 N ( st ) , 0 ) ∗ 1 { Ne ( st+1 ) = 1 } ( 2 ) Ne here stands for episodic state count and is reset every episode . In contrast , the visitation count N is a life-long memory bank counting state visitation across all of training . Inverse visitation counts as prediction difference . We use the difference between a teacher φ and a student network φ′ to approximate visitation counts : N ( st+1 ) ≈ 1\|\|φ ( ot+1 ) −φ′ ( ot+1 ) \|\|2 , here ot+1 is the observation of the agent in state st+1 . This yields the following implementation of BeBold : ri ( st , at , st+1 ) =max ( \|\|φ ( ot+1 ) − φ′ ( ot+1 ) \|\|2 − \|\|φ ( ot ) − φ′ ( ot ) \|\|2 , 0 ) ∗ 1 { Ne ( ot+1 ) = 1 } ) ( 3 ) Shared visitation counts N ( st ) in the training of Procedurally-Generated ( PG ) Environments . During training , the environment changes constantly ( e.g. , blue keys becomes red ) , while the semantic links of these objects remain the same . We use a shared RND ( φ , φ′ ) across different PG environments , and treat these semantically similar states as new without using domain knowledge ( e.g. , image downsampling like in Go-Explore ( Ecoffet et al. , 2019 ) ) . Partial observability and generalization of neural network φ handles these differences and leads to count-sharing . For episodic count Ne ( ot+1 ) , since it is not shared across episodes ( and environments ) , we use a hash table . 4 CONCEPTUAL ADVANTAGES OF BEBOLD OVER EXISTING CRITERIA . Short-sightedness and Detachment . One issue in the count-based approach is its short-sightedness . Let ’ s assume in a simple environment , there are M corridors { τj } Mj=1 starting at s0 and extending to different parts of the environment . The corridor τj has a length of Tj . The agent starts at s0 . For each visited state , the agent receives the reward of 1N ( s ) where N ( · ) is the visitation count , and learns with Q-learning . Then with some calculation ( See Appendix ) , we see that the agent has a strong preference on exploring the longest corridor first ( say τ1 ) , and only after a long period does it start to explore the second longest . This is because the agent initially receives high IR in τ1 due to its length , which makes the policy π visit τ1 more often , until it depletes the IR in τ1 . This behavior of “ dedication ” could lead to serious issues . If M ≥ 3 and 2 corridors are long enough ( say τ1 and τ2 are long ) , then before the agent is able to explore other corridors , its policy π has already been trained long enough so that it only remembers how to get into τ1 and τ2 . When τ1 has depleted its IR , the agent goes to τ2 following the policy . After that , the IR in τ1 revives since the visitation counts in τ1 is now comparable or even smaller than τ2 , which lures the agent to explore τ1 again following the policy . This leaves other corridors ( e.g. , τ3 ) unexplored for a very long time . Note that using a neural-network-approximated IR ( RND ) instead of tabular IR could potentially alleviate this issue , but it is often far less than enough in complex environments . As mentioned in Go-Explore series ( Ecoffet et al. , 2019 ; 2020 ) , count-based approaches also suffer from detachment : if the agent by chance starts exploring τ2 after briefly exploring the first few states of τ1 , it would not return and explore τ1 further since τ1 is now “ shorter ” than τ2 and has lower IR than τ2 for a long period . Go-Explore tries to resolve this dilemma between “ dedication ” and “ exploration ” by using a two-stage approach with many hand-tuned parameters . In contrast , IR of BeBold depends on the difference of the visitation counts along the trajectory , and is insensitive to the length of the corridor . This leads to simultaneous exploration of multiple corridors and yields a diverse policy π ( See Sec . 5.2 for empirical evidence ) . Moreover , the IR focuses on the boundary between explored and unexplored regions , where the two goals ( dedication and exploration ) align , yielding a much cleaner , one-stage method . Asymptotic Inconsistency . Approaches that define IR as the difference between state representations ‖ψ ( s ) − ψ ( s′ ) ‖ ( ψ is a learned embedding network ) ( Zhang et al. , 2019 ; Marino et al. , 2019 ) suffer from asymptotic inconsistency . In other words , their IR does not vanish even after sufficient exploration : ri 6→ 0 whenN →∞ . This is because when the embedding network ψ converges after sufficient exploration , the agent can always obtain non-zero IR if a major change in state representation occurs ( e.g. , opening a door or picking up a key in MiniGrid ) . Therefore , the learned policy does not maximize the extrinsic reward re , deviating from the goal of RL . Automatic curriculum approaches ( Campero et al. , 2020 ) ) have similar issues due to an ever-present IR . For this , ( Zhang et al. , 2019 ) proposes to learn a separate scheduler to switch between intrinsic and extrinsic rewards , and ( Raileanu and Rocktäschel , 2020 ) divides the state representation difference by the square root of visitation counts . In comparison , BeBold does not require any extra stage and is a much simpler solution .	This paper proposes BeBold, a new definition of intrinsic reward to guide exploration in sparse reward problems. This intrinsic reward combines the ideas behind count-based approaches and state-diff approaches. They demonstrate the success of BeBold by comparing their algorithm to a set of state-of-the-art exploration methods using intrinsic rewards on a set of tasks from the MiniGrid and NetHack environments.	SP:cb17cc8e64068c1b5294af47cc07ccc3ebcada5b
BeBold: Exploration Beyond the Boundary of Explored Regions	1 INTRODUCTION . Deep reinforcement learning ( RL ) has experienced significant progress over the last several years , with impressive performance in games like Atari ( Mnih et al. , 2015 ; Badia et al. , 2020a ) , StarCraft ( Vinyals et al. , 2019 ) and Chess ( Silver et al. , 2016 ; 2017 ; 2018 ) . However , most work requires either a manually-designed dense reward ( Brockman et al. , 2016 ) or a perfect environment model ( Silver et al. , 2017 ; Moravčı́k et al. , 2017 ) . This is impractical for real-world settings , where the reward is sparse ; in fact , the proper reward function for a task is often even unknown due to lack of domain knowledge . Random exploration ( e.g. , -greedy ) in these environments is often insufficient and leads to poor performance ( Bellemare et al. , 2016 ) . Recent approaches have proposed to use intrinsic rewards ( IR ) ( Schmidhuber , 2010 ) to motivate agents for exploration before any extrinsic rewards are obtained . Various criteria have been proposed , including curiosity/surprise-driven ( Pathak et al. , 2017 ) , count-based ( Bellemare et al. , 2016 ; Burda et al. , 2018a ; b ; Ostrovski et al. , 2017 ; Badia et al. , 2020b ) , and state-diff approaches ( Zhang et al. , 2019 ; Marino et al. , 2019 ) . Each approach has its upsides and downsides : Curiosity-driven approaches look for prediction errors in the learned dynamics model and may be misled by the noisy TV ( Burda et al. , 2018b ) problem , where environment dynamics are inherently stochastic . Count-based approaches favor novel states in the environment but suffer from detachment and derailment ( Ecoffet et al. , 2019 ) , in which the agent gets trapped into one ( long ) corridor and fails to try other choices . Count-based approaches are also short-sighted : the agent often settles in local minima , sometimes oscillating between two states that alternately feature lower visitation counts ( Burda et al. , 2018b ) . Finally , state-diff approaches offer rewards if , for each trajectory , representations of consecutive states differ significantly . While these approaches consider the entire trajectory of the agent rather than a local state , it is asymptotically inconsistent : the intrinsic reward remains positive when the visitation counts approach infinity . As a result , the final policy does not necessarily maximize the cumulative extrinsic reward . In this paper , we propose a novel exploration criterion that combines count-based and state-diff approaches : instead of using the difference of state representations , we use the regulated difference of inverse visitation counts of consecutive states in a trajectory . The inverse visitation count is approximated by Random Network Distillation ( Burda et al. , 2018b ) . Our IR provides two benefits : ( 1 ) This addresses asymptotic inconsistency in the state-diff , since the inverse visitation count vanishes with sufficient explorations . ( 2 ) Our IR is large at the end of a trajectory and at the boundary between the explored and the unexplored regions ( Fig . 1 ) . This motivates the agent to move Beyond the Boundary of the explored regions and step into the unknown , mitigating the short-sighted issue in count-based approaches . Following this simple criterion , we propose a novel algorithm BeBold and evaluate it on two very challenging procedurally-generated ( PG ) environments : MiniGrid ( Chevalier-Boisvert et al. , 2018 ) and NetHack ( Küttler et al. , 2020 ) . MiniGrid is a popular benchmark for evaluating exploration algorithms ( Raileanu and Rocktäschel , 2020 ; Campero et al. , 2020 ; Goyal et al. , 2019 ) and NetHack is a much more realistic environment with complex goals and skills . BeBold manages to solve the 12 most challenging environments in MiniGrid within 120M environment steps , without curriculum learning . In contrast , ( Campero et al. , 2020 ) solves 50 % of the tasks , which were categorized as “ easy ” and “ medium ” , by training a separate goal-generating teacher network in 500M steps . In NetHack , a more challenging procedurally-generated environment , BeBold also outperforms all baselines with a significant margin on various tasks . In addition , we analyze BeBold extensively in MiniGrid . The quantitative results show that BeBold largely mitigates the detachment problem , with a much simpler design than Go-Explore ( Ecoffet et al. , 2020 ) which contains multiple handtune stages and hyper-parameters . Most Related Works . RIDE ( Raileanu and Rocktäschel , 2020 ) also combines multiple criteria together . RIDE learns the state representation with curiosity-driven approaches , and then uses the difference of learned representation along a trajectory as the reward , weighted by pseudo counts of the state . However , as a two-stage approach , RIDE heavily relies on the quality of generalization of the learned representation on novel states . As a result , BeBold shows substantially better performance in the same procedurally-generated environments . Go-Explore ( Ecoffet et al. , 2020 ) stores many visited states ( including boundaries ) , reaches these states without exploration , and explores from them . BeBold focuses on boundaries , perform exploration without human-designed cell representation ( e.g. , image downsampling ) and is end-to-end . Frontier-based exploration ( Yamauchi , 1997 ; 1998 ; Topiwala et al. , 2018 ) is used to help specific robots explore the map by maximizing the information gain . The “ frontier ” is defined as the 2D spatial regions out of the explored parts . No automatic policy optimization with deep models is used . In contrast , BeBold can be applied to more general partial observable MDPs with deep policies . 2 BACKGROUND . Following single agent Markov Decision Process ( MDP ) , we define a state space S , an action space A , and a ( non-deterministic ) transition function T : S × A → P ( S ) where P ( S ) is the probability of next state given the current state and action . The goal is to maximize the expected reward R = E [ ∑T k=0 γ krt+k=1 ] where rt is the reward , γ is the discount factor , and the expectation is taken w.r.t . the policy π and MDP transition P ( S ) . In this paper , the total reward received at time step t is given by rt = ret +αr i t , where r e t is the extrinsic reward given by the environment , r i t is the intrinsic reward from the exploration criterion , and α is a scaling hyperparameter . 3 EXPLORATION BEYOND THE BOUNDARY . Our new exploration criterion combines both counting-based and state-diff-based criteria . Our criterion doesn ’ t suffer from short-sightedness and is asymptomatically consistent . We ’ ll first introduce BeBold and then analyse the advantages of BeBold over existing criteria in Sec . 4 . Exploration Beyond the Boundary . BeBold gives intrinsic reward ( IR ) to the agent when it explores beyond the boundary of explored regions , i.e. , along a trajectory , the previous state st has been sufficiently explored but st+1 is new : ri ( st , at , st+1 ) = max ( 1 N ( st+1 ) − 1 N ( st ) , 0 ) , ( 1 ) Here N is the visitation counts . We clip the IR here because we don ’ t want to give a negative IR to the agent if it transits back from a novel state to a familiar state . From the equation , only crossing the frontier matters to the intrinsic reward ; if both N ( st ) and N ( st+1 ) are high or low , their difference would be small . As we will show in Sec . 4 , for each trajectory going towards the frontier/boundary , BeBold assigns an approximately equal IR , regardless of their length . As a result , the agent will continue pushing the frontier of exploration in a much more uniform manner than RND and won ’ t suffer from short-sightedness . This motivates the agent to explore different trajectories uniformly . Also Eq . 1 is asymptotically consistent as ri → 0 when N →∞ . Like RIDE ( Raileanu and Rocktäschel , 2020 ) , in our implementation , partial observation ot are used instead of the real state st , when st is not available . Episodic Restriction on Intrinsic Reward ( ERIR ) . In many environments where the state transition is reversible , simply using intrinsic reward to guide exploration would result in the agent going back and forth between novel states st+1 and their previous states st. RIDE ( Raileanu and Rocktäschel , 2020 ) avoids this by scaling the intrinsic reward r ( s ) by the inverse of the state visitation counts . BeBold puts a more aggressive restriction : the agent is only rewarded when it visits the state s for the first time in an episode . Thus , the intrinsic reward of BeBold becomes : ri ( st , at , st+1 ) = max ( 1 N ( st+1 ) − 1 N ( st ) , 0 ) ∗ 1 { Ne ( st+1 ) = 1 } ( 2 ) Ne here stands for episodic state count and is reset every episode . In contrast , the visitation count N is a life-long memory bank counting state visitation across all of training . Inverse visitation counts as prediction difference . We use the difference between a teacher φ and a student network φ′ to approximate visitation counts : N ( st+1 ) ≈ 1\|\|φ ( ot+1 ) −φ′ ( ot+1 ) \|\|2 , here ot+1 is the observation of the agent in state st+1 . This yields the following implementation of BeBold : ri ( st , at , st+1 ) =max ( \|\|φ ( ot+1 ) − φ′ ( ot+1 ) \|\|2 − \|\|φ ( ot ) − φ′ ( ot ) \|\|2 , 0 ) ∗ 1 { Ne ( ot+1 ) = 1 } ) ( 3 ) Shared visitation counts N ( st ) in the training of Procedurally-Generated ( PG ) Environments . During training , the environment changes constantly ( e.g. , blue keys becomes red ) , while the semantic links of these objects remain the same . We use a shared RND ( φ , φ′ ) across different PG environments , and treat these semantically similar states as new without using domain knowledge ( e.g. , image downsampling like in Go-Explore ( Ecoffet et al. , 2019 ) ) . Partial observability and generalization of neural network φ handles these differences and leads to count-sharing . For episodic count Ne ( ot+1 ) , since it is not shared across episodes ( and environments ) , we use a hash table . 4 CONCEPTUAL ADVANTAGES OF BEBOLD OVER EXISTING CRITERIA . Short-sightedness and Detachment . One issue in the count-based approach is its short-sightedness . Let ’ s assume in a simple environment , there are M corridors { τj } Mj=1 starting at s0 and extending to different parts of the environment . The corridor τj has a length of Tj . The agent starts at s0 . For each visited state , the agent receives the reward of 1N ( s ) where N ( · ) is the visitation count , and learns with Q-learning . Then with some calculation ( See Appendix ) , we see that the agent has a strong preference on exploring the longest corridor first ( say τ1 ) , and only after a long period does it start to explore the second longest . This is because the agent initially receives high IR in τ1 due to its length , which makes the policy π visit τ1 more often , until it depletes the IR in τ1 . This behavior of “ dedication ” could lead to serious issues . If M ≥ 3 and 2 corridors are long enough ( say τ1 and τ2 are long ) , then before the agent is able to explore other corridors , its policy π has already been trained long enough so that it only remembers how to get into τ1 and τ2 . When τ1 has depleted its IR , the agent goes to τ2 following the policy . After that , the IR in τ1 revives since the visitation counts in τ1 is now comparable or even smaller than τ2 , which lures the agent to explore τ1 again following the policy . This leaves other corridors ( e.g. , τ3 ) unexplored for a very long time . Note that using a neural-network-approximated IR ( RND ) instead of tabular IR could potentially alleviate this issue , but it is often far less than enough in complex environments . As mentioned in Go-Explore series ( Ecoffet et al. , 2019 ; 2020 ) , count-based approaches also suffer from detachment : if the agent by chance starts exploring τ2 after briefly exploring the first few states of τ1 , it would not return and explore τ1 further since τ1 is now “ shorter ” than τ2 and has lower IR than τ2 for a long period . Go-Explore tries to resolve this dilemma between “ dedication ” and “ exploration ” by using a two-stage approach with many hand-tuned parameters . In contrast , IR of BeBold depends on the difference of the visitation counts along the trajectory , and is insensitive to the length of the corridor . This leads to simultaneous exploration of multiple corridors and yields a diverse policy π ( See Sec . 5.2 for empirical evidence ) . Moreover , the IR focuses on the boundary between explored and unexplored regions , where the two goals ( dedication and exploration ) align , yielding a much cleaner , one-stage method . Asymptotic Inconsistency . Approaches that define IR as the difference between state representations ‖ψ ( s ) − ψ ( s′ ) ‖ ( ψ is a learned embedding network ) ( Zhang et al. , 2019 ; Marino et al. , 2019 ) suffer from asymptotic inconsistency . In other words , their IR does not vanish even after sufficient exploration : ri 6→ 0 whenN →∞ . This is because when the embedding network ψ converges after sufficient exploration , the agent can always obtain non-zero IR if a major change in state representation occurs ( e.g. , opening a door or picking up a key in MiniGrid ) . Therefore , the learned policy does not maximize the extrinsic reward re , deviating from the goal of RL . Automatic curriculum approaches ( Campero et al. , 2020 ) ) have similar issues due to an ever-present IR . For this , ( Zhang et al. , 2019 ) proposes to learn a separate scheduler to switch between intrinsic and extrinsic rewards , and ( Raileanu and Rocktäschel , 2020 ) divides the state representation difference by the square root of visitation counts . In comparison , BeBold does not require any extra stage and is a much simpler solution .	The authors propose a novel intrinsic reward based on the difference of inverse visitation counts for consecutive states. This reward encourages the agent to explore beyond the boundary of already explored regions. Using a few simple examples, they show that the proposed intrinsic reward mitigates the problems of detachment and short-sightedness which are common for count-based methods. The method shows superior performance on a number of tasks from two procedurally-generated benchmarks, MiniGrid and NetHack. The paper also contains comparisons with a few strong baselines (including SOTA on these benchmarks), analysis of the learned behavior and intrinsic reward, as well as ablations of the proposed approach.	SP:cb17cc8e64068c1b5294af47cc07ccc3ebcada5b
How Benign is Benign Overfitting ?	1 INTRODUCTION . Modern machine learning methods achieve a very high accuracy on wide range of tasks , e.g . in computer vision , natural language processing etc . However , especially in vision tasks , they have been shown to be highly vulnerable to small adversarial perturbations that are imperceptible to the human eye ( Dalvi et al. , 2004 ; Biggio & Roli , 2018 ; Goodfellow et al. , 2014 ) . This vulnerability poses serious security concerns when these models are deployed in real-world tasks ( cf . ( Papernot et al. , 2017 ; Schönherr et al. , 2018 ; Hendrycks et al. , 2019b ; Li et al. , 2019a ) ) . A large body of research has been devoted to crafting defences to protect neural networks from adversarial attacks ( e.g . ( Goodfellow et al. , 2014 ; Papernot et al. , 2015 ; Tramèr et al. , 2018 ; Madry et al. , 2018 ; Zhang et al. , 2019 ) ) . However , such defences have usually been broken by future attacks ( Athalye et al. , 2018 ; Tramer et al. , 2020 ) . This arms race between attacks and defenses suggests that to create a truly robust model would require a deeper understanding of the source of this vulnerability . Our goal in this paper is not to propose new defenses , but to provide better answers to the question : what causes adversarial vulnerability ? In doing so , we also seek to understand how existing methods designed to achieve adversarial robustness overcome some of the hurdles pointed out by our work . We identify two sources of adversarial vulnerability that , to the best of our knowledge , have not been properly studied before : a ) memorization of label noise , and b ) improper representation learning . Overfitting Label Noise : Starting with the celebrated work of Zhang et al . ( 2016 ) it has been observed that neural networks trained with SGD are capable of memorizing large amounts of label noise . Recent theoretical work ( e.g . ( Liang & Rakhlin , 2018 ; Belkin et al. , 2018b ; a ; Hastie et al. , 2019 ; Belkin et al. , 2019a ; b ; Bartlett et al. , 2020 ; Muthukumar et al. , 2020 ; Chatterji & Long , 2020 ) ) has also sought to explain why fitting training data perfectly does not lead to a large drop in test accuracy , as the classical notion of overfitting might suggest . This is commonly referred to as memorization or interpolation . We show through simple theoretical models , as well as experiments on standard datasets , that there are scenarios where label noise causes significant adversarial vulnerability , even when high natural ( test ) accuracy can be achieved . Surprisingly , we find that label noise is not at all uncommon in datasets such as MNIST and CIFAR-10 ( see Figure 1 ) . Our experiments show that robust training methods like Adversarial training ( AT ) ( Madry et al. , 2018 ) and TRADES ( Zhang et al. , 2019 ) produce models that incur training error on at least some of the noisy examples , but also on atypical examples from the classes ( Zhang & Feldman , 2020 ) . Viewed differently , robust training methods are unable to differentiate between atypical correctly labelled examples ( rare dog ) and a mislabelled example ( cat labelled as dog ) and end up not memorizing either ; interestingly , the lack of memorizing these atypical examples has been pointed out as an explanation for slight drops in test accuracy , as the test set often contains similarly atypical ( or even identical ) examples in some cases Feldman ( 2019 ) ; Zhang & Feldman ( 2020 ) . We point out this phenomenon for robust models through visual examples on MNIST , CIFAR10 , and Imagenet ( c.f . Figure 4 ) . Representation Learning and Robustness : Recent works ( Tsipras et al. , 2019 ) and Zhang et al . ( 2019 ) have argued that the trade-off between robustness and accuracy might be unavoidable . However , their setting involves a distribution that is not robustly separable by any classifier . In such a situation there is indeed a trade-off between robustness and accuracy . In this paper , we focus on settings where robust classifiers exist , which is a more realistic scenario for real-world data . At least for vision , one may well argue that “ humans ” are robust classifiers , and as a result we would expect that classes are well-separated at least in some representation space . In fact , Yang et al . ( 2020 ) show that classes are already well-separated in the input space . In such situations , there is no need for robustness to be at odds with accuracy . A more plausible scenario which we posit , and provide theoretical evidence in support of in Theorem 2 , is that depending on the choice of representations , the trade-off may exist or can be avoided . Recent empirical work ( Sanyal et al. , 2020a ; Mao et al. , 2020 ) has also established that modifying the training objective to favour certain properties in the learned representations can automatically lead to improved robustness . However , we show in Section 3.2 that some training algorithms can create an apparent trade-off even though the trade-off might not necessarily be fundamental to the problem . On a related note , it has been suggested in recent works that adversarially robust learning may require more “ complex ” decision boundaries , and as a result may require more data ( Shah et al . ; Schmidt et al. , 2018 ; Yin et al. , 2019 ; Nakkiran , 2019 ; Madry et al. , 2018 ) . However , the question of decision boundaries in neural networks is subtle as the network learns a feature representation as well as a decision boundary on top of it . We develop concrete theoretical examples in Theorem 2 and 3 to establish that choosing one feature representation over another may lead to visually more complex decision boundaries on the input space , though these are not necessarily more complex in terms of statistical learning theoretic concepts such as VC dimension . Summary of Theoretical Contributions . 1 . We provide simple sufficient conditions on the data distribution under which any classifier that fits the training data with label noise perfectly is adversarially vulnerable . 2 . There exists data distributions and training algorithms , which when trained with ( some fraction of ) random label noise have the following property : ( i ) using one representation , it is possible to have high natural and robust test accuracies but at the cost of having training error ; ( ii ) using another representation , it is possible to have no training error ( including fitting noise ) and high test accuracy , but low robust accuracy . ( See Theorem 2 ) . The second example shows that the choice of representation matters significantly when it comes to adversarial accuracy , and that memorizing label noise directly leads to loss of robust accuracy . Summary of Experimental Contributions . 1 . As predicted theoretically , neural nets trained to convergence with label noise have greater adversarial vulnerability . ( See Section 3.1 ) . 2 . Robust training methods , such as AT and TRADES that have higher robust accuracy , avoid overfitting ( some ) label noise . This behaviour is also partly responsible for their decrease in natural test accuracy . ( See Section 3.2 ) . 3 . To demonstrate the benefit of representation learning for adversarial robustness , we show that learning richer representation by training with more fine-grained labels , subclasses within each class , leads to higher robust accuracy . ( Due to lack of space we moved this to Appendix C.3 ) . While our primary contribution is showing the effect of overfitting label noise on adversarial robustness , we hope our theoretical and experimental evidences on the importance of representation learning for robustness will inspire further research in this direction . 2 THEORETICAL SETTING . We develop a simple theoretical framework to demonstrate how overfitting , even very minimal , label noise causes significant adversarial vulnerability . We also show in Theorem 2 and 3 how the choice of representation can significantly affect robust accuracy . Although we state the results for binary classification , they can easily be generalized to multi-class problems . We formally define the notions of natural ( test ) error and adversarial error . Definition 1 ( Natural and Adversarial Error ) . For any distribution D defined over ( x , y ) ∈ Rd × { 0 , 1 } and any binary classifier f : Rd → { 0 , 1 } , • the natural error is R ( f ; D ) = P ( x , y ) ∼D [ f ( x ) 6= y ] , ( 1 ) • if Bγ ( x ) is a ball of radius γ ≥ 0 around x under some norm1 , the γ-adversarial error is RAdv , γ ( f ; D ) = P ( x , y ) ∼D [ ∃z ∈ Bγ ( x ) ; f ( z ) 6= y ] , ( 2 ) In the rest of the section , we provide theoretical results to show the effect of overfitting label noise on the robustness of classifiers . 2.1 OVERFITTING LABEL NOISE . The following result provides a sufficient condition under which even a small amount of label noise causes any classifier that fits the training data perfectly to have significant adversarial error . Informally , Theorem 1 states that if the data distribution has significant probability mass in a union of ( a relatively small number of , and possibly overlapping ) balls , each of which has roughly the same probability mass ( cf . Eq . ( 3 ) ) , then even a small amount of label noise renders this entire region vulnerable to adversarial attacks to classifiers that fit the training data perfectly . Theorem 1 . Let c be the target classifier , and let D be a distribution over ( x , y ) , such that y = c ( x ) in its support . Using the notation PD [ A ] to denote P ( x , y ) ∼D [ x ∈ A ] for any measurable subset A ⊆ Rd , suppose that there exist c1 ≥ c2 > 0 , ρ > 0 , and a finite set ζ ⊂ Rd satisfying PD ⋃ s∈ζ Bpρ ( s ) ≥ c1 and ∀s ∈ ζ , PD [ Bpρ ( s ) ] ≥ c2\|ζ\| ( 3 ) 1Throughout , we will mostly use the ( most commonly used ) ` ∞ norm , but the results hold for other norms . ( a ) Toy-MNIST , = 64 255 ( b ) Full-MNIST where Bpρ ( s ) represents a ` p-ball of radius ρ around s. Further , suppose that each of these balls contain points from a single class i.e . for all s ∈ ζ , for all x , z ∈ Bpρ ( s ) : c ( x ) = c ( z ) . Let Sm be a dataset of m i.i.d . samples drawn from D , which subsequently has each label flipped independently with probability η . For any classifier f that perfectly fits the training data Sm i.e . ∀ x , y ∈ Sm , f ( x ) = y , ∀δ > 0 and m ≥ \|ζ\|ηc2 log ( \|ζ\| δ ) , with probability at least 1 − δ , RAdv,2ρ ( f ; D ) ≥ c1 . The goal is to find a relatively small set ζ that satisfies the condition as this will mean that even for modest sample sizes , the trained models have significant adversarial error . We remark that it is easy to construct concrete instantiations of problems that satisfy the conditions of the theorem , e.g . each class represented by a spherical ( truncated ) Gaussian with radius ρ , with the classes being well-separated satisfies Eq . ( 3 ) . The main idea of the proof is that there is sufficient probability mass for points which are within distance 2ρ of a training datum that was mislabelled . We note that the generality of the result , namely that any classifier ( including neural networks ) that fits the training data must be vulnerable irrespective of its structure , requires a result like Theorem 1 . For instance , one could construct the classifier h , where h ( x ) = c ( x ) , if ( x , b ) 6∈ Sm for b = 0 , 1 , and h ( x ) = y if ( x , y ) ∈ Sm . Note that the classifier h agrees with the target c on every point of Rd except the mislabelled training examples , and as a result these examples are the only source of vulnerability . The complete proof is presented in Appendix B.1 . There are a few things to note about Theorem 1 . First , the lower bound on adversarial error applies to any classifier f that fits the training data Sm perfectly and is agnostic to the type of model f is . Second , for a given c1 , there maybe multiple ζs that satisfy the bounds in ( 3 ) and the adversarial risk holds for all of them . Thus , smaller the value of \|ζ\| the smaller the size of the training data it needs to fit and it can be done by simpler classifiers . Third , if the distribution of the data is such that it is concentrated around some points then for a fixed c1 , c2 , a smaller value of ρ would be required to satisfy ( 3 ) and thus a weaker adversary ( smaller perturbation budget 2ρ ) can cause a much larger adversarial error . In practice , classifiers exhibit much greater vulnerability than purely arising from the presence of memorized noisy data . Experiments in Section 3.1 show how label noise causes vulnerability in a toy MNIST model , the full MNIST and CIFAR10 for a variety of architectures .	The main contribution of the paper is to study the connection between adversarial robustness, on the one hand, and label noise & data representation on the other hand. Here, an algorithm is said to be robust if for every training example xi with label yi, one cannot find an instance x within a small distance of xi that is assigned a different label from yi by the model. This is a standard definition of robustness in the literature.	SP:c95614e6cad71a73222413ff12a67e96ec487e40
How Benign is Benign Overfitting ?	1 INTRODUCTION . Modern machine learning methods achieve a very high accuracy on wide range of tasks , e.g . in computer vision , natural language processing etc . However , especially in vision tasks , they have been shown to be highly vulnerable to small adversarial perturbations that are imperceptible to the human eye ( Dalvi et al. , 2004 ; Biggio & Roli , 2018 ; Goodfellow et al. , 2014 ) . This vulnerability poses serious security concerns when these models are deployed in real-world tasks ( cf . ( Papernot et al. , 2017 ; Schönherr et al. , 2018 ; Hendrycks et al. , 2019b ; Li et al. , 2019a ) ) . A large body of research has been devoted to crafting defences to protect neural networks from adversarial attacks ( e.g . ( Goodfellow et al. , 2014 ; Papernot et al. , 2015 ; Tramèr et al. , 2018 ; Madry et al. , 2018 ; Zhang et al. , 2019 ) ) . However , such defences have usually been broken by future attacks ( Athalye et al. , 2018 ; Tramer et al. , 2020 ) . This arms race between attacks and defenses suggests that to create a truly robust model would require a deeper understanding of the source of this vulnerability . Our goal in this paper is not to propose new defenses , but to provide better answers to the question : what causes adversarial vulnerability ? In doing so , we also seek to understand how existing methods designed to achieve adversarial robustness overcome some of the hurdles pointed out by our work . We identify two sources of adversarial vulnerability that , to the best of our knowledge , have not been properly studied before : a ) memorization of label noise , and b ) improper representation learning . Overfitting Label Noise : Starting with the celebrated work of Zhang et al . ( 2016 ) it has been observed that neural networks trained with SGD are capable of memorizing large amounts of label noise . Recent theoretical work ( e.g . ( Liang & Rakhlin , 2018 ; Belkin et al. , 2018b ; a ; Hastie et al. , 2019 ; Belkin et al. , 2019a ; b ; Bartlett et al. , 2020 ; Muthukumar et al. , 2020 ; Chatterji & Long , 2020 ) ) has also sought to explain why fitting training data perfectly does not lead to a large drop in test accuracy , as the classical notion of overfitting might suggest . This is commonly referred to as memorization or interpolation . We show through simple theoretical models , as well as experiments on standard datasets , that there are scenarios where label noise causes significant adversarial vulnerability , even when high natural ( test ) accuracy can be achieved . Surprisingly , we find that label noise is not at all uncommon in datasets such as MNIST and CIFAR-10 ( see Figure 1 ) . Our experiments show that robust training methods like Adversarial training ( AT ) ( Madry et al. , 2018 ) and TRADES ( Zhang et al. , 2019 ) produce models that incur training error on at least some of the noisy examples , but also on atypical examples from the classes ( Zhang & Feldman , 2020 ) . Viewed differently , robust training methods are unable to differentiate between atypical correctly labelled examples ( rare dog ) and a mislabelled example ( cat labelled as dog ) and end up not memorizing either ; interestingly , the lack of memorizing these atypical examples has been pointed out as an explanation for slight drops in test accuracy , as the test set often contains similarly atypical ( or even identical ) examples in some cases Feldman ( 2019 ) ; Zhang & Feldman ( 2020 ) . We point out this phenomenon for robust models through visual examples on MNIST , CIFAR10 , and Imagenet ( c.f . Figure 4 ) . Representation Learning and Robustness : Recent works ( Tsipras et al. , 2019 ) and Zhang et al . ( 2019 ) have argued that the trade-off between robustness and accuracy might be unavoidable . However , their setting involves a distribution that is not robustly separable by any classifier . In such a situation there is indeed a trade-off between robustness and accuracy . In this paper , we focus on settings where robust classifiers exist , which is a more realistic scenario for real-world data . At least for vision , one may well argue that “ humans ” are robust classifiers , and as a result we would expect that classes are well-separated at least in some representation space . In fact , Yang et al . ( 2020 ) show that classes are already well-separated in the input space . In such situations , there is no need for robustness to be at odds with accuracy . A more plausible scenario which we posit , and provide theoretical evidence in support of in Theorem 2 , is that depending on the choice of representations , the trade-off may exist or can be avoided . Recent empirical work ( Sanyal et al. , 2020a ; Mao et al. , 2020 ) has also established that modifying the training objective to favour certain properties in the learned representations can automatically lead to improved robustness . However , we show in Section 3.2 that some training algorithms can create an apparent trade-off even though the trade-off might not necessarily be fundamental to the problem . On a related note , it has been suggested in recent works that adversarially robust learning may require more “ complex ” decision boundaries , and as a result may require more data ( Shah et al . ; Schmidt et al. , 2018 ; Yin et al. , 2019 ; Nakkiran , 2019 ; Madry et al. , 2018 ) . However , the question of decision boundaries in neural networks is subtle as the network learns a feature representation as well as a decision boundary on top of it . We develop concrete theoretical examples in Theorem 2 and 3 to establish that choosing one feature representation over another may lead to visually more complex decision boundaries on the input space , though these are not necessarily more complex in terms of statistical learning theoretic concepts such as VC dimension . Summary of Theoretical Contributions . 1 . We provide simple sufficient conditions on the data distribution under which any classifier that fits the training data with label noise perfectly is adversarially vulnerable . 2 . There exists data distributions and training algorithms , which when trained with ( some fraction of ) random label noise have the following property : ( i ) using one representation , it is possible to have high natural and robust test accuracies but at the cost of having training error ; ( ii ) using another representation , it is possible to have no training error ( including fitting noise ) and high test accuracy , but low robust accuracy . ( See Theorem 2 ) . The second example shows that the choice of representation matters significantly when it comes to adversarial accuracy , and that memorizing label noise directly leads to loss of robust accuracy . Summary of Experimental Contributions . 1 . As predicted theoretically , neural nets trained to convergence with label noise have greater adversarial vulnerability . ( See Section 3.1 ) . 2 . Robust training methods , such as AT and TRADES that have higher robust accuracy , avoid overfitting ( some ) label noise . This behaviour is also partly responsible for their decrease in natural test accuracy . ( See Section 3.2 ) . 3 . To demonstrate the benefit of representation learning for adversarial robustness , we show that learning richer representation by training with more fine-grained labels , subclasses within each class , leads to higher robust accuracy . ( Due to lack of space we moved this to Appendix C.3 ) . While our primary contribution is showing the effect of overfitting label noise on adversarial robustness , we hope our theoretical and experimental evidences on the importance of representation learning for robustness will inspire further research in this direction . 2 THEORETICAL SETTING . We develop a simple theoretical framework to demonstrate how overfitting , even very minimal , label noise causes significant adversarial vulnerability . We also show in Theorem 2 and 3 how the choice of representation can significantly affect robust accuracy . Although we state the results for binary classification , they can easily be generalized to multi-class problems . We formally define the notions of natural ( test ) error and adversarial error . Definition 1 ( Natural and Adversarial Error ) . For any distribution D defined over ( x , y ) ∈ Rd × { 0 , 1 } and any binary classifier f : Rd → { 0 , 1 } , • the natural error is R ( f ; D ) = P ( x , y ) ∼D [ f ( x ) 6= y ] , ( 1 ) • if Bγ ( x ) is a ball of radius γ ≥ 0 around x under some norm1 , the γ-adversarial error is RAdv , γ ( f ; D ) = P ( x , y ) ∼D [ ∃z ∈ Bγ ( x ) ; f ( z ) 6= y ] , ( 2 ) In the rest of the section , we provide theoretical results to show the effect of overfitting label noise on the robustness of classifiers . 2.1 OVERFITTING LABEL NOISE . The following result provides a sufficient condition under which even a small amount of label noise causes any classifier that fits the training data perfectly to have significant adversarial error . Informally , Theorem 1 states that if the data distribution has significant probability mass in a union of ( a relatively small number of , and possibly overlapping ) balls , each of which has roughly the same probability mass ( cf . Eq . ( 3 ) ) , then even a small amount of label noise renders this entire region vulnerable to adversarial attacks to classifiers that fit the training data perfectly . Theorem 1 . Let c be the target classifier , and let D be a distribution over ( x , y ) , such that y = c ( x ) in its support . Using the notation PD [ A ] to denote P ( x , y ) ∼D [ x ∈ A ] for any measurable subset A ⊆ Rd , suppose that there exist c1 ≥ c2 > 0 , ρ > 0 , and a finite set ζ ⊂ Rd satisfying PD ⋃ s∈ζ Bpρ ( s ) ≥ c1 and ∀s ∈ ζ , PD [ Bpρ ( s ) ] ≥ c2\|ζ\| ( 3 ) 1Throughout , we will mostly use the ( most commonly used ) ` ∞ norm , but the results hold for other norms . ( a ) Toy-MNIST , = 64 255 ( b ) Full-MNIST where Bpρ ( s ) represents a ` p-ball of radius ρ around s. Further , suppose that each of these balls contain points from a single class i.e . for all s ∈ ζ , for all x , z ∈ Bpρ ( s ) : c ( x ) = c ( z ) . Let Sm be a dataset of m i.i.d . samples drawn from D , which subsequently has each label flipped independently with probability η . For any classifier f that perfectly fits the training data Sm i.e . ∀ x , y ∈ Sm , f ( x ) = y , ∀δ > 0 and m ≥ \|ζ\|ηc2 log ( \|ζ\| δ ) , with probability at least 1 − δ , RAdv,2ρ ( f ; D ) ≥ c1 . The goal is to find a relatively small set ζ that satisfies the condition as this will mean that even for modest sample sizes , the trained models have significant adversarial error . We remark that it is easy to construct concrete instantiations of problems that satisfy the conditions of the theorem , e.g . each class represented by a spherical ( truncated ) Gaussian with radius ρ , with the classes being well-separated satisfies Eq . ( 3 ) . The main idea of the proof is that there is sufficient probability mass for points which are within distance 2ρ of a training datum that was mislabelled . We note that the generality of the result , namely that any classifier ( including neural networks ) that fits the training data must be vulnerable irrespective of its structure , requires a result like Theorem 1 . For instance , one could construct the classifier h , where h ( x ) = c ( x ) , if ( x , b ) 6∈ Sm for b = 0 , 1 , and h ( x ) = y if ( x , y ) ∈ Sm . Note that the classifier h agrees with the target c on every point of Rd except the mislabelled training examples , and as a result these examples are the only source of vulnerability . The complete proof is presented in Appendix B.1 . There are a few things to note about Theorem 1 . First , the lower bound on adversarial error applies to any classifier f that fits the training data Sm perfectly and is agnostic to the type of model f is . Second , for a given c1 , there maybe multiple ζs that satisfy the bounds in ( 3 ) and the adversarial risk holds for all of them . Thus , smaller the value of \|ζ\| the smaller the size of the training data it needs to fit and it can be done by simpler classifiers . Third , if the distribution of the data is such that it is concentrated around some points then for a fixed c1 , c2 , a smaller value of ρ would be required to satisfy ( 3 ) and thus a weaker adversary ( smaller perturbation budget 2ρ ) can cause a much larger adversarial error . In practice , classifiers exhibit much greater vulnerability than purely arising from the presence of memorized noisy data . Experiments in Section 3.1 show how label noise causes vulnerability in a toy MNIST model , the full MNIST and CIFAR10 for a variety of architectures .	The goal of the paper is to investigate both theoretically and empirically the reasons of vulnerability of overparameterized classifiers obtained by the so called “benign overfitting”. More precisely, two causes of adversarial vulnerability are underlined: label noise memorization and sub-optimal representation learning. The first theorem of the paper shows that for some data generating distributions, even a small fraction of label noise leads to an adversarial prediction risk bounded away from zero for any classifier having zero training error and for any sufficiently large sample size. The second theorem shows that in the presence of label noise the choice of the overparameterized family (the representation) is very important. Namely, while for a good representation one may have “training error = test error = adversarial error = 0”, for another representation it holds that “training error = test error = 0” but “adversarial error > 0.1”. This theoretical results are illustrated by extensive experimental results.	SP:c95614e6cad71a73222413ff12a67e96ec487e40
How Benign is Benign Overfitting ?	1 INTRODUCTION . Modern machine learning methods achieve a very high accuracy on wide range of tasks , e.g . in computer vision , natural language processing etc . However , especially in vision tasks , they have been shown to be highly vulnerable to small adversarial perturbations that are imperceptible to the human eye ( Dalvi et al. , 2004 ; Biggio & Roli , 2018 ; Goodfellow et al. , 2014 ) . This vulnerability poses serious security concerns when these models are deployed in real-world tasks ( cf . ( Papernot et al. , 2017 ; Schönherr et al. , 2018 ; Hendrycks et al. , 2019b ; Li et al. , 2019a ) ) . A large body of research has been devoted to crafting defences to protect neural networks from adversarial attacks ( e.g . ( Goodfellow et al. , 2014 ; Papernot et al. , 2015 ; Tramèr et al. , 2018 ; Madry et al. , 2018 ; Zhang et al. , 2019 ) ) . However , such defences have usually been broken by future attacks ( Athalye et al. , 2018 ; Tramer et al. , 2020 ) . This arms race between attacks and defenses suggests that to create a truly robust model would require a deeper understanding of the source of this vulnerability . Our goal in this paper is not to propose new defenses , but to provide better answers to the question : what causes adversarial vulnerability ? In doing so , we also seek to understand how existing methods designed to achieve adversarial robustness overcome some of the hurdles pointed out by our work . We identify two sources of adversarial vulnerability that , to the best of our knowledge , have not been properly studied before : a ) memorization of label noise , and b ) improper representation learning . Overfitting Label Noise : Starting with the celebrated work of Zhang et al . ( 2016 ) it has been observed that neural networks trained with SGD are capable of memorizing large amounts of label noise . Recent theoretical work ( e.g . ( Liang & Rakhlin , 2018 ; Belkin et al. , 2018b ; a ; Hastie et al. , 2019 ; Belkin et al. , 2019a ; b ; Bartlett et al. , 2020 ; Muthukumar et al. , 2020 ; Chatterji & Long , 2020 ) ) has also sought to explain why fitting training data perfectly does not lead to a large drop in test accuracy , as the classical notion of overfitting might suggest . This is commonly referred to as memorization or interpolation . We show through simple theoretical models , as well as experiments on standard datasets , that there are scenarios where label noise causes significant adversarial vulnerability , even when high natural ( test ) accuracy can be achieved . Surprisingly , we find that label noise is not at all uncommon in datasets such as MNIST and CIFAR-10 ( see Figure 1 ) . Our experiments show that robust training methods like Adversarial training ( AT ) ( Madry et al. , 2018 ) and TRADES ( Zhang et al. , 2019 ) produce models that incur training error on at least some of the noisy examples , but also on atypical examples from the classes ( Zhang & Feldman , 2020 ) . Viewed differently , robust training methods are unable to differentiate between atypical correctly labelled examples ( rare dog ) and a mislabelled example ( cat labelled as dog ) and end up not memorizing either ; interestingly , the lack of memorizing these atypical examples has been pointed out as an explanation for slight drops in test accuracy , as the test set often contains similarly atypical ( or even identical ) examples in some cases Feldman ( 2019 ) ; Zhang & Feldman ( 2020 ) . We point out this phenomenon for robust models through visual examples on MNIST , CIFAR10 , and Imagenet ( c.f . Figure 4 ) . Representation Learning and Robustness : Recent works ( Tsipras et al. , 2019 ) and Zhang et al . ( 2019 ) have argued that the trade-off between robustness and accuracy might be unavoidable . However , their setting involves a distribution that is not robustly separable by any classifier . In such a situation there is indeed a trade-off between robustness and accuracy . In this paper , we focus on settings where robust classifiers exist , which is a more realistic scenario for real-world data . At least for vision , one may well argue that “ humans ” are robust classifiers , and as a result we would expect that classes are well-separated at least in some representation space . In fact , Yang et al . ( 2020 ) show that classes are already well-separated in the input space . In such situations , there is no need for robustness to be at odds with accuracy . A more plausible scenario which we posit , and provide theoretical evidence in support of in Theorem 2 , is that depending on the choice of representations , the trade-off may exist or can be avoided . Recent empirical work ( Sanyal et al. , 2020a ; Mao et al. , 2020 ) has also established that modifying the training objective to favour certain properties in the learned representations can automatically lead to improved robustness . However , we show in Section 3.2 that some training algorithms can create an apparent trade-off even though the trade-off might not necessarily be fundamental to the problem . On a related note , it has been suggested in recent works that adversarially robust learning may require more “ complex ” decision boundaries , and as a result may require more data ( Shah et al . ; Schmidt et al. , 2018 ; Yin et al. , 2019 ; Nakkiran , 2019 ; Madry et al. , 2018 ) . However , the question of decision boundaries in neural networks is subtle as the network learns a feature representation as well as a decision boundary on top of it . We develop concrete theoretical examples in Theorem 2 and 3 to establish that choosing one feature representation over another may lead to visually more complex decision boundaries on the input space , though these are not necessarily more complex in terms of statistical learning theoretic concepts such as VC dimension . Summary of Theoretical Contributions . 1 . We provide simple sufficient conditions on the data distribution under which any classifier that fits the training data with label noise perfectly is adversarially vulnerable . 2 . There exists data distributions and training algorithms , which when trained with ( some fraction of ) random label noise have the following property : ( i ) using one representation , it is possible to have high natural and robust test accuracies but at the cost of having training error ; ( ii ) using another representation , it is possible to have no training error ( including fitting noise ) and high test accuracy , but low robust accuracy . ( See Theorem 2 ) . The second example shows that the choice of representation matters significantly when it comes to adversarial accuracy , and that memorizing label noise directly leads to loss of robust accuracy . Summary of Experimental Contributions . 1 . As predicted theoretically , neural nets trained to convergence with label noise have greater adversarial vulnerability . ( See Section 3.1 ) . 2 . Robust training methods , such as AT and TRADES that have higher robust accuracy , avoid overfitting ( some ) label noise . This behaviour is also partly responsible for their decrease in natural test accuracy . ( See Section 3.2 ) . 3 . To demonstrate the benefit of representation learning for adversarial robustness , we show that learning richer representation by training with more fine-grained labels , subclasses within each class , leads to higher robust accuracy . ( Due to lack of space we moved this to Appendix C.3 ) . While our primary contribution is showing the effect of overfitting label noise on adversarial robustness , we hope our theoretical and experimental evidences on the importance of representation learning for robustness will inspire further research in this direction . 2 THEORETICAL SETTING . We develop a simple theoretical framework to demonstrate how overfitting , even very minimal , label noise causes significant adversarial vulnerability . We also show in Theorem 2 and 3 how the choice of representation can significantly affect robust accuracy . Although we state the results for binary classification , they can easily be generalized to multi-class problems . We formally define the notions of natural ( test ) error and adversarial error . Definition 1 ( Natural and Adversarial Error ) . For any distribution D defined over ( x , y ) ∈ Rd × { 0 , 1 } and any binary classifier f : Rd → { 0 , 1 } , • the natural error is R ( f ; D ) = P ( x , y ) ∼D [ f ( x ) 6= y ] , ( 1 ) • if Bγ ( x ) is a ball of radius γ ≥ 0 around x under some norm1 , the γ-adversarial error is RAdv , γ ( f ; D ) = P ( x , y ) ∼D [ ∃z ∈ Bγ ( x ) ; f ( z ) 6= y ] , ( 2 ) In the rest of the section , we provide theoretical results to show the effect of overfitting label noise on the robustness of classifiers . 2.1 OVERFITTING LABEL NOISE . The following result provides a sufficient condition under which even a small amount of label noise causes any classifier that fits the training data perfectly to have significant adversarial error . Informally , Theorem 1 states that if the data distribution has significant probability mass in a union of ( a relatively small number of , and possibly overlapping ) balls , each of which has roughly the same probability mass ( cf . Eq . ( 3 ) ) , then even a small amount of label noise renders this entire region vulnerable to adversarial attacks to classifiers that fit the training data perfectly . Theorem 1 . Let c be the target classifier , and let D be a distribution over ( x , y ) , such that y = c ( x ) in its support . Using the notation PD [ A ] to denote P ( x , y ) ∼D [ x ∈ A ] for any measurable subset A ⊆ Rd , suppose that there exist c1 ≥ c2 > 0 , ρ > 0 , and a finite set ζ ⊂ Rd satisfying PD ⋃ s∈ζ Bpρ ( s ) ≥ c1 and ∀s ∈ ζ , PD [ Bpρ ( s ) ] ≥ c2\|ζ\| ( 3 ) 1Throughout , we will mostly use the ( most commonly used ) ` ∞ norm , but the results hold for other norms . ( a ) Toy-MNIST , = 64 255 ( b ) Full-MNIST where Bpρ ( s ) represents a ` p-ball of radius ρ around s. Further , suppose that each of these balls contain points from a single class i.e . for all s ∈ ζ , for all x , z ∈ Bpρ ( s ) : c ( x ) = c ( z ) . Let Sm be a dataset of m i.i.d . samples drawn from D , which subsequently has each label flipped independently with probability η . For any classifier f that perfectly fits the training data Sm i.e . ∀ x , y ∈ Sm , f ( x ) = y , ∀δ > 0 and m ≥ \|ζ\|ηc2 log ( \|ζ\| δ ) , with probability at least 1 − δ , RAdv,2ρ ( f ; D ) ≥ c1 . The goal is to find a relatively small set ζ that satisfies the condition as this will mean that even for modest sample sizes , the trained models have significant adversarial error . We remark that it is easy to construct concrete instantiations of problems that satisfy the conditions of the theorem , e.g . each class represented by a spherical ( truncated ) Gaussian with radius ρ , with the classes being well-separated satisfies Eq . ( 3 ) . The main idea of the proof is that there is sufficient probability mass for points which are within distance 2ρ of a training datum that was mislabelled . We note that the generality of the result , namely that any classifier ( including neural networks ) that fits the training data must be vulnerable irrespective of its structure , requires a result like Theorem 1 . For instance , one could construct the classifier h , where h ( x ) = c ( x ) , if ( x , b ) 6∈ Sm for b = 0 , 1 , and h ( x ) = y if ( x , y ) ∈ Sm . Note that the classifier h agrees with the target c on every point of Rd except the mislabelled training examples , and as a result these examples are the only source of vulnerability . The complete proof is presented in Appendix B.1 . There are a few things to note about Theorem 1 . First , the lower bound on adversarial error applies to any classifier f that fits the training data Sm perfectly and is agnostic to the type of model f is . Second , for a given c1 , there maybe multiple ζs that satisfy the bounds in ( 3 ) and the adversarial risk holds for all of them . Thus , smaller the value of \|ζ\| the smaller the size of the training data it needs to fit and it can be done by simpler classifiers . Third , if the distribution of the data is such that it is concentrated around some points then for a fixed c1 , c2 , a smaller value of ρ would be required to satisfy ( 3 ) and thus a weaker adversary ( smaller perturbation budget 2ρ ) can cause a much larger adversarial error . In practice , classifiers exhibit much greater vulnerability than purely arising from the presence of memorized noisy data . Experiments in Section 3.1 show how label noise causes vulnerability in a toy MNIST model , the full MNIST and CIFAR10 for a variety of architectures .	The generalization ability of networks with zero training error has been heavily studied. This paper extends beyond generalization to test sets to study the network's robustness to adversarial examples. The paper provides two theoretical contributions demonstrating that a very low training error can indicate poor robustness under reasonable conditions. They illustrate this with experiments using label noise, demonstrating that adversarially robust networks spurn overfitting on incorrectly labelled data. They additionally experimentally demonstrate that unusual training examples, even if correctly labelled, are unlikely to be correctly predicted by adversarially robust networks.	SP:c95614e6cad71a73222413ff12a67e96ec487e40
Optimal allocation of data across training tasks in meta-learning	Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones . They are tested on benchmarks with a fixed number of data-points for each training task , and this number is usually arbitrary , for example , 5 instances per class in few-shot classification . It is unknown how the performance of meta-learning is affected by the distribution of data across training tasks . Since labelling of data is expensive , finding the optimal allocation of labels across training tasks may reduce costs . Given a fixed budget b of labels to distribute across tasks , should we use a small number of highly labelled tasks , or many tasks with few labels each ? In MAML applied to mixed linear regression , we prove that the optimal number of tasks follows the scaling law √ b . We develop an online algorithm for data allocation across tasks , and show that the same scaling law applies to nonlinear regression . We also show preliminary experiments on few-shot image classification . Our work provides a theoretical guide for allocating labels across tasks in meta-learning , which we believe will prove useful in a large number of applications . 1 INTRODUCTION . Deep learning ( DL ) models require a large amount of data in order to perform well , when trained from scratch , but labeling data is expensive and time consuming . An effective approach to avoid the costs of collecting and labeling large amount of data is transfer learning : train a model on one big dataset , or a few related datasets that are already available , and then fine-tune the model on the target dataset , which can be of much smaller size ( Donahue et al . ( 2014 ) ) . In this context , there has been a recent surge of interest in the field of meta-learning , which is inspired by the ability of humans to learn how to learn Hospedales et al . ( 2020 ) . A model is meta-trained on a large number of tasks , each characterized by a small dataset , and meta-tested on the target dataset . The number of data points per task is usually set to an arbitrary number in standard meta-learning benchmarks . For example , in few-shot image classification benchmarks , such as mini-ImageNet ( Vinyals et al . ( 2017 ) , Ravi & Larochelle ( 2017 ) ) and CIFAR-FS ( Bertinetto et al . ( 2019 ) ) , this number is usually set to 1 or 5 . So far , there has not been any reason to optimize this number , as in most circumstances the performance of a model will improve with the number of data points ( see Nakkiran et al . ( 2019 ) for exceptions ) . However , if the total number of labels across training tasks is limited , is it better to have a large number of tasks with very small data in each , or a relatively smaller number of highly labelled tasks ? Since data-labeling is costly , the answer to this question may inform the design of new meta-learning datasets and benchmarks . In this work , to our knowledge , we answer this question for the first time , for a specific meta-learning algorithm : MAML ( Finn et al . ( 2017 ) ) . We study the problem of optimizing the number of metatraining tasks , with a fixed budget b of total data-points to distribute across tasks . We study the application of MAML to three datasets : mixed linear regression , sinusoid regression , and CIFAR . In the case of mixed linear regression , we derive an approximation for the meta-test loss , and according to which the optimal number of tasks follows the scaling rule √ b . In order to optimize the number of tasks empirically , we design an algorithm for online allocation of data across training tasks , and we validate the algorithm by performing a grid search over a large set of possible allocations . In summary , our contributions are : • We introduce and formalize the problem of optimizing data allocation with a fixed budget b in meta-learning . • We prove that the optimal scaling of the number of tasks is √ b in mixed linear regression , and confirm this scaling empirically in nolinear regression . • We introduce an algorithm for online allocation of data across tasks , to find the optimal number of tasks during meta-training , and validate the algorithm by grid search . • We perform preliminary experiments on few-shot image classification . 2 RELATED WORK . A couple of recent papers investigated a problem similar to ours . In the context of meta-learning and mixed linear regression , Kong et al . ( 2020 ) asks whether many tasks with small data can compensate for a lack of tasks with big data . However , they do not address the problem of finding the optimal number of tasks within a fixed budget . The work of Shekhar et al . ( 2020 ) studies exactly the problem of allocating a fixed budget of data points , but to the problem of estimating a finite set of discrete distributions , therefore they do not study the meta-learning problem and their data has no labels . An alternative approach to avoid labelling a large amount of data is active learning , where a model learns with fewer labels by accurately selecting which data to learn from ( Settles ( 2010 ) ) . In the context of meta-learning , the option of implementing active learning has been considered in a few recent studies ( Bachman et al . ( 2017 ) , Garcia & Bruna ( 2018 ) , Kim et al . ( 2018 ) , Finn et al . ( 2019 ) , Requeima et al . ( 2020 ) ) . However , they considered the active labeling of data within a given task , for the purpose of improving performance in that task only . Instead , we ask how data should be distributed across tasks . In the context of recommender systems and text classification , a few studies considered whether labeling a data point , within a given task , may increase performance not only in that task but also in all other tasks . This problem has been referred to as multi-task active learning ( Reichart et al . ( 2008 ) , Zhang ( 2010 ) , Saha et al . ( 2011 ) , Harpale ( 2012 ) , Fang et al . ( 2017 ) ) , or multi-domain active learning ( Li et al . ( 2012 ) , Zhang et al . ( 2016 ) ) . However , none of these studies consider the problem of meta-learning with a fixed budget . A few studies have looked into actively choosing the next task in a sequence of tasks ( Ruvolo & Eaton ( 2013 ) , Pentina et al . ( 2015 ) , Pentina & Lampert ( 2017 ) , Sun et al . ( 2018 ) ) , but they do not look at how to distribute data across tasks . 3 THE PROBLEM OF DATA ALLOCATION FOR META-LEARNING . In the cross-task setting , we are presented with a hierarchically structured dataset , with task parameters ( τ ( i ) ) mi=1 sampled from T ∼ p ( τ ) and data ( xτj ) nτ j=1 sampled from Dτ : = ( D\|T ) ∼ p ( x\|T = τ ) . Our problem is minimizing the following loss function with respect to a parameter ω : L ( ω ) = E T E Dτ L ( ω ; xτ ) ( 1 ) The empirical risk minimization principle ( see Vapnik ( 1998 ) ) ensures that the optimum of the empirical risk converges to that of the true risk with an increase in samples from the joint distribution of ( D , T ) . 3.1 META-LEARNING ACROSS TASKS . In the meta-learning problem , we are given the opportunity to adjust the objective function to each task . This adjustment is given by the adaptation step of meta-learning ( Hospedales et al . ( 2020 ) ) , which represents a transformation on the parameters ω , which is task-dependent and which we refer to as θτ ( ω ) . The loss function Lmeta is defined as an average across both distribution of tasks and data points . The goal of meta-learning is to minimize the loss function with respect to a vector of metaparameters ω Lmeta ( ω ) = E T E Dτ Lτ ( θτ ( ω ) ; xτ ) ( 2 ) Different meta-learning algorithms correspond to a different choice of θτ ( ω ) and we allow for each task τ to have its own specific loss function . The dependence on τ built in through the composition with θτ ( ω ) makes the loss function a sample from a random function field , when considered as a deterministic function of the data and model parameter ω . 3.2 MODEL-AGNOSTIC META-LEARNING . Our case-study for this paper is the meta-learning algorithm MAML developed in Finn et al . ( 2017 ) . MAML employs a base learner , which parametrizes an estimator family by ω . The algorithm ’ s adaptation step , inspired from fine-tuning , performs a fixed number of SGD steps with respect to the data for each task . Thus , the adaptation step maps into the same parameter space as ω . In MAML with a single gradient step , if we denote the data for task τ ( i ) by ( x ( i ) j ) ni j=1 , and the SGD learning rate by α , this transformation is equal to : θ ( i ) ( ω ) = ω − α ni ni∑ j=1 ∇ωL\|ω ; x ( i ) j ( 3 ) This formula corresponds to a full-batch update , employing all the data for task τ ( i ) , but minibatch gradient updates can be performed as well . During meta-training , the loss is evaluated on a sample of m tasks , and a sample of validation data points ni for each task , leading to the following optimization objective : Lmeta ( ω ) = 1 m m∑ i=1 1 ni ni∑ j=1 L ( θ ( i ) ( ω ) ; x ( i ) j ) ( 4 ) During meta-testing , a new ( target ) task is given and the parameters θ are learned by a set of target data points following the same equation 3 . The final performance of the model is computed on test data of the target task . 3.3 DATA ALLOCATION . In this work , we study the problem of finding values of m and ni , such that , under some constraint , the meta-learning loss 2 is minimized given the available data . In this section we make the terms allocation and budget explicit . The budget of a meta-learning problem is defined as the value b =∑m i=1 ni , i.e . the total number of data points available for meta-training . A meta-training set which respects a budget b is a collection of m tasks sampled independently from the task distribution , each composed of a set of ni data points sampled independently from their respective data-generating distributions , such that b = ∑m i=1 ni . Conversely , given a value of the budget b , and a set of tasks τ ( 1 ) , ... τ ( m ) , a data-allocation of the budget b to these tasks is a partition of b into ni such that b = ∑m i=1 ni . If n = ni for all i , this definition is independent of the task samples drawn , and we call this the uniform allocation on m tasks for the budget b . In this work , we only consider the family of uniform allocations , and we leave the study of non-uniform allocations for future work . We denote a meta-dataset which respects the uniform allocation with m tasks and n datapoints per task , drawn independently from task distributions T and data distributionsDτ byM ( T , Dτ ; m , n ) . The optimal uniform data allocation is given by the values ofm and n which minimize the expected test meta-loss ( 2 ) , after optimizing the meta-parameter ω on a meta-training setM ( T , Dτ ; m , n ) . Formally , expanding notation to make the dependence of Lmeta onM explicit in its arguments : ω∗ ( M ( T , Dτ ; m , n ) ) = arg min ω Lmeta ( ω ; M ( T , Dτ ; m , n ) ) ( 5 ) Then the optimal data allocation is ( m∗ ( b ) , n∗ ( b ) ) = arg min m·n=b E T E Dτ Lmeta ( ω∗ ( M ( T , Dτ ; m , n ) ) ; Mtest ) ( 6 ) Remark . Notice that in equation ( 6 ) , Lmeta is already an expectation taken over the data distribution which generated the test set . In the definition , this expectation is taken again over the distribution of meta training setsM ( T , Dτ ; m , n ) . It will be useful to writeLmeta ( m , n ; T , Dτ ) for the expression minimized in equation 6 .	In most popular meta-learning approaches, there are usually a pre-defined number of data per task. For example, 1-shot or 5-shot learning. It has shown as the number of data increases in such methods, model performance improves. In this paper, they try to analyze the effect of having different number of tasks with different number of data points in meta-learning benchmarks. Specifically, given a fixed number of data across different tasks as a budget, they want to see if having a large number of tasks with small data in each works better (or worse) than having small number of tasks with more data in each. They focus on MAML as a meta-learning method, and analyzed the results on mixed linear regression, sinusoid regression, and CIFAR. For mixed linear regression, they showed that the optimal number of tasks is \sqrt(b). They also provide an online algorithm for finding optimal number of tasks in meta-learning.	SP:2dd943d37d914575aa6ac4f3948a2d5d9f53a8d9
Optimal allocation of data across training tasks in meta-learning	Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones . They are tested on benchmarks with a fixed number of data-points for each training task , and this number is usually arbitrary , for example , 5 instances per class in few-shot classification . It is unknown how the performance of meta-learning is affected by the distribution of data across training tasks . Since labelling of data is expensive , finding the optimal allocation of labels across training tasks may reduce costs . Given a fixed budget b of labels to distribute across tasks , should we use a small number of highly labelled tasks , or many tasks with few labels each ? In MAML applied to mixed linear regression , we prove that the optimal number of tasks follows the scaling law √ b . We develop an online algorithm for data allocation across tasks , and show that the same scaling law applies to nonlinear regression . We also show preliminary experiments on few-shot image classification . Our work provides a theoretical guide for allocating labels across tasks in meta-learning , which we believe will prove useful in a large number of applications . 1 INTRODUCTION . Deep learning ( DL ) models require a large amount of data in order to perform well , when trained from scratch , but labeling data is expensive and time consuming . An effective approach to avoid the costs of collecting and labeling large amount of data is transfer learning : train a model on one big dataset , or a few related datasets that are already available , and then fine-tune the model on the target dataset , which can be of much smaller size ( Donahue et al . ( 2014 ) ) . In this context , there has been a recent surge of interest in the field of meta-learning , which is inspired by the ability of humans to learn how to learn Hospedales et al . ( 2020 ) . A model is meta-trained on a large number of tasks , each characterized by a small dataset , and meta-tested on the target dataset . The number of data points per task is usually set to an arbitrary number in standard meta-learning benchmarks . For example , in few-shot image classification benchmarks , such as mini-ImageNet ( Vinyals et al . ( 2017 ) , Ravi & Larochelle ( 2017 ) ) and CIFAR-FS ( Bertinetto et al . ( 2019 ) ) , this number is usually set to 1 or 5 . So far , there has not been any reason to optimize this number , as in most circumstances the performance of a model will improve with the number of data points ( see Nakkiran et al . ( 2019 ) for exceptions ) . However , if the total number of labels across training tasks is limited , is it better to have a large number of tasks with very small data in each , or a relatively smaller number of highly labelled tasks ? Since data-labeling is costly , the answer to this question may inform the design of new meta-learning datasets and benchmarks . In this work , to our knowledge , we answer this question for the first time , for a specific meta-learning algorithm : MAML ( Finn et al . ( 2017 ) ) . We study the problem of optimizing the number of metatraining tasks , with a fixed budget b of total data-points to distribute across tasks . We study the application of MAML to three datasets : mixed linear regression , sinusoid regression , and CIFAR . In the case of mixed linear regression , we derive an approximation for the meta-test loss , and according to which the optimal number of tasks follows the scaling rule √ b . In order to optimize the number of tasks empirically , we design an algorithm for online allocation of data across training tasks , and we validate the algorithm by performing a grid search over a large set of possible allocations . In summary , our contributions are : • We introduce and formalize the problem of optimizing data allocation with a fixed budget b in meta-learning . • We prove that the optimal scaling of the number of tasks is √ b in mixed linear regression , and confirm this scaling empirically in nolinear regression . • We introduce an algorithm for online allocation of data across tasks , to find the optimal number of tasks during meta-training , and validate the algorithm by grid search . • We perform preliminary experiments on few-shot image classification . 2 RELATED WORK . A couple of recent papers investigated a problem similar to ours . In the context of meta-learning and mixed linear regression , Kong et al . ( 2020 ) asks whether many tasks with small data can compensate for a lack of tasks with big data . However , they do not address the problem of finding the optimal number of tasks within a fixed budget . The work of Shekhar et al . ( 2020 ) studies exactly the problem of allocating a fixed budget of data points , but to the problem of estimating a finite set of discrete distributions , therefore they do not study the meta-learning problem and their data has no labels . An alternative approach to avoid labelling a large amount of data is active learning , where a model learns with fewer labels by accurately selecting which data to learn from ( Settles ( 2010 ) ) . In the context of meta-learning , the option of implementing active learning has been considered in a few recent studies ( Bachman et al . ( 2017 ) , Garcia & Bruna ( 2018 ) , Kim et al . ( 2018 ) , Finn et al . ( 2019 ) , Requeima et al . ( 2020 ) ) . However , they considered the active labeling of data within a given task , for the purpose of improving performance in that task only . Instead , we ask how data should be distributed across tasks . In the context of recommender systems and text classification , a few studies considered whether labeling a data point , within a given task , may increase performance not only in that task but also in all other tasks . This problem has been referred to as multi-task active learning ( Reichart et al . ( 2008 ) , Zhang ( 2010 ) , Saha et al . ( 2011 ) , Harpale ( 2012 ) , Fang et al . ( 2017 ) ) , or multi-domain active learning ( Li et al . ( 2012 ) , Zhang et al . ( 2016 ) ) . However , none of these studies consider the problem of meta-learning with a fixed budget . A few studies have looked into actively choosing the next task in a sequence of tasks ( Ruvolo & Eaton ( 2013 ) , Pentina et al . ( 2015 ) , Pentina & Lampert ( 2017 ) , Sun et al . ( 2018 ) ) , but they do not look at how to distribute data across tasks . 3 THE PROBLEM OF DATA ALLOCATION FOR META-LEARNING . In the cross-task setting , we are presented with a hierarchically structured dataset , with task parameters ( τ ( i ) ) mi=1 sampled from T ∼ p ( τ ) and data ( xτj ) nτ j=1 sampled from Dτ : = ( D\|T ) ∼ p ( x\|T = τ ) . Our problem is minimizing the following loss function with respect to a parameter ω : L ( ω ) = E T E Dτ L ( ω ; xτ ) ( 1 ) The empirical risk minimization principle ( see Vapnik ( 1998 ) ) ensures that the optimum of the empirical risk converges to that of the true risk with an increase in samples from the joint distribution of ( D , T ) . 3.1 META-LEARNING ACROSS TASKS . In the meta-learning problem , we are given the opportunity to adjust the objective function to each task . This adjustment is given by the adaptation step of meta-learning ( Hospedales et al . ( 2020 ) ) , which represents a transformation on the parameters ω , which is task-dependent and which we refer to as θτ ( ω ) . The loss function Lmeta is defined as an average across both distribution of tasks and data points . The goal of meta-learning is to minimize the loss function with respect to a vector of metaparameters ω Lmeta ( ω ) = E T E Dτ Lτ ( θτ ( ω ) ; xτ ) ( 2 ) Different meta-learning algorithms correspond to a different choice of θτ ( ω ) and we allow for each task τ to have its own specific loss function . The dependence on τ built in through the composition with θτ ( ω ) makes the loss function a sample from a random function field , when considered as a deterministic function of the data and model parameter ω . 3.2 MODEL-AGNOSTIC META-LEARNING . Our case-study for this paper is the meta-learning algorithm MAML developed in Finn et al . ( 2017 ) . MAML employs a base learner , which parametrizes an estimator family by ω . The algorithm ’ s adaptation step , inspired from fine-tuning , performs a fixed number of SGD steps with respect to the data for each task . Thus , the adaptation step maps into the same parameter space as ω . In MAML with a single gradient step , if we denote the data for task τ ( i ) by ( x ( i ) j ) ni j=1 , and the SGD learning rate by α , this transformation is equal to : θ ( i ) ( ω ) = ω − α ni ni∑ j=1 ∇ωL\|ω ; x ( i ) j ( 3 ) This formula corresponds to a full-batch update , employing all the data for task τ ( i ) , but minibatch gradient updates can be performed as well . During meta-training , the loss is evaluated on a sample of m tasks , and a sample of validation data points ni for each task , leading to the following optimization objective : Lmeta ( ω ) = 1 m m∑ i=1 1 ni ni∑ j=1 L ( θ ( i ) ( ω ) ; x ( i ) j ) ( 4 ) During meta-testing , a new ( target ) task is given and the parameters θ are learned by a set of target data points following the same equation 3 . The final performance of the model is computed on test data of the target task . 3.3 DATA ALLOCATION . In this work , we study the problem of finding values of m and ni , such that , under some constraint , the meta-learning loss 2 is minimized given the available data . In this section we make the terms allocation and budget explicit . The budget of a meta-learning problem is defined as the value b =∑m i=1 ni , i.e . the total number of data points available for meta-training . A meta-training set which respects a budget b is a collection of m tasks sampled independently from the task distribution , each composed of a set of ni data points sampled independently from their respective data-generating distributions , such that b = ∑m i=1 ni . Conversely , given a value of the budget b , and a set of tasks τ ( 1 ) , ... τ ( m ) , a data-allocation of the budget b to these tasks is a partition of b into ni such that b = ∑m i=1 ni . If n = ni for all i , this definition is independent of the task samples drawn , and we call this the uniform allocation on m tasks for the budget b . In this work , we only consider the family of uniform allocations , and we leave the study of non-uniform allocations for future work . We denote a meta-dataset which respects the uniform allocation with m tasks and n datapoints per task , drawn independently from task distributions T and data distributionsDτ byM ( T , Dτ ; m , n ) . The optimal uniform data allocation is given by the values ofm and n which minimize the expected test meta-loss ( 2 ) , after optimizing the meta-parameter ω on a meta-training setM ( T , Dτ ; m , n ) . Formally , expanding notation to make the dependence of Lmeta onM explicit in its arguments : ω∗ ( M ( T , Dτ ; m , n ) ) = arg min ω Lmeta ( ω ; M ( T , Dτ ; m , n ) ) ( 5 ) Then the optimal data allocation is ( m∗ ( b ) , n∗ ( b ) ) = arg min m·n=b E T E Dτ Lmeta ( ω∗ ( M ( T , Dτ ; m , n ) ) ; Mtest ) ( 6 ) Remark . Notice that in equation ( 6 ) , Lmeta is already an expectation taken over the data distribution which generated the test set . In the definition , this expectation is taken again over the distribution of meta training setsM ( T , Dτ ; m , n ) . It will be useful to writeLmeta ( m , n ; T , Dτ ) for the expression minimized in equation 6 .	The authors study the problem of finding the optimal allocation of labels across training tasks given a fixed budget. The authors mainly want to answer the question that "if the total number of labels across training tasks is limited, it is better to have a large number of tasks with very small data in each or a relatively smaller number of highly labeled tasks?" Although the authors prove that the optimal scaling of the number of tasks in synthetic tasks, the conclusions remain unclear while handling real-world challenging tasks. More specifically, the authors get the result on CIFAR-FS that the optimum allocation is close to the maximal number of tasks, which is weird and not consistent with previous studies in the paper.	SP:2dd943d37d914575aa6ac4f3948a2d5d9f53a8d9
Optimal allocation of data across training tasks in meta-learning	Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones . They are tested on benchmarks with a fixed number of data-points for each training task , and this number is usually arbitrary , for example , 5 instances per class in few-shot classification . It is unknown how the performance of meta-learning is affected by the distribution of data across training tasks . Since labelling of data is expensive , finding the optimal allocation of labels across training tasks may reduce costs . Given a fixed budget b of labels to distribute across tasks , should we use a small number of highly labelled tasks , or many tasks with few labels each ? In MAML applied to mixed linear regression , we prove that the optimal number of tasks follows the scaling law √ b . We develop an online algorithm for data allocation across tasks , and show that the same scaling law applies to nonlinear regression . We also show preliminary experiments on few-shot image classification . Our work provides a theoretical guide for allocating labels across tasks in meta-learning , which we believe will prove useful in a large number of applications . 1 INTRODUCTION . Deep learning ( DL ) models require a large amount of data in order to perform well , when trained from scratch , but labeling data is expensive and time consuming . An effective approach to avoid the costs of collecting and labeling large amount of data is transfer learning : train a model on one big dataset , or a few related datasets that are already available , and then fine-tune the model on the target dataset , which can be of much smaller size ( Donahue et al . ( 2014 ) ) . In this context , there has been a recent surge of interest in the field of meta-learning , which is inspired by the ability of humans to learn how to learn Hospedales et al . ( 2020 ) . A model is meta-trained on a large number of tasks , each characterized by a small dataset , and meta-tested on the target dataset . The number of data points per task is usually set to an arbitrary number in standard meta-learning benchmarks . For example , in few-shot image classification benchmarks , such as mini-ImageNet ( Vinyals et al . ( 2017 ) , Ravi & Larochelle ( 2017 ) ) and CIFAR-FS ( Bertinetto et al . ( 2019 ) ) , this number is usually set to 1 or 5 . So far , there has not been any reason to optimize this number , as in most circumstances the performance of a model will improve with the number of data points ( see Nakkiran et al . ( 2019 ) for exceptions ) . However , if the total number of labels across training tasks is limited , is it better to have a large number of tasks with very small data in each , or a relatively smaller number of highly labelled tasks ? Since data-labeling is costly , the answer to this question may inform the design of new meta-learning datasets and benchmarks . In this work , to our knowledge , we answer this question for the first time , for a specific meta-learning algorithm : MAML ( Finn et al . ( 2017 ) ) . We study the problem of optimizing the number of metatraining tasks , with a fixed budget b of total data-points to distribute across tasks . We study the application of MAML to three datasets : mixed linear regression , sinusoid regression , and CIFAR . In the case of mixed linear regression , we derive an approximation for the meta-test loss , and according to which the optimal number of tasks follows the scaling rule √ b . In order to optimize the number of tasks empirically , we design an algorithm for online allocation of data across training tasks , and we validate the algorithm by performing a grid search over a large set of possible allocations . In summary , our contributions are : • We introduce and formalize the problem of optimizing data allocation with a fixed budget b in meta-learning . • We prove that the optimal scaling of the number of tasks is √ b in mixed linear regression , and confirm this scaling empirically in nolinear regression . • We introduce an algorithm for online allocation of data across tasks , to find the optimal number of tasks during meta-training , and validate the algorithm by grid search . • We perform preliminary experiments on few-shot image classification . 2 RELATED WORK . A couple of recent papers investigated a problem similar to ours . In the context of meta-learning and mixed linear regression , Kong et al . ( 2020 ) asks whether many tasks with small data can compensate for a lack of tasks with big data . However , they do not address the problem of finding the optimal number of tasks within a fixed budget . The work of Shekhar et al . ( 2020 ) studies exactly the problem of allocating a fixed budget of data points , but to the problem of estimating a finite set of discrete distributions , therefore they do not study the meta-learning problem and their data has no labels . An alternative approach to avoid labelling a large amount of data is active learning , where a model learns with fewer labels by accurately selecting which data to learn from ( Settles ( 2010 ) ) . In the context of meta-learning , the option of implementing active learning has been considered in a few recent studies ( Bachman et al . ( 2017 ) , Garcia & Bruna ( 2018 ) , Kim et al . ( 2018 ) , Finn et al . ( 2019 ) , Requeima et al . ( 2020 ) ) . However , they considered the active labeling of data within a given task , for the purpose of improving performance in that task only . Instead , we ask how data should be distributed across tasks . In the context of recommender systems and text classification , a few studies considered whether labeling a data point , within a given task , may increase performance not only in that task but also in all other tasks . This problem has been referred to as multi-task active learning ( Reichart et al . ( 2008 ) , Zhang ( 2010 ) , Saha et al . ( 2011 ) , Harpale ( 2012 ) , Fang et al . ( 2017 ) ) , or multi-domain active learning ( Li et al . ( 2012 ) , Zhang et al . ( 2016 ) ) . However , none of these studies consider the problem of meta-learning with a fixed budget . A few studies have looked into actively choosing the next task in a sequence of tasks ( Ruvolo & Eaton ( 2013 ) , Pentina et al . ( 2015 ) , Pentina & Lampert ( 2017 ) , Sun et al . ( 2018 ) ) , but they do not look at how to distribute data across tasks . 3 THE PROBLEM OF DATA ALLOCATION FOR META-LEARNING . In the cross-task setting , we are presented with a hierarchically structured dataset , with task parameters ( τ ( i ) ) mi=1 sampled from T ∼ p ( τ ) and data ( xτj ) nτ j=1 sampled from Dτ : = ( D\|T ) ∼ p ( x\|T = τ ) . Our problem is minimizing the following loss function with respect to a parameter ω : L ( ω ) = E T E Dτ L ( ω ; xτ ) ( 1 ) The empirical risk minimization principle ( see Vapnik ( 1998 ) ) ensures that the optimum of the empirical risk converges to that of the true risk with an increase in samples from the joint distribution of ( D , T ) . 3.1 META-LEARNING ACROSS TASKS . In the meta-learning problem , we are given the opportunity to adjust the objective function to each task . This adjustment is given by the adaptation step of meta-learning ( Hospedales et al . ( 2020 ) ) , which represents a transformation on the parameters ω , which is task-dependent and which we refer to as θτ ( ω ) . The loss function Lmeta is defined as an average across both distribution of tasks and data points . The goal of meta-learning is to minimize the loss function with respect to a vector of metaparameters ω Lmeta ( ω ) = E T E Dτ Lτ ( θτ ( ω ) ; xτ ) ( 2 ) Different meta-learning algorithms correspond to a different choice of θτ ( ω ) and we allow for each task τ to have its own specific loss function . The dependence on τ built in through the composition with θτ ( ω ) makes the loss function a sample from a random function field , when considered as a deterministic function of the data and model parameter ω . 3.2 MODEL-AGNOSTIC META-LEARNING . Our case-study for this paper is the meta-learning algorithm MAML developed in Finn et al . ( 2017 ) . MAML employs a base learner , which parametrizes an estimator family by ω . The algorithm ’ s adaptation step , inspired from fine-tuning , performs a fixed number of SGD steps with respect to the data for each task . Thus , the adaptation step maps into the same parameter space as ω . In MAML with a single gradient step , if we denote the data for task τ ( i ) by ( x ( i ) j ) ni j=1 , and the SGD learning rate by α , this transformation is equal to : θ ( i ) ( ω ) = ω − α ni ni∑ j=1 ∇ωL\|ω ; x ( i ) j ( 3 ) This formula corresponds to a full-batch update , employing all the data for task τ ( i ) , but minibatch gradient updates can be performed as well . During meta-training , the loss is evaluated on a sample of m tasks , and a sample of validation data points ni for each task , leading to the following optimization objective : Lmeta ( ω ) = 1 m m∑ i=1 1 ni ni∑ j=1 L ( θ ( i ) ( ω ) ; x ( i ) j ) ( 4 ) During meta-testing , a new ( target ) task is given and the parameters θ are learned by a set of target data points following the same equation 3 . The final performance of the model is computed on test data of the target task . 3.3 DATA ALLOCATION . In this work , we study the problem of finding values of m and ni , such that , under some constraint , the meta-learning loss 2 is minimized given the available data . In this section we make the terms allocation and budget explicit . The budget of a meta-learning problem is defined as the value b =∑m i=1 ni , i.e . the total number of data points available for meta-training . A meta-training set which respects a budget b is a collection of m tasks sampled independently from the task distribution , each composed of a set of ni data points sampled independently from their respective data-generating distributions , such that b = ∑m i=1 ni . Conversely , given a value of the budget b , and a set of tasks τ ( 1 ) , ... τ ( m ) , a data-allocation of the budget b to these tasks is a partition of b into ni such that b = ∑m i=1 ni . If n = ni for all i , this definition is independent of the task samples drawn , and we call this the uniform allocation on m tasks for the budget b . In this work , we only consider the family of uniform allocations , and we leave the study of non-uniform allocations for future work . We denote a meta-dataset which respects the uniform allocation with m tasks and n datapoints per task , drawn independently from task distributions T and data distributionsDτ byM ( T , Dτ ; m , n ) . The optimal uniform data allocation is given by the values ofm and n which minimize the expected test meta-loss ( 2 ) , after optimizing the meta-parameter ω on a meta-training setM ( T , Dτ ; m , n ) . Formally , expanding notation to make the dependence of Lmeta onM explicit in its arguments : ω∗ ( M ( T , Dτ ; m , n ) ) = arg min ω Lmeta ( ω ; M ( T , Dτ ; m , n ) ) ( 5 ) Then the optimal data allocation is ( m∗ ( b ) , n∗ ( b ) ) = arg min m·n=b E T E Dτ Lmeta ( ω∗ ( M ( T , Dτ ; m , n ) ) ; Mtest ) ( 6 ) Remark . Notice that in equation ( 6 ) , Lmeta is already an expectation taken over the data distribution which generated the test set . In the definition , this expectation is taken again over the distribution of meta training setsM ( T , Dτ ; m , n ) . It will be useful to writeLmeta ( m , n ; T , Dτ ) for the expression minimized in equation 6 .	This paper proposes a data allocation scheme for meta-learning. The authors argue that it is important to consider the number of total tasks versus the number of datapoints per task given a fixed budget of the total number of datapoints since labeling is expensive for large datasets. The paper presents an algorithm that does the data allocation using a sequential decision process (SDM) and models the problem as a two-armed contextual bandit where for each fixed number of iterations, the agent can choose to add 1 task or 1 datapoint per task. There are some theoretical results on the optimal data allocation in the linear regression setting. The authors also empirically show that the number of the fixed budget and the data allocation can affect performance fairly a bit in a sinusoid regression dataset and the CIFAR-FS dataset. The proposed SDM algorithm can also recover the optimal data allocation in the sinusoid regression setting.	SP:2dd943d37d914575aa6ac4f3948a2d5d9f53a8d9
Modelling Hierarchical Structure between Dialogue Policy and Natural Language Generator with Option Framework for Task-oriented Dialogue System	Designing task-oriented dialogue systems is a challenging research topic , since it needs not only to generate utterances fulfilling user requests but also to guarantee the comprehensibility . Many previous works trained end-to-end ( E2E ) models with supervised learning ( SL ) , however , the bias in annotated system utterances remains as a bottleneck . Reinforcement learning ( RL ) deals with the problem through using non-differentiable evaluation metrics ( e.g. , the success rate ) as rewards . Nonetheless , existing works with RL showed that the comprehensibility of generated system utterances could be corrupted when improving the performance on fulfilling user requests . In our work , we ( 1 ) propose modelling the hierarchical structure between dialogue policy and natural language generator ( NLG ) with the option framework , called HDNO , where the latent dialogue act is applied to avoid designing specific dialogue act representations ; ( 2 ) train HDNO via hierarchical reinforcement learning ( HRL ) , as well as suggest the asynchronous updates between dialogue policy and NLG during training to theoretically guarantee their convergence to a local maximizer ; and ( 3 ) propose using a discriminator modelled with language models as an additional reward to further improve the comprehensibility . We test HDNO on MultiWoz 2.0 and MultiWoz 2.1 , the datasets on multi-domain dialogues , in comparison with word-level E2E model trained with RL , LaRL and HDSA , showing improvements on the performance evaluated by automatic evaluation metrics and human evaluation . Finally , we demonstrate the semantic meanings of latent dialogue acts to show the explanability for HDNO . 1 INTRODUCTION . Designing a task-oriented dialogue system is a popular and challenging research topic in the recent decades . In contrast to the open-domain dialogue system ( Ritter et al. , 2011 ) , it aims to help people complete real-life tasks through dialogues without human service ( e.g. , booking tickets ) ( Young , 2006 ) . In a task-oriented dialogue task , each dialogue is defined with a goal which includes user requests ( i.e. , represented as a set of key words known as slot values ) . The conventional taskoriented dialogue system is comprised of 4 modules ( see Appendix 3.1 ) , each of which used to be implemented with handcrafted rules ( Chen et al. , 2017 ) . Given user utterances , it gives responses in turn to fulfill the requests via mentioning corresponding slot values . Recently , several works focused on training a task-oriented dialogue system in end-to-end fashion ( E2E ) ( Bordes et al. , 2016 ; Wen et al. , 2017 ) for generalizing dialogues outside corpora . To train a E2E model via supervised learning ( SL ) , generated system utterances are forced to fit the oracle responses collected from human-to-human conversations ( Budzianowski et al. , 2017a ) . The oracle responses contain faults by humans thus being inaccurate , which leads to biased SL . On the other hand , the goal is absolutely clear , though the criterion of success rate that evaluates the goal completion is non-differentiable and can not be used as a loss for SL . ∗Imperial College London . †Laiye Network Technology Co. Ltd .. ♣KAIST . ♠University of Bath . Correspondence to Yunjie Gu : yg934 @ bath.ac.uk . To tackle this problem , reinforcement learning ( RL ) is applied to train a task-oriented dialogue system ( Williams and Young , 2007 ; Zhao and Eskénazi , 2016 ; Peng et al. , 2018 ; Zhao et al. , 2019 ) . Specifically , some works merely optimized dialogue policy while other modules , e.g. , the natural language generator ( NLG ) , were fixed ( Peng et al. , 2018 ; Zhao et al. , 2019 ; Su et al. , 2018 ) . In contrast , other works extended the dialogue policy to NLG and applied RL on the entire E2E dialogue system , regarding each generated word in a response as an action ( Zhao and Eskénazi , 2016 ) . Although previous works enhanced the performance on fulfilling user requests , the comprehensibility of generated system utterances are corrupted ( Peng et al. , 2018 ; Zhao et al. , 2019 ; Tang et al. , 2018a ) . The possible reasons are : ( 1 ) solely optimizing dialogue policy could easily cause the biased improvement on fulfilling user requests , ignoring the comprehensibility of generated utterances ( see Section 3.1 ) ; ( 2 ) the state space and action space ( represented as a vocabulary ) in E2E fashion is so huge that learning to generate comprehensible utterances becomes difficult ( Lewis et al. , 2017 ) ; and ( 3 ) dialogue system in E2E fashion may lack explanation during the procedure of decision . In our work , we propose to model the hierarchical structure between dialogue policy and NLG with the option framework , i.e. , a hierarchical reinforcement learning ( HRL ) framework ( Sutton et al. , 1999 ) called HDNO ( see Section 4.1 ) so that the high-level temporal abstraction can provide the ability of explanation during the procedure of decision . Specifically , dialogue policy works as a highlevel policy over dialogue acts ( i.e . options ) and NLG works as a low-level policy over generated words ( i.e . primitive actions ) . Therefore , these two modules are decoupled during optimization with the smaller state space for NLG and the smaller action space for dialogue policy ( see Appendix F ) . To reduce the efforts on designing dialogue act representations , we represent a dialogue act as latent factors . During training , we suggest the asynchronous updates between dialogue policy and NLG to theoretically guarantee their convergence to a local maximizer ( see Section 4.2 ) . Finally , we propose using a discriminator modelled with language models ( Yang et al. , 2018 ) as an additional reward to further improve the comprehensibility ( see Section 5 ) . We evaluate HDNO on two datasets with dialogues in multiple domains : MultiWOZ 2.0 ( Budzianowski et al. , 2018 ) and MultiWOZ 2.1 ( Eric et al. , 2019 ) , compared with word-level E2E ( Budzianowski et al. , 2018 ) trained with RL , LaRL ( Zhao et al. , 2019 ) and HDSA ( Chen et al. , 2019 ) . The experiments show that HDNO works best in the total performance evaluated with automatic metrics ( see Section 6.2.1 ) and the human evaluation ( see Section B.1 ) . Furthermore , we study the latent dialogue acts and show the ability of explanation for HDNO ( see Section 6.4 ) . 2 RELATED WORK . Firstly , we go through the previous works on studying the dialogue act representation for taskoriented dialogue systems . Some previous works optimized dialogue policy with reinforcement learning ( RL ) , which made decision via selecting from handcrafted dialogue acts represented as ontology ( Peng et al. , 2018 ; Young et al. , 2007 ; Walker , 2000 ; He et al. , 2018 ) . Such a representation method is easily understood by human beings , while the dialogue act space becomes limited in representation . To deal with this problem , some researchers investigated training dialogue acts via fitting oracle dialogue acts represented in sequence ( Chen et al. , 2019 ; Zhang et al. , 2019 ; Lei et al. , 2018 ) . This representation method generalized dialogue acts , however , designing a good representation is effort demanding . To handle this problem , learning a latent representation of dialogue act was attempted ( Zhao et al. , 2019 ; Yarats and Lewis , 2018 ) . In our work , similar to ( Zhao et al. , 2019 ) we learn latent dialogue acts without any labels of dialogue acts . By this view , our work can be regarded as an extension of LaRL ( Zhao et al. , 2019 ) on learning strategy . Then , we review the previous works modelling a dialogue system with a hierarchical structure . In the field of task-oriented dialogue systems , many works lay on modelling dialogue acts or the state space with a hierarchical structure to tackle the decision problem for dialogues with multi-domain tasks ( Cuayáhuitl et al. , 2009 ; Peng et al. , 2017 ; Chen et al. , 2019 ; Tang et al. , 2018b ; Budzianowski et al. , 2017b ) . Distinguished from these works , our work views the relationship between dialogue policy and natural language generator ( NLG ) as a natural hierarchical structure and models it with the option framework ( Sutton et al. , 1999 ) . In the field of open-domain dialogue system , a similar hierarchical structure was proposed ( Serban et al. , 2017 ; Saleh et al. , 2019 ) but with a different motivation from ours . In this sense , these two fields are possible to be unified . Finally , among the works training with hierarchical reinforcement learning ( HRL ) , some of them set up an extrinsic reward for high-level policy and an intrinsic reward for low-level policy respectively to encourage the convergence ( Peng et al. , 2017 ; Budzianowski et al. , 2017b ) . In our work , we train both high-level policy and low-level policy with identical rewards to guarantee the consistency between two policies ( Sutton et al. , 1999 ) . On the other hand , in the field of open-domain dialogue system , Saleh et al . ( 2019 ) represented the joint generated utterances over a turn as a low-level action such that both high-level policy and low-level policy were in identical time scales . Besides , its lowlevel policy gradients flew through high-level policy during training , which degraded hierarchical policies to an E2E policy with a word-level action space . In our work , ( 1 ) dialogue policy and NLG are decoupled during optimization and no gradients are allowed to flow between them ; ( 2 ) these two policies are asynchronously updated to theoretically guarantee the convergence to a local maximizer ; and ( 3 ) each generated word is regarded as a low-level action . 3 BACKGROUND . 3.1 TASK-ORIENTED DIALOGUE SYSTEM . Brief Introduction : A task-oriented dialogue system aims to help fulfill a user ’ s task through conversation in turns . In general , each dialogue is modelled with an ontology called goal which includes inform slots and request slots . The traditional modular dialogue system is constituted of natural language understanding ( NLU ) , dialogue state tracker ( DST ) , dialogue policy and natural language generator ( NLG ) . For a dialogue system , it needs to infer inform slots from user utterances and transform them to a dialogue state , which is completed by NLU and DST ( Chen et al. , 2017 ) . In this work , we focus on optimizing dialogue policy and NLG , leveraging oracle dialogue states and database search results to produce dialogue acts and then responses ( that should include as many request slots as possible ) in turns . For optimizing dialogue policy , it is modelled with Markov decision process ( MDP ) ( Williams and Young , 2007 ) . Existing Challenges : We identify the main challenges of task-oriented dialogue systems : ( 1 ) A dialogue with a single domain ( i.e . completing one task in a dialogue ) has been broadly studied , however , handling a dialogue with multiple domains is more challenging and needs more studies on it ( Budzianowski et al. , 2018 ) ; ( 2 ) If ignoring the syntactic structure of generated system utterances ( i.e . losing comprehensibility ) , the mission of task-oriented dialogues will be simplified to generating corresponding labels ( i.e. , slots ) for user utterances . Several existing algorithms already reached high scores on request slots acquisition but low scores on the comprehensibility of generated system utterances ( Zhao et al. , 2019 ; Mehri et al. , 2019 ) , so the simplified task has been well-addressed . Reversely , if only focusing on the comprehensibility , the score on request slots acquisition could be drastically affected ( Chen et al. , 2019 ; Hosseini-Asl et al. , 2020 ) . In this work , we investigate the trade-off between the comprehensibility and request slots acquisition ; ( 3 ) Designing and annotating a dialogue act structure is effort demanding ( Budzianowski et al. , 2018 ) . Therefore , learning a meaningful latent dialogue act becomes a new challenge ( Zhao et al. , 2019 ) .	The paper looks the problem of lack of comprehensibility that arises when we use RL to train a E2E dialog system to maximise a given reward function. The paper proposes a HRL/options framework based method to learn a dialog policy over learned latent dialog acts which can then guide the lower level NLG. This along with a regularization reward using language model the paper aims to improve comprehensibility. The show improved performance in MultiWoz dataset.	SP:e8863d56eb4be6ed7aa17241af9ee376570d0770
Modelling Hierarchical Structure between Dialogue Policy and Natural Language Generator with Option Framework for Task-oriented Dialogue System	Designing task-oriented dialogue systems is a challenging research topic , since it needs not only to generate utterances fulfilling user requests but also to guarantee the comprehensibility . Many previous works trained end-to-end ( E2E ) models with supervised learning ( SL ) , however , the bias in annotated system utterances remains as a bottleneck . Reinforcement learning ( RL ) deals with the problem through using non-differentiable evaluation metrics ( e.g. , the success rate ) as rewards . Nonetheless , existing works with RL showed that the comprehensibility of generated system utterances could be corrupted when improving the performance on fulfilling user requests . In our work , we ( 1 ) propose modelling the hierarchical structure between dialogue policy and natural language generator ( NLG ) with the option framework , called HDNO , where the latent dialogue act is applied to avoid designing specific dialogue act representations ; ( 2 ) train HDNO via hierarchical reinforcement learning ( HRL ) , as well as suggest the asynchronous updates between dialogue policy and NLG during training to theoretically guarantee their convergence to a local maximizer ; and ( 3 ) propose using a discriminator modelled with language models as an additional reward to further improve the comprehensibility . We test HDNO on MultiWoz 2.0 and MultiWoz 2.1 , the datasets on multi-domain dialogues , in comparison with word-level E2E model trained with RL , LaRL and HDSA , showing improvements on the performance evaluated by automatic evaluation metrics and human evaluation . Finally , we demonstrate the semantic meanings of latent dialogue acts to show the explanability for HDNO . 1 INTRODUCTION . Designing a task-oriented dialogue system is a popular and challenging research topic in the recent decades . In contrast to the open-domain dialogue system ( Ritter et al. , 2011 ) , it aims to help people complete real-life tasks through dialogues without human service ( e.g. , booking tickets ) ( Young , 2006 ) . In a task-oriented dialogue task , each dialogue is defined with a goal which includes user requests ( i.e. , represented as a set of key words known as slot values ) . The conventional taskoriented dialogue system is comprised of 4 modules ( see Appendix 3.1 ) , each of which used to be implemented with handcrafted rules ( Chen et al. , 2017 ) . Given user utterances , it gives responses in turn to fulfill the requests via mentioning corresponding slot values . Recently , several works focused on training a task-oriented dialogue system in end-to-end fashion ( E2E ) ( Bordes et al. , 2016 ; Wen et al. , 2017 ) for generalizing dialogues outside corpora . To train a E2E model via supervised learning ( SL ) , generated system utterances are forced to fit the oracle responses collected from human-to-human conversations ( Budzianowski et al. , 2017a ) . The oracle responses contain faults by humans thus being inaccurate , which leads to biased SL . On the other hand , the goal is absolutely clear , though the criterion of success rate that evaluates the goal completion is non-differentiable and can not be used as a loss for SL . ∗Imperial College London . †Laiye Network Technology Co. Ltd .. ♣KAIST . ♠University of Bath . Correspondence to Yunjie Gu : yg934 @ bath.ac.uk . To tackle this problem , reinforcement learning ( RL ) is applied to train a task-oriented dialogue system ( Williams and Young , 2007 ; Zhao and Eskénazi , 2016 ; Peng et al. , 2018 ; Zhao et al. , 2019 ) . Specifically , some works merely optimized dialogue policy while other modules , e.g. , the natural language generator ( NLG ) , were fixed ( Peng et al. , 2018 ; Zhao et al. , 2019 ; Su et al. , 2018 ) . In contrast , other works extended the dialogue policy to NLG and applied RL on the entire E2E dialogue system , regarding each generated word in a response as an action ( Zhao and Eskénazi , 2016 ) . Although previous works enhanced the performance on fulfilling user requests , the comprehensibility of generated system utterances are corrupted ( Peng et al. , 2018 ; Zhao et al. , 2019 ; Tang et al. , 2018a ) . The possible reasons are : ( 1 ) solely optimizing dialogue policy could easily cause the biased improvement on fulfilling user requests , ignoring the comprehensibility of generated utterances ( see Section 3.1 ) ; ( 2 ) the state space and action space ( represented as a vocabulary ) in E2E fashion is so huge that learning to generate comprehensible utterances becomes difficult ( Lewis et al. , 2017 ) ; and ( 3 ) dialogue system in E2E fashion may lack explanation during the procedure of decision . In our work , we propose to model the hierarchical structure between dialogue policy and NLG with the option framework , i.e. , a hierarchical reinforcement learning ( HRL ) framework ( Sutton et al. , 1999 ) called HDNO ( see Section 4.1 ) so that the high-level temporal abstraction can provide the ability of explanation during the procedure of decision . Specifically , dialogue policy works as a highlevel policy over dialogue acts ( i.e . options ) and NLG works as a low-level policy over generated words ( i.e . primitive actions ) . Therefore , these two modules are decoupled during optimization with the smaller state space for NLG and the smaller action space for dialogue policy ( see Appendix F ) . To reduce the efforts on designing dialogue act representations , we represent a dialogue act as latent factors . During training , we suggest the asynchronous updates between dialogue policy and NLG to theoretically guarantee their convergence to a local maximizer ( see Section 4.2 ) . Finally , we propose using a discriminator modelled with language models ( Yang et al. , 2018 ) as an additional reward to further improve the comprehensibility ( see Section 5 ) . We evaluate HDNO on two datasets with dialogues in multiple domains : MultiWOZ 2.0 ( Budzianowski et al. , 2018 ) and MultiWOZ 2.1 ( Eric et al. , 2019 ) , compared with word-level E2E ( Budzianowski et al. , 2018 ) trained with RL , LaRL ( Zhao et al. , 2019 ) and HDSA ( Chen et al. , 2019 ) . The experiments show that HDNO works best in the total performance evaluated with automatic metrics ( see Section 6.2.1 ) and the human evaluation ( see Section B.1 ) . Furthermore , we study the latent dialogue acts and show the ability of explanation for HDNO ( see Section 6.4 ) . 2 RELATED WORK . Firstly , we go through the previous works on studying the dialogue act representation for taskoriented dialogue systems . Some previous works optimized dialogue policy with reinforcement learning ( RL ) , which made decision via selecting from handcrafted dialogue acts represented as ontology ( Peng et al. , 2018 ; Young et al. , 2007 ; Walker , 2000 ; He et al. , 2018 ) . Such a representation method is easily understood by human beings , while the dialogue act space becomes limited in representation . To deal with this problem , some researchers investigated training dialogue acts via fitting oracle dialogue acts represented in sequence ( Chen et al. , 2019 ; Zhang et al. , 2019 ; Lei et al. , 2018 ) . This representation method generalized dialogue acts , however , designing a good representation is effort demanding . To handle this problem , learning a latent representation of dialogue act was attempted ( Zhao et al. , 2019 ; Yarats and Lewis , 2018 ) . In our work , similar to ( Zhao et al. , 2019 ) we learn latent dialogue acts without any labels of dialogue acts . By this view , our work can be regarded as an extension of LaRL ( Zhao et al. , 2019 ) on learning strategy . Then , we review the previous works modelling a dialogue system with a hierarchical structure . In the field of task-oriented dialogue systems , many works lay on modelling dialogue acts or the state space with a hierarchical structure to tackle the decision problem for dialogues with multi-domain tasks ( Cuayáhuitl et al. , 2009 ; Peng et al. , 2017 ; Chen et al. , 2019 ; Tang et al. , 2018b ; Budzianowski et al. , 2017b ) . Distinguished from these works , our work views the relationship between dialogue policy and natural language generator ( NLG ) as a natural hierarchical structure and models it with the option framework ( Sutton et al. , 1999 ) . In the field of open-domain dialogue system , a similar hierarchical structure was proposed ( Serban et al. , 2017 ; Saleh et al. , 2019 ) but with a different motivation from ours . In this sense , these two fields are possible to be unified . Finally , among the works training with hierarchical reinforcement learning ( HRL ) , some of them set up an extrinsic reward for high-level policy and an intrinsic reward for low-level policy respectively to encourage the convergence ( Peng et al. , 2017 ; Budzianowski et al. , 2017b ) . In our work , we train both high-level policy and low-level policy with identical rewards to guarantee the consistency between two policies ( Sutton et al. , 1999 ) . On the other hand , in the field of open-domain dialogue system , Saleh et al . ( 2019 ) represented the joint generated utterances over a turn as a low-level action such that both high-level policy and low-level policy were in identical time scales . Besides , its lowlevel policy gradients flew through high-level policy during training , which degraded hierarchical policies to an E2E policy with a word-level action space . In our work , ( 1 ) dialogue policy and NLG are decoupled during optimization and no gradients are allowed to flow between them ; ( 2 ) these two policies are asynchronously updated to theoretically guarantee the convergence to a local maximizer ; and ( 3 ) each generated word is regarded as a low-level action . 3 BACKGROUND . 3.1 TASK-ORIENTED DIALOGUE SYSTEM . Brief Introduction : A task-oriented dialogue system aims to help fulfill a user ’ s task through conversation in turns . In general , each dialogue is modelled with an ontology called goal which includes inform slots and request slots . The traditional modular dialogue system is constituted of natural language understanding ( NLU ) , dialogue state tracker ( DST ) , dialogue policy and natural language generator ( NLG ) . For a dialogue system , it needs to infer inform slots from user utterances and transform them to a dialogue state , which is completed by NLU and DST ( Chen et al. , 2017 ) . In this work , we focus on optimizing dialogue policy and NLG , leveraging oracle dialogue states and database search results to produce dialogue acts and then responses ( that should include as many request slots as possible ) in turns . For optimizing dialogue policy , it is modelled with Markov decision process ( MDP ) ( Williams and Young , 2007 ) . Existing Challenges : We identify the main challenges of task-oriented dialogue systems : ( 1 ) A dialogue with a single domain ( i.e . completing one task in a dialogue ) has been broadly studied , however , handling a dialogue with multiple domains is more challenging and needs more studies on it ( Budzianowski et al. , 2018 ) ; ( 2 ) If ignoring the syntactic structure of generated system utterances ( i.e . losing comprehensibility ) , the mission of task-oriented dialogues will be simplified to generating corresponding labels ( i.e. , slots ) for user utterances . Several existing algorithms already reached high scores on request slots acquisition but low scores on the comprehensibility of generated system utterances ( Zhao et al. , 2019 ; Mehri et al. , 2019 ) , so the simplified task has been well-addressed . Reversely , if only focusing on the comprehensibility , the score on request slots acquisition could be drastically affected ( Chen et al. , 2019 ; Hosseini-Asl et al. , 2020 ) . In this work , we investigate the trade-off between the comprehensibility and request slots acquisition ; ( 3 ) Designing and annotating a dialogue act structure is effort demanding ( Budzianowski et al. , 2018 ) . Therefore , learning a meaningful latent dialogue act becomes a new challenge ( Zhao et al. , 2019 ) .	This paper proposes modeling the hierarchical structure between dialog policy and natural language generator with option network and train it with HRL. It also introduces a discriminator modeled with language models as an additional reward, which further improves the learning procedure's comprehensibility. Besides, this paper has demonstrated the interpretability of the latent dialog act via clustering methods.	SP:e8863d56eb4be6ed7aa17241af9ee376570d0770
Modelling Hierarchical Structure between Dialogue Policy and Natural Language Generator with Option Framework for Task-oriented Dialogue System	Designing task-oriented dialogue systems is a challenging research topic , since it needs not only to generate utterances fulfilling user requests but also to guarantee the comprehensibility . Many previous works trained end-to-end ( E2E ) models with supervised learning ( SL ) , however , the bias in annotated system utterances remains as a bottleneck . Reinforcement learning ( RL ) deals with the problem through using non-differentiable evaluation metrics ( e.g. , the success rate ) as rewards . Nonetheless , existing works with RL showed that the comprehensibility of generated system utterances could be corrupted when improving the performance on fulfilling user requests . In our work , we ( 1 ) propose modelling the hierarchical structure between dialogue policy and natural language generator ( NLG ) with the option framework , called HDNO , where the latent dialogue act is applied to avoid designing specific dialogue act representations ; ( 2 ) train HDNO via hierarchical reinforcement learning ( HRL ) , as well as suggest the asynchronous updates between dialogue policy and NLG during training to theoretically guarantee their convergence to a local maximizer ; and ( 3 ) propose using a discriminator modelled with language models as an additional reward to further improve the comprehensibility . We test HDNO on MultiWoz 2.0 and MultiWoz 2.1 , the datasets on multi-domain dialogues , in comparison with word-level E2E model trained with RL , LaRL and HDSA , showing improvements on the performance evaluated by automatic evaluation metrics and human evaluation . Finally , we demonstrate the semantic meanings of latent dialogue acts to show the explanability for HDNO . 1 INTRODUCTION . Designing a task-oriented dialogue system is a popular and challenging research topic in the recent decades . In contrast to the open-domain dialogue system ( Ritter et al. , 2011 ) , it aims to help people complete real-life tasks through dialogues without human service ( e.g. , booking tickets ) ( Young , 2006 ) . In a task-oriented dialogue task , each dialogue is defined with a goal which includes user requests ( i.e. , represented as a set of key words known as slot values ) . The conventional taskoriented dialogue system is comprised of 4 modules ( see Appendix 3.1 ) , each of which used to be implemented with handcrafted rules ( Chen et al. , 2017 ) . Given user utterances , it gives responses in turn to fulfill the requests via mentioning corresponding slot values . Recently , several works focused on training a task-oriented dialogue system in end-to-end fashion ( E2E ) ( Bordes et al. , 2016 ; Wen et al. , 2017 ) for generalizing dialogues outside corpora . To train a E2E model via supervised learning ( SL ) , generated system utterances are forced to fit the oracle responses collected from human-to-human conversations ( Budzianowski et al. , 2017a ) . The oracle responses contain faults by humans thus being inaccurate , which leads to biased SL . On the other hand , the goal is absolutely clear , though the criterion of success rate that evaluates the goal completion is non-differentiable and can not be used as a loss for SL . ∗Imperial College London . †Laiye Network Technology Co. Ltd .. ♣KAIST . ♠University of Bath . Correspondence to Yunjie Gu : yg934 @ bath.ac.uk . To tackle this problem , reinforcement learning ( RL ) is applied to train a task-oriented dialogue system ( Williams and Young , 2007 ; Zhao and Eskénazi , 2016 ; Peng et al. , 2018 ; Zhao et al. , 2019 ) . Specifically , some works merely optimized dialogue policy while other modules , e.g. , the natural language generator ( NLG ) , were fixed ( Peng et al. , 2018 ; Zhao et al. , 2019 ; Su et al. , 2018 ) . In contrast , other works extended the dialogue policy to NLG and applied RL on the entire E2E dialogue system , regarding each generated word in a response as an action ( Zhao and Eskénazi , 2016 ) . Although previous works enhanced the performance on fulfilling user requests , the comprehensibility of generated system utterances are corrupted ( Peng et al. , 2018 ; Zhao et al. , 2019 ; Tang et al. , 2018a ) . The possible reasons are : ( 1 ) solely optimizing dialogue policy could easily cause the biased improvement on fulfilling user requests , ignoring the comprehensibility of generated utterances ( see Section 3.1 ) ; ( 2 ) the state space and action space ( represented as a vocabulary ) in E2E fashion is so huge that learning to generate comprehensible utterances becomes difficult ( Lewis et al. , 2017 ) ; and ( 3 ) dialogue system in E2E fashion may lack explanation during the procedure of decision . In our work , we propose to model the hierarchical structure between dialogue policy and NLG with the option framework , i.e. , a hierarchical reinforcement learning ( HRL ) framework ( Sutton et al. , 1999 ) called HDNO ( see Section 4.1 ) so that the high-level temporal abstraction can provide the ability of explanation during the procedure of decision . Specifically , dialogue policy works as a highlevel policy over dialogue acts ( i.e . options ) and NLG works as a low-level policy over generated words ( i.e . primitive actions ) . Therefore , these two modules are decoupled during optimization with the smaller state space for NLG and the smaller action space for dialogue policy ( see Appendix F ) . To reduce the efforts on designing dialogue act representations , we represent a dialogue act as latent factors . During training , we suggest the asynchronous updates between dialogue policy and NLG to theoretically guarantee their convergence to a local maximizer ( see Section 4.2 ) . Finally , we propose using a discriminator modelled with language models ( Yang et al. , 2018 ) as an additional reward to further improve the comprehensibility ( see Section 5 ) . We evaluate HDNO on two datasets with dialogues in multiple domains : MultiWOZ 2.0 ( Budzianowski et al. , 2018 ) and MultiWOZ 2.1 ( Eric et al. , 2019 ) , compared with word-level E2E ( Budzianowski et al. , 2018 ) trained with RL , LaRL ( Zhao et al. , 2019 ) and HDSA ( Chen et al. , 2019 ) . The experiments show that HDNO works best in the total performance evaluated with automatic metrics ( see Section 6.2.1 ) and the human evaluation ( see Section B.1 ) . Furthermore , we study the latent dialogue acts and show the ability of explanation for HDNO ( see Section 6.4 ) . 2 RELATED WORK . Firstly , we go through the previous works on studying the dialogue act representation for taskoriented dialogue systems . Some previous works optimized dialogue policy with reinforcement learning ( RL ) , which made decision via selecting from handcrafted dialogue acts represented as ontology ( Peng et al. , 2018 ; Young et al. , 2007 ; Walker , 2000 ; He et al. , 2018 ) . Such a representation method is easily understood by human beings , while the dialogue act space becomes limited in representation . To deal with this problem , some researchers investigated training dialogue acts via fitting oracle dialogue acts represented in sequence ( Chen et al. , 2019 ; Zhang et al. , 2019 ; Lei et al. , 2018 ) . This representation method generalized dialogue acts , however , designing a good representation is effort demanding . To handle this problem , learning a latent representation of dialogue act was attempted ( Zhao et al. , 2019 ; Yarats and Lewis , 2018 ) . In our work , similar to ( Zhao et al. , 2019 ) we learn latent dialogue acts without any labels of dialogue acts . By this view , our work can be regarded as an extension of LaRL ( Zhao et al. , 2019 ) on learning strategy . Then , we review the previous works modelling a dialogue system with a hierarchical structure . In the field of task-oriented dialogue systems , many works lay on modelling dialogue acts or the state space with a hierarchical structure to tackle the decision problem for dialogues with multi-domain tasks ( Cuayáhuitl et al. , 2009 ; Peng et al. , 2017 ; Chen et al. , 2019 ; Tang et al. , 2018b ; Budzianowski et al. , 2017b ) . Distinguished from these works , our work views the relationship between dialogue policy and natural language generator ( NLG ) as a natural hierarchical structure and models it with the option framework ( Sutton et al. , 1999 ) . In the field of open-domain dialogue system , a similar hierarchical structure was proposed ( Serban et al. , 2017 ; Saleh et al. , 2019 ) but with a different motivation from ours . In this sense , these two fields are possible to be unified . Finally , among the works training with hierarchical reinforcement learning ( HRL ) , some of them set up an extrinsic reward for high-level policy and an intrinsic reward for low-level policy respectively to encourage the convergence ( Peng et al. , 2017 ; Budzianowski et al. , 2017b ) . In our work , we train both high-level policy and low-level policy with identical rewards to guarantee the consistency between two policies ( Sutton et al. , 1999 ) . On the other hand , in the field of open-domain dialogue system , Saleh et al . ( 2019 ) represented the joint generated utterances over a turn as a low-level action such that both high-level policy and low-level policy were in identical time scales . Besides , its lowlevel policy gradients flew through high-level policy during training , which degraded hierarchical policies to an E2E policy with a word-level action space . In our work , ( 1 ) dialogue policy and NLG are decoupled during optimization and no gradients are allowed to flow between them ; ( 2 ) these two policies are asynchronously updated to theoretically guarantee the convergence to a local maximizer ; and ( 3 ) each generated word is regarded as a low-level action . 3 BACKGROUND . 3.1 TASK-ORIENTED DIALOGUE SYSTEM . Brief Introduction : A task-oriented dialogue system aims to help fulfill a user ’ s task through conversation in turns . In general , each dialogue is modelled with an ontology called goal which includes inform slots and request slots . The traditional modular dialogue system is constituted of natural language understanding ( NLU ) , dialogue state tracker ( DST ) , dialogue policy and natural language generator ( NLG ) . For a dialogue system , it needs to infer inform slots from user utterances and transform them to a dialogue state , which is completed by NLU and DST ( Chen et al. , 2017 ) . In this work , we focus on optimizing dialogue policy and NLG , leveraging oracle dialogue states and database search results to produce dialogue acts and then responses ( that should include as many request slots as possible ) in turns . For optimizing dialogue policy , it is modelled with Markov decision process ( MDP ) ( Williams and Young , 2007 ) . Existing Challenges : We identify the main challenges of task-oriented dialogue systems : ( 1 ) A dialogue with a single domain ( i.e . completing one task in a dialogue ) has been broadly studied , however , handling a dialogue with multiple domains is more challenging and needs more studies on it ( Budzianowski et al. , 2018 ) ; ( 2 ) If ignoring the syntactic structure of generated system utterances ( i.e . losing comprehensibility ) , the mission of task-oriented dialogues will be simplified to generating corresponding labels ( i.e. , slots ) for user utterances . Several existing algorithms already reached high scores on request slots acquisition but low scores on the comprehensibility of generated system utterances ( Zhao et al. , 2019 ; Mehri et al. , 2019 ) , so the simplified task has been well-addressed . Reversely , if only focusing on the comprehensibility , the score on request slots acquisition could be drastically affected ( Chen et al. , 2019 ; Hosseini-Asl et al. , 2020 ) . In this work , we investigate the trade-off between the comprehensibility and request slots acquisition ; ( 3 ) Designing and annotating a dialogue act structure is effort demanding ( Budzianowski et al. , 2018 ) . Therefore , learning a meaningful latent dialogue act becomes a new challenge ( Zhao et al. , 2019 ) .	Authors applied reinforcement learning framework to the problem of task-oriented dialog. In particular, they used the option framework to represent the connection between the dialog policy and the natural language generation. Theoretically, they showed that synchronized updates to the low-level and high-level policy may never converge, yet asynchronized updates guarantees convergence. Authors also used a discriminator reward signal to cope with sparse reward (dialog success rate) and better representation of the human evaluation.	SP:e8863d56eb4be6ed7aa17241af9ee376570d0770
On Self-Supervised Image Representations for GAN Evaluation	1 INTRODUCTION . Generative adversarial networks ( GANs ) are an extremely active research direction in machine learning . The intensive development of the field requires established quantitative measures to assess constantly appearing models . While a large number of evaluation protocols were proposed ( Borji , 2019 ; Xu et al. , 2018 ; Zhou et al. , 2019 ; Naeem et al. , 2020 ) , there is still no consensus regarding the best evaluation measure . Across the existing measures , the Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) and Precision/Recall ( Kynkäänniemi et al. , 2019 ) are the most widely adopted due to their simplicity and decent consistency with human judgments . FID and Precision/Recall quantify the discrepancy between distributions of real and generated images . Since these distributions are complicated to describe in the original RGB space , the images are represented by embeddings , typically extracted with CNNs pretrained on the Imagenet classification ( Deng et al. , 2009 ) . While FID computed with these embeddings was shown to correlate with human evaluation ( Heusel et al. , 2017 ) , these observations were mostly obtained on datasets , semantically close to Imagenet . Meanwhile , on non-Imagenet datasets , FID can result in inadequate evaluation , as widely reported in the literature ( Rosca et al. , 2017 ; Barratt & Sharma , 2018 ; Zhou et al. , 2019 ) . In this work , we propose to employ the state-of-the-art self-supervised models ( Chen et al. , 2020a ; He et al. , 2020 ; Caron et al. , 2020 ) to extract image embeddings for GAN evaluation . These models were shown to produce features that transfer better to new tasks , hence , they become a promising candidate to provide a more universal representation . Intuitively , classification-pretrained embeddings by design can suppress the information , irrelevant for the Imagenet class labels , which , however , can be crucial for other domains , like human faces . On the contrary , self-supervised models , mostly trained via contrastive or clustering-based learning , do not have such a bias since their main goal is typically to learn invariances to common image augmentations . To justify the usage of self-supervised embeddings , we perform a thorough comparison of the recent GAN models trained on the five most common benchmark datasets . We demonstrate that classification-pretrained embeddings can lead to incorrect ranking in terms of FID , Precision , and Recall , which are the most popular metrics . On the other hand , self-supervised representations produce more sensible ranking , advocating their advantage over “ classification-oriented ” counterparts . Since all the checkpoints needed to compute self-supervised embeddings are publicly available , they can serve as a handy instrument for GAN comparison , consistent between different papers . We release the code for the “ self-supervised ” GAN evaluation along with data and human labeling reported in the paper online1 . To sum up , the contributions of this paper are as follows : 1 . To the best of our knowledge , our work is the first to employ self-supervised image representations to evaluate GANs trained on natural images . 2 . By extensive experiments on the standard non-Imagenet benchmarks , we demonstrate that the usage of self-supervised representations provides a more reliable GAN comparison . 3 . We show that the FID measure computed with self-supervised representations often has higher sample-efficiency and analyze the sources of this advantage . 2 RELATED WORK . GAN evaluation measures . Over the last years , a variety of quantitative GAN evaluation methods have been developed by the community , and the development process has yet to converge since all the measures possess specific weaknesses ( Borji , 2019 ; Xu et al. , 2018 ) . The Inception Score ( Salimans et al. , 2016 ) was the first widely adopted measure but was shown to be hardly applicable for non-Imagenet domains ( Barratt & Sharma , 2018 ) . The Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) quantifies dissimilarity of real and generated distributions , computing the Wasserstein distance between their Gaussian approximations , and is currently the most popular scalar measure of GAN ’ s quality . Several recent measures were proposed ( Sajjadi et al. , 2018 ; Kynkäänniemi et al. , 2019 ; Naeem et al. , 2020 ) that separately evaluate fidelity and diversity of GAN-produced images . All of them mostly use the embeddings produced by the Imagenet classification CNN . A recent work ( Zhou et al. , 2019 ) has introduced a human-in-the-loop measure , which is more reliable compared to automated ones but can not be used , e.g. , for monitoring the training process . We focus on three the most widely used measures : FID , Precision , and Recall , which are discussed briefly below . Fréchet Inception Distance quantifies the discrepancy between the distributions of real and generated images , denoted by pD and pG . Both pD and pG are defined on the high-dimensional image space forming nontrivial manifolds , which are challenging to approximate by simple functions . To be practical , FID operates in the lower-dimensional space of image embeddings . Formally , the embeddings are defined by a map f : RN → Rd , where N and d correspond to the dimensionalities of the images and embeddings spaces , respectively . By design , FID measures the dissimilarity between the induced distributions fpD , fpG as follows . First , fpD and fpG are approximated by Gaussian distributions . Then the Wasserstein distance between these distributions is evaluated . As was shown in ( Dowson & Landau , 1982 ) , for distributions defined by the means µD , µG and the covariance matrices ΣD , ΣG , this quantity equals to ‖µD − µG‖22 + tr ( ΣD + ΣG − 2 ( ΣDΣG ) 1 2 ) . Lower FID values correspond to higher similarity between pG and pD ; hence , can be used to evaluate the performance of generative models . As a common practice in the FID computation , one typically uses the activations from the InceptionV3 ( Szegedy et al. , 2016 ) pretrained on Imagenet classification . Precision and Recall . When assessing generative models , it is important to quantify both the visual quality of generated images and the model diversity , e.g. , to diagnose mode collapsing . However , the scalar FID values were shown ( Sajjadi et al. , 2018 ; Kynkäänniemi et al. , 2019 ) to sacrifice diversity in favor of visual quality , therefore FID can not serve as the only sufficient metric . To this end , ( Sajjadi et al. , 2018 ) introduced Precision and Recall , which aim to measure the image realism and the model diversity , respectively . A recent follow-up ( Kynkäänniemi et al. , 2019 ) elaborates on these metrics and proposes a reasonable procedure to quantify both precision and recall based only on the image embeddings . In a nutshell , ( Kynkäänniemi et al. , 2019 ) assumes that the visual quality of a particular sample is high if its embedding is neighboring for the embeddings of the real images . On 1https : //github.com/stanis-morozov/self-supervised-gan-eval the other hand , a given real image is considered covered by the model if its embedding belongs to the neighborhood of embeddings of the generated images . Self-supervised representations . Self-supervised learning is currently attracting much research attention , especially to contrastive learning and clustering-based methods ( Chen et al. , 2020a ; He et al. , 2020 ; Caron et al. , 2020 ) . The common idea behind these methods is to construct representations that are invariant to a wide range of common image augmentations . The recent self-supervised methods were shown to provide more transferrable ( He et al. , 2020 ; Caron et al. , 2020 ) and robust ( Hendrycks et al. , 2019 ) features , which implies their usage as more universal representations . In this paper , we show them being a better alternative compared to established classifier-produced embeddings in the context of GAN assessment . 3 GAN EVALUATION . Here we systematically compare the publicly available GANs to highlight the cases of misleading comparison with classification-pretrained embeddings . Our goal is to demonstrate that selfsupervised embeddings are a better alternative in these cases , while in other cases , the rankings with both types of embeddings are mostly consistent . We examine open-sourced GAN models2 trained on five popular benchmarks : • CelebaHQ 1024x1024 ( Karras et al. , 2017 ) with the following GAN models : StyleGAN with truncation 0.7 ( Karras et al. , 2019a ) and without it , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it , PGGAN ( Karras et al. , 2017 ) . To compute the metrics , we use 30k real and synthetic images ; • FFHQ 1024x1024 ( Karras et al. , 2019a ) with the following GAN models : StyleGAN ( Karras et al. , 2019a ) , StyleGAN2 ( Karras et al. , 2019b ) , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it . To compute the metrics , we use 30k real and synthetic images ; • LSUN Bedroom 256x256 ( Yu et al. , 2015 ) with the following GAN models : StyleGAN ( Karras et al. , 2019a ) with truncation 0.7 and without it , PGGAN ( Karras et al. , 2017 ) , COCO-GAN ( Lin et al. , 2019 ) , RPGAN ( Voynov & Babenko , 2019 ) , RPGAN with high diversity ( RPGAN div. ) . RPGAN generates 128x128 images , so we upscale them to 256x256 . To compute the metrics , we use 30k real and synthetic images ; • LSUN Church 256x256 ( Yu et al. , 2015 ) with the models : StyleGAN2 ( Karras et al. , 2019b ) with truncation 0.5 and without it , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it , PGGAN ( Karras et al. , 2017 ) , SNGAN ( Miyato et al. , 2018 ) . SNGAN generates 128x128 images , so we upscale them to 256x256 . To compute the metrics , we use 100k real and synthetic images ; • Imagenet 128x128 ( Deng et al. , 2009 ) with the following GAN models : BigGAN ( Brock et al. , 2019 ) , BigGAN-deep ( Brock et al. , 2019 ) ( both with truncation 2.0 ) , S3GAN ( Lucic et al. , 2019 ) , Your Local GAN ( YLG ) ( Daras et al. , 2020 ) . To compute the metrics , we use 50k images ( 50 per class ) . We include this dataset to demonstrate that for Imagenet , the proposed self-supervised representations provide consistent ranking with commonly used InceptionV3 embeddings . To compute image embeddings , we use the following publicly available models : • InceptionV3 ( Szegedy et al. , 2016 ) pretrained on the ILSVRC-2012 task ( Deng et al. , 2009 ) ; • Resnet50 ( He et al. , 2016 ) pretrained on the ILSVRC-2012 task . We include this model since self-supervised models employ Resnet50 , therefore , it is important to demonstrate that better GAN ranking comes from the training objective rather than the deeper architecture ; • Imagenet21k ( Kolesnikov et al. , 2019 ) pretrained on the multi-label classification task on approximately 14M images from the full Imagenet . Kolesnikov et al . ( 2019 ) have shown that supervised pretraining on huge datasets provides more transferrable features , therefore , Imagenet21k can also potentially provide more universal representations . The model architecture is Resnet50 ; • SwAV ( Caron et al. , 2020 ) is the state-of-the-art self-supervised image representation model trained on ILSVRC-2012 . The idea of SwAV is to simultaneously cluster the images while enforcing consistency between cluster assignments produced for different augmentations of the same image . The model architecture is Resnet50 ; 2The URLs for all models are provided in Appendix . • DeepClusterV2 ( Caron et al. , 2020 ) is another self-supervised model obtained by alternating between pseudo-labels generation via k-means clustering and training the network with a classification loss supervised by these pseudo-labels . The model architecture is Resnet50 ; • MoCoV2 ( Chen et al. , 2020b ) is the state-of-the-art contrastive learning approach , which training objective enforces the closeness of representations produced for different augmentations of the same image while pushing apart the representations of unrelated images . The model architecture is Resnet50 . Three self-supervised models listed above outperform supervised pretraining on a number of transfer tasks ( He et al. , 2020 ; Caron et al. , 2020 ) , which implies that their embeddings capture more information relevant for these tasks , compared to supervised models pretrained on Imagenet . Below , for a large number of publicly available GANs , we present the values of FID , Precision , and Recall metrics computed with different embeddings . For the cases where the GANs ranking is inconsistent , we aim to show that the ranking obtained with the self-supervised representations is more reasonable .	Overview of paper: this work compares supervised feature extractors vs. two types of self-supervised feature extractors for the task of GAN model evaluation. It shows that the ranking provided by self-supervised features is different from that of supervised features, and claims it corresponds better with human judgement. Experiments are conducted of multiple large GANs and datasets.	SP:c43f864a3d2c7be9c5aa4c2d0f30d3678c80376b
On Self-Supervised Image Representations for GAN Evaluation	1 INTRODUCTION . Generative adversarial networks ( GANs ) are an extremely active research direction in machine learning . The intensive development of the field requires established quantitative measures to assess constantly appearing models . While a large number of evaluation protocols were proposed ( Borji , 2019 ; Xu et al. , 2018 ; Zhou et al. , 2019 ; Naeem et al. , 2020 ) , there is still no consensus regarding the best evaluation measure . Across the existing measures , the Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) and Precision/Recall ( Kynkäänniemi et al. , 2019 ) are the most widely adopted due to their simplicity and decent consistency with human judgments . FID and Precision/Recall quantify the discrepancy between distributions of real and generated images . Since these distributions are complicated to describe in the original RGB space , the images are represented by embeddings , typically extracted with CNNs pretrained on the Imagenet classification ( Deng et al. , 2009 ) . While FID computed with these embeddings was shown to correlate with human evaluation ( Heusel et al. , 2017 ) , these observations were mostly obtained on datasets , semantically close to Imagenet . Meanwhile , on non-Imagenet datasets , FID can result in inadequate evaluation , as widely reported in the literature ( Rosca et al. , 2017 ; Barratt & Sharma , 2018 ; Zhou et al. , 2019 ) . In this work , we propose to employ the state-of-the-art self-supervised models ( Chen et al. , 2020a ; He et al. , 2020 ; Caron et al. , 2020 ) to extract image embeddings for GAN evaluation . These models were shown to produce features that transfer better to new tasks , hence , they become a promising candidate to provide a more universal representation . Intuitively , classification-pretrained embeddings by design can suppress the information , irrelevant for the Imagenet class labels , which , however , can be crucial for other domains , like human faces . On the contrary , self-supervised models , mostly trained via contrastive or clustering-based learning , do not have such a bias since their main goal is typically to learn invariances to common image augmentations . To justify the usage of self-supervised embeddings , we perform a thorough comparison of the recent GAN models trained on the five most common benchmark datasets . We demonstrate that classification-pretrained embeddings can lead to incorrect ranking in terms of FID , Precision , and Recall , which are the most popular metrics . On the other hand , self-supervised representations produce more sensible ranking , advocating their advantage over “ classification-oriented ” counterparts . Since all the checkpoints needed to compute self-supervised embeddings are publicly available , they can serve as a handy instrument for GAN comparison , consistent between different papers . We release the code for the “ self-supervised ” GAN evaluation along with data and human labeling reported in the paper online1 . To sum up , the contributions of this paper are as follows : 1 . To the best of our knowledge , our work is the first to employ self-supervised image representations to evaluate GANs trained on natural images . 2 . By extensive experiments on the standard non-Imagenet benchmarks , we demonstrate that the usage of self-supervised representations provides a more reliable GAN comparison . 3 . We show that the FID measure computed with self-supervised representations often has higher sample-efficiency and analyze the sources of this advantage . 2 RELATED WORK . GAN evaluation measures . Over the last years , a variety of quantitative GAN evaluation methods have been developed by the community , and the development process has yet to converge since all the measures possess specific weaknesses ( Borji , 2019 ; Xu et al. , 2018 ) . The Inception Score ( Salimans et al. , 2016 ) was the first widely adopted measure but was shown to be hardly applicable for non-Imagenet domains ( Barratt & Sharma , 2018 ) . The Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) quantifies dissimilarity of real and generated distributions , computing the Wasserstein distance between their Gaussian approximations , and is currently the most popular scalar measure of GAN ’ s quality . Several recent measures were proposed ( Sajjadi et al. , 2018 ; Kynkäänniemi et al. , 2019 ; Naeem et al. , 2020 ) that separately evaluate fidelity and diversity of GAN-produced images . All of them mostly use the embeddings produced by the Imagenet classification CNN . A recent work ( Zhou et al. , 2019 ) has introduced a human-in-the-loop measure , which is more reliable compared to automated ones but can not be used , e.g. , for monitoring the training process . We focus on three the most widely used measures : FID , Precision , and Recall , which are discussed briefly below . Fréchet Inception Distance quantifies the discrepancy between the distributions of real and generated images , denoted by pD and pG . Both pD and pG are defined on the high-dimensional image space forming nontrivial manifolds , which are challenging to approximate by simple functions . To be practical , FID operates in the lower-dimensional space of image embeddings . Formally , the embeddings are defined by a map f : RN → Rd , where N and d correspond to the dimensionalities of the images and embeddings spaces , respectively . By design , FID measures the dissimilarity between the induced distributions fpD , fpG as follows . First , fpD and fpG are approximated by Gaussian distributions . Then the Wasserstein distance between these distributions is evaluated . As was shown in ( Dowson & Landau , 1982 ) , for distributions defined by the means µD , µG and the covariance matrices ΣD , ΣG , this quantity equals to ‖µD − µG‖22 + tr ( ΣD + ΣG − 2 ( ΣDΣG ) 1 2 ) . Lower FID values correspond to higher similarity between pG and pD ; hence , can be used to evaluate the performance of generative models . As a common practice in the FID computation , one typically uses the activations from the InceptionV3 ( Szegedy et al. , 2016 ) pretrained on Imagenet classification . Precision and Recall . When assessing generative models , it is important to quantify both the visual quality of generated images and the model diversity , e.g. , to diagnose mode collapsing . However , the scalar FID values were shown ( Sajjadi et al. , 2018 ; Kynkäänniemi et al. , 2019 ) to sacrifice diversity in favor of visual quality , therefore FID can not serve as the only sufficient metric . To this end , ( Sajjadi et al. , 2018 ) introduced Precision and Recall , which aim to measure the image realism and the model diversity , respectively . A recent follow-up ( Kynkäänniemi et al. , 2019 ) elaborates on these metrics and proposes a reasonable procedure to quantify both precision and recall based only on the image embeddings . In a nutshell , ( Kynkäänniemi et al. , 2019 ) assumes that the visual quality of a particular sample is high if its embedding is neighboring for the embeddings of the real images . On 1https : //github.com/stanis-morozov/self-supervised-gan-eval the other hand , a given real image is considered covered by the model if its embedding belongs to the neighborhood of embeddings of the generated images . Self-supervised representations . Self-supervised learning is currently attracting much research attention , especially to contrastive learning and clustering-based methods ( Chen et al. , 2020a ; He et al. , 2020 ; Caron et al. , 2020 ) . The common idea behind these methods is to construct representations that are invariant to a wide range of common image augmentations . The recent self-supervised methods were shown to provide more transferrable ( He et al. , 2020 ; Caron et al. , 2020 ) and robust ( Hendrycks et al. , 2019 ) features , which implies their usage as more universal representations . In this paper , we show them being a better alternative compared to established classifier-produced embeddings in the context of GAN assessment . 3 GAN EVALUATION . Here we systematically compare the publicly available GANs to highlight the cases of misleading comparison with classification-pretrained embeddings . Our goal is to demonstrate that selfsupervised embeddings are a better alternative in these cases , while in other cases , the rankings with both types of embeddings are mostly consistent . We examine open-sourced GAN models2 trained on five popular benchmarks : • CelebaHQ 1024x1024 ( Karras et al. , 2017 ) with the following GAN models : StyleGAN with truncation 0.7 ( Karras et al. , 2019a ) and without it , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it , PGGAN ( Karras et al. , 2017 ) . To compute the metrics , we use 30k real and synthetic images ; • FFHQ 1024x1024 ( Karras et al. , 2019a ) with the following GAN models : StyleGAN ( Karras et al. , 2019a ) , StyleGAN2 ( Karras et al. , 2019b ) , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it . To compute the metrics , we use 30k real and synthetic images ; • LSUN Bedroom 256x256 ( Yu et al. , 2015 ) with the following GAN models : StyleGAN ( Karras et al. , 2019a ) with truncation 0.7 and without it , PGGAN ( Karras et al. , 2017 ) , COCO-GAN ( Lin et al. , 2019 ) , RPGAN ( Voynov & Babenko , 2019 ) , RPGAN with high diversity ( RPGAN div. ) . RPGAN generates 128x128 images , so we upscale them to 256x256 . To compute the metrics , we use 30k real and synthetic images ; • LSUN Church 256x256 ( Yu et al. , 2015 ) with the models : StyleGAN2 ( Karras et al. , 2019b ) with truncation 0.5 and without it , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it , PGGAN ( Karras et al. , 2017 ) , SNGAN ( Miyato et al. , 2018 ) . SNGAN generates 128x128 images , so we upscale them to 256x256 . To compute the metrics , we use 100k real and synthetic images ; • Imagenet 128x128 ( Deng et al. , 2009 ) with the following GAN models : BigGAN ( Brock et al. , 2019 ) , BigGAN-deep ( Brock et al. , 2019 ) ( both with truncation 2.0 ) , S3GAN ( Lucic et al. , 2019 ) , Your Local GAN ( YLG ) ( Daras et al. , 2020 ) . To compute the metrics , we use 50k images ( 50 per class ) . We include this dataset to demonstrate that for Imagenet , the proposed self-supervised representations provide consistent ranking with commonly used InceptionV3 embeddings . To compute image embeddings , we use the following publicly available models : • InceptionV3 ( Szegedy et al. , 2016 ) pretrained on the ILSVRC-2012 task ( Deng et al. , 2009 ) ; • Resnet50 ( He et al. , 2016 ) pretrained on the ILSVRC-2012 task . We include this model since self-supervised models employ Resnet50 , therefore , it is important to demonstrate that better GAN ranking comes from the training objective rather than the deeper architecture ; • Imagenet21k ( Kolesnikov et al. , 2019 ) pretrained on the multi-label classification task on approximately 14M images from the full Imagenet . Kolesnikov et al . ( 2019 ) have shown that supervised pretraining on huge datasets provides more transferrable features , therefore , Imagenet21k can also potentially provide more universal representations . The model architecture is Resnet50 ; • SwAV ( Caron et al. , 2020 ) is the state-of-the-art self-supervised image representation model trained on ILSVRC-2012 . The idea of SwAV is to simultaneously cluster the images while enforcing consistency between cluster assignments produced for different augmentations of the same image . The model architecture is Resnet50 ; 2The URLs for all models are provided in Appendix . • DeepClusterV2 ( Caron et al. , 2020 ) is another self-supervised model obtained by alternating between pseudo-labels generation via k-means clustering and training the network with a classification loss supervised by these pseudo-labels . The model architecture is Resnet50 ; • MoCoV2 ( Chen et al. , 2020b ) is the state-of-the-art contrastive learning approach , which training objective enforces the closeness of representations produced for different augmentations of the same image while pushing apart the representations of unrelated images . The model architecture is Resnet50 . Three self-supervised models listed above outperform supervised pretraining on a number of transfer tasks ( He et al. , 2020 ; Caron et al. , 2020 ) , which implies that their embeddings capture more information relevant for these tasks , compared to supervised models pretrained on Imagenet . Below , for a large number of publicly available GANs , we present the values of FID , Precision , and Recall metrics computed with different embeddings . For the cases where the GANs ranking is inconsistent , we aim to show that the ranking obtained with the self-supervised representations is more reasonable .	This paper proposes to use image representations from trained self-supervised models to evaluate GANs more accurately. Compared to the currently used representations from supervised-pretrained models e.g. InceptionV3, the authors claim, that such embeddings suppress information not critical for the classification process which, however, are assumed to be crucial for assessing the full distributions of real and generated images. The authors use 5 datasets and their respective representations from 5 models, 3 supervised and 2 self-supervised, to show that representations from self-supervised models lead to better GAN evaluations. The representations were used to evaluate 6 GAN models with 3 metrics, namely FID, Precision and Recall. A ranking of the GAN models shows inconsistencies between supervised and self-supervised based representations. By visual inspection, prediction accuracy tests, and a comparison of representation invariances the authors show that rankings via self-supervised embeddings are more plausible.	SP:c43f864a3d2c7be9c5aa4c2d0f30d3678c80376b
On Self-Supervised Image Representations for GAN Evaluation	1 INTRODUCTION . Generative adversarial networks ( GANs ) are an extremely active research direction in machine learning . The intensive development of the field requires established quantitative measures to assess constantly appearing models . While a large number of evaluation protocols were proposed ( Borji , 2019 ; Xu et al. , 2018 ; Zhou et al. , 2019 ; Naeem et al. , 2020 ) , there is still no consensus regarding the best evaluation measure . Across the existing measures , the Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) and Precision/Recall ( Kynkäänniemi et al. , 2019 ) are the most widely adopted due to their simplicity and decent consistency with human judgments . FID and Precision/Recall quantify the discrepancy between distributions of real and generated images . Since these distributions are complicated to describe in the original RGB space , the images are represented by embeddings , typically extracted with CNNs pretrained on the Imagenet classification ( Deng et al. , 2009 ) . While FID computed with these embeddings was shown to correlate with human evaluation ( Heusel et al. , 2017 ) , these observations were mostly obtained on datasets , semantically close to Imagenet . Meanwhile , on non-Imagenet datasets , FID can result in inadequate evaluation , as widely reported in the literature ( Rosca et al. , 2017 ; Barratt & Sharma , 2018 ; Zhou et al. , 2019 ) . In this work , we propose to employ the state-of-the-art self-supervised models ( Chen et al. , 2020a ; He et al. , 2020 ; Caron et al. , 2020 ) to extract image embeddings for GAN evaluation . These models were shown to produce features that transfer better to new tasks , hence , they become a promising candidate to provide a more universal representation . Intuitively , classification-pretrained embeddings by design can suppress the information , irrelevant for the Imagenet class labels , which , however , can be crucial for other domains , like human faces . On the contrary , self-supervised models , mostly trained via contrastive or clustering-based learning , do not have such a bias since their main goal is typically to learn invariances to common image augmentations . To justify the usage of self-supervised embeddings , we perform a thorough comparison of the recent GAN models trained on the five most common benchmark datasets . We demonstrate that classification-pretrained embeddings can lead to incorrect ranking in terms of FID , Precision , and Recall , which are the most popular metrics . On the other hand , self-supervised representations produce more sensible ranking , advocating their advantage over “ classification-oriented ” counterparts . Since all the checkpoints needed to compute self-supervised embeddings are publicly available , they can serve as a handy instrument for GAN comparison , consistent between different papers . We release the code for the “ self-supervised ” GAN evaluation along with data and human labeling reported in the paper online1 . To sum up , the contributions of this paper are as follows : 1 . To the best of our knowledge , our work is the first to employ self-supervised image representations to evaluate GANs trained on natural images . 2 . By extensive experiments on the standard non-Imagenet benchmarks , we demonstrate that the usage of self-supervised representations provides a more reliable GAN comparison . 3 . We show that the FID measure computed with self-supervised representations often has higher sample-efficiency and analyze the sources of this advantage . 2 RELATED WORK . GAN evaluation measures . Over the last years , a variety of quantitative GAN evaluation methods have been developed by the community , and the development process has yet to converge since all the measures possess specific weaknesses ( Borji , 2019 ; Xu et al. , 2018 ) . The Inception Score ( Salimans et al. , 2016 ) was the first widely adopted measure but was shown to be hardly applicable for non-Imagenet domains ( Barratt & Sharma , 2018 ) . The Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) quantifies dissimilarity of real and generated distributions , computing the Wasserstein distance between their Gaussian approximations , and is currently the most popular scalar measure of GAN ’ s quality . Several recent measures were proposed ( Sajjadi et al. , 2018 ; Kynkäänniemi et al. , 2019 ; Naeem et al. , 2020 ) that separately evaluate fidelity and diversity of GAN-produced images . All of them mostly use the embeddings produced by the Imagenet classification CNN . A recent work ( Zhou et al. , 2019 ) has introduced a human-in-the-loop measure , which is more reliable compared to automated ones but can not be used , e.g. , for monitoring the training process . We focus on three the most widely used measures : FID , Precision , and Recall , which are discussed briefly below . Fréchet Inception Distance quantifies the discrepancy between the distributions of real and generated images , denoted by pD and pG . Both pD and pG are defined on the high-dimensional image space forming nontrivial manifolds , which are challenging to approximate by simple functions . To be practical , FID operates in the lower-dimensional space of image embeddings . Formally , the embeddings are defined by a map f : RN → Rd , where N and d correspond to the dimensionalities of the images and embeddings spaces , respectively . By design , FID measures the dissimilarity between the induced distributions fpD , fpG as follows . First , fpD and fpG are approximated by Gaussian distributions . Then the Wasserstein distance between these distributions is evaluated . As was shown in ( Dowson & Landau , 1982 ) , for distributions defined by the means µD , µG and the covariance matrices ΣD , ΣG , this quantity equals to ‖µD − µG‖22 + tr ( ΣD + ΣG − 2 ( ΣDΣG ) 1 2 ) . Lower FID values correspond to higher similarity between pG and pD ; hence , can be used to evaluate the performance of generative models . As a common practice in the FID computation , one typically uses the activations from the InceptionV3 ( Szegedy et al. , 2016 ) pretrained on Imagenet classification . Precision and Recall . When assessing generative models , it is important to quantify both the visual quality of generated images and the model diversity , e.g. , to diagnose mode collapsing . However , the scalar FID values were shown ( Sajjadi et al. , 2018 ; Kynkäänniemi et al. , 2019 ) to sacrifice diversity in favor of visual quality , therefore FID can not serve as the only sufficient metric . To this end , ( Sajjadi et al. , 2018 ) introduced Precision and Recall , which aim to measure the image realism and the model diversity , respectively . A recent follow-up ( Kynkäänniemi et al. , 2019 ) elaborates on these metrics and proposes a reasonable procedure to quantify both precision and recall based only on the image embeddings . In a nutshell , ( Kynkäänniemi et al. , 2019 ) assumes that the visual quality of a particular sample is high if its embedding is neighboring for the embeddings of the real images . On 1https : //github.com/stanis-morozov/self-supervised-gan-eval the other hand , a given real image is considered covered by the model if its embedding belongs to the neighborhood of embeddings of the generated images . Self-supervised representations . Self-supervised learning is currently attracting much research attention , especially to contrastive learning and clustering-based methods ( Chen et al. , 2020a ; He et al. , 2020 ; Caron et al. , 2020 ) . The common idea behind these methods is to construct representations that are invariant to a wide range of common image augmentations . The recent self-supervised methods were shown to provide more transferrable ( He et al. , 2020 ; Caron et al. , 2020 ) and robust ( Hendrycks et al. , 2019 ) features , which implies their usage as more universal representations . In this paper , we show them being a better alternative compared to established classifier-produced embeddings in the context of GAN assessment . 3 GAN EVALUATION . Here we systematically compare the publicly available GANs to highlight the cases of misleading comparison with classification-pretrained embeddings . Our goal is to demonstrate that selfsupervised embeddings are a better alternative in these cases , while in other cases , the rankings with both types of embeddings are mostly consistent . We examine open-sourced GAN models2 trained on five popular benchmarks : • CelebaHQ 1024x1024 ( Karras et al. , 2017 ) with the following GAN models : StyleGAN with truncation 0.7 ( Karras et al. , 2019a ) and without it , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it , PGGAN ( Karras et al. , 2017 ) . To compute the metrics , we use 30k real and synthetic images ; • FFHQ 1024x1024 ( Karras et al. , 2019a ) with the following GAN models : StyleGAN ( Karras et al. , 2019a ) , StyleGAN2 ( Karras et al. , 2019b ) , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it . To compute the metrics , we use 30k real and synthetic images ; • LSUN Bedroom 256x256 ( Yu et al. , 2015 ) with the following GAN models : StyleGAN ( Karras et al. , 2019a ) with truncation 0.7 and without it , PGGAN ( Karras et al. , 2017 ) , COCO-GAN ( Lin et al. , 2019 ) , RPGAN ( Voynov & Babenko , 2019 ) , RPGAN with high diversity ( RPGAN div. ) . RPGAN generates 128x128 images , so we upscale them to 256x256 . To compute the metrics , we use 30k real and synthetic images ; • LSUN Church 256x256 ( Yu et al. , 2015 ) with the models : StyleGAN2 ( Karras et al. , 2019b ) with truncation 0.5 and without it , MSG ( Karnewar & Wang , 2020 ) with truncation 0.6 and without it , PGGAN ( Karras et al. , 2017 ) , SNGAN ( Miyato et al. , 2018 ) . SNGAN generates 128x128 images , so we upscale them to 256x256 . To compute the metrics , we use 100k real and synthetic images ; • Imagenet 128x128 ( Deng et al. , 2009 ) with the following GAN models : BigGAN ( Brock et al. , 2019 ) , BigGAN-deep ( Brock et al. , 2019 ) ( both with truncation 2.0 ) , S3GAN ( Lucic et al. , 2019 ) , Your Local GAN ( YLG ) ( Daras et al. , 2020 ) . To compute the metrics , we use 50k images ( 50 per class ) . We include this dataset to demonstrate that for Imagenet , the proposed self-supervised representations provide consistent ranking with commonly used InceptionV3 embeddings . To compute image embeddings , we use the following publicly available models : • InceptionV3 ( Szegedy et al. , 2016 ) pretrained on the ILSVRC-2012 task ( Deng et al. , 2009 ) ; • Resnet50 ( He et al. , 2016 ) pretrained on the ILSVRC-2012 task . We include this model since self-supervised models employ Resnet50 , therefore , it is important to demonstrate that better GAN ranking comes from the training objective rather than the deeper architecture ; • Imagenet21k ( Kolesnikov et al. , 2019 ) pretrained on the multi-label classification task on approximately 14M images from the full Imagenet . Kolesnikov et al . ( 2019 ) have shown that supervised pretraining on huge datasets provides more transferrable features , therefore , Imagenet21k can also potentially provide more universal representations . The model architecture is Resnet50 ; • SwAV ( Caron et al. , 2020 ) is the state-of-the-art self-supervised image representation model trained on ILSVRC-2012 . The idea of SwAV is to simultaneously cluster the images while enforcing consistency between cluster assignments produced for different augmentations of the same image . The model architecture is Resnet50 ; 2The URLs for all models are provided in Appendix . • DeepClusterV2 ( Caron et al. , 2020 ) is another self-supervised model obtained by alternating between pseudo-labels generation via k-means clustering and training the network with a classification loss supervised by these pseudo-labels . The model architecture is Resnet50 ; • MoCoV2 ( Chen et al. , 2020b ) is the state-of-the-art contrastive learning approach , which training objective enforces the closeness of representations produced for different augmentations of the same image while pushing apart the representations of unrelated images . The model architecture is Resnet50 . Three self-supervised models listed above outperform supervised pretraining on a number of transfer tasks ( He et al. , 2020 ; Caron et al. , 2020 ) , which implies that their embeddings capture more information relevant for these tasks , compared to supervised models pretrained on Imagenet . Below , for a large number of publicly available GANs , we present the values of FID , Precision , and Recall metrics computed with different embeddings . For the cases where the GANs ranking is inconsistent , we aim to show that the ranking obtained with the self-supervised representations is more reasonable .	The papers looks at the problem of evaluating GAN samples. Current methods, such as FID/PR with Inception v3 are problematic because they generally depend on using the features of a model discriminatively trained on (a super set of) ImageNet. The authors show that these type of models ignore details that are meaningful when for example comparing results on CelebaHQ.	SP:c43f864a3d2c7be9c5aa4c2d0f30d3678c80376b
Data Instance Prior for Transfer Learning in GANs	1 INTRODUCTION . Generative Adversarial Networks ( GANs ) are at the forefront of modern high-quality image synthesis in recent years ( Brock et al. , 2018 ; Karras et al. , 2020b ; 2019 ) . GANs have also demonstrated excellent performance on many related computer vision tasks such as image manipulation ( Zhu et al. , 2017 ; Isola et al. , 2017 ) , image editing ( Plumerault et al. , 2020 ; Shen et al. , 2020 ; Jahanian et al. , 2020 ) and compression ( Tschannen et al. , 2018 ) . Despite the success in large-scale image synthesis , GAN training suffers from a number of drawbacks that arise in practice , such as training instability and mode collapse ( Goodfellow et al. , 2016 ; Arora et al. , 2017 ) . It has been observed that the issue of unstable training can be mitigated to an extent by using conditional GANs . However , this is expected as learning the conditional model for each class is easier than learning the joint distribution . The disadvantages of GAN training have prompted research in several non-adversarial generative models ( Hoshen et al. , 2019 ; Bojanowski et al. , 2018 ; Li & Malik , 2018 ; Kingma & Welling , 2014 ) . These techniques are implicitly designed to overcome the mode collapse problem , however , the quality of generated samples are still not on par with GANs . Current state-of-the-art deep generative models require a large volume of data and computation resources . The collection of large datasets of images suitable for training - especially labeled data in case of conditional GANs - can easily become a daunting task due to issues such as copyright , image quality and also the training time required to get state-of-the-art image generation performance . To curb these limitations , researchers have recently proposed techniques inspired by transfer learning ( Noguchi & Harada , 2019 ; Wang et al. , 2018 ; Mo et al. , 2020 ) and data augmentation methods ( Karras et al. , 2020a ; Zhao et al. , 2020b ; Zhang et al. , 2019 ) . Advancements in data and computation efficiency for image synthesis can enable its applications in data-deficient fields such as medicine ( Yi et al. , 2019 ) where labeled data generation can be difficult to obtain . Transfer learning is a promising area of research ( Oquab et al. , 2014 ; Pan & Yang , 2009 ) that leverages prior information acquired from large datasets to help in training models on a target dataset under limited data and resource constraints . There has been extensive exploration of transfer learning in classification problems that have shown excellent performance on various downstream data-deficient domains . Similar extensions of reusing pre-trained networks for transfer learning ( i.e . fine-tuning a subset of pre-trained network weights from a data-rich domain ) have also been recently employed for image synthesis in GANs ( Wang et al. , 2018 ; Noguchi & Harada , 2019 ; Mo et al. , 2020 ; Wang et al. , 2020 ; Zhao et al. , 2020a ) in the limited data regime . However , these approaches are still prone to overfitting on the sparse target data , and hence suffer from degraded image quality and diversity . In this work , we propose a simple yet effective way of transferring prior knowledge in unsupervised image generation given a small sample size ( ∼ 100-2000 ) of the target data distribution . Our approach is motivated by the formulation of the IMLE technique ( Li & Malik , 2018 ) that seeks to obtain mode coverage of target data distribution by learning a mapping between latent and target distributions using a maximum likelihood criterion . We instead propose the use of data priors in GANs to match the representation of the generated samples to real modes of data . In contrast to ( Li & Malik , 2018 ) , we use the images generated using data priors to find the nearest neighbor match to real modes in the generator ’ s learned distribution . In particular , we show that using an informative data instance prior in limited and large-scale unsupervised image generation substantially improves the performance in image synthesis . We show that these data priors can be derived from commonly used computer vision pre-trained networks ( Simonyan & Zisserman , 2014 ; Zhang et al. , 2018 ; Noguchi & Harada , 2019 ; Hoshen et al. , 2019 ) or self-supervised data representations ( Chen et al. , 2020 ) ( without any violation of the target setting ’ s requirements , i.e . ensuring that the pre-trained network has not been trained on few-shot classes in the few-shot learning setting , for instance ) . In case of sparse training data , our approach of using data instance priors leverages a model pre-trained on a rich source domain to the learn the target distribution . Different from previous works ( Noguchi & Harada , 2019 ; Wang et al. , 2020 ; 2018 ) which rely on fine-tuning models trained on a data-rich domain , we propose to leverage the feature representations of our source model as data instance priors , to distill knowledge ( Romero et al. , 2015 ; Hinton et al. , 2015 ) into our target generative model . We note that our technique of using data instance priors for transfer learning becomes fully unsupervised in case the data priors are extracted from self-supervised pre-trained networks . Furthermore , in addition to image generation in low data domain , we also achieve state-of-the-art Fréchet inception distance ( FID ) score ( Heusel et al. , 2017 ) on large-scale unsupervised image generation and also show how this framework of transfer learning supports several image editing tasks . We summarize our main contributions as follows : • We propose Data Instance Prior ( DIP ) , a novel transfer learning technique for GAN image synthesis in low-data regime . We show that employing DIP in conjunction with existing few-shot image generation methods outperforms state-of-the-art results . We show with as little as 100 images our approach DIP results in generation of diverse and high quality images ( see Figure 3 ) . • We demonstrate the utility of our approach in large-scale unsupervised GANs ( Miyato et al. , 2018 ; Brock et al. , 2018 ) achieving the new state-of-the-art in terms of image quality ( Heusel et al. , 2017 ) and diversity ( Sajjadi et al. , 2018 ; Metz et al. , 2017 ) . • We show how our framework of DIP by construction enables inversion of images and common image editing tasks ( such as cutmix , in-painting , image translation ) in GANs . We call our method a data instance prior ( and not just data prior ) , since it uses representations of instances as a prior , and not a data distribution itself . 2 RELATED WORK . Deep Generative Models In recent years , there has been a surge in the research of deep generative models . Some of the popular approaches include variational auto-encoders ( VAEs ) ( Rezende et al. , 2014 ; Kingma & Welling , 2014 ) , auto-regressive ( AR ) models ( Van Oord et al. , 2016 ; Van den Oord et al. , 2016 ) and GANs ( Goodfellow et al. , 2014 ) . VAE models learn by maximizing the variational lower bound of likelihood of generating data from a given distribution . Auto-regressive approaches model the data distribution as a product of the conditional probabilities to sequentially generate data . GANs comprise of two networks , a generator and a discriminator that train in a min-max optimization . Specifically , the generator aims to generate samples to fool the discriminator , while the discriminator learns distinguish these generated samples from the real samples . Several research efforts in GANs have focused around improving the performance ( Karras et al. , 2018 ; Denton et al. , 2015 ; Radford et al. , 2016 ; Karras et al. , 2020b ; 2019 ; Brock et al. , 2018 ; Zhang et al. , 2019 ) and training stability ( Salimans et al. , 2016b ; Gulrajani et al. , 2017 ; Arjovsky et al. , 2017 ; Miyato et al. , 2018 ; Mao et al. , 2017 ; Chen et al. , 2019 ) . Recently , the areas of latent space manipulation for semantic editing ( Shen et al. , 2020 ; Jahanian et al. , 2020 ; Zhu et al. , 2020 ; Plumerault et al. , 2020 ) and few-shot image generation ( Wang et al. , 2020 ; Mo et al. , 2020 ; Noguchi & Harada , 2019 ) have gained traction in an effort to mitigate the practical challenges while deploying GANs . Several other non-adversarial training approaches such as ( Hoshen et al. , 2019 ; Bojanowski et al. , 2018 ; Li & Malik , 2018 ) have also been explored for generative modeling , which leverage supervised learning along with perceptual loss ( Zhang et al. , 2018 ) for training such models . Transfer Learning in GANs While there has been extensive research in the area of transfer learning for classification models ( Yosinski et al. , 2014 ; Oquab et al. , 2014 ; Tzeng et al. , 2015 ; Pan & Yang , 2009 ; Donahue et al. , 2014 ) , relatively fewer efforts have explored this on the task of data generation ( Wang et al. , 2018 ; 2020 ; Noguchi & Harada , 2019 ; Zhao et al. , 2020a ; Mo et al. , 2020 ) . ( Wang et al. , 2018 ) proposed to fine-tune a pre-trained GAN model ( often having millions of parameters ) from a data-rich source to adapt to the target domain with limited samples . This approach , however , often suffers from overfitting as the final model parameters are updated using only few samples of the target domain . To counter overfitting , the work of ( Noguchi & Harada , 2019 ) proposes to update only the batch normalization parameters of the pre-trained GAN model . In this approach , however , the generator is not adversarially trained and uses supervised L1 pixel distance loss and perceptual loss ( Johnson et al. , 2016 ; Zhang et al. , 2018 ) which often leads to generation of blurry images in the target domain . Based on the assumption that source and target domain support sets are similar , ( Wang et al. , 2020 ) recently proposed to learn an additional mapping network that transforms the latent code suitable for generating images of target domain while keeping the other parameters frozen . We compare against all leading baselines including ( Noguchi & Harada , 2019 ; Wang et al. , 2020 ) on their respective tasks , and show that our method DIP outperforms them substantially , while being simple to implement . A related line of recent research aims to improve large-scale unsupervised image generation in GANs by employing self-supervision - in particular , an auxiliary task of rotation prediction ( Chen et al. , 2019 ) or using one-hot labels obtained by clustering in the discriminator ’ s ( Liu et al. , 2020 ) or ImageNet classifier feature space ( Sage et al. , 2018 ) . In contrast , our method utilizes data instance priors derived from the feature activations of self-supervised/supervised pre-trained networks to improve unsupervised few-shot and large-scale image generation , leading to simpler formulation and higher performance as shown in our experiments in Section 5.3 and Appendix A . Recently , some methods ( Karras et al. , 2020a ; Zhao et al. , 2020b ; Zhang et al. , 2019 ; Zhao et al. , 2020c ) have leveraged data augmentation to effectively increase the number of samples and prevent overfitting in GAN training . However , data augmentation techniques often times alter the true data distribution and there is a leakage of these augmentations to the generated image , as shown in ( Zhao et al. , 2020c ; b ) . To overcome this , ( Zhao et al. , 2020b ) recently proposed to use differential augmentation and ( Karras et al. , 2020a ) leveraged an adaptive discriminator augmentation mechanism . We instead focus on leveraging informative data instance priors , and in fact , show how our DIP method can be used in conjunction with ( Zhao et al. , 2020b ) to improve performance .	The paper focuses on improving the performance of training generative adversarial networks (GANs) with limited target data. With the low diversity and quality when traing GANs with few data, the paper proposes to use data instance prior to reduce the overfitting. Specially, taking the target sample as input, the data prior is extracted by a pre-trained network / self-supervised model , and then mapped into the embedding by both G_emb and D_emb. The former acts as the class embedding, and the latter is the image embedding combined with the discriminator. Authors also extend the proposed method into the large dataset, and provide the cluster method or a Gaussian Mixture Model. The quantitative and qualitative result support the proposed method	SP:f5ac44287ac769114d4b4d8dce60c61bfc43ef69
Data Instance Prior for Transfer Learning in GANs	1 INTRODUCTION . Generative Adversarial Networks ( GANs ) are at the forefront of modern high-quality image synthesis in recent years ( Brock et al. , 2018 ; Karras et al. , 2020b ; 2019 ) . GANs have also demonstrated excellent performance on many related computer vision tasks such as image manipulation ( Zhu et al. , 2017 ; Isola et al. , 2017 ) , image editing ( Plumerault et al. , 2020 ; Shen et al. , 2020 ; Jahanian et al. , 2020 ) and compression ( Tschannen et al. , 2018 ) . Despite the success in large-scale image synthesis , GAN training suffers from a number of drawbacks that arise in practice , such as training instability and mode collapse ( Goodfellow et al. , 2016 ; Arora et al. , 2017 ) . It has been observed that the issue of unstable training can be mitigated to an extent by using conditional GANs . However , this is expected as learning the conditional model for each class is easier than learning the joint distribution . The disadvantages of GAN training have prompted research in several non-adversarial generative models ( Hoshen et al. , 2019 ; Bojanowski et al. , 2018 ; Li & Malik , 2018 ; Kingma & Welling , 2014 ) . These techniques are implicitly designed to overcome the mode collapse problem , however , the quality of generated samples are still not on par with GANs . Current state-of-the-art deep generative models require a large volume of data and computation resources . The collection of large datasets of images suitable for training - especially labeled data in case of conditional GANs - can easily become a daunting task due to issues such as copyright , image quality and also the training time required to get state-of-the-art image generation performance . To curb these limitations , researchers have recently proposed techniques inspired by transfer learning ( Noguchi & Harada , 2019 ; Wang et al. , 2018 ; Mo et al. , 2020 ) and data augmentation methods ( Karras et al. , 2020a ; Zhao et al. , 2020b ; Zhang et al. , 2019 ) . Advancements in data and computation efficiency for image synthesis can enable its applications in data-deficient fields such as medicine ( Yi et al. , 2019 ) where labeled data generation can be difficult to obtain . Transfer learning is a promising area of research ( Oquab et al. , 2014 ; Pan & Yang , 2009 ) that leverages prior information acquired from large datasets to help in training models on a target dataset under limited data and resource constraints . There has been extensive exploration of transfer learning in classification problems that have shown excellent performance on various downstream data-deficient domains . Similar extensions of reusing pre-trained networks for transfer learning ( i.e . fine-tuning a subset of pre-trained network weights from a data-rich domain ) have also been recently employed for image synthesis in GANs ( Wang et al. , 2018 ; Noguchi & Harada , 2019 ; Mo et al. , 2020 ; Wang et al. , 2020 ; Zhao et al. , 2020a ) in the limited data regime . However , these approaches are still prone to overfitting on the sparse target data , and hence suffer from degraded image quality and diversity . In this work , we propose a simple yet effective way of transferring prior knowledge in unsupervised image generation given a small sample size ( ∼ 100-2000 ) of the target data distribution . Our approach is motivated by the formulation of the IMLE technique ( Li & Malik , 2018 ) that seeks to obtain mode coverage of target data distribution by learning a mapping between latent and target distributions using a maximum likelihood criterion . We instead propose the use of data priors in GANs to match the representation of the generated samples to real modes of data . In contrast to ( Li & Malik , 2018 ) , we use the images generated using data priors to find the nearest neighbor match to real modes in the generator ’ s learned distribution . In particular , we show that using an informative data instance prior in limited and large-scale unsupervised image generation substantially improves the performance in image synthesis . We show that these data priors can be derived from commonly used computer vision pre-trained networks ( Simonyan & Zisserman , 2014 ; Zhang et al. , 2018 ; Noguchi & Harada , 2019 ; Hoshen et al. , 2019 ) or self-supervised data representations ( Chen et al. , 2020 ) ( without any violation of the target setting ’ s requirements , i.e . ensuring that the pre-trained network has not been trained on few-shot classes in the few-shot learning setting , for instance ) . In case of sparse training data , our approach of using data instance priors leverages a model pre-trained on a rich source domain to the learn the target distribution . Different from previous works ( Noguchi & Harada , 2019 ; Wang et al. , 2020 ; 2018 ) which rely on fine-tuning models trained on a data-rich domain , we propose to leverage the feature representations of our source model as data instance priors , to distill knowledge ( Romero et al. , 2015 ; Hinton et al. , 2015 ) into our target generative model . We note that our technique of using data instance priors for transfer learning becomes fully unsupervised in case the data priors are extracted from self-supervised pre-trained networks . Furthermore , in addition to image generation in low data domain , we also achieve state-of-the-art Fréchet inception distance ( FID ) score ( Heusel et al. , 2017 ) on large-scale unsupervised image generation and also show how this framework of transfer learning supports several image editing tasks . We summarize our main contributions as follows : • We propose Data Instance Prior ( DIP ) , a novel transfer learning technique for GAN image synthesis in low-data regime . We show that employing DIP in conjunction with existing few-shot image generation methods outperforms state-of-the-art results . We show with as little as 100 images our approach DIP results in generation of diverse and high quality images ( see Figure 3 ) . • We demonstrate the utility of our approach in large-scale unsupervised GANs ( Miyato et al. , 2018 ; Brock et al. , 2018 ) achieving the new state-of-the-art in terms of image quality ( Heusel et al. , 2017 ) and diversity ( Sajjadi et al. , 2018 ; Metz et al. , 2017 ) . • We show how our framework of DIP by construction enables inversion of images and common image editing tasks ( such as cutmix , in-painting , image translation ) in GANs . We call our method a data instance prior ( and not just data prior ) , since it uses representations of instances as a prior , and not a data distribution itself . 2 RELATED WORK . Deep Generative Models In recent years , there has been a surge in the research of deep generative models . Some of the popular approaches include variational auto-encoders ( VAEs ) ( Rezende et al. , 2014 ; Kingma & Welling , 2014 ) , auto-regressive ( AR ) models ( Van Oord et al. , 2016 ; Van den Oord et al. , 2016 ) and GANs ( Goodfellow et al. , 2014 ) . VAE models learn by maximizing the variational lower bound of likelihood of generating data from a given distribution . Auto-regressive approaches model the data distribution as a product of the conditional probabilities to sequentially generate data . GANs comprise of two networks , a generator and a discriminator that train in a min-max optimization . Specifically , the generator aims to generate samples to fool the discriminator , while the discriminator learns distinguish these generated samples from the real samples . Several research efforts in GANs have focused around improving the performance ( Karras et al. , 2018 ; Denton et al. , 2015 ; Radford et al. , 2016 ; Karras et al. , 2020b ; 2019 ; Brock et al. , 2018 ; Zhang et al. , 2019 ) and training stability ( Salimans et al. , 2016b ; Gulrajani et al. , 2017 ; Arjovsky et al. , 2017 ; Miyato et al. , 2018 ; Mao et al. , 2017 ; Chen et al. , 2019 ) . Recently , the areas of latent space manipulation for semantic editing ( Shen et al. , 2020 ; Jahanian et al. , 2020 ; Zhu et al. , 2020 ; Plumerault et al. , 2020 ) and few-shot image generation ( Wang et al. , 2020 ; Mo et al. , 2020 ; Noguchi & Harada , 2019 ) have gained traction in an effort to mitigate the practical challenges while deploying GANs . Several other non-adversarial training approaches such as ( Hoshen et al. , 2019 ; Bojanowski et al. , 2018 ; Li & Malik , 2018 ) have also been explored for generative modeling , which leverage supervised learning along with perceptual loss ( Zhang et al. , 2018 ) for training such models . Transfer Learning in GANs While there has been extensive research in the area of transfer learning for classification models ( Yosinski et al. , 2014 ; Oquab et al. , 2014 ; Tzeng et al. , 2015 ; Pan & Yang , 2009 ; Donahue et al. , 2014 ) , relatively fewer efforts have explored this on the task of data generation ( Wang et al. , 2018 ; 2020 ; Noguchi & Harada , 2019 ; Zhao et al. , 2020a ; Mo et al. , 2020 ) . ( Wang et al. , 2018 ) proposed to fine-tune a pre-trained GAN model ( often having millions of parameters ) from a data-rich source to adapt to the target domain with limited samples . This approach , however , often suffers from overfitting as the final model parameters are updated using only few samples of the target domain . To counter overfitting , the work of ( Noguchi & Harada , 2019 ) proposes to update only the batch normalization parameters of the pre-trained GAN model . In this approach , however , the generator is not adversarially trained and uses supervised L1 pixel distance loss and perceptual loss ( Johnson et al. , 2016 ; Zhang et al. , 2018 ) which often leads to generation of blurry images in the target domain . Based on the assumption that source and target domain support sets are similar , ( Wang et al. , 2020 ) recently proposed to learn an additional mapping network that transforms the latent code suitable for generating images of target domain while keeping the other parameters frozen . We compare against all leading baselines including ( Noguchi & Harada , 2019 ; Wang et al. , 2020 ) on their respective tasks , and show that our method DIP outperforms them substantially , while being simple to implement . A related line of recent research aims to improve large-scale unsupervised image generation in GANs by employing self-supervision - in particular , an auxiliary task of rotation prediction ( Chen et al. , 2019 ) or using one-hot labels obtained by clustering in the discriminator ’ s ( Liu et al. , 2020 ) or ImageNet classifier feature space ( Sage et al. , 2018 ) . In contrast , our method utilizes data instance priors derived from the feature activations of self-supervised/supervised pre-trained networks to improve unsupervised few-shot and large-scale image generation , leading to simpler formulation and higher performance as shown in our experiments in Section 5.3 and Appendix A . Recently , some methods ( Karras et al. , 2020a ; Zhao et al. , 2020b ; Zhang et al. , 2019 ; Zhao et al. , 2020c ) have leveraged data augmentation to effectively increase the number of samples and prevent overfitting in GAN training . However , data augmentation techniques often times alter the true data distribution and there is a leakage of these augmentations to the generated image , as shown in ( Zhao et al. , 2020c ; b ) . To overcome this , ( Zhao et al. , 2020b ) recently proposed to use differential augmentation and ( Karras et al. , 2020a ) leveraged an adaptive discriminator augmentation mechanism . We instead focus on leveraging informative data instance priors , and in fact , show how our DIP method can be used in conjunction with ( Zhao et al. , 2020b ) to improve performance .	This paper illustrates how they train GANs with small sample sizes with the help of Transfer Learning. The paper tackled a very specific problem: what should we do with a small sample training size if we want to train a GAN. The authors have supported their arguments by a proof in Data In Prior and experiment results. They illustrated well in both aspects.	SP:f5ac44287ac769114d4b4d8dce60c61bfc43ef69
Data Instance Prior for Transfer Learning in GANs	1 INTRODUCTION . Generative Adversarial Networks ( GANs ) are at the forefront of modern high-quality image synthesis in recent years ( Brock et al. , 2018 ; Karras et al. , 2020b ; 2019 ) . GANs have also demonstrated excellent performance on many related computer vision tasks such as image manipulation ( Zhu et al. , 2017 ; Isola et al. , 2017 ) , image editing ( Plumerault et al. , 2020 ; Shen et al. , 2020 ; Jahanian et al. , 2020 ) and compression ( Tschannen et al. , 2018 ) . Despite the success in large-scale image synthesis , GAN training suffers from a number of drawbacks that arise in practice , such as training instability and mode collapse ( Goodfellow et al. , 2016 ; Arora et al. , 2017 ) . It has been observed that the issue of unstable training can be mitigated to an extent by using conditional GANs . However , this is expected as learning the conditional model for each class is easier than learning the joint distribution . The disadvantages of GAN training have prompted research in several non-adversarial generative models ( Hoshen et al. , 2019 ; Bojanowski et al. , 2018 ; Li & Malik , 2018 ; Kingma & Welling , 2014 ) . These techniques are implicitly designed to overcome the mode collapse problem , however , the quality of generated samples are still not on par with GANs . Current state-of-the-art deep generative models require a large volume of data and computation resources . The collection of large datasets of images suitable for training - especially labeled data in case of conditional GANs - can easily become a daunting task due to issues such as copyright , image quality and also the training time required to get state-of-the-art image generation performance . To curb these limitations , researchers have recently proposed techniques inspired by transfer learning ( Noguchi & Harada , 2019 ; Wang et al. , 2018 ; Mo et al. , 2020 ) and data augmentation methods ( Karras et al. , 2020a ; Zhao et al. , 2020b ; Zhang et al. , 2019 ) . Advancements in data and computation efficiency for image synthesis can enable its applications in data-deficient fields such as medicine ( Yi et al. , 2019 ) where labeled data generation can be difficult to obtain . Transfer learning is a promising area of research ( Oquab et al. , 2014 ; Pan & Yang , 2009 ) that leverages prior information acquired from large datasets to help in training models on a target dataset under limited data and resource constraints . There has been extensive exploration of transfer learning in classification problems that have shown excellent performance on various downstream data-deficient domains . Similar extensions of reusing pre-trained networks for transfer learning ( i.e . fine-tuning a subset of pre-trained network weights from a data-rich domain ) have also been recently employed for image synthesis in GANs ( Wang et al. , 2018 ; Noguchi & Harada , 2019 ; Mo et al. , 2020 ; Wang et al. , 2020 ; Zhao et al. , 2020a ) in the limited data regime . However , these approaches are still prone to overfitting on the sparse target data , and hence suffer from degraded image quality and diversity . In this work , we propose a simple yet effective way of transferring prior knowledge in unsupervised image generation given a small sample size ( ∼ 100-2000 ) of the target data distribution . Our approach is motivated by the formulation of the IMLE technique ( Li & Malik , 2018 ) that seeks to obtain mode coverage of target data distribution by learning a mapping between latent and target distributions using a maximum likelihood criterion . We instead propose the use of data priors in GANs to match the representation of the generated samples to real modes of data . In contrast to ( Li & Malik , 2018 ) , we use the images generated using data priors to find the nearest neighbor match to real modes in the generator ’ s learned distribution . In particular , we show that using an informative data instance prior in limited and large-scale unsupervised image generation substantially improves the performance in image synthesis . We show that these data priors can be derived from commonly used computer vision pre-trained networks ( Simonyan & Zisserman , 2014 ; Zhang et al. , 2018 ; Noguchi & Harada , 2019 ; Hoshen et al. , 2019 ) or self-supervised data representations ( Chen et al. , 2020 ) ( without any violation of the target setting ’ s requirements , i.e . ensuring that the pre-trained network has not been trained on few-shot classes in the few-shot learning setting , for instance ) . In case of sparse training data , our approach of using data instance priors leverages a model pre-trained on a rich source domain to the learn the target distribution . Different from previous works ( Noguchi & Harada , 2019 ; Wang et al. , 2020 ; 2018 ) which rely on fine-tuning models trained on a data-rich domain , we propose to leverage the feature representations of our source model as data instance priors , to distill knowledge ( Romero et al. , 2015 ; Hinton et al. , 2015 ) into our target generative model . We note that our technique of using data instance priors for transfer learning becomes fully unsupervised in case the data priors are extracted from self-supervised pre-trained networks . Furthermore , in addition to image generation in low data domain , we also achieve state-of-the-art Fréchet inception distance ( FID ) score ( Heusel et al. , 2017 ) on large-scale unsupervised image generation and also show how this framework of transfer learning supports several image editing tasks . We summarize our main contributions as follows : • We propose Data Instance Prior ( DIP ) , a novel transfer learning technique for GAN image synthesis in low-data regime . We show that employing DIP in conjunction with existing few-shot image generation methods outperforms state-of-the-art results . We show with as little as 100 images our approach DIP results in generation of diverse and high quality images ( see Figure 3 ) . • We demonstrate the utility of our approach in large-scale unsupervised GANs ( Miyato et al. , 2018 ; Brock et al. , 2018 ) achieving the new state-of-the-art in terms of image quality ( Heusel et al. , 2017 ) and diversity ( Sajjadi et al. , 2018 ; Metz et al. , 2017 ) . • We show how our framework of DIP by construction enables inversion of images and common image editing tasks ( such as cutmix , in-painting , image translation ) in GANs . We call our method a data instance prior ( and not just data prior ) , since it uses representations of instances as a prior , and not a data distribution itself . 2 RELATED WORK . Deep Generative Models In recent years , there has been a surge in the research of deep generative models . Some of the popular approaches include variational auto-encoders ( VAEs ) ( Rezende et al. , 2014 ; Kingma & Welling , 2014 ) , auto-regressive ( AR ) models ( Van Oord et al. , 2016 ; Van den Oord et al. , 2016 ) and GANs ( Goodfellow et al. , 2014 ) . VAE models learn by maximizing the variational lower bound of likelihood of generating data from a given distribution . Auto-regressive approaches model the data distribution as a product of the conditional probabilities to sequentially generate data . GANs comprise of two networks , a generator and a discriminator that train in a min-max optimization . Specifically , the generator aims to generate samples to fool the discriminator , while the discriminator learns distinguish these generated samples from the real samples . Several research efforts in GANs have focused around improving the performance ( Karras et al. , 2018 ; Denton et al. , 2015 ; Radford et al. , 2016 ; Karras et al. , 2020b ; 2019 ; Brock et al. , 2018 ; Zhang et al. , 2019 ) and training stability ( Salimans et al. , 2016b ; Gulrajani et al. , 2017 ; Arjovsky et al. , 2017 ; Miyato et al. , 2018 ; Mao et al. , 2017 ; Chen et al. , 2019 ) . Recently , the areas of latent space manipulation for semantic editing ( Shen et al. , 2020 ; Jahanian et al. , 2020 ; Zhu et al. , 2020 ; Plumerault et al. , 2020 ) and few-shot image generation ( Wang et al. , 2020 ; Mo et al. , 2020 ; Noguchi & Harada , 2019 ) have gained traction in an effort to mitigate the practical challenges while deploying GANs . Several other non-adversarial training approaches such as ( Hoshen et al. , 2019 ; Bojanowski et al. , 2018 ; Li & Malik , 2018 ) have also been explored for generative modeling , which leverage supervised learning along with perceptual loss ( Zhang et al. , 2018 ) for training such models . Transfer Learning in GANs While there has been extensive research in the area of transfer learning for classification models ( Yosinski et al. , 2014 ; Oquab et al. , 2014 ; Tzeng et al. , 2015 ; Pan & Yang , 2009 ; Donahue et al. , 2014 ) , relatively fewer efforts have explored this on the task of data generation ( Wang et al. , 2018 ; 2020 ; Noguchi & Harada , 2019 ; Zhao et al. , 2020a ; Mo et al. , 2020 ) . ( Wang et al. , 2018 ) proposed to fine-tune a pre-trained GAN model ( often having millions of parameters ) from a data-rich source to adapt to the target domain with limited samples . This approach , however , often suffers from overfitting as the final model parameters are updated using only few samples of the target domain . To counter overfitting , the work of ( Noguchi & Harada , 2019 ) proposes to update only the batch normalization parameters of the pre-trained GAN model . In this approach , however , the generator is not adversarially trained and uses supervised L1 pixel distance loss and perceptual loss ( Johnson et al. , 2016 ; Zhang et al. , 2018 ) which often leads to generation of blurry images in the target domain . Based on the assumption that source and target domain support sets are similar , ( Wang et al. , 2020 ) recently proposed to learn an additional mapping network that transforms the latent code suitable for generating images of target domain while keeping the other parameters frozen . We compare against all leading baselines including ( Noguchi & Harada , 2019 ; Wang et al. , 2020 ) on their respective tasks , and show that our method DIP outperforms them substantially , while being simple to implement . A related line of recent research aims to improve large-scale unsupervised image generation in GANs by employing self-supervision - in particular , an auxiliary task of rotation prediction ( Chen et al. , 2019 ) or using one-hot labels obtained by clustering in the discriminator ’ s ( Liu et al. , 2020 ) or ImageNet classifier feature space ( Sage et al. , 2018 ) . In contrast , our method utilizes data instance priors derived from the feature activations of self-supervised/supervised pre-trained networks to improve unsupervised few-shot and large-scale image generation , leading to simpler formulation and higher performance as shown in our experiments in Section 5.3 and Appendix A . Recently , some methods ( Karras et al. , 2020a ; Zhao et al. , 2020b ; Zhang et al. , 2019 ; Zhao et al. , 2020c ) have leveraged data augmentation to effectively increase the number of samples and prevent overfitting in GAN training . However , data augmentation techniques often times alter the true data distribution and there is a leakage of these augmentations to the generated image , as shown in ( Zhao et al. , 2020c ; b ) . To overcome this , ( Zhao et al. , 2020b ) recently proposed to use differential augmentation and ( Karras et al. , 2020a ) leveraged an adaptive discriminator augmentation mechanism . We instead focus on leveraging informative data instance priors , and in fact , show how our DIP method can be used in conjunction with ( Zhao et al. , 2020b ) to improve performance .	This submission deals with transfer learning for training GANs with limited label data. The challenge is that training with limited data can result in mode collapse. This submission proposes to use data priors for each instance of the target distribution, transformed through knowledge from a source domain, as conditional information in GAN to ensure mode coverage of the target data distribution. A pre-trained feature extractor is used to provide the information to condition the GAN. A range of experiments is performed with the features extracted from VGG16, SIMCLR. They show consistent improvements in the image quality and diversity, measured via FID and precision-recall, for few-shot, limited data, and even large scale data settings.	SP:f5ac44287ac769114d4b4d8dce60c61bfc43ef69
Network Architecture Search for Domain Adaptation	1 INTRODUCTION . Supervised machine learning models ( Φ ) aim to minimize the empirical test error ( ( Φ ( x ) , y ) ) by optimizing Φ on training data ( x ) and ground truth labels ( y ) , assuming that the training and testing data are sampled i.i.d from the same distribution . While in practical , the training and testing data are typically collected from related domains under different distributions , a phenomenon known as domain shift ( or domain discrepancy ) ( Quionero-Candela et al. , 2009 ) . To avoid the cost of annotating each new test data , Unsupervised Domain Adaptation ( UDA ) tackles domain shift by transferring the knowledge learned from a rich-labeled source domain ( P ( xs , ys ) ) to the unlabeled target domain ( Q ( xt ) ) . Recently unsupervised domain adaptation research has achieved significant progress with techniques like discrepancy alignment ( Long et al. , 2017 ; Tzeng et al. , 2014 ; Ghifary et al. , 2014 ; Peng & Saenko , 2018 ; Long et al. , 2015 ; Sun & Saenko , 2016 ) , adversarial alignment ( Xu et al. , 2019a ; Liu & Tuzel , 2016 ; Tzeng et al. , 2017 ; Liu et al. , 2018a ; Ganin & Lempitsky , 2015 ; Saito et al. , 2018 ; Long et al. , 2018 ) , and reconstruction-based alignment ( Yi et al. , 2017 ; Zhu et al. , 2017 ; Hoffman et al. , 2018 ; Kim et al. , 2017 ) . While such models typically learn feature mapping from one domain ( Φ ( xs ) ) to another ( Φ ( xt ) ) or derive a joint representation across domains ( Φ ( xs ) ⊗ Φ ( xt ) ) , the developed models have limited capacities in deriving an optimal neural architecture specific for domain transfer . To advance network designs , neural architecture search ( NAS ) automates the net architecture engineering process by reinforcement supervision ( Zoph & Le , 2017 ) or through neuro-evlolution ( Real et al. , 2019a ) . Conventional NAS models aim to derive neural architecture α along with the network parameters w , by solving a bilevel optimization problem ( Anandalingam & Friesz , 1992 ) : Φα , w = arg minα Lval ( w∗ ( α ) , α ) s.t . w∗ ( α ) = argminwLtrain ( w , α ) , where Ltrain and Lval indicate the training and validation loss , respectively . While recent works demonstrate competitive performance on tasks such as image classification ( Zoph et al. , 2018 ; Liu et al. , 2018c ; b ; Real et al. , 2019b ) and object detection ( Zoph & Le , 2017 ) , designs of existing NAS algorithms typically assume that the training and testing domain are sampled from the same distribution , neglecting the scenario where two data domains or multiple feature distributions are of interest . To efficiently devise a neural architecture across different data domains , we propose a novel learning task called Neural Architecture Search for Domain Adaptation ( NASDA ) . The ultimate goal of NASDA is to minimize the validation loss of the target domain ( Ltval ) . We postulate that a solution to NASDA should not only minimize validation loss of the source domain ( Lsval ) , but should also reduce the domain gap between the source and target . To this end , we propose a new NAS learning schema : Φα , w = argminαLsval ( w∗ ( α ) , α ) + disc ( Φ∗ ( xs ) , Φ∗ ( xt ) ) ( 1 ) s.t . w∗ ( α ) = argminw Lstrain ( w , α ) ( 2 ) where Φ∗ = Φα , w∗ ( α ) , and disc ( Φ∗ ( xs ) , Φ∗ ( xt ) ) denotes the domain discrepancy between the source and target . Note that in unsupervised domain adaptation , Lttrain and Ltval can not be computed directly due to the lack of label in the target domain . Inspired by the past works in NAS and unsupervised domain adaptation , we propose in this paper an instantiated NASDA model , which comprises of two training phases , as shown in Figure 1 . The first is the neural architecture searching phase , aiming to derive an optimal neural architecture ( α∗ ) , following the learning schema of Equation 1,2 . Inspired by Differentiable ARchiTecture Search ( DARTS ) ( Liu et al. , 2019a ) , we relax the search space to be continuous so that α can be optimized with respect to Lsval and disc ( Φ ( xs ) , Φ ( xt ) ) by gradient descent . Specifically , we enhance the feature transferability by embedding the hidden representations of the task-specific layers to a reproducing kernel Hilbert space where the mean embeddings can be explicitly matched by minimizing disc ( Φ ( xs ) , Φ ( xt ) ) . We use multi-kernel Maximum Mean Discrepancy ( MK-MMD ) ( Gretton et al. , 2007 ) to evaluate the domain discrepancy . The second training phase aims to learn a good feature generator with task-specific loss , based on the derived α∗ from the first phase . To establish this goal , we use the derived deep neural network ( Φα∗ ) as the feature generator ( G ) and devise an adversarial training process between G and a batch of classifiers C. The high-level intuition is to first diversify C in the training process , and train G to generate features such that the diversified C can have similar outputs . The training process is similar to Maximum Classifier Discrepancy framework ( MCD ) ( Saito et al. , 2018 ) except that we extend the dual-classifier in MCD to an ensembling of multiple classifiers . Experiments on standard UDA benchmarks demonstrate the effectiveness of our derived NASDA model in achieving significant improvements over state-of-the-art methods . Our contributions of this paper are highlighted as follows : • We formulate a novel dual-objective task of Neural Architecture Search for Domain Adaptation ( NASDA ) , which optimize neural architecture for unsupervised domain adaptation , concerning both source performance objective and transfer learning objective . • We propose an instantiated NASDA model that comprises two training stages , aiming to derive optimal architecture parameters α∗ and feature extractor G , respectively . We are the first to show the effectiveness of MK-MMD in NAS process specified for domain adaptation . • Extensive experiments on multiple cross-domain recognition tasks demonstrate that NASDA achieves significant improvements over traditional unsupervised domain adaptation models as well as state-of-the-art NAS-based methods . 2 RELATED WORK . Deep convolutional neural network has been dominating image recognition task . In recent years , many handcrafted architectures have been proposed , including VGG ( Simonyan & Zisserman , 2014 ) , ResNet ( He et al. , 2016 ) , Inception ( Szegedy et al. , 2015 ) , etc. , all of which verifies the importance of human expertise in network design . Our work bridges domain adaptation and the emerging field of neural architecture search ( NAS ) , a process of automating architecture engineering technique . Neural Architecture Search Neural Architecture Search has become the mainstream approach to discover efficient and powerful network structures ( Zoph & Le , 2017 ; Zoph et al. , 2018 ) . The automatically searched architectures have achieved highly competitive performance in tasks such as image classification ( Liu et al. , 2018c ; b ) , object detection ( Zoph et al. , 2018 ) , and semantic segmentation ( Chen et al. , 2018 ) . Reinforce learning based NAS methods ( Zoph & Le , 2017 ; Tan et al. , 2019 ; Tan & Le , 2019 ) are usually computational intensive , thus hampering its usage with limited computational budget . To accelerate the search procedure , many techniques has been proposed and they mainly follow four directions : ( 1 ) estimating the actual performance with lower fidelities . Such lower fidelities include shorter training times ( Zoph et al. , 2018 ; Zela et al. , 2018 ) , training on a subset of the data ( Klein et al. , 2017 ) , or on lower-resolution images . ( 2 ) estimating the performance based on the learning curve extrapolation . Domhan et al . ( 2015 ) propose to extrapolate initial learning curves and terminate those predicted to perform poorly . ( 3 ) initializing the novel architectures based on other well-trained architectures . Wei et al . ( 2016 ) introduce network morphisms to modify an architecture without changing the network objects , resulting in methods that only require a few GPU days ( Elsken et al. , 2017 ; Cai et al. , 2018a ; Jin et al. , 2019 ; Cai et al. , 2018b ) . ( 4 ) one-shot architecture search . One-shot NAS treats all architectures as different subgraphs of a supergraph and shares weights between architectures that have edges of this supergraph in common ( Saxena & Verbeek , 2016 ; Liu et al. , 2019b ; Bender , 2018 ) . DARTS ( Liu et al. , 2019a ) places a mixture of candidate operations on each edge of the one-shot model and optimizes the weights of the candidate operations with a continuous relaxation of the search space . Inspired by DARTS ( Liu et al. , 2019a ) , our model employs differentiable architecture search to derive the optimal feature extractor for unsupervised domain adaptation . Domain Adaptation Unsupervised domain adaptation ( UDA ) aims to transfer the knowledge learned from one or more labeled source domains to an unlabeled target domain . Various methods have been proposed , including discrepancy-based UDA approaches ( Long et al. , 2017 ; Tzeng et al. , 2014 ; Ghifary et al. , 2014 ; Peng & Saenko , 2018 ) , adversary-based approaches ( Liu & Tuzel , 2016 ; Tzeng et al. , 2017 ; Liu et al. , 2018a ) , and reconstruction-based approaches ( Yi et al. , 2017 ; Zhu et al. , 2017 ; Hoffman et al. , 2018 ; Kim et al. , 2017 ) . These models are typically designed to tackle single source to single target adaptation . Compared with single source adaptation , multi-source domain adaptation ( MSDA ) assumes that training data are collected from multiple sources . Originating from the theoretical analysis in ( Ben-David et al. , 2010 ; Mansour et al. , 2009 ; Crammer et al. , 2008 ) , MSDA has been applied to many practical applications ( Xu et al. , 2018 ; Duan et al. , 2012 ; Peng et al. , 2019 ) . Specifically , Ben-David et al . ( 2010 ) introduce an H∆H-divergence between the weighted combination of source domains and a target domain . These models are developed using the existing hand-crafted network architecture . This property limits the capacity and versatility of domain adaptation as the backbones to extract the features are fixed . In contrast , we tackle the UDA from a different perspective , not yet considered in the UDA literature . We propose a novel dual-objective model of NASDA , which optimize neural architecture for unsupervised domain adaptation . We are the first to show the effectiveness of MK-MMD in NAS process which is designed specifically for domain adaptation . 3 NEURAL ARCHITECTURE SEARCH FOR DOMAIN ADAPTATION . In unsupervised domain adaptation , we are given a source domain Ds = { ( xsi , ysi ) } ns i=1 of ns labeled examples and a target domainDt = { xtj } nt j=1 of nt unlabeled examples . The source domain and target domain are sampled from joint distributions P ( xs , ys ) and Q ( xt , yt ) , respectively . The goal of this paper is to leverage NAS to derive a deep networkG : x 7→ y , which is optimal for reducing the shifts in data distributions across domains , such that the target risk t ( G ) = E ( xt , yt ) ∼Q [ G ( xt ) 6= yt ] is minimized . We will start by introducing some preliminary background in Section 3.1 . We then describe how to incorporate the MK-MMD into the neural architecture searching framework in Section 3.2 . Finally , we introduce the adversarial training between our derived deep network and a batch of classifiers in Section 3.3 . An overview of our model can be seen in Algorithm 1 .	This paper introduces an approach to search for the best network architecture for a domain adaptation task. This is achieved by following a differentiable architecture search strategy in which an additional loss function is included to account for the domain shift. Specifically, the loss function aims to minimize the discrepancy between feature representations from the two domains.	SP:81a8951f6c1d60ee080b72d0e5c5c33425002ee5
Network Architecture Search for Domain Adaptation	1 INTRODUCTION . Supervised machine learning models ( Φ ) aim to minimize the empirical test error ( ( Φ ( x ) , y ) ) by optimizing Φ on training data ( x ) and ground truth labels ( y ) , assuming that the training and testing data are sampled i.i.d from the same distribution . While in practical , the training and testing data are typically collected from related domains under different distributions , a phenomenon known as domain shift ( or domain discrepancy ) ( Quionero-Candela et al. , 2009 ) . To avoid the cost of annotating each new test data , Unsupervised Domain Adaptation ( UDA ) tackles domain shift by transferring the knowledge learned from a rich-labeled source domain ( P ( xs , ys ) ) to the unlabeled target domain ( Q ( xt ) ) . Recently unsupervised domain adaptation research has achieved significant progress with techniques like discrepancy alignment ( Long et al. , 2017 ; Tzeng et al. , 2014 ; Ghifary et al. , 2014 ; Peng & Saenko , 2018 ; Long et al. , 2015 ; Sun & Saenko , 2016 ) , adversarial alignment ( Xu et al. , 2019a ; Liu & Tuzel , 2016 ; Tzeng et al. , 2017 ; Liu et al. , 2018a ; Ganin & Lempitsky , 2015 ; Saito et al. , 2018 ; Long et al. , 2018 ) , and reconstruction-based alignment ( Yi et al. , 2017 ; Zhu et al. , 2017 ; Hoffman et al. , 2018 ; Kim et al. , 2017 ) . While such models typically learn feature mapping from one domain ( Φ ( xs ) ) to another ( Φ ( xt ) ) or derive a joint representation across domains ( Φ ( xs ) ⊗ Φ ( xt ) ) , the developed models have limited capacities in deriving an optimal neural architecture specific for domain transfer . To advance network designs , neural architecture search ( NAS ) automates the net architecture engineering process by reinforcement supervision ( Zoph & Le , 2017 ) or through neuro-evlolution ( Real et al. , 2019a ) . Conventional NAS models aim to derive neural architecture α along with the network parameters w , by solving a bilevel optimization problem ( Anandalingam & Friesz , 1992 ) : Φα , w = arg minα Lval ( w∗ ( α ) , α ) s.t . w∗ ( α ) = argminwLtrain ( w , α ) , where Ltrain and Lval indicate the training and validation loss , respectively . While recent works demonstrate competitive performance on tasks such as image classification ( Zoph et al. , 2018 ; Liu et al. , 2018c ; b ; Real et al. , 2019b ) and object detection ( Zoph & Le , 2017 ) , designs of existing NAS algorithms typically assume that the training and testing domain are sampled from the same distribution , neglecting the scenario where two data domains or multiple feature distributions are of interest . To efficiently devise a neural architecture across different data domains , we propose a novel learning task called Neural Architecture Search for Domain Adaptation ( NASDA ) . The ultimate goal of NASDA is to minimize the validation loss of the target domain ( Ltval ) . We postulate that a solution to NASDA should not only minimize validation loss of the source domain ( Lsval ) , but should also reduce the domain gap between the source and target . To this end , we propose a new NAS learning schema : Φα , w = argminαLsval ( w∗ ( α ) , α ) + disc ( Φ∗ ( xs ) , Φ∗ ( xt ) ) ( 1 ) s.t . w∗ ( α ) = argminw Lstrain ( w , α ) ( 2 ) where Φ∗ = Φα , w∗ ( α ) , and disc ( Φ∗ ( xs ) , Φ∗ ( xt ) ) denotes the domain discrepancy between the source and target . Note that in unsupervised domain adaptation , Lttrain and Ltval can not be computed directly due to the lack of label in the target domain . Inspired by the past works in NAS and unsupervised domain adaptation , we propose in this paper an instantiated NASDA model , which comprises of two training phases , as shown in Figure 1 . The first is the neural architecture searching phase , aiming to derive an optimal neural architecture ( α∗ ) , following the learning schema of Equation 1,2 . Inspired by Differentiable ARchiTecture Search ( DARTS ) ( Liu et al. , 2019a ) , we relax the search space to be continuous so that α can be optimized with respect to Lsval and disc ( Φ ( xs ) , Φ ( xt ) ) by gradient descent . Specifically , we enhance the feature transferability by embedding the hidden representations of the task-specific layers to a reproducing kernel Hilbert space where the mean embeddings can be explicitly matched by minimizing disc ( Φ ( xs ) , Φ ( xt ) ) . We use multi-kernel Maximum Mean Discrepancy ( MK-MMD ) ( Gretton et al. , 2007 ) to evaluate the domain discrepancy . The second training phase aims to learn a good feature generator with task-specific loss , based on the derived α∗ from the first phase . To establish this goal , we use the derived deep neural network ( Φα∗ ) as the feature generator ( G ) and devise an adversarial training process between G and a batch of classifiers C. The high-level intuition is to first diversify C in the training process , and train G to generate features such that the diversified C can have similar outputs . The training process is similar to Maximum Classifier Discrepancy framework ( MCD ) ( Saito et al. , 2018 ) except that we extend the dual-classifier in MCD to an ensembling of multiple classifiers . Experiments on standard UDA benchmarks demonstrate the effectiveness of our derived NASDA model in achieving significant improvements over state-of-the-art methods . Our contributions of this paper are highlighted as follows : • We formulate a novel dual-objective task of Neural Architecture Search for Domain Adaptation ( NASDA ) , which optimize neural architecture for unsupervised domain adaptation , concerning both source performance objective and transfer learning objective . • We propose an instantiated NASDA model that comprises two training stages , aiming to derive optimal architecture parameters α∗ and feature extractor G , respectively . We are the first to show the effectiveness of MK-MMD in NAS process specified for domain adaptation . • Extensive experiments on multiple cross-domain recognition tasks demonstrate that NASDA achieves significant improvements over traditional unsupervised domain adaptation models as well as state-of-the-art NAS-based methods . 2 RELATED WORK . Deep convolutional neural network has been dominating image recognition task . In recent years , many handcrafted architectures have been proposed , including VGG ( Simonyan & Zisserman , 2014 ) , ResNet ( He et al. , 2016 ) , Inception ( Szegedy et al. , 2015 ) , etc. , all of which verifies the importance of human expertise in network design . Our work bridges domain adaptation and the emerging field of neural architecture search ( NAS ) , a process of automating architecture engineering technique . Neural Architecture Search Neural Architecture Search has become the mainstream approach to discover efficient and powerful network structures ( Zoph & Le , 2017 ; Zoph et al. , 2018 ) . The automatically searched architectures have achieved highly competitive performance in tasks such as image classification ( Liu et al. , 2018c ; b ) , object detection ( Zoph et al. , 2018 ) , and semantic segmentation ( Chen et al. , 2018 ) . Reinforce learning based NAS methods ( Zoph & Le , 2017 ; Tan et al. , 2019 ; Tan & Le , 2019 ) are usually computational intensive , thus hampering its usage with limited computational budget . To accelerate the search procedure , many techniques has been proposed and they mainly follow four directions : ( 1 ) estimating the actual performance with lower fidelities . Such lower fidelities include shorter training times ( Zoph et al. , 2018 ; Zela et al. , 2018 ) , training on a subset of the data ( Klein et al. , 2017 ) , or on lower-resolution images . ( 2 ) estimating the performance based on the learning curve extrapolation . Domhan et al . ( 2015 ) propose to extrapolate initial learning curves and terminate those predicted to perform poorly . ( 3 ) initializing the novel architectures based on other well-trained architectures . Wei et al . ( 2016 ) introduce network morphisms to modify an architecture without changing the network objects , resulting in methods that only require a few GPU days ( Elsken et al. , 2017 ; Cai et al. , 2018a ; Jin et al. , 2019 ; Cai et al. , 2018b ) . ( 4 ) one-shot architecture search . One-shot NAS treats all architectures as different subgraphs of a supergraph and shares weights between architectures that have edges of this supergraph in common ( Saxena & Verbeek , 2016 ; Liu et al. , 2019b ; Bender , 2018 ) . DARTS ( Liu et al. , 2019a ) places a mixture of candidate operations on each edge of the one-shot model and optimizes the weights of the candidate operations with a continuous relaxation of the search space . Inspired by DARTS ( Liu et al. , 2019a ) , our model employs differentiable architecture search to derive the optimal feature extractor for unsupervised domain adaptation . Domain Adaptation Unsupervised domain adaptation ( UDA ) aims to transfer the knowledge learned from one or more labeled source domains to an unlabeled target domain . Various methods have been proposed , including discrepancy-based UDA approaches ( Long et al. , 2017 ; Tzeng et al. , 2014 ; Ghifary et al. , 2014 ; Peng & Saenko , 2018 ) , adversary-based approaches ( Liu & Tuzel , 2016 ; Tzeng et al. , 2017 ; Liu et al. , 2018a ) , and reconstruction-based approaches ( Yi et al. , 2017 ; Zhu et al. , 2017 ; Hoffman et al. , 2018 ; Kim et al. , 2017 ) . These models are typically designed to tackle single source to single target adaptation . Compared with single source adaptation , multi-source domain adaptation ( MSDA ) assumes that training data are collected from multiple sources . Originating from the theoretical analysis in ( Ben-David et al. , 2010 ; Mansour et al. , 2009 ; Crammer et al. , 2008 ) , MSDA has been applied to many practical applications ( Xu et al. , 2018 ; Duan et al. , 2012 ; Peng et al. , 2019 ) . Specifically , Ben-David et al . ( 2010 ) introduce an H∆H-divergence between the weighted combination of source domains and a target domain . These models are developed using the existing hand-crafted network architecture . This property limits the capacity and versatility of domain adaptation as the backbones to extract the features are fixed . In contrast , we tackle the UDA from a different perspective , not yet considered in the UDA literature . We propose a novel dual-objective model of NASDA , which optimize neural architecture for unsupervised domain adaptation . We are the first to show the effectiveness of MK-MMD in NAS process which is designed specifically for domain adaptation . 3 NEURAL ARCHITECTURE SEARCH FOR DOMAIN ADAPTATION . In unsupervised domain adaptation , we are given a source domain Ds = { ( xsi , ysi ) } ns i=1 of ns labeled examples and a target domainDt = { xtj } nt j=1 of nt unlabeled examples . The source domain and target domain are sampled from joint distributions P ( xs , ys ) and Q ( xt , yt ) , respectively . The goal of this paper is to leverage NAS to derive a deep networkG : x 7→ y , which is optimal for reducing the shifts in data distributions across domains , such that the target risk t ( G ) = E ( xt , yt ) ∼Q [ G ( xt ) 6= yt ] is minimized . We will start by introducing some preliminary background in Section 3.1 . We then describe how to incorporate the MK-MMD into the neural architecture searching framework in Section 3.2 . Finally , we introduce the adversarial training between our derived deep network and a batch of classifiers in Section 3.3 . An overview of our model can be seen in Algorithm 1 .	In the work, the authors aim at improving the transferability of domain adaptation models from the perspective of neural architecture search. It consists of two phases, in particular, the first phase searches a neural architecture for domain adaptation based on a famous differentiable NAS method named DARTs, and the second phase develops an adversarial training method for domain adaptation by extending MCD to a multiple classifiers version. The empirical study evaluates the performance on some UDA tasks and shows the effectiveness of the introduced method.	SP:81a8951f6c1d60ee080b72d0e5c5c33425002ee5
Network Architecture Search for Domain Adaptation	1 INTRODUCTION . Supervised machine learning models ( Φ ) aim to minimize the empirical test error ( ( Φ ( x ) , y ) ) by optimizing Φ on training data ( x ) and ground truth labels ( y ) , assuming that the training and testing data are sampled i.i.d from the same distribution . While in practical , the training and testing data are typically collected from related domains under different distributions , a phenomenon known as domain shift ( or domain discrepancy ) ( Quionero-Candela et al. , 2009 ) . To avoid the cost of annotating each new test data , Unsupervised Domain Adaptation ( UDA ) tackles domain shift by transferring the knowledge learned from a rich-labeled source domain ( P ( xs , ys ) ) to the unlabeled target domain ( Q ( xt ) ) . Recently unsupervised domain adaptation research has achieved significant progress with techniques like discrepancy alignment ( Long et al. , 2017 ; Tzeng et al. , 2014 ; Ghifary et al. , 2014 ; Peng & Saenko , 2018 ; Long et al. , 2015 ; Sun & Saenko , 2016 ) , adversarial alignment ( Xu et al. , 2019a ; Liu & Tuzel , 2016 ; Tzeng et al. , 2017 ; Liu et al. , 2018a ; Ganin & Lempitsky , 2015 ; Saito et al. , 2018 ; Long et al. , 2018 ) , and reconstruction-based alignment ( Yi et al. , 2017 ; Zhu et al. , 2017 ; Hoffman et al. , 2018 ; Kim et al. , 2017 ) . While such models typically learn feature mapping from one domain ( Φ ( xs ) ) to another ( Φ ( xt ) ) or derive a joint representation across domains ( Φ ( xs ) ⊗ Φ ( xt ) ) , the developed models have limited capacities in deriving an optimal neural architecture specific for domain transfer . To advance network designs , neural architecture search ( NAS ) automates the net architecture engineering process by reinforcement supervision ( Zoph & Le , 2017 ) or through neuro-evlolution ( Real et al. , 2019a ) . Conventional NAS models aim to derive neural architecture α along with the network parameters w , by solving a bilevel optimization problem ( Anandalingam & Friesz , 1992 ) : Φα , w = arg minα Lval ( w∗ ( α ) , α ) s.t . w∗ ( α ) = argminwLtrain ( w , α ) , where Ltrain and Lval indicate the training and validation loss , respectively . While recent works demonstrate competitive performance on tasks such as image classification ( Zoph et al. , 2018 ; Liu et al. , 2018c ; b ; Real et al. , 2019b ) and object detection ( Zoph & Le , 2017 ) , designs of existing NAS algorithms typically assume that the training and testing domain are sampled from the same distribution , neglecting the scenario where two data domains or multiple feature distributions are of interest . To efficiently devise a neural architecture across different data domains , we propose a novel learning task called Neural Architecture Search for Domain Adaptation ( NASDA ) . The ultimate goal of NASDA is to minimize the validation loss of the target domain ( Ltval ) . We postulate that a solution to NASDA should not only minimize validation loss of the source domain ( Lsval ) , but should also reduce the domain gap between the source and target . To this end , we propose a new NAS learning schema : Φα , w = argminαLsval ( w∗ ( α ) , α ) + disc ( Φ∗ ( xs ) , Φ∗ ( xt ) ) ( 1 ) s.t . w∗ ( α ) = argminw Lstrain ( w , α ) ( 2 ) where Φ∗ = Φα , w∗ ( α ) , and disc ( Φ∗ ( xs ) , Φ∗ ( xt ) ) denotes the domain discrepancy between the source and target . Note that in unsupervised domain adaptation , Lttrain and Ltval can not be computed directly due to the lack of label in the target domain . Inspired by the past works in NAS and unsupervised domain adaptation , we propose in this paper an instantiated NASDA model , which comprises of two training phases , as shown in Figure 1 . The first is the neural architecture searching phase , aiming to derive an optimal neural architecture ( α∗ ) , following the learning schema of Equation 1,2 . Inspired by Differentiable ARchiTecture Search ( DARTS ) ( Liu et al. , 2019a ) , we relax the search space to be continuous so that α can be optimized with respect to Lsval and disc ( Φ ( xs ) , Φ ( xt ) ) by gradient descent . Specifically , we enhance the feature transferability by embedding the hidden representations of the task-specific layers to a reproducing kernel Hilbert space where the mean embeddings can be explicitly matched by minimizing disc ( Φ ( xs ) , Φ ( xt ) ) . We use multi-kernel Maximum Mean Discrepancy ( MK-MMD ) ( Gretton et al. , 2007 ) to evaluate the domain discrepancy . The second training phase aims to learn a good feature generator with task-specific loss , based on the derived α∗ from the first phase . To establish this goal , we use the derived deep neural network ( Φα∗ ) as the feature generator ( G ) and devise an adversarial training process between G and a batch of classifiers C. The high-level intuition is to first diversify C in the training process , and train G to generate features such that the diversified C can have similar outputs . The training process is similar to Maximum Classifier Discrepancy framework ( MCD ) ( Saito et al. , 2018 ) except that we extend the dual-classifier in MCD to an ensembling of multiple classifiers . Experiments on standard UDA benchmarks demonstrate the effectiveness of our derived NASDA model in achieving significant improvements over state-of-the-art methods . Our contributions of this paper are highlighted as follows : • We formulate a novel dual-objective task of Neural Architecture Search for Domain Adaptation ( NASDA ) , which optimize neural architecture for unsupervised domain adaptation , concerning both source performance objective and transfer learning objective . • We propose an instantiated NASDA model that comprises two training stages , aiming to derive optimal architecture parameters α∗ and feature extractor G , respectively . We are the first to show the effectiveness of MK-MMD in NAS process specified for domain adaptation . • Extensive experiments on multiple cross-domain recognition tasks demonstrate that NASDA achieves significant improvements over traditional unsupervised domain adaptation models as well as state-of-the-art NAS-based methods . 2 RELATED WORK . Deep convolutional neural network has been dominating image recognition task . In recent years , many handcrafted architectures have been proposed , including VGG ( Simonyan & Zisserman , 2014 ) , ResNet ( He et al. , 2016 ) , Inception ( Szegedy et al. , 2015 ) , etc. , all of which verifies the importance of human expertise in network design . Our work bridges domain adaptation and the emerging field of neural architecture search ( NAS ) , a process of automating architecture engineering technique . Neural Architecture Search Neural Architecture Search has become the mainstream approach to discover efficient and powerful network structures ( Zoph & Le , 2017 ; Zoph et al. , 2018 ) . The automatically searched architectures have achieved highly competitive performance in tasks such as image classification ( Liu et al. , 2018c ; b ) , object detection ( Zoph et al. , 2018 ) , and semantic segmentation ( Chen et al. , 2018 ) . Reinforce learning based NAS methods ( Zoph & Le , 2017 ; Tan et al. , 2019 ; Tan & Le , 2019 ) are usually computational intensive , thus hampering its usage with limited computational budget . To accelerate the search procedure , many techniques has been proposed and they mainly follow four directions : ( 1 ) estimating the actual performance with lower fidelities . Such lower fidelities include shorter training times ( Zoph et al. , 2018 ; Zela et al. , 2018 ) , training on a subset of the data ( Klein et al. , 2017 ) , or on lower-resolution images . ( 2 ) estimating the performance based on the learning curve extrapolation . Domhan et al . ( 2015 ) propose to extrapolate initial learning curves and terminate those predicted to perform poorly . ( 3 ) initializing the novel architectures based on other well-trained architectures . Wei et al . ( 2016 ) introduce network morphisms to modify an architecture without changing the network objects , resulting in methods that only require a few GPU days ( Elsken et al. , 2017 ; Cai et al. , 2018a ; Jin et al. , 2019 ; Cai et al. , 2018b ) . ( 4 ) one-shot architecture search . One-shot NAS treats all architectures as different subgraphs of a supergraph and shares weights between architectures that have edges of this supergraph in common ( Saxena & Verbeek , 2016 ; Liu et al. , 2019b ; Bender , 2018 ) . DARTS ( Liu et al. , 2019a ) places a mixture of candidate operations on each edge of the one-shot model and optimizes the weights of the candidate operations with a continuous relaxation of the search space . Inspired by DARTS ( Liu et al. , 2019a ) , our model employs differentiable architecture search to derive the optimal feature extractor for unsupervised domain adaptation . Domain Adaptation Unsupervised domain adaptation ( UDA ) aims to transfer the knowledge learned from one or more labeled source domains to an unlabeled target domain . Various methods have been proposed , including discrepancy-based UDA approaches ( Long et al. , 2017 ; Tzeng et al. , 2014 ; Ghifary et al. , 2014 ; Peng & Saenko , 2018 ) , adversary-based approaches ( Liu & Tuzel , 2016 ; Tzeng et al. , 2017 ; Liu et al. , 2018a ) , and reconstruction-based approaches ( Yi et al. , 2017 ; Zhu et al. , 2017 ; Hoffman et al. , 2018 ; Kim et al. , 2017 ) . These models are typically designed to tackle single source to single target adaptation . Compared with single source adaptation , multi-source domain adaptation ( MSDA ) assumes that training data are collected from multiple sources . Originating from the theoretical analysis in ( Ben-David et al. , 2010 ; Mansour et al. , 2009 ; Crammer et al. , 2008 ) , MSDA has been applied to many practical applications ( Xu et al. , 2018 ; Duan et al. , 2012 ; Peng et al. , 2019 ) . Specifically , Ben-David et al . ( 2010 ) introduce an H∆H-divergence between the weighted combination of source domains and a target domain . These models are developed using the existing hand-crafted network architecture . This property limits the capacity and versatility of domain adaptation as the backbones to extract the features are fixed . In contrast , we tackle the UDA from a different perspective , not yet considered in the UDA literature . We propose a novel dual-objective model of NASDA , which optimize neural architecture for unsupervised domain adaptation . We are the first to show the effectiveness of MK-MMD in NAS process which is designed specifically for domain adaptation . 3 NEURAL ARCHITECTURE SEARCH FOR DOMAIN ADAPTATION . In unsupervised domain adaptation , we are given a source domain Ds = { ( xsi , ysi ) } ns i=1 of ns labeled examples and a target domainDt = { xtj } nt j=1 of nt unlabeled examples . The source domain and target domain are sampled from joint distributions P ( xs , ys ) and Q ( xt , yt ) , respectively . The goal of this paper is to leverage NAS to derive a deep networkG : x 7→ y , which is optimal for reducing the shifts in data distributions across domains , such that the target risk t ( G ) = E ( xt , yt ) ∼Q [ G ( xt ) 6= yt ] is minimized . We will start by introducing some preliminary background in Section 3.1 . We then describe how to incorporate the MK-MMD into the neural architecture searching framework in Section 3.2 . Finally , we introduce the adversarial training between our derived deep network and a batch of classifiers in Section 3.3 . An overview of our model can be seen in Algorithm 1 .	This work devises a two step process for searching optimal models for unsupervised domain adaptation. The first step involves a modification of DARTS where a discrepancy term between features for source and target domain is added to the negative-reward. The obtained feature transformer is then re-trained with an adversarial objective in order to ensure that it performs well across multiple classifiers.	SP:81a8951f6c1d60ee080b72d0e5c5c33425002ee5
Learning Neural Event Functions for Ordinary Differential Equations	1 INTRODUCTION . Event handling in the context of solving ordinary differential equations ( Shampine & Thompson , 2000 ) allows the user to specify a termination criteria using an event function . Part of the reason is to introduce discontinuous changes to a system that can not be modeled by an ODE alone . Examples being collision in physical systems , chemical reactions , or switching dynamics ( Ackerson & Fu , 1970 ) . Another part of the motivation is to create discrete outputs from a continuous-time process ; such is the case in point processes and event-driven sampling ( e.g . Steinbrecher & Shaw ( 2008 ) ; Peters et al . ( 2012 ) ; Bouchard-Côté et al . ( 2018 ) ) . In general , an event function is a tool for monitoring a continuous-time system and performing instantaneous interventions when events occur . The use of ordinary differential equation ( ODE ) solvers within deep learning frameworks has allowed end-to-end training of Neural ODEs ( Chen et al. , 2018 ) in a variety of settings . Examples include graphics ( Yang et al. , 2019 ; Rempe et al. , 2020 ; Gupta & Chandraker , 2020 ) , generative modeling ( Grathwohl et al. , 2018 ; Zhang et al. , 2018 ; Chen & Duvenaud , 2019 ; Onken et al. , 2020 ) , time series modeling ( Rubanova et al. , 2019 ; De Brouwer et al. , 2019 ; Jia & Benson , 2019 ; Kidger et al. , 2020 ) , and physics-based models ( Zhong et al. , 2019 ; Greydanus et al. , 2019 ) . However , these existing models are defined with a fixed termination time . To further expand the applications of Neural ODEs , we investigate the parameterization and learning of a termination criteria , such that the termination time is only implicitly defined and will depend on changes in the continuous-time state . For this , we make use of event handling in ODE solvers and derive the gradients necessarily for training event functions that are parameterized with neural networks . By introducing differentiable termination criteria in Neural ODEs , our approach allows the model to efficiently and automatically handle state discontinuities . 1.1 EVENT HANDLING . Suppose we have a continuous-time state z ( t ) that follows an ODE dzdt = f ( t , z ( t ) , θ ) —where θ are parameters of f—with an initial state z ( t0 ) = z0 . The solution at a time value τ can be written as ODESolve ( z0 , f , t0 , τ , θ ) , z ( τ ) = z0 + ∫ τ t0 f ( t , z ( t ) , θ ) dt . ( 1 ) In the context of a Neural ODE , f can be defined using a Lipschitz-continuous neural network . However , since the state z ( t ) is defined through infinitesimal changes , z ( t ) is always continuous in ∗Work done while at Facebook AI Research . t. While smooth trajectories can be a desirable property in some settings , trajectories modeled by an ODE can have limited representation capabilities ( Dupont et al. , 2019 ; Zhang et al. , 2020 ) and in some applications , it is desirable to model discontinuities in the state . Bouncing ball example As a motivating example of a system with discontinuous transitions , consider modeling a bouncing ball with classical mechanics . In an environment with constant gravity , a Markov state for representing the ball is a combination of position x ( t ) ∈ R and velocity v ( t ) ∈ R , z ( t ) = [ x ( t ) , v ( t ) ] , dz ( t ) dt = [ v ( t ) , a ] , ( 2 ) where a is a scalar for acceleration , in our context a gravitational constant . To simulate this system , we need to be mindful that the ball will eventually pass through the ground—when x ( t ) ≤ r for some r that is the radius of the ball—but when it hits the ground , it bounces back up . At the moment of impact , the sign of the velocity is changed instantaneously . However , no such ODE can model this behavior because v ( t ) needs to change discontinuously at the moment of impact . This simple bouncing ball is an example of a scenario that would be ill-suited for a Neural ODE alone to model . In order to model this discontinuity in the state , we can make use of event functions . Event functions allow the ODE to be terminated when a criteria is satisfied , at which point we can instantaneously modify the state and then resume solving the ODE with this new state . Concretely , let g ( t , z ( t ) , φ ) be an event function with φ denoting a set of parameters . An ODE solver with event handling capabilities will terminate at the first occurrence when the event function crosses zero , i.e . time t∗ such that g ( t∗ , z ( t∗ ) , φ ) = 0 , conditioned on some initial value . We express this relationship as t∗ , z ( t∗ ) = ODESolveEvent ( z0 , f , g , t0 , θ , φ ) . ( 3 ) Note that in contrast to eq . ( 1 ) , there is no predetermined termination time . The time of termination t∗ has to be solved alongside the initial value problem as it depends on the trajectory z ( t ) . Nevertheless , ODESolveEvent strictly generalizes ODESolve since the event function can simply encode an explicit termination time and is reduced back into an ODESolve . The benefits of using ODESolveEvent lie in being able to define event functions that depend on the evolving state . Going back to the bouncing ball example , we can simply introduce an event function to detect when the ball hits the ground , i.e . g ( t , z ( t ) , φ ) = x ( t ) − r. We can then instantaneously modify the state so that z′ ( t∗ ) = [ x ( t∗ ) , − ( 1 − α ) v ( t∗ ) ] , where α is the fraction of momentum that is absorbed by the contact , and then resume solving the ODE in eq . ( 2 ) with this new state z′ ( t∗ ) . Figure 1 shows the bouncing ball example being fit by a Neural ODE and a Neural Event ODE where both f and g are neural networks . The Neural ODE model parameterizes a non-linear function for f while the Neural Event ODE parameterizes f and g as linear functions of z ( t ) . We see that the Neural Event ODE can perfectly recover the underlying physics and extrapolate seamlessly . Meanwhile , the Neural ODE has trouble fitting to the sudden changes in dynamics when the ball bounces off the ground , and furthermore , does not generalize because the true model requires the trajectory to be discontinuous . The Neural Event ODE , while being capable of modeling discontinuities in t , is a continuous function of the parameters and hence can be trained with gradient descent . Going forwards , we will discuss how to differentiate t∗ w.r.t . the variables that depend on it , such as z ( t∗ ) , φ and θ . Before this , we briefly summarize how gradients can be computed through any black-box ODE solver . 2 BACKGROUND : DIFFERENTIATING THROUGH ODE SOLUTIONS . Consider a scalar-valued loss function L that depends on the output of an ODE solver , L ( z ( τ ) ) = L ( ODESolve ( z0 , f , t0 , τ , θ ) ) ( 4 ) where f ( t , z , θ ) describes the dynamics . To optimize L , we require the gradients with respect to each of the inputs : z0 , t0 , τ and θ . All of these inputs influence the loss function through the intermediate states z ( t ) , for t ∈ [ t0 , τ ] , and their gradients can be expressed in relation to the adjoint state a ( t ) , dLdz ( t ) which contains the gradient of all intermediate states . The adjoint method ( see e.g . Pontryagin et al. , 1962 ; Le Cun , 1988 ; Giles & Pierce , 2000 ; Chen et al. , 2018 ) provides an identity that quantifies the instantaneous change in the adjoint state : da ( t ) dt = −a ( t ) T ∂f ( t , z ( t ) , θ ) ∂z ( 5 ) which when combined with z ( t ) is an ordinary differential equation that—by solving the adjoint state backwards-in-time , similar to a continuous-time chain rule—allows us to compute vectorJacobian products of the form vT [ ∂z ( τ ) ∂ξ ] , where ξ is any of the inputs z0 , t0 , τ , θ . For instance , with v = dLdz ( τ ) , the product dL dz0 = dLdz ( τ ) T dz ( τ ) dz0 effectively propagates the gradient vector from the final state , dLdz ( τ ) , to the intial state , dL dz0 . The ability to propagate gradients allows ODESolve to be used within reverse-mode automatic differentiation ( Baydin et al. , 2018 ) . We use the method in Chen et al . ( 2018 ) , which solves the adjoint state and parameter gradients jointly backwards-in-time alongside the state z ( t ) . This method does not require intermediate values of z ( t ) to be stored and only invokes ODESolve once for gradient computation . There exist other notable approaches for solving the adjoint equations with different memorycompute trade-offs , such as storing all intermediate quantities ( also known as discrete adjoint ) ( e.g . Zhang & Sandu 2014 ) , more sophisciated methods of checkpointing ( Chen et al. , 2016 ; Gholami et al. , 2019 ; Zhuang et al. , 2020 ) , the use of interpolation schemes ( Hindmarsh et al. , 2005 ; Daulbaev et al. , 2020 ) , and symplectic integration ( Zhuang et al. , 2021 ; Matsubara et al. , 2021 ) . Any of these approaches can be used and is tangential to our contributions . 3 DIFFERENTIATING THROUGH EVENT HANDLING . In an event-terminated ODE solve , the final time value t∗ is not an input argument but a function of the other inputs . As such , for gradient-based optimization , we would need to propagate gradients from t∗ to the input arguments of ODESolveEvent ( eq . ( 3 ) ) . Consider a loss function L that depends on the outputs of ODESolveEvent , L ( t∗ , z ( t∗ ) ) where t∗ , z ( t∗ ) = ODESolveEvent ( z0 , f , g , t0 , θ , φ ) . ( 6 ) Without loss of generality , we can move the parameters φ inside the state z0 and set dφ ( t ) dt = 0 . As long as we can compute gradients w.r.t z0 , these will include gradients w.r.t . φ . This simplifies the event function to g ( t , z ) . Furthermore , we can interpret the event function to be solving an ODE at every evaluation ( as opposed to passing the event function as an input to an ODE solver ) conditioned on the input arguments . This simplifies the event handling procedure to finding the root of groot ( t , z0 , t0 , θ ) , g ( t , z = ODESolve ( z0 , f , t0 , t , θ ) ) ( 7 ) and factorizes the ODESolveEvent procedure into two steps : { t∗ = arg mint t ≥ t0 such that groot ( t , z0 , t0 , θ ) = 0 z ( t∗ ) = ODESolve ( z0 , f , t0 , t∗ , θ ) . ( 8 ) It is obviously computationally infeasible to numerically solve an ODE within the inner loop of a root finding procedure , but this re-interpretation allows us to use existing tools to derive the gradients for ODESolveEvent which can be simplified later to just solving one ODE backwards-in-time . First , the implicit function theorem ( Krantz & Parks , 2012 ) gives us the derivative from t∗ through the root finding procedure . Let ξ denote any of the inputs ( z0 , t0 , θ ) . Then the gradient satisfies dt∗ dξ = − ( dgroot ( t ∗ , z0 , t0 , θ ) dt ) −1 [ ∂g ( t∗ , z ( t∗ ) ) ∂z ∂z ( t∗ ) ∂ξ ] . ( 9 ) Though groot requires solving an ODE , the derivative of z ( t∗ ) w.r.t . t∗ is just f∗ , f ( t∗ , z ( t∗ ) ) , so dgroot ( t ∗ , z0 , t0 , θ ) dt = ∂g ( t∗ , z ( t∗ ) ) ∂t + ∂g ( t∗ , z ( t∗ ) ) ∂z T f∗ . ( 10 ) Taking into account that the loss function may directly depend on both t∗ and z ( t∗ ) , the gradient from the loss function L w.r.t . an input ξ is dL dξ = ( ∂L ∂t∗ + ∂L ∂z ( t∗ ) T f∗ ) ︸︷︷︸ dL dt∗ ( −dgroot ( t ∗ , z0 , t0 , θ ) dt −1 [ ∂g ( t∗ , z ( t∗ ) ) ∂z T ∂z ( t∗ ) ∂ξ ] ) ︸︷︷︸ dt∗ dξ + ∂L ∂z ( t∗ ) T [ ∂z ( t∗ ) ∂ξ ] . ( 11 ) Re-organizing this equation , we can reduce this to dL dξ = vT [ ∂z ( t∗ ) ∂ξ ] ( 12 ) where v = ( ∂L ∂t∗ + ∂L ∂z ( t∗ ) T f∗ ) ( −dgroot ( t ∗ , z0 , t0 , θ ) dt −1 ∂g ( t∗ , z ( t∗ ) ) ∂z ) + ∂L∂z ( t∗ ) . All quantities in v can be computed efficiently since g and t∗ are scalar quantities . As they only require gradients from g , there is no need to differentiate through the ODE simulation to compute v. Finally , the vector-Jacobian product in eq . ( 12 ) can be computed with a single ODESolve . We implemented our method in the torchdiffeq ( Chen , 2018 ) library written in the PyTorch ( Paszke et al. , 2019a ) framework , allowing us to make use of GPU-enabled ODE solvers . We implemented event handling capabilities for all ODE solvers in the library along with gradient computation for event functions . Differentiable event handling generalizes many numerical methods that often have specialized methods for gradient computation , such as ray tracing ( Li et al. , 2018 ) , physics engines ( de Avila BelbutePeres et al. , 2018 ; Hu et al. , 2020 ) , and spiking neural networks ( Wunderlich & Pehle , 2020 ) .	This authors extend neural ODEs to implicitly defined termination criteria modelled by 'neural event functions'. This allows neural ODEs to model abrupt changes in the dynamics (such as collisions or switching dynamics). The authors present how the even handling can be differentiated through, and include a representative set of example studies in the experiments.	SP:aa1d40b52346a894713fb9d4114811a5cff237df
Learning Neural Event Functions for Ordinary Differential Equations	1 INTRODUCTION . Event handling in the context of solving ordinary differential equations ( Shampine & Thompson , 2000 ) allows the user to specify a termination criteria using an event function . Part of the reason is to introduce discontinuous changes to a system that can not be modeled by an ODE alone . Examples being collision in physical systems , chemical reactions , or switching dynamics ( Ackerson & Fu , 1970 ) . Another part of the motivation is to create discrete outputs from a continuous-time process ; such is the case in point processes and event-driven sampling ( e.g . Steinbrecher & Shaw ( 2008 ) ; Peters et al . ( 2012 ) ; Bouchard-Côté et al . ( 2018 ) ) . In general , an event function is a tool for monitoring a continuous-time system and performing instantaneous interventions when events occur . The use of ordinary differential equation ( ODE ) solvers within deep learning frameworks has allowed end-to-end training of Neural ODEs ( Chen et al. , 2018 ) in a variety of settings . Examples include graphics ( Yang et al. , 2019 ; Rempe et al. , 2020 ; Gupta & Chandraker , 2020 ) , generative modeling ( Grathwohl et al. , 2018 ; Zhang et al. , 2018 ; Chen & Duvenaud , 2019 ; Onken et al. , 2020 ) , time series modeling ( Rubanova et al. , 2019 ; De Brouwer et al. , 2019 ; Jia & Benson , 2019 ; Kidger et al. , 2020 ) , and physics-based models ( Zhong et al. , 2019 ; Greydanus et al. , 2019 ) . However , these existing models are defined with a fixed termination time . To further expand the applications of Neural ODEs , we investigate the parameterization and learning of a termination criteria , such that the termination time is only implicitly defined and will depend on changes in the continuous-time state . For this , we make use of event handling in ODE solvers and derive the gradients necessarily for training event functions that are parameterized with neural networks . By introducing differentiable termination criteria in Neural ODEs , our approach allows the model to efficiently and automatically handle state discontinuities . 1.1 EVENT HANDLING . Suppose we have a continuous-time state z ( t ) that follows an ODE dzdt = f ( t , z ( t ) , θ ) —where θ are parameters of f—with an initial state z ( t0 ) = z0 . The solution at a time value τ can be written as ODESolve ( z0 , f , t0 , τ , θ ) , z ( τ ) = z0 + ∫ τ t0 f ( t , z ( t ) , θ ) dt . ( 1 ) In the context of a Neural ODE , f can be defined using a Lipschitz-continuous neural network . However , since the state z ( t ) is defined through infinitesimal changes , z ( t ) is always continuous in ∗Work done while at Facebook AI Research . t. While smooth trajectories can be a desirable property in some settings , trajectories modeled by an ODE can have limited representation capabilities ( Dupont et al. , 2019 ; Zhang et al. , 2020 ) and in some applications , it is desirable to model discontinuities in the state . Bouncing ball example As a motivating example of a system with discontinuous transitions , consider modeling a bouncing ball with classical mechanics . In an environment with constant gravity , a Markov state for representing the ball is a combination of position x ( t ) ∈ R and velocity v ( t ) ∈ R , z ( t ) = [ x ( t ) , v ( t ) ] , dz ( t ) dt = [ v ( t ) , a ] , ( 2 ) where a is a scalar for acceleration , in our context a gravitational constant . To simulate this system , we need to be mindful that the ball will eventually pass through the ground—when x ( t ) ≤ r for some r that is the radius of the ball—but when it hits the ground , it bounces back up . At the moment of impact , the sign of the velocity is changed instantaneously . However , no such ODE can model this behavior because v ( t ) needs to change discontinuously at the moment of impact . This simple bouncing ball is an example of a scenario that would be ill-suited for a Neural ODE alone to model . In order to model this discontinuity in the state , we can make use of event functions . Event functions allow the ODE to be terminated when a criteria is satisfied , at which point we can instantaneously modify the state and then resume solving the ODE with this new state . Concretely , let g ( t , z ( t ) , φ ) be an event function with φ denoting a set of parameters . An ODE solver with event handling capabilities will terminate at the first occurrence when the event function crosses zero , i.e . time t∗ such that g ( t∗ , z ( t∗ ) , φ ) = 0 , conditioned on some initial value . We express this relationship as t∗ , z ( t∗ ) = ODESolveEvent ( z0 , f , g , t0 , θ , φ ) . ( 3 ) Note that in contrast to eq . ( 1 ) , there is no predetermined termination time . The time of termination t∗ has to be solved alongside the initial value problem as it depends on the trajectory z ( t ) . Nevertheless , ODESolveEvent strictly generalizes ODESolve since the event function can simply encode an explicit termination time and is reduced back into an ODESolve . The benefits of using ODESolveEvent lie in being able to define event functions that depend on the evolving state . Going back to the bouncing ball example , we can simply introduce an event function to detect when the ball hits the ground , i.e . g ( t , z ( t ) , φ ) = x ( t ) − r. We can then instantaneously modify the state so that z′ ( t∗ ) = [ x ( t∗ ) , − ( 1 − α ) v ( t∗ ) ] , where α is the fraction of momentum that is absorbed by the contact , and then resume solving the ODE in eq . ( 2 ) with this new state z′ ( t∗ ) . Figure 1 shows the bouncing ball example being fit by a Neural ODE and a Neural Event ODE where both f and g are neural networks . The Neural ODE model parameterizes a non-linear function for f while the Neural Event ODE parameterizes f and g as linear functions of z ( t ) . We see that the Neural Event ODE can perfectly recover the underlying physics and extrapolate seamlessly . Meanwhile , the Neural ODE has trouble fitting to the sudden changes in dynamics when the ball bounces off the ground , and furthermore , does not generalize because the true model requires the trajectory to be discontinuous . The Neural Event ODE , while being capable of modeling discontinuities in t , is a continuous function of the parameters and hence can be trained with gradient descent . Going forwards , we will discuss how to differentiate t∗ w.r.t . the variables that depend on it , such as z ( t∗ ) , φ and θ . Before this , we briefly summarize how gradients can be computed through any black-box ODE solver . 2 BACKGROUND : DIFFERENTIATING THROUGH ODE SOLUTIONS . Consider a scalar-valued loss function L that depends on the output of an ODE solver , L ( z ( τ ) ) = L ( ODESolve ( z0 , f , t0 , τ , θ ) ) ( 4 ) where f ( t , z , θ ) describes the dynamics . To optimize L , we require the gradients with respect to each of the inputs : z0 , t0 , τ and θ . All of these inputs influence the loss function through the intermediate states z ( t ) , for t ∈ [ t0 , τ ] , and their gradients can be expressed in relation to the adjoint state a ( t ) , dLdz ( t ) which contains the gradient of all intermediate states . The adjoint method ( see e.g . Pontryagin et al. , 1962 ; Le Cun , 1988 ; Giles & Pierce , 2000 ; Chen et al. , 2018 ) provides an identity that quantifies the instantaneous change in the adjoint state : da ( t ) dt = −a ( t ) T ∂f ( t , z ( t ) , θ ) ∂z ( 5 ) which when combined with z ( t ) is an ordinary differential equation that—by solving the adjoint state backwards-in-time , similar to a continuous-time chain rule—allows us to compute vectorJacobian products of the form vT [ ∂z ( τ ) ∂ξ ] , where ξ is any of the inputs z0 , t0 , τ , θ . For instance , with v = dLdz ( τ ) , the product dL dz0 = dLdz ( τ ) T dz ( τ ) dz0 effectively propagates the gradient vector from the final state , dLdz ( τ ) , to the intial state , dL dz0 . The ability to propagate gradients allows ODESolve to be used within reverse-mode automatic differentiation ( Baydin et al. , 2018 ) . We use the method in Chen et al . ( 2018 ) , which solves the adjoint state and parameter gradients jointly backwards-in-time alongside the state z ( t ) . This method does not require intermediate values of z ( t ) to be stored and only invokes ODESolve once for gradient computation . There exist other notable approaches for solving the adjoint equations with different memorycompute trade-offs , such as storing all intermediate quantities ( also known as discrete adjoint ) ( e.g . Zhang & Sandu 2014 ) , more sophisciated methods of checkpointing ( Chen et al. , 2016 ; Gholami et al. , 2019 ; Zhuang et al. , 2020 ) , the use of interpolation schemes ( Hindmarsh et al. , 2005 ; Daulbaev et al. , 2020 ) , and symplectic integration ( Zhuang et al. , 2021 ; Matsubara et al. , 2021 ) . Any of these approaches can be used and is tangential to our contributions . 3 DIFFERENTIATING THROUGH EVENT HANDLING . In an event-terminated ODE solve , the final time value t∗ is not an input argument but a function of the other inputs . As such , for gradient-based optimization , we would need to propagate gradients from t∗ to the input arguments of ODESolveEvent ( eq . ( 3 ) ) . Consider a loss function L that depends on the outputs of ODESolveEvent , L ( t∗ , z ( t∗ ) ) where t∗ , z ( t∗ ) = ODESolveEvent ( z0 , f , g , t0 , θ , φ ) . ( 6 ) Without loss of generality , we can move the parameters φ inside the state z0 and set dφ ( t ) dt = 0 . As long as we can compute gradients w.r.t z0 , these will include gradients w.r.t . φ . This simplifies the event function to g ( t , z ) . Furthermore , we can interpret the event function to be solving an ODE at every evaluation ( as opposed to passing the event function as an input to an ODE solver ) conditioned on the input arguments . This simplifies the event handling procedure to finding the root of groot ( t , z0 , t0 , θ ) , g ( t , z = ODESolve ( z0 , f , t0 , t , θ ) ) ( 7 ) and factorizes the ODESolveEvent procedure into two steps : { t∗ = arg mint t ≥ t0 such that groot ( t , z0 , t0 , θ ) = 0 z ( t∗ ) = ODESolve ( z0 , f , t0 , t∗ , θ ) . ( 8 ) It is obviously computationally infeasible to numerically solve an ODE within the inner loop of a root finding procedure , but this re-interpretation allows us to use existing tools to derive the gradients for ODESolveEvent which can be simplified later to just solving one ODE backwards-in-time . First , the implicit function theorem ( Krantz & Parks , 2012 ) gives us the derivative from t∗ through the root finding procedure . Let ξ denote any of the inputs ( z0 , t0 , θ ) . Then the gradient satisfies dt∗ dξ = − ( dgroot ( t ∗ , z0 , t0 , θ ) dt ) −1 [ ∂g ( t∗ , z ( t∗ ) ) ∂z ∂z ( t∗ ) ∂ξ ] . ( 9 ) Though groot requires solving an ODE , the derivative of z ( t∗ ) w.r.t . t∗ is just f∗ , f ( t∗ , z ( t∗ ) ) , so dgroot ( t ∗ , z0 , t0 , θ ) dt = ∂g ( t∗ , z ( t∗ ) ) ∂t + ∂g ( t∗ , z ( t∗ ) ) ∂z T f∗ . ( 10 ) Taking into account that the loss function may directly depend on both t∗ and z ( t∗ ) , the gradient from the loss function L w.r.t . an input ξ is dL dξ = ( ∂L ∂t∗ + ∂L ∂z ( t∗ ) T f∗ ) ︸︷︷︸ dL dt∗ ( −dgroot ( t ∗ , z0 , t0 , θ ) dt −1 [ ∂g ( t∗ , z ( t∗ ) ) ∂z T ∂z ( t∗ ) ∂ξ ] ) ︸︷︷︸ dt∗ dξ + ∂L ∂z ( t∗ ) T [ ∂z ( t∗ ) ∂ξ ] . ( 11 ) Re-organizing this equation , we can reduce this to dL dξ = vT [ ∂z ( t∗ ) ∂ξ ] ( 12 ) where v = ( ∂L ∂t∗ + ∂L ∂z ( t∗ ) T f∗ ) ( −dgroot ( t ∗ , z0 , t0 , θ ) dt −1 ∂g ( t∗ , z ( t∗ ) ) ∂z ) + ∂L∂z ( t∗ ) . All quantities in v can be computed efficiently since g and t∗ are scalar quantities . As they only require gradients from g , there is no need to differentiate through the ODE simulation to compute v. Finally , the vector-Jacobian product in eq . ( 12 ) can be computed with a single ODESolve . We implemented our method in the torchdiffeq ( Chen , 2018 ) library written in the PyTorch ( Paszke et al. , 2019a ) framework , allowing us to make use of GPU-enabled ODE solvers . We implemented event handling capabilities for all ODE solvers in the library along with gradient computation for event functions . Differentiable event handling generalizes many numerical methods that often have specialized methods for gradient computation , such as ray tracing ( Li et al. , 2018 ) , physics engines ( de Avila BelbutePeres et al. , 2018 ; Hu et al. , 2020 ) , and spiking neural networks ( Wunderlich & Pehle , 2020 ) .	This work provides an extension to the neural ODEs framework to include discrete changes (i.e. switching) in continuous-time dynamics. The authors provide a few examples of such systems (bouncing balls, collisions of particles, discrete control systems) and derive formally the gradients with respect to the unknown switching time (which is a solution to the so-called event function), where a discontinuity (the switch) happens. The authors implement their method in the torchdiffeq library of Chen et al. 2018, and provide an extensive experimental evaluation in this manuscript.	SP:aa1d40b52346a894713fb9d4114811a5cff237df
Learning Neural Event Functions for Ordinary Differential Equations	1 INTRODUCTION . Event handling in the context of solving ordinary differential equations ( Shampine & Thompson , 2000 ) allows the user to specify a termination criteria using an event function . Part of the reason is to introduce discontinuous changes to a system that can not be modeled by an ODE alone . Examples being collision in physical systems , chemical reactions , or switching dynamics ( Ackerson & Fu , 1970 ) . Another part of the motivation is to create discrete outputs from a continuous-time process ; such is the case in point processes and event-driven sampling ( e.g . Steinbrecher & Shaw ( 2008 ) ; Peters et al . ( 2012 ) ; Bouchard-Côté et al . ( 2018 ) ) . In general , an event function is a tool for monitoring a continuous-time system and performing instantaneous interventions when events occur . The use of ordinary differential equation ( ODE ) solvers within deep learning frameworks has allowed end-to-end training of Neural ODEs ( Chen et al. , 2018 ) in a variety of settings . Examples include graphics ( Yang et al. , 2019 ; Rempe et al. , 2020 ; Gupta & Chandraker , 2020 ) , generative modeling ( Grathwohl et al. , 2018 ; Zhang et al. , 2018 ; Chen & Duvenaud , 2019 ; Onken et al. , 2020 ) , time series modeling ( Rubanova et al. , 2019 ; De Brouwer et al. , 2019 ; Jia & Benson , 2019 ; Kidger et al. , 2020 ) , and physics-based models ( Zhong et al. , 2019 ; Greydanus et al. , 2019 ) . However , these existing models are defined with a fixed termination time . To further expand the applications of Neural ODEs , we investigate the parameterization and learning of a termination criteria , such that the termination time is only implicitly defined and will depend on changes in the continuous-time state . For this , we make use of event handling in ODE solvers and derive the gradients necessarily for training event functions that are parameterized with neural networks . By introducing differentiable termination criteria in Neural ODEs , our approach allows the model to efficiently and automatically handle state discontinuities . 1.1 EVENT HANDLING . Suppose we have a continuous-time state z ( t ) that follows an ODE dzdt = f ( t , z ( t ) , θ ) —where θ are parameters of f—with an initial state z ( t0 ) = z0 . The solution at a time value τ can be written as ODESolve ( z0 , f , t0 , τ , θ ) , z ( τ ) = z0 + ∫ τ t0 f ( t , z ( t ) , θ ) dt . ( 1 ) In the context of a Neural ODE , f can be defined using a Lipschitz-continuous neural network . However , since the state z ( t ) is defined through infinitesimal changes , z ( t ) is always continuous in ∗Work done while at Facebook AI Research . t. While smooth trajectories can be a desirable property in some settings , trajectories modeled by an ODE can have limited representation capabilities ( Dupont et al. , 2019 ; Zhang et al. , 2020 ) and in some applications , it is desirable to model discontinuities in the state . Bouncing ball example As a motivating example of a system with discontinuous transitions , consider modeling a bouncing ball with classical mechanics . In an environment with constant gravity , a Markov state for representing the ball is a combination of position x ( t ) ∈ R and velocity v ( t ) ∈ R , z ( t ) = [ x ( t ) , v ( t ) ] , dz ( t ) dt = [ v ( t ) , a ] , ( 2 ) where a is a scalar for acceleration , in our context a gravitational constant . To simulate this system , we need to be mindful that the ball will eventually pass through the ground—when x ( t ) ≤ r for some r that is the radius of the ball—but when it hits the ground , it bounces back up . At the moment of impact , the sign of the velocity is changed instantaneously . However , no such ODE can model this behavior because v ( t ) needs to change discontinuously at the moment of impact . This simple bouncing ball is an example of a scenario that would be ill-suited for a Neural ODE alone to model . In order to model this discontinuity in the state , we can make use of event functions . Event functions allow the ODE to be terminated when a criteria is satisfied , at which point we can instantaneously modify the state and then resume solving the ODE with this new state . Concretely , let g ( t , z ( t ) , φ ) be an event function with φ denoting a set of parameters . An ODE solver with event handling capabilities will terminate at the first occurrence when the event function crosses zero , i.e . time t∗ such that g ( t∗ , z ( t∗ ) , φ ) = 0 , conditioned on some initial value . We express this relationship as t∗ , z ( t∗ ) = ODESolveEvent ( z0 , f , g , t0 , θ , φ ) . ( 3 ) Note that in contrast to eq . ( 1 ) , there is no predetermined termination time . The time of termination t∗ has to be solved alongside the initial value problem as it depends on the trajectory z ( t ) . Nevertheless , ODESolveEvent strictly generalizes ODESolve since the event function can simply encode an explicit termination time and is reduced back into an ODESolve . The benefits of using ODESolveEvent lie in being able to define event functions that depend on the evolving state . Going back to the bouncing ball example , we can simply introduce an event function to detect when the ball hits the ground , i.e . g ( t , z ( t ) , φ ) = x ( t ) − r. We can then instantaneously modify the state so that z′ ( t∗ ) = [ x ( t∗ ) , − ( 1 − α ) v ( t∗ ) ] , where α is the fraction of momentum that is absorbed by the contact , and then resume solving the ODE in eq . ( 2 ) with this new state z′ ( t∗ ) . Figure 1 shows the bouncing ball example being fit by a Neural ODE and a Neural Event ODE where both f and g are neural networks . The Neural ODE model parameterizes a non-linear function for f while the Neural Event ODE parameterizes f and g as linear functions of z ( t ) . We see that the Neural Event ODE can perfectly recover the underlying physics and extrapolate seamlessly . Meanwhile , the Neural ODE has trouble fitting to the sudden changes in dynamics when the ball bounces off the ground , and furthermore , does not generalize because the true model requires the trajectory to be discontinuous . The Neural Event ODE , while being capable of modeling discontinuities in t , is a continuous function of the parameters and hence can be trained with gradient descent . Going forwards , we will discuss how to differentiate t∗ w.r.t . the variables that depend on it , such as z ( t∗ ) , φ and θ . Before this , we briefly summarize how gradients can be computed through any black-box ODE solver . 2 BACKGROUND : DIFFERENTIATING THROUGH ODE SOLUTIONS . Consider a scalar-valued loss function L that depends on the output of an ODE solver , L ( z ( τ ) ) = L ( ODESolve ( z0 , f , t0 , τ , θ ) ) ( 4 ) where f ( t , z , θ ) describes the dynamics . To optimize L , we require the gradients with respect to each of the inputs : z0 , t0 , τ and θ . All of these inputs influence the loss function through the intermediate states z ( t ) , for t ∈ [ t0 , τ ] , and their gradients can be expressed in relation to the adjoint state a ( t ) , dLdz ( t ) which contains the gradient of all intermediate states . The adjoint method ( see e.g . Pontryagin et al. , 1962 ; Le Cun , 1988 ; Giles & Pierce , 2000 ; Chen et al. , 2018 ) provides an identity that quantifies the instantaneous change in the adjoint state : da ( t ) dt = −a ( t ) T ∂f ( t , z ( t ) , θ ) ∂z ( 5 ) which when combined with z ( t ) is an ordinary differential equation that—by solving the adjoint state backwards-in-time , similar to a continuous-time chain rule—allows us to compute vectorJacobian products of the form vT [ ∂z ( τ ) ∂ξ ] , where ξ is any of the inputs z0 , t0 , τ , θ . For instance , with v = dLdz ( τ ) , the product dL dz0 = dLdz ( τ ) T dz ( τ ) dz0 effectively propagates the gradient vector from the final state , dLdz ( τ ) , to the intial state , dL dz0 . The ability to propagate gradients allows ODESolve to be used within reverse-mode automatic differentiation ( Baydin et al. , 2018 ) . We use the method in Chen et al . ( 2018 ) , which solves the adjoint state and parameter gradients jointly backwards-in-time alongside the state z ( t ) . This method does not require intermediate values of z ( t ) to be stored and only invokes ODESolve once for gradient computation . There exist other notable approaches for solving the adjoint equations with different memorycompute trade-offs , such as storing all intermediate quantities ( also known as discrete adjoint ) ( e.g . Zhang & Sandu 2014 ) , more sophisciated methods of checkpointing ( Chen et al. , 2016 ; Gholami et al. , 2019 ; Zhuang et al. , 2020 ) , the use of interpolation schemes ( Hindmarsh et al. , 2005 ; Daulbaev et al. , 2020 ) , and symplectic integration ( Zhuang et al. , 2021 ; Matsubara et al. , 2021 ) . Any of these approaches can be used and is tangential to our contributions . 3 DIFFERENTIATING THROUGH EVENT HANDLING . In an event-terminated ODE solve , the final time value t∗ is not an input argument but a function of the other inputs . As such , for gradient-based optimization , we would need to propagate gradients from t∗ to the input arguments of ODESolveEvent ( eq . ( 3 ) ) . Consider a loss function L that depends on the outputs of ODESolveEvent , L ( t∗ , z ( t∗ ) ) where t∗ , z ( t∗ ) = ODESolveEvent ( z0 , f , g , t0 , θ , φ ) . ( 6 ) Without loss of generality , we can move the parameters φ inside the state z0 and set dφ ( t ) dt = 0 . As long as we can compute gradients w.r.t z0 , these will include gradients w.r.t . φ . This simplifies the event function to g ( t , z ) . Furthermore , we can interpret the event function to be solving an ODE at every evaluation ( as opposed to passing the event function as an input to an ODE solver ) conditioned on the input arguments . This simplifies the event handling procedure to finding the root of groot ( t , z0 , t0 , θ ) , g ( t , z = ODESolve ( z0 , f , t0 , t , θ ) ) ( 7 ) and factorizes the ODESolveEvent procedure into two steps : { t∗ = arg mint t ≥ t0 such that groot ( t , z0 , t0 , θ ) = 0 z ( t∗ ) = ODESolve ( z0 , f , t0 , t∗ , θ ) . ( 8 ) It is obviously computationally infeasible to numerically solve an ODE within the inner loop of a root finding procedure , but this re-interpretation allows us to use existing tools to derive the gradients for ODESolveEvent which can be simplified later to just solving one ODE backwards-in-time . First , the implicit function theorem ( Krantz & Parks , 2012 ) gives us the derivative from t∗ through the root finding procedure . Let ξ denote any of the inputs ( z0 , t0 , θ ) . Then the gradient satisfies dt∗ dξ = − ( dgroot ( t ∗ , z0 , t0 , θ ) dt ) −1 [ ∂g ( t∗ , z ( t∗ ) ) ∂z ∂z ( t∗ ) ∂ξ ] . ( 9 ) Though groot requires solving an ODE , the derivative of z ( t∗ ) w.r.t . t∗ is just f∗ , f ( t∗ , z ( t∗ ) ) , so dgroot ( t ∗ , z0 , t0 , θ ) dt = ∂g ( t∗ , z ( t∗ ) ) ∂t + ∂g ( t∗ , z ( t∗ ) ) ∂z T f∗ . ( 10 ) Taking into account that the loss function may directly depend on both t∗ and z ( t∗ ) , the gradient from the loss function L w.r.t . an input ξ is dL dξ = ( ∂L ∂t∗ + ∂L ∂z ( t∗ ) T f∗ ) ︸︷︷︸ dL dt∗ ( −dgroot ( t ∗ , z0 , t0 , θ ) dt −1 [ ∂g ( t∗ , z ( t∗ ) ) ∂z T ∂z ( t∗ ) ∂ξ ] ) ︸︷︷︸ dt∗ dξ + ∂L ∂z ( t∗ ) T [ ∂z ( t∗ ) ∂ξ ] . ( 11 ) Re-organizing this equation , we can reduce this to dL dξ = vT [ ∂z ( t∗ ) ∂ξ ] ( 12 ) where v = ( ∂L ∂t∗ + ∂L ∂z ( t∗ ) T f∗ ) ( −dgroot ( t ∗ , z0 , t0 , θ ) dt −1 ∂g ( t∗ , z ( t∗ ) ) ∂z ) + ∂L∂z ( t∗ ) . All quantities in v can be computed efficiently since g and t∗ are scalar quantities . As they only require gradients from g , there is no need to differentiate through the ODE simulation to compute v. Finally , the vector-Jacobian product in eq . ( 12 ) can be computed with a single ODESolve . We implemented our method in the torchdiffeq ( Chen , 2018 ) library written in the PyTorch ( Paszke et al. , 2019a ) framework , allowing us to make use of GPU-enabled ODE solvers . We implemented event handling capabilities for all ODE solvers in the library along with gradient computation for event functions . Differentiable event handling generalizes many numerical methods that often have specialized methods for gradient computation , such as ray tracing ( Li et al. , 2018 ) , physics engines ( de Avila BelbutePeres et al. , 2018 ; Hu et al. , 2020 ) , and spiking neural networks ( Wunderlich & Pehle , 2020 ) .	This paper presents Neural Event ODEs, a method to extend Neural ODEs for modeling discontinuous dynamics in a continuous-time system. Neural Event ODEs allow to learn termination criteria dependent on the system's state while being fully differentiable. Experiments on time series and temporal point processes validate the benefits of Neural Event ODEs on discountinuous dynamics.	SP:aa1d40b52346a894713fb9d4114811a5cff237df
Encoded Prior Sliced Wasserstein AutoEncoder for learning latent manifold representations	1 INTRODUCTION . Generative models have the potential to capture rich representations of data and use them to generate realistic outputs . In particular , Variational AutoEncoders ( VAEs ) ( Kingma & Welling , 2014 ) can capture important properties of high-dimensional data in their latent embeddings , and sample from a prior distribution to generate realistic images . Whille VAEs have been very successful in a variety of tasks , the use of a simplistic standard normal prior is known to cause problems such as under-fitting and over-regularization , and fails to use the network ’ s entire modeling capacity ( Burda et al. , 2016 ) . Gaussian or Gaussian mixture model ( GMM ) priors are also limited in their ability to represent geometric and topological properties of the underlying data manifold . High-dimensional data can typically be modeled as lying on or near an embedded low-dimensional , nonlinear manifold ( Fefferman et al. , 2016 ) . Learning improved latent representations of this nonlinear manifold is an important problem , for which a more flexible prior may be desirable . Conventional variational inference uses Kullback-Leibler ( KL ) divergence as a measure of distance between the posterior and the prior , restricting the prior distribution to cases that have tractable approximations of the KL divergence . Many works such as Guo et al . ( 2020 ) ; Tomczak & Welling ( 2018 ) ; Rezende & Mohamed ( 2015 ) etc . have investigated the use of more complicated priors ( notably GMMs ) which lead to improved latent representation and generation compared to a single Gaussion prior . Alternate approaches such as adversarial training learn arbitrary priors by using a discriminator network to compute a divergence ( Wang et al. , 2020 ) , however they are reported to be harder to train and are computationally expensive . In this work , we introduce the Encoded Prior Sliced Wasserstein AutoEncoder ( EPSWAE ) , which consists of a conventional autoencoder architecture and an additional prior-encoder network that learns an unconstrained prior distribution that encodes the geometry and topology of any data manifold . We use a type of Sliced Wasserstein ( SW ) distance ( Bonnotte , 2013 ; Bonneel et al. , 2015 ) , a concept from optimal transport theory that is a simple and convenient alternative to the KL divergence for any sampleable distributions . A Sliced Wasserstein AutoEncoder ( SWAE ) that regularizes an autoencoder using SW distance was proposed in Kolouri et al . ( 2018a ) . Several works improve the SW distance through additional optimizations ( Deshpande et al. , 2019 ; Chen et al. , 2020b ; Deshpande et al. , 2018 ) , and show improved generation , however involve additional training and use a fixed ( usually Gaussian ) prior . Kolouri et al . ( 2019 ) presents a comparison between max-SW distance , polynomial generalized SW distances , and their combinations . In contrast , we use a simple and efficient nonlinear shearing which requires no additional optimization . Additionally , we introduce a structural consistency term that encourages the latent space to be isometric to the feature space , which is typically measured at the output of the convolutional layers of the data encoder . Variants of this penalty have previously been used to encourage isometry between the latent space and data space ( Yu et al. , 2013 ; Benaim & Wolf , 2017 ; Sainburg et al. , 2018 ) . The structural consistency term further encourages the prior to match the encoded data manifold by preserving feature-isometry , which in turn is expected to assist with encoding the geometry of the data manifold , thus leading to improved latent representations . A key contribution of our work is the graph-based geodesic-interpolation algorithm . Conventionally , VAEs use Euclidean interpolation between two points in latent space . However , since manifolds typically have curvature , this is an unintuitive distance metric that can lead to unrealistic intermediate points . Our goal is to learn a true representation of the underlying data manifold , hence it is natural to interpolate along the manifold geodesics in latent space . Several works such as Shao et al . ( 2018 ) ; Miolane & Holmes ( 2020b ) endow the latent space with a Riemannian geometry and measure corresponding distances , however these are difficult and involve explicitly solving expensive ordinary differential equations . In this work , we introduce ‘ network-geodesics ’ , a graph-based method for interpolating along a manifold in latent space , that maximizes sample density along paths while minimizing total energy . This involves first generating a distance graph between samples from the prior . Then this network is non-uniformly thresholded such that the set of allowable paths from a given sample traverse high density regions through short hops . Lastly , we use a shortest path algorithm like Dijkstra ’ s algorithm ( Dijkstra , 1959 ) to identify the lowest ‘ energy ’ path between two samples through the allowable paths . Since the prior is trained to learn the data manifold , the resulting network-geodesic curves give a notion of distance on the manifold and can be used to generate realistic interpolation points with relatively few prior samples . The novel contributions of this work are : • We introduce a novel architecture , EPSWAE , that consists of a prior-encoder network that is efficiently trained ( without expensive adversarial methods ) to generate a prior that encodes the geometric and topological properties of data . • We introduce a novel graph-based method for interpolating along network-geodesics in latent space through maximizing sample density while minimizing total energy . We show that it generates natural interpolations through realistic images . • Improvements to the latent space representation are obtained by using a structural consistency term in the loss that encourages isometry between feature space and latent space and by using a simple and efficient nonlinear variant of the SW distance . 2 BACKGROUND AND RELATED WORK . Several works have attempted to increase the complexity of the prior in order to obtain better latent representations . Most data can mathematically be thought of as living on a high dimensional manifold . In an image dataset , for instance , if images in high dimensional pixel space are effectively parametrized using a small number of continuous variables , they will lie on or near a low dimensional manifold ( Lu et al. , 1998 ) . Many works such as Weinberger & Saul ( 2006 ) ; Rahimi et al . ( 2005 ) have investigated image and video manifolds . Encoding and exploiting the topological and geometric properties of such data is a question of increasing interest . Some representative works are Dilokthanakul et al . ( 2016 ) which shows improved unsupervised clustering of latent space by using GMM priors , Takahashi et al . ( 2019 ) which uses the density ratio trick to calculate the KL divergence using implicit priors , Rezende & Mohamed ( 2015 ) , which maps a Gaussian prior by a series of explicit transformations , Guo et al . ( 2020 ) which learns a GMM prior using an approximate ELBO , Goyal et al . ( 2017 ) which uses a hierarchical Bayesian framework , and VampPrior Tomczak & Welling ( 2018 ) which learns a two-layer hierarchical GMM from aggregated posteriors . Yin & Zhou ( 2018 ) ; Molchanov et al . ( 2019 ) ; Liutkus et al . ( 2019 ) expand the variational family to incorporate implicit variational distributions while providing exact theoretical support . The KL divergence is tractable only for Gaussian distributions , however the Wasserstein distance of 1D projections ( also called Sliced Wasserstein ( SW ) distance ) has a closed-form for any arbitrary distribution ( Kolouri et al. , 2018b ) . Wasserstein distances ( Villani , 2003 ) have several of the same properties as KL divergence , but often lead to better sampling ( Gulrajani et al. , 2017 ; Tolstikhin et al. , 2018 ) , and have been used in several machine learning applications ( e.g . Arjovsky et al . ( 2017 ) ; Tolstikhin et al . ( 2018 ) ; Kolouri et al . ( 2018b ) ) . A Sliced Wasserstein AutoEncoder ( SWAE ) that regularizes an autoencoder using SW distance was proposed in Kolouri et al . ( 2018a ) . There exist several variants of SW distance that have been successful at improving generative ability - for example , Deshpande et al . ( 2018 ) ; Chen et al . ( 2020b ) train discriminator-like networks to perform nonlinear transformations ( instead of purely linear projections ) whereas Deshpande et al . ( 2019 ) introduces max-SW distance , which looks for the best linear projection , generalized in Nguyen et al . ( 2020 ) to finding optimal distributions of projections . Wu et al . ( 2019 ) uses a different optimization approach to computing the SW distance based on the Kantorovich dual formulation . All of the above use some form of additional training . In contrast , we use a simple and efficient nonlinear shearing which requires no additional optimization and is sufficient for improving latent representation , however , our method can easily incorporate other SW distances if desired . Alternatively , many works such as Wang et al . ( 2020 ) ; Makhzani et al . ( 2015 ) ; Arjovsky et al . ( 2017 ) ; Sainburg et al . ( 2018 ) replace the KL divergence with an adversarial loss , however adversarial methods tend to be significantly more expensive and difficult to optimize . In higher dimensions , using a discriminator network as in adversarial approaches is a natural way of implicitly computing an equivalent of the Wasserstein-1 distance ( Arjovsky et al. , 2017 ; Tolstikhin et al. , 2018 ) . However , using the SW distance is much simpler and more efficient . Adversarial training for interpolation in Sainburg et al . ( 2018 ) uses a structural consistency term that encourages relative distances in data space to be preserved in latent space . Similar distance-preserving terms have also been used successfully in Yu et al . ( 2013 ) ; Benaim & Wolf ( 2017 ) , however Euclidean distances in data space can be a poor measure of the natural geometry of the data . In our work , we preserve relative distances in latent space and feature space to improve latent encoding , where features are extracted at the output of the last convolutional layer in the data encoder . Several manifold learning techniques such as Dollár et al . ( 2007 ) ; Weinberger & Saul ( 2006 ) compute embeddings of high-dimensional data but are less general and lack generative or interpolation abilities . The hyperspherical VAE Davidson et al . ( 2018 ) outperforms the standard VAE for data residing on a hyperspherical manifold . Miolane & Holmes ( 2020a ) formulates a Riemannian VAE , however computing its ELBO is challenging . Along similar lines , Arvanitidis et al . ( 2017 ) shows that under certain conditions , a Riemannian metric is naturally induced in the latent space , and uses it to compute geodesics . In Chen et al . ( 2020a ) , a flat manifold is approximated by penalizing curvature in latent space , and geodesics are defined through Euclidean distances on the flat manifold . There exists limited work on integrating graphical structures with generative models ( Kipf & Welling , 2016 ) . Hadjeres et al . ( 2017 ) studies monophonic music using a latent space geodesic regularization that allows interpolations in the latent space to be more meaningful , giving a notion of geodesic distance . Several approaches such as Tenenbaum et al . ( 2000 ) ; Bernstein et al . ( 2000 ) ; Klein & Zachmann ( 2004 ) ; Mémoli & Sapiro ( 2005 ) ; Luo & Hu ( 2020 ) approximate geodesics on point clouds by , for instance , building a nearest-neighbor network on the manifold and applying a shortest path algorithm , however , these often require many samples in order generate reasonable approximations and aren ’ t robust to noise ( Sober et al. , 2020 ) . Inspired by this , we introduce an energy-based network algorithm to identify network-geodesics in latent space with relatively few , noisy samples .	The paper extends the variational autoencoder framework with a richer prior distribution to model more complex correlations in the latent variable distribution. They start with a Gaussian mixture distribution as the prior for the latent variables, and add an encoder network to allow richer correlation structure in the latent variables. Training the prior distribution requires an optimization between the prior distribution and the latent encoded distribution of the training data set. The paper starts with an existing method of optimizing the prior by computing an approximation of the Wasserstein distance between prior and encoded training distribution that uses an average over slices through the prior and encoded training distribution. The paper replaces linear projections used in prior work with a non-linear projection. The paper also employs a structural consistency term which has been used in prior work, however, the paper employs this term differently than prior work by applying it between encoder features and latent variables rather than inputs and latent variables. Since the latent variable space is now a complex and possibly a nonconvex submanifold, points in the latent space R^D lying between points corresponding to training data points may not actually fall in the training distribution. The paper therefore proposes a method of interpolating between points in the manifold by constructing a graph between points sampled from the manifold and then choosing points lying along lines in the graph. The paper tests the method on three datasets, a synthetic 40 dimensional spiral dataset, the venerable MNIST dataset and a scaled down CELEB A dataset. Plots of the latent space trained on the spiral dataset shows that the latent space can in fact have complex internal structure.	SP:8cf8825b4a9cf5611a0477a5c18ed60d8f0b052a
Encoded Prior Sliced Wasserstein AutoEncoder for learning latent manifold representations	1 INTRODUCTION . Generative models have the potential to capture rich representations of data and use them to generate realistic outputs . In particular , Variational AutoEncoders ( VAEs ) ( Kingma & Welling , 2014 ) can capture important properties of high-dimensional data in their latent embeddings , and sample from a prior distribution to generate realistic images . Whille VAEs have been very successful in a variety of tasks , the use of a simplistic standard normal prior is known to cause problems such as under-fitting and over-regularization , and fails to use the network ’ s entire modeling capacity ( Burda et al. , 2016 ) . Gaussian or Gaussian mixture model ( GMM ) priors are also limited in their ability to represent geometric and topological properties of the underlying data manifold . High-dimensional data can typically be modeled as lying on or near an embedded low-dimensional , nonlinear manifold ( Fefferman et al. , 2016 ) . Learning improved latent representations of this nonlinear manifold is an important problem , for which a more flexible prior may be desirable . Conventional variational inference uses Kullback-Leibler ( KL ) divergence as a measure of distance between the posterior and the prior , restricting the prior distribution to cases that have tractable approximations of the KL divergence . Many works such as Guo et al . ( 2020 ) ; Tomczak & Welling ( 2018 ) ; Rezende & Mohamed ( 2015 ) etc . have investigated the use of more complicated priors ( notably GMMs ) which lead to improved latent representation and generation compared to a single Gaussion prior . Alternate approaches such as adversarial training learn arbitrary priors by using a discriminator network to compute a divergence ( Wang et al. , 2020 ) , however they are reported to be harder to train and are computationally expensive . In this work , we introduce the Encoded Prior Sliced Wasserstein AutoEncoder ( EPSWAE ) , which consists of a conventional autoencoder architecture and an additional prior-encoder network that learns an unconstrained prior distribution that encodes the geometry and topology of any data manifold . We use a type of Sliced Wasserstein ( SW ) distance ( Bonnotte , 2013 ; Bonneel et al. , 2015 ) , a concept from optimal transport theory that is a simple and convenient alternative to the KL divergence for any sampleable distributions . A Sliced Wasserstein AutoEncoder ( SWAE ) that regularizes an autoencoder using SW distance was proposed in Kolouri et al . ( 2018a ) . Several works improve the SW distance through additional optimizations ( Deshpande et al. , 2019 ; Chen et al. , 2020b ; Deshpande et al. , 2018 ) , and show improved generation , however involve additional training and use a fixed ( usually Gaussian ) prior . Kolouri et al . ( 2019 ) presents a comparison between max-SW distance , polynomial generalized SW distances , and their combinations . In contrast , we use a simple and efficient nonlinear shearing which requires no additional optimization . Additionally , we introduce a structural consistency term that encourages the latent space to be isometric to the feature space , which is typically measured at the output of the convolutional layers of the data encoder . Variants of this penalty have previously been used to encourage isometry between the latent space and data space ( Yu et al. , 2013 ; Benaim & Wolf , 2017 ; Sainburg et al. , 2018 ) . The structural consistency term further encourages the prior to match the encoded data manifold by preserving feature-isometry , which in turn is expected to assist with encoding the geometry of the data manifold , thus leading to improved latent representations . A key contribution of our work is the graph-based geodesic-interpolation algorithm . Conventionally , VAEs use Euclidean interpolation between two points in latent space . However , since manifolds typically have curvature , this is an unintuitive distance metric that can lead to unrealistic intermediate points . Our goal is to learn a true representation of the underlying data manifold , hence it is natural to interpolate along the manifold geodesics in latent space . Several works such as Shao et al . ( 2018 ) ; Miolane & Holmes ( 2020b ) endow the latent space with a Riemannian geometry and measure corresponding distances , however these are difficult and involve explicitly solving expensive ordinary differential equations . In this work , we introduce ‘ network-geodesics ’ , a graph-based method for interpolating along a manifold in latent space , that maximizes sample density along paths while minimizing total energy . This involves first generating a distance graph between samples from the prior . Then this network is non-uniformly thresholded such that the set of allowable paths from a given sample traverse high density regions through short hops . Lastly , we use a shortest path algorithm like Dijkstra ’ s algorithm ( Dijkstra , 1959 ) to identify the lowest ‘ energy ’ path between two samples through the allowable paths . Since the prior is trained to learn the data manifold , the resulting network-geodesic curves give a notion of distance on the manifold and can be used to generate realistic interpolation points with relatively few prior samples . The novel contributions of this work are : • We introduce a novel architecture , EPSWAE , that consists of a prior-encoder network that is efficiently trained ( without expensive adversarial methods ) to generate a prior that encodes the geometric and topological properties of data . • We introduce a novel graph-based method for interpolating along network-geodesics in latent space through maximizing sample density while minimizing total energy . We show that it generates natural interpolations through realistic images . • Improvements to the latent space representation are obtained by using a structural consistency term in the loss that encourages isometry between feature space and latent space and by using a simple and efficient nonlinear variant of the SW distance . 2 BACKGROUND AND RELATED WORK . Several works have attempted to increase the complexity of the prior in order to obtain better latent representations . Most data can mathematically be thought of as living on a high dimensional manifold . In an image dataset , for instance , if images in high dimensional pixel space are effectively parametrized using a small number of continuous variables , they will lie on or near a low dimensional manifold ( Lu et al. , 1998 ) . Many works such as Weinberger & Saul ( 2006 ) ; Rahimi et al . ( 2005 ) have investigated image and video manifolds . Encoding and exploiting the topological and geometric properties of such data is a question of increasing interest . Some representative works are Dilokthanakul et al . ( 2016 ) which shows improved unsupervised clustering of latent space by using GMM priors , Takahashi et al . ( 2019 ) which uses the density ratio trick to calculate the KL divergence using implicit priors , Rezende & Mohamed ( 2015 ) , which maps a Gaussian prior by a series of explicit transformations , Guo et al . ( 2020 ) which learns a GMM prior using an approximate ELBO , Goyal et al . ( 2017 ) which uses a hierarchical Bayesian framework , and VampPrior Tomczak & Welling ( 2018 ) which learns a two-layer hierarchical GMM from aggregated posteriors . Yin & Zhou ( 2018 ) ; Molchanov et al . ( 2019 ) ; Liutkus et al . ( 2019 ) expand the variational family to incorporate implicit variational distributions while providing exact theoretical support . The KL divergence is tractable only for Gaussian distributions , however the Wasserstein distance of 1D projections ( also called Sliced Wasserstein ( SW ) distance ) has a closed-form for any arbitrary distribution ( Kolouri et al. , 2018b ) . Wasserstein distances ( Villani , 2003 ) have several of the same properties as KL divergence , but often lead to better sampling ( Gulrajani et al. , 2017 ; Tolstikhin et al. , 2018 ) , and have been used in several machine learning applications ( e.g . Arjovsky et al . ( 2017 ) ; Tolstikhin et al . ( 2018 ) ; Kolouri et al . ( 2018b ) ) . A Sliced Wasserstein AutoEncoder ( SWAE ) that regularizes an autoencoder using SW distance was proposed in Kolouri et al . ( 2018a ) . There exist several variants of SW distance that have been successful at improving generative ability - for example , Deshpande et al . ( 2018 ) ; Chen et al . ( 2020b ) train discriminator-like networks to perform nonlinear transformations ( instead of purely linear projections ) whereas Deshpande et al . ( 2019 ) introduces max-SW distance , which looks for the best linear projection , generalized in Nguyen et al . ( 2020 ) to finding optimal distributions of projections . Wu et al . ( 2019 ) uses a different optimization approach to computing the SW distance based on the Kantorovich dual formulation . All of the above use some form of additional training . In contrast , we use a simple and efficient nonlinear shearing which requires no additional optimization and is sufficient for improving latent representation , however , our method can easily incorporate other SW distances if desired . Alternatively , many works such as Wang et al . ( 2020 ) ; Makhzani et al . ( 2015 ) ; Arjovsky et al . ( 2017 ) ; Sainburg et al . ( 2018 ) replace the KL divergence with an adversarial loss , however adversarial methods tend to be significantly more expensive and difficult to optimize . In higher dimensions , using a discriminator network as in adversarial approaches is a natural way of implicitly computing an equivalent of the Wasserstein-1 distance ( Arjovsky et al. , 2017 ; Tolstikhin et al. , 2018 ) . However , using the SW distance is much simpler and more efficient . Adversarial training for interpolation in Sainburg et al . ( 2018 ) uses a structural consistency term that encourages relative distances in data space to be preserved in latent space . Similar distance-preserving terms have also been used successfully in Yu et al . ( 2013 ) ; Benaim & Wolf ( 2017 ) , however Euclidean distances in data space can be a poor measure of the natural geometry of the data . In our work , we preserve relative distances in latent space and feature space to improve latent encoding , where features are extracted at the output of the last convolutional layer in the data encoder . Several manifold learning techniques such as Dollár et al . ( 2007 ) ; Weinberger & Saul ( 2006 ) compute embeddings of high-dimensional data but are less general and lack generative or interpolation abilities . The hyperspherical VAE Davidson et al . ( 2018 ) outperforms the standard VAE for data residing on a hyperspherical manifold . Miolane & Holmes ( 2020a ) formulates a Riemannian VAE , however computing its ELBO is challenging . Along similar lines , Arvanitidis et al . ( 2017 ) shows that under certain conditions , a Riemannian metric is naturally induced in the latent space , and uses it to compute geodesics . In Chen et al . ( 2020a ) , a flat manifold is approximated by penalizing curvature in latent space , and geodesics are defined through Euclidean distances on the flat manifold . There exists limited work on integrating graphical structures with generative models ( Kipf & Welling , 2016 ) . Hadjeres et al . ( 2017 ) studies monophonic music using a latent space geodesic regularization that allows interpolations in the latent space to be more meaningful , giving a notion of geodesic distance . Several approaches such as Tenenbaum et al . ( 2000 ) ; Bernstein et al . ( 2000 ) ; Klein & Zachmann ( 2004 ) ; Mémoli & Sapiro ( 2005 ) ; Luo & Hu ( 2020 ) approximate geodesics on point clouds by , for instance , building a nearest-neighbor network on the manifold and applying a shortest path algorithm , however , these often require many samples in order generate reasonable approximations and aren ’ t robust to noise ( Sober et al. , 2020 ) . Inspired by this , we introduce an energy-based network algorithm to identify network-geodesics in latent space with relatively few , noisy samples .	The paper introduces an additional prior-encoder network to autoencoders to learn an unconstrained prior. The autoencoder and prior-encoder networks are iteratively trained with the sliced Wasserstein distance (SWD). To strengthen SWD, this paper further applies nonlinear transformations with a structural consistency term for better match between two distributions. For better interpolation on the latent space, it also introduces a graph-based algorithm.	SP:8cf8825b4a9cf5611a0477a5c18ed60d8f0b052a
Encoded Prior Sliced Wasserstein AutoEncoder for learning latent manifold representations	1 INTRODUCTION . Generative models have the potential to capture rich representations of data and use them to generate realistic outputs . In particular , Variational AutoEncoders ( VAEs ) ( Kingma & Welling , 2014 ) can capture important properties of high-dimensional data in their latent embeddings , and sample from a prior distribution to generate realistic images . Whille VAEs have been very successful in a variety of tasks , the use of a simplistic standard normal prior is known to cause problems such as under-fitting and over-regularization , and fails to use the network ’ s entire modeling capacity ( Burda et al. , 2016 ) . Gaussian or Gaussian mixture model ( GMM ) priors are also limited in their ability to represent geometric and topological properties of the underlying data manifold . High-dimensional data can typically be modeled as lying on or near an embedded low-dimensional , nonlinear manifold ( Fefferman et al. , 2016 ) . Learning improved latent representations of this nonlinear manifold is an important problem , for which a more flexible prior may be desirable . Conventional variational inference uses Kullback-Leibler ( KL ) divergence as a measure of distance between the posterior and the prior , restricting the prior distribution to cases that have tractable approximations of the KL divergence . Many works such as Guo et al . ( 2020 ) ; Tomczak & Welling ( 2018 ) ; Rezende & Mohamed ( 2015 ) etc . have investigated the use of more complicated priors ( notably GMMs ) which lead to improved latent representation and generation compared to a single Gaussion prior . Alternate approaches such as adversarial training learn arbitrary priors by using a discriminator network to compute a divergence ( Wang et al. , 2020 ) , however they are reported to be harder to train and are computationally expensive . In this work , we introduce the Encoded Prior Sliced Wasserstein AutoEncoder ( EPSWAE ) , which consists of a conventional autoencoder architecture and an additional prior-encoder network that learns an unconstrained prior distribution that encodes the geometry and topology of any data manifold . We use a type of Sliced Wasserstein ( SW ) distance ( Bonnotte , 2013 ; Bonneel et al. , 2015 ) , a concept from optimal transport theory that is a simple and convenient alternative to the KL divergence for any sampleable distributions . A Sliced Wasserstein AutoEncoder ( SWAE ) that regularizes an autoencoder using SW distance was proposed in Kolouri et al . ( 2018a ) . Several works improve the SW distance through additional optimizations ( Deshpande et al. , 2019 ; Chen et al. , 2020b ; Deshpande et al. , 2018 ) , and show improved generation , however involve additional training and use a fixed ( usually Gaussian ) prior . Kolouri et al . ( 2019 ) presents a comparison between max-SW distance , polynomial generalized SW distances , and their combinations . In contrast , we use a simple and efficient nonlinear shearing which requires no additional optimization . Additionally , we introduce a structural consistency term that encourages the latent space to be isometric to the feature space , which is typically measured at the output of the convolutional layers of the data encoder . Variants of this penalty have previously been used to encourage isometry between the latent space and data space ( Yu et al. , 2013 ; Benaim & Wolf , 2017 ; Sainburg et al. , 2018 ) . The structural consistency term further encourages the prior to match the encoded data manifold by preserving feature-isometry , which in turn is expected to assist with encoding the geometry of the data manifold , thus leading to improved latent representations . A key contribution of our work is the graph-based geodesic-interpolation algorithm . Conventionally , VAEs use Euclidean interpolation between two points in latent space . However , since manifolds typically have curvature , this is an unintuitive distance metric that can lead to unrealistic intermediate points . Our goal is to learn a true representation of the underlying data manifold , hence it is natural to interpolate along the manifold geodesics in latent space . Several works such as Shao et al . ( 2018 ) ; Miolane & Holmes ( 2020b ) endow the latent space with a Riemannian geometry and measure corresponding distances , however these are difficult and involve explicitly solving expensive ordinary differential equations . In this work , we introduce ‘ network-geodesics ’ , a graph-based method for interpolating along a manifold in latent space , that maximizes sample density along paths while minimizing total energy . This involves first generating a distance graph between samples from the prior . Then this network is non-uniformly thresholded such that the set of allowable paths from a given sample traverse high density regions through short hops . Lastly , we use a shortest path algorithm like Dijkstra ’ s algorithm ( Dijkstra , 1959 ) to identify the lowest ‘ energy ’ path between two samples through the allowable paths . Since the prior is trained to learn the data manifold , the resulting network-geodesic curves give a notion of distance on the manifold and can be used to generate realistic interpolation points with relatively few prior samples . The novel contributions of this work are : • We introduce a novel architecture , EPSWAE , that consists of a prior-encoder network that is efficiently trained ( without expensive adversarial methods ) to generate a prior that encodes the geometric and topological properties of data . • We introduce a novel graph-based method for interpolating along network-geodesics in latent space through maximizing sample density while minimizing total energy . We show that it generates natural interpolations through realistic images . • Improvements to the latent space representation are obtained by using a structural consistency term in the loss that encourages isometry between feature space and latent space and by using a simple and efficient nonlinear variant of the SW distance . 2 BACKGROUND AND RELATED WORK . Several works have attempted to increase the complexity of the prior in order to obtain better latent representations . Most data can mathematically be thought of as living on a high dimensional manifold . In an image dataset , for instance , if images in high dimensional pixel space are effectively parametrized using a small number of continuous variables , they will lie on or near a low dimensional manifold ( Lu et al. , 1998 ) . Many works such as Weinberger & Saul ( 2006 ) ; Rahimi et al . ( 2005 ) have investigated image and video manifolds . Encoding and exploiting the topological and geometric properties of such data is a question of increasing interest . Some representative works are Dilokthanakul et al . ( 2016 ) which shows improved unsupervised clustering of latent space by using GMM priors , Takahashi et al . ( 2019 ) which uses the density ratio trick to calculate the KL divergence using implicit priors , Rezende & Mohamed ( 2015 ) , which maps a Gaussian prior by a series of explicit transformations , Guo et al . ( 2020 ) which learns a GMM prior using an approximate ELBO , Goyal et al . ( 2017 ) which uses a hierarchical Bayesian framework , and VampPrior Tomczak & Welling ( 2018 ) which learns a two-layer hierarchical GMM from aggregated posteriors . Yin & Zhou ( 2018 ) ; Molchanov et al . ( 2019 ) ; Liutkus et al . ( 2019 ) expand the variational family to incorporate implicit variational distributions while providing exact theoretical support . The KL divergence is tractable only for Gaussian distributions , however the Wasserstein distance of 1D projections ( also called Sliced Wasserstein ( SW ) distance ) has a closed-form for any arbitrary distribution ( Kolouri et al. , 2018b ) . Wasserstein distances ( Villani , 2003 ) have several of the same properties as KL divergence , but often lead to better sampling ( Gulrajani et al. , 2017 ; Tolstikhin et al. , 2018 ) , and have been used in several machine learning applications ( e.g . Arjovsky et al . ( 2017 ) ; Tolstikhin et al . ( 2018 ) ; Kolouri et al . ( 2018b ) ) . A Sliced Wasserstein AutoEncoder ( SWAE ) that regularizes an autoencoder using SW distance was proposed in Kolouri et al . ( 2018a ) . There exist several variants of SW distance that have been successful at improving generative ability - for example , Deshpande et al . ( 2018 ) ; Chen et al . ( 2020b ) train discriminator-like networks to perform nonlinear transformations ( instead of purely linear projections ) whereas Deshpande et al . ( 2019 ) introduces max-SW distance , which looks for the best linear projection , generalized in Nguyen et al . ( 2020 ) to finding optimal distributions of projections . Wu et al . ( 2019 ) uses a different optimization approach to computing the SW distance based on the Kantorovich dual formulation . All of the above use some form of additional training . In contrast , we use a simple and efficient nonlinear shearing which requires no additional optimization and is sufficient for improving latent representation , however , our method can easily incorporate other SW distances if desired . Alternatively , many works such as Wang et al . ( 2020 ) ; Makhzani et al . ( 2015 ) ; Arjovsky et al . ( 2017 ) ; Sainburg et al . ( 2018 ) replace the KL divergence with an adversarial loss , however adversarial methods tend to be significantly more expensive and difficult to optimize . In higher dimensions , using a discriminator network as in adversarial approaches is a natural way of implicitly computing an equivalent of the Wasserstein-1 distance ( Arjovsky et al. , 2017 ; Tolstikhin et al. , 2018 ) . However , using the SW distance is much simpler and more efficient . Adversarial training for interpolation in Sainburg et al . ( 2018 ) uses a structural consistency term that encourages relative distances in data space to be preserved in latent space . Similar distance-preserving terms have also been used successfully in Yu et al . ( 2013 ) ; Benaim & Wolf ( 2017 ) , however Euclidean distances in data space can be a poor measure of the natural geometry of the data . In our work , we preserve relative distances in latent space and feature space to improve latent encoding , where features are extracted at the output of the last convolutional layer in the data encoder . Several manifold learning techniques such as Dollár et al . ( 2007 ) ; Weinberger & Saul ( 2006 ) compute embeddings of high-dimensional data but are less general and lack generative or interpolation abilities . The hyperspherical VAE Davidson et al . ( 2018 ) outperforms the standard VAE for data residing on a hyperspherical manifold . Miolane & Holmes ( 2020a ) formulates a Riemannian VAE , however computing its ELBO is challenging . Along similar lines , Arvanitidis et al . ( 2017 ) shows that under certain conditions , a Riemannian metric is naturally induced in the latent space , and uses it to compute geodesics . In Chen et al . ( 2020a ) , a flat manifold is approximated by penalizing curvature in latent space , and geodesics are defined through Euclidean distances on the flat manifold . There exists limited work on integrating graphical structures with generative models ( Kipf & Welling , 2016 ) . Hadjeres et al . ( 2017 ) studies monophonic music using a latent space geodesic regularization that allows interpolations in the latent space to be more meaningful , giving a notion of geodesic distance . Several approaches such as Tenenbaum et al . ( 2000 ) ; Bernstein et al . ( 2000 ) ; Klein & Zachmann ( 2004 ) ; Mémoli & Sapiro ( 2005 ) ; Luo & Hu ( 2020 ) approximate geodesics on point clouds by , for instance , building a nearest-neighbor network on the manifold and applying a shortest path algorithm , however , these often require many samples in order generate reasonable approximations and aren ’ t robust to noise ( Sober et al. , 2020 ) . Inspired by this , we introduce an energy-based network algorithm to identify network-geodesics in latent space with relatively few , noisy samples .	This paper addresses the issues of representation learning with VAEs and propose EPSWAE as a solution. EPSWAE applies a prior encoder to construct an implicit prior, which is more flexible. Moreover, the authors apply the sliced Wasserstein distance for the matching between the posterior and the prior, enhance the conventional SWD with non-linear transformations and make the latent space similar to the feature space with a structural consistency loss. This paper also proposes a graph-based algorithm for minimizing the pathwise energy to achieve the manifold walking to improve the interpolation in the latent space.	SP:8cf8825b4a9cf5611a0477a5c18ed60d8f0b052a
Channel-Directed Gradients for Optimization of Convolutional Neural Networks	1 INTRODUCTION . Stochastic gradient descent ( SGD ) is currently the dominant algorithm for optimizing large-scale convolutional neural networks ( CNNs ) ( LeCun et al . ( 1998 ) ; Simonyan & Zisserman ( 2014 ) ; He et al . ( 2016b ) ) . Although there has been large activity in optimization methods seeking to improve performance , SGD still dominates in terms of its generalization ability . Despite SGD ’ s dominance , there is still often a gap between training and real-world test accuracy performance , which motivates research in improved optimization methods . In this paper , we derive new optimization methods that are simple modifications of SGD . The methods implicitly induce correlation in the output direction of parameter tensors in CNNs . This is based on the empirical observation that parameter tensors in trained networks typically exhibit correlation over output channel dimension ( see Figure 1 ) . We thus explore encoding correlation by constructing smooth gradients in the output direction , which we show improves generalization accuracy . This is done by introducing new Riemmanian metrics on the parameter tensors , which changes the underlying geometry of the space of tensors , and reformulating the gradient with respect to those metrics . Our contributions are as follows . First , we formulate output channel-directed Riemannian metrics ( a re-weighted version of the standard L2 metric and another that is a Sobolev metric ) over the space of parameter tensors . This encodes channel-directed correlation in the gradient optimization without changing the loss . Second , we compute Riemannian gradients with respect to the metrics showing linear complexity ( in the number of parameters ) over standard gradient computation , and thus derive new optimization methods for CNN training . Finally , we apply the methodology to training CNNs and show the empirical advantage in generalization accuracy , especially with small batch sizes , over standard optimizers ( SGD , Adam ) on numerous applications ( image classification , semantic segmentation , generative adversarial networks ) with simple modification of existing optimizers . 1.1 RELATED WORK . We discuss related work in deep network optimization ; for a detailed survey , see Bottou et al . ( 2018 ) . SGD , e.g. , Bottou ( 2012 ) , samples a batch of data to tractably estimate the gradient of the loss function . As the stochastic gradient is a noisy version of the gradient , learning rates must follow a decay schedule in order to converge . Many methods have been formulated to choose learning rate over epochs and components of the gradient , including adaptive learning rates ( e.g. , Duchi et al . ( 2011 ) ; Zeiler ( 2012 ) ; Kingma & Ba ( 2014 ) ; Bengio ( 2015 ) ; Loshchilov & Hutter ( 2017 ) ; Luo et al . ( 2019 ) ) . For instance , Adam Kingma & Ba ( 2014 ) adaptively adjusts the learning rate so that parameters that have changed infrequently based on historical gradients are updated more quickly than parameters that have changed frequently . Another way to interpret such methods is that they change the underlying metric on the space on which the loss function is defined to an iso-tropically scaled version of the L2 metric given by a simple diagonal matrix ; we change the metrics an-isotropically . We show that our method can be used in conjunction with such methods by simply using the stochastic gradient computed with our metrics to boost performance . As the stochastic gradient is computed based on sampling , different runs of the algorithm can result in different local optima . To reduce the variance , several methods have been been formulated , e.g. , Defazio et al . ( 2014 ) ; Johnson & Zhang ( 2013 ) . We are not motivated by variance reduction , rather , inducing correlation in the parameter tensor to improve generalization . However , as our method smooths the gradient , our experiments show reduced variance with our metrics compared to SGD . Another method motivated by variance reduction is Osher et al . ( 2018 ) ( see applications Wang et al . ( 2019 ) ; Liang et al . ( 2020 ) ; Wang et al . ( 2020 ) ) , where the stochastic gradient is pre-multiplied with an inverse Laplacian smoothing matrix . For CNNs , the gradient with respect to parameters is rasterized in row or column order of network filters before smoothing . Our work is inspired by Osher et al . ( 2018 ) , though we are motivated by correlation in the parameter tensor . Osher et al . ( 2018 ) can be interpreted as using the gradient of the loss with respect to a Sobolev metric . One insight over Osher et al . ( 2018 ) is that keeping the structure of the parameter tensor and defining the Sobolev metric with respect to the output-channel direction boosts accuracy , while other directions do not . Secondly , we introduce a re-weightedH0 metric that preferentially treats the output-channel direction , and can be implemented with a line of Pytorch code , has linear ( in parameter size ) complexity , and performs comparably ( in many cases ) to our channel-directed Sobolev metric , boosting accuracy of SGD . Third , our Sobolev gradient , a variant of the ordinary one , has linear complexity rather than quasi-linear ( not requiring FFT as Osher et al . ( 2018 ) ) . Sobolev gradients have been used in computer vision Sundaramoorthi et al . ( 2007 ) ; Charpiat et al . ( 2007 ) for their coarse-to-fine evolution Sundaramoorthi et al . ( 2008 ) ; we adapt that formulation to CNNs . We formulate Sobolev gradients by considering the space of parameter tensors as a Riemannian manifold , and choosing the Sobolev metric on the tangent space . By choosing a metric , gradients intrinsic to the manifold can be computed and gradient flows are guaranteed to decrease loss . Other Riemannian metrics have been used for optimization in neural networks , e.g. , Amari ( 1998 ) ; MarceauCaron & Ollivier ( 2016 ) ; Hoffman et al . ( 2013 ) ; Gunasekar et al . ( 2020 ) and tangentially relate to our work . These works are based on Amari ’ s Amari ( 1998 ) information geometry on probability measures , and the metric considered is the Fisher information metric . The motivation for these methods is re-parametrization invariance of optimization , whereas our motivation is imposing correlation in the parameter space . Other works Gunasekar et al . ( 2020 ) use the Hessian metric ( in the convex case ) , but these metrics are data-dependent and the gradient is challenging to compute , requiring ( a large ) inverse matrix computation . 2 CHANNEL-DIRECTED GRADIENTS . We now present the theory to define channel-directed gradients . To do this , we formulate new metrics on the space of tensors , and then derive analytic formulas for channel-directed gradients in terms of the standard L2 gradient . As we show , our channel-directed gradients effectively smooth the components of the L2 gradient across the output direction of the parameter tensors of the CNN , which induces correlation in that direction in the gradient and thus also the parameter tensor . Another interpretation is we are changing the geometry of the loss landscape ( without changing the loss ) to a more smooth one by changing the metric of the space on which the loss is defined . 2.1 BACKGROUND ON RIEMANNIAN GRADIENTS . We present the definition of gradient on a Riemannian manifold , and show the dependence of the gradient on the chosen metric on the manifold ( see Carmo ( 1992 ) ; Abraham et al . ( 2012 ) for more details ) . A manifold X is a space that is locally linear around each point X ∈ X ; this linear space is the tangent space , denoted TXX . A Riemannian manifold has a smoothly varying positive definite bilinear form 〈· , ·〉 ( called the metric ) on the tangent space . This metric allows one to define the notion of lengths of curves on the space , in addition to other operations , including gradients . Definition 1 ( Gradient of a Function ) Let X be a Riemannian manifold , and f : X → R be a function . The directional derivative of f at X ∈ X along a direction k ∈ TXX is defined as df ( X ) · k = ddε f ( X + εk ) \|ε=0 . The gradient of f at X ∈ X is the vector , ∇f ( X ) ∈ TXX , that satisfies the relation df ( X ) · k = 〈∇f ( X ) , k〉 , for all k ∈ TXX . ( 1 ) Note that “ the ” gradient will depend on the choice of the metric on the manifold . We note that any such gradient will decrease the the function f by moving infinitesimally in the tangent space in the direction of negative the gradient as df ( X ) · k = −‖∇f ( X ) ‖2 < 0 when k = −∇f ( X ) , where ‖ · ‖ is the norm induced from the metric . The gradient flow , defined by the differential equation Ẋt = −∇f ( Xt ) , will converge to a local minimum . In our application of this theory to CNN optimization , f will be the loss function , and X will be the space of parameter tensors . A consequence of this definition is that the gradient is the direction ( up to a scale factor ) in the tangent space that optimizes the following problem : arg max k∈TXX\ { 0 } \|df ( X ) · k\| ‖k‖ . ( 2 ) Thus , the gradient can be regarded as the most efficient direction as it maximizes the ratio of the change in energy by perturbing in a direction k over the cost ( defined by the metric ) of k. Thus , by constructing the metric to have small costs for perturbations ( directions ) that we prefer for gradients , the gradient flow will move in these preferential directions while minimizng the function , and thus land in favorable local minima . 2.2 CHANNEL-DIRECTED METRICS . In existing deep network gradient-based optimization schemes , the underlying metric on the loss function is assumed to be the standard Euclidean L2 metric . We will consider a re-weighted version of the L2 metric and a Sobolev metric that favor correlation in the output channel direction of the gradient and thus the parameter tensors . To formulate the methodology , we start from a continuum formulation , where we treat weight tensors in the continuum , formulate the metrics in the continuum and then in the next sub-section derive the gradients with respect to these metrics . Finally , we discretize gradient flows in the implementation to derive iterative schemes . Let X : O × I × H × W → R denote a parameter tensor of a layer of a convolutional neural network . Here O = [ 0 , O ] denotes indices to the output channel dimension of the tensor , I = [ 0 , I ] Figure 2 : Visualization of kernels applied to the H0 gradient under different metrics for λ = 1 . This illustrates the smoothing effect of the metrics . In computation , linear cost formulas are applied to compute the gradients not using the convolution interpretation . denote the indices to the input channel , and H = [ 0 , H ] , W = [ 0 , W ] denote indices to the height and width dimension of the spatial filters of the tensor . The metric is defined on the tangent space to the space of such X . An element of the tangent space will have the same form of the tensor , i.e. , k : O × I ×H×W → R. The L2 ( called H0 from now on ) metric is defined as 〈k1 , k2〉 H0 = ∫ O , I , H , W k1 ( o , i , h , w ) · k2 ( o , i , h , w ) do didhdw , ( 3 ) where k1 , k2 are in the tangent space of tensors . We now define a re-weighted version of H0 that favors tangent vectors that have global smoothness in the direction of the O dimension : 〈k1 , k2〉H0λ = ∫ I , H , W k̄1 ( i , h , w ) · k̄2 ( i , h , w ) didhdw + λ O 〈 k1 − k̄1 , k2 − k̄2 〉 H0 , ( 4 ) where λ > 0 is a hyper-parameter , and k̄ is the average value in the output channel direction , i.e. , k̄ ( i , h , w ) = 1 O ∫ O k ( o , i , h , w ) do . ( 5 ) The metric in ( 4 ) splits the tangent vector into global translations in the output channel direction and its orthogonal complement , i.e. , the deformation . The weight λ is used to control the weighting between the translation and deformation components , i.e. , larger values of λ means that deformations more heavily influence the norm of the perturbation . As shown in the next sub-section that means gradients with respect to this metric have higher weighted channel-directed translations than deformations . Next , we introduce a channel-directed version of a Sobolev metric , defined as follows : 〈k1 , k2〉H̃1 = ∫ I , H , W k̄1 ( i , h , w ) · k̄2 ( i , h , w ) di dhdw + λO 〈 ∂k1 ∂o , ∂k2 ∂o 〉 H0 , ( 6 ) where ∂∂o indicates the partial derivative with respect to the the output channel direction . The partial derivative in the o-direction implies that tensor perturbations that are smooth along the o-direction are close with respect to these metrics , which will imply that the corresponding gradients will exhibit smoothness in this direction , i.e. , convolution filters that are nearby in the output direction will exhibit correlation . The metric is a weighted combination of the H0 metric of the derivative in the output direction , and the H0 metric of the output-directed translation . Note that the traditional Sobolev metric uses the H0 metric of the perturbation rather than the translation . Our choice is motivated by computational efficiency of the corresponding gradient , to be discussed below . The scale factors of O in the expressions above are so that the metric is scale invariant with respect to different sizes of output channels . The part of the metric with the partial derivative component implies that tensors that differ in the output channel direction by a non-smooth perturbation are far away in distance . Tensors that differ by just a channel-directed translation are close .	In this paper propose a stochastic optimization method for CNNs, which can obtain an improvement in terms of generation error. The proposed method is used to compute the gradient with respect to output-channel directed re-weighted matrices or Sobolev metrics, and some anabasis is also provided. Some experimental results show that the proposed method can be improved in generalization error.	SP:dfb0838d0bd978aeff5b8b6b165d86b7e28ceccc
Channel-Directed Gradients for Optimization of Convolutional Neural Networks	1 INTRODUCTION . Stochastic gradient descent ( SGD ) is currently the dominant algorithm for optimizing large-scale convolutional neural networks ( CNNs ) ( LeCun et al . ( 1998 ) ; Simonyan & Zisserman ( 2014 ) ; He et al . ( 2016b ) ) . Although there has been large activity in optimization methods seeking to improve performance , SGD still dominates in terms of its generalization ability . Despite SGD ’ s dominance , there is still often a gap between training and real-world test accuracy performance , which motivates research in improved optimization methods . In this paper , we derive new optimization methods that are simple modifications of SGD . The methods implicitly induce correlation in the output direction of parameter tensors in CNNs . This is based on the empirical observation that parameter tensors in trained networks typically exhibit correlation over output channel dimension ( see Figure 1 ) . We thus explore encoding correlation by constructing smooth gradients in the output direction , which we show improves generalization accuracy . This is done by introducing new Riemmanian metrics on the parameter tensors , which changes the underlying geometry of the space of tensors , and reformulating the gradient with respect to those metrics . Our contributions are as follows . First , we formulate output channel-directed Riemannian metrics ( a re-weighted version of the standard L2 metric and another that is a Sobolev metric ) over the space of parameter tensors . This encodes channel-directed correlation in the gradient optimization without changing the loss . Second , we compute Riemannian gradients with respect to the metrics showing linear complexity ( in the number of parameters ) over standard gradient computation , and thus derive new optimization methods for CNN training . Finally , we apply the methodology to training CNNs and show the empirical advantage in generalization accuracy , especially with small batch sizes , over standard optimizers ( SGD , Adam ) on numerous applications ( image classification , semantic segmentation , generative adversarial networks ) with simple modification of existing optimizers . 1.1 RELATED WORK . We discuss related work in deep network optimization ; for a detailed survey , see Bottou et al . ( 2018 ) . SGD , e.g. , Bottou ( 2012 ) , samples a batch of data to tractably estimate the gradient of the loss function . As the stochastic gradient is a noisy version of the gradient , learning rates must follow a decay schedule in order to converge . Many methods have been formulated to choose learning rate over epochs and components of the gradient , including adaptive learning rates ( e.g. , Duchi et al . ( 2011 ) ; Zeiler ( 2012 ) ; Kingma & Ba ( 2014 ) ; Bengio ( 2015 ) ; Loshchilov & Hutter ( 2017 ) ; Luo et al . ( 2019 ) ) . For instance , Adam Kingma & Ba ( 2014 ) adaptively adjusts the learning rate so that parameters that have changed infrequently based on historical gradients are updated more quickly than parameters that have changed frequently . Another way to interpret such methods is that they change the underlying metric on the space on which the loss function is defined to an iso-tropically scaled version of the L2 metric given by a simple diagonal matrix ; we change the metrics an-isotropically . We show that our method can be used in conjunction with such methods by simply using the stochastic gradient computed with our metrics to boost performance . As the stochastic gradient is computed based on sampling , different runs of the algorithm can result in different local optima . To reduce the variance , several methods have been been formulated , e.g. , Defazio et al . ( 2014 ) ; Johnson & Zhang ( 2013 ) . We are not motivated by variance reduction , rather , inducing correlation in the parameter tensor to improve generalization . However , as our method smooths the gradient , our experiments show reduced variance with our metrics compared to SGD . Another method motivated by variance reduction is Osher et al . ( 2018 ) ( see applications Wang et al . ( 2019 ) ; Liang et al . ( 2020 ) ; Wang et al . ( 2020 ) ) , where the stochastic gradient is pre-multiplied with an inverse Laplacian smoothing matrix . For CNNs , the gradient with respect to parameters is rasterized in row or column order of network filters before smoothing . Our work is inspired by Osher et al . ( 2018 ) , though we are motivated by correlation in the parameter tensor . Osher et al . ( 2018 ) can be interpreted as using the gradient of the loss with respect to a Sobolev metric . One insight over Osher et al . ( 2018 ) is that keeping the structure of the parameter tensor and defining the Sobolev metric with respect to the output-channel direction boosts accuracy , while other directions do not . Secondly , we introduce a re-weightedH0 metric that preferentially treats the output-channel direction , and can be implemented with a line of Pytorch code , has linear ( in parameter size ) complexity , and performs comparably ( in many cases ) to our channel-directed Sobolev metric , boosting accuracy of SGD . Third , our Sobolev gradient , a variant of the ordinary one , has linear complexity rather than quasi-linear ( not requiring FFT as Osher et al . ( 2018 ) ) . Sobolev gradients have been used in computer vision Sundaramoorthi et al . ( 2007 ) ; Charpiat et al . ( 2007 ) for their coarse-to-fine evolution Sundaramoorthi et al . ( 2008 ) ; we adapt that formulation to CNNs . We formulate Sobolev gradients by considering the space of parameter tensors as a Riemannian manifold , and choosing the Sobolev metric on the tangent space . By choosing a metric , gradients intrinsic to the manifold can be computed and gradient flows are guaranteed to decrease loss . Other Riemannian metrics have been used for optimization in neural networks , e.g. , Amari ( 1998 ) ; MarceauCaron & Ollivier ( 2016 ) ; Hoffman et al . ( 2013 ) ; Gunasekar et al . ( 2020 ) and tangentially relate to our work . These works are based on Amari ’ s Amari ( 1998 ) information geometry on probability measures , and the metric considered is the Fisher information metric . The motivation for these methods is re-parametrization invariance of optimization , whereas our motivation is imposing correlation in the parameter space . Other works Gunasekar et al . ( 2020 ) use the Hessian metric ( in the convex case ) , but these metrics are data-dependent and the gradient is challenging to compute , requiring ( a large ) inverse matrix computation . 2 CHANNEL-DIRECTED GRADIENTS . We now present the theory to define channel-directed gradients . To do this , we formulate new metrics on the space of tensors , and then derive analytic formulas for channel-directed gradients in terms of the standard L2 gradient . As we show , our channel-directed gradients effectively smooth the components of the L2 gradient across the output direction of the parameter tensors of the CNN , which induces correlation in that direction in the gradient and thus also the parameter tensor . Another interpretation is we are changing the geometry of the loss landscape ( without changing the loss ) to a more smooth one by changing the metric of the space on which the loss is defined . 2.1 BACKGROUND ON RIEMANNIAN GRADIENTS . We present the definition of gradient on a Riemannian manifold , and show the dependence of the gradient on the chosen metric on the manifold ( see Carmo ( 1992 ) ; Abraham et al . ( 2012 ) for more details ) . A manifold X is a space that is locally linear around each point X ∈ X ; this linear space is the tangent space , denoted TXX . A Riemannian manifold has a smoothly varying positive definite bilinear form 〈· , ·〉 ( called the metric ) on the tangent space . This metric allows one to define the notion of lengths of curves on the space , in addition to other operations , including gradients . Definition 1 ( Gradient of a Function ) Let X be a Riemannian manifold , and f : X → R be a function . The directional derivative of f at X ∈ X along a direction k ∈ TXX is defined as df ( X ) · k = ddε f ( X + εk ) \|ε=0 . The gradient of f at X ∈ X is the vector , ∇f ( X ) ∈ TXX , that satisfies the relation df ( X ) · k = 〈∇f ( X ) , k〉 , for all k ∈ TXX . ( 1 ) Note that “ the ” gradient will depend on the choice of the metric on the manifold . We note that any such gradient will decrease the the function f by moving infinitesimally in the tangent space in the direction of negative the gradient as df ( X ) · k = −‖∇f ( X ) ‖2 < 0 when k = −∇f ( X ) , where ‖ · ‖ is the norm induced from the metric . The gradient flow , defined by the differential equation Ẋt = −∇f ( Xt ) , will converge to a local minimum . In our application of this theory to CNN optimization , f will be the loss function , and X will be the space of parameter tensors . A consequence of this definition is that the gradient is the direction ( up to a scale factor ) in the tangent space that optimizes the following problem : arg max k∈TXX\ { 0 } \|df ( X ) · k\| ‖k‖ . ( 2 ) Thus , the gradient can be regarded as the most efficient direction as it maximizes the ratio of the change in energy by perturbing in a direction k over the cost ( defined by the metric ) of k. Thus , by constructing the metric to have small costs for perturbations ( directions ) that we prefer for gradients , the gradient flow will move in these preferential directions while minimizng the function , and thus land in favorable local minima . 2.2 CHANNEL-DIRECTED METRICS . In existing deep network gradient-based optimization schemes , the underlying metric on the loss function is assumed to be the standard Euclidean L2 metric . We will consider a re-weighted version of the L2 metric and a Sobolev metric that favor correlation in the output channel direction of the gradient and thus the parameter tensors . To formulate the methodology , we start from a continuum formulation , where we treat weight tensors in the continuum , formulate the metrics in the continuum and then in the next sub-section derive the gradients with respect to these metrics . Finally , we discretize gradient flows in the implementation to derive iterative schemes . Let X : O × I × H × W → R denote a parameter tensor of a layer of a convolutional neural network . Here O = [ 0 , O ] denotes indices to the output channel dimension of the tensor , I = [ 0 , I ] Figure 2 : Visualization of kernels applied to the H0 gradient under different metrics for λ = 1 . This illustrates the smoothing effect of the metrics . In computation , linear cost formulas are applied to compute the gradients not using the convolution interpretation . denote the indices to the input channel , and H = [ 0 , H ] , W = [ 0 , W ] denote indices to the height and width dimension of the spatial filters of the tensor . The metric is defined on the tangent space to the space of such X . An element of the tangent space will have the same form of the tensor , i.e. , k : O × I ×H×W → R. The L2 ( called H0 from now on ) metric is defined as 〈k1 , k2〉 H0 = ∫ O , I , H , W k1 ( o , i , h , w ) · k2 ( o , i , h , w ) do didhdw , ( 3 ) where k1 , k2 are in the tangent space of tensors . We now define a re-weighted version of H0 that favors tangent vectors that have global smoothness in the direction of the O dimension : 〈k1 , k2〉H0λ = ∫ I , H , W k̄1 ( i , h , w ) · k̄2 ( i , h , w ) didhdw + λ O 〈 k1 − k̄1 , k2 − k̄2 〉 H0 , ( 4 ) where λ > 0 is a hyper-parameter , and k̄ is the average value in the output channel direction , i.e. , k̄ ( i , h , w ) = 1 O ∫ O k ( o , i , h , w ) do . ( 5 ) The metric in ( 4 ) splits the tangent vector into global translations in the output channel direction and its orthogonal complement , i.e. , the deformation . The weight λ is used to control the weighting between the translation and deformation components , i.e. , larger values of λ means that deformations more heavily influence the norm of the perturbation . As shown in the next sub-section that means gradients with respect to this metric have higher weighted channel-directed translations than deformations . Next , we introduce a channel-directed version of a Sobolev metric , defined as follows : 〈k1 , k2〉H̃1 = ∫ I , H , W k̄1 ( i , h , w ) · k̄2 ( i , h , w ) di dhdw + λO 〈 ∂k1 ∂o , ∂k2 ∂o 〉 H0 , ( 6 ) where ∂∂o indicates the partial derivative with respect to the the output channel direction . The partial derivative in the o-direction implies that tensor perturbations that are smooth along the o-direction are close with respect to these metrics , which will imply that the corresponding gradients will exhibit smoothness in this direction , i.e. , convolution filters that are nearby in the output direction will exhibit correlation . The metric is a weighted combination of the H0 metric of the derivative in the output direction , and the H0 metric of the output-directed translation . Note that the traditional Sobolev metric uses the H0 metric of the perturbation rather than the translation . Our choice is motivated by computational efficiency of the corresponding gradient , to be discussed below . The scale factors of O in the expressions above are so that the metric is scale invariant with respect to different sizes of output channels . The part of the metric with the partial derivative component implies that tensors that differ in the output channel direction by a non-smooth perturbation are far away in distance . Tensors that differ by just a channel-directed translation are close .	The paper proposes a modification of the gradients of objective function wrt model parameters by changing the underling metric on the manifold of model parameters. It proposes two metrics, a reweighed L2 metric, and a specific Sobolev metric, and presents how to compute modified gradients in a compute efficient manner. Motivated by empirical observation that parameters of trained CNN tensors tend to be correlated along output dimension, the paper focuses on metrics that promote such correlation.	SP:dfb0838d0bd978aeff5b8b6b165d86b7e28ceccc
Channel-Directed Gradients for Optimization of Convolutional Neural Networks	1 INTRODUCTION . Stochastic gradient descent ( SGD ) is currently the dominant algorithm for optimizing large-scale convolutional neural networks ( CNNs ) ( LeCun et al . ( 1998 ) ; Simonyan & Zisserman ( 2014 ) ; He et al . ( 2016b ) ) . Although there has been large activity in optimization methods seeking to improve performance , SGD still dominates in terms of its generalization ability . Despite SGD ’ s dominance , there is still often a gap between training and real-world test accuracy performance , which motivates research in improved optimization methods . In this paper , we derive new optimization methods that are simple modifications of SGD . The methods implicitly induce correlation in the output direction of parameter tensors in CNNs . This is based on the empirical observation that parameter tensors in trained networks typically exhibit correlation over output channel dimension ( see Figure 1 ) . We thus explore encoding correlation by constructing smooth gradients in the output direction , which we show improves generalization accuracy . This is done by introducing new Riemmanian metrics on the parameter tensors , which changes the underlying geometry of the space of tensors , and reformulating the gradient with respect to those metrics . Our contributions are as follows . First , we formulate output channel-directed Riemannian metrics ( a re-weighted version of the standard L2 metric and another that is a Sobolev metric ) over the space of parameter tensors . This encodes channel-directed correlation in the gradient optimization without changing the loss . Second , we compute Riemannian gradients with respect to the metrics showing linear complexity ( in the number of parameters ) over standard gradient computation , and thus derive new optimization methods for CNN training . Finally , we apply the methodology to training CNNs and show the empirical advantage in generalization accuracy , especially with small batch sizes , over standard optimizers ( SGD , Adam ) on numerous applications ( image classification , semantic segmentation , generative adversarial networks ) with simple modification of existing optimizers . 1.1 RELATED WORK . We discuss related work in deep network optimization ; for a detailed survey , see Bottou et al . ( 2018 ) . SGD , e.g. , Bottou ( 2012 ) , samples a batch of data to tractably estimate the gradient of the loss function . As the stochastic gradient is a noisy version of the gradient , learning rates must follow a decay schedule in order to converge . Many methods have been formulated to choose learning rate over epochs and components of the gradient , including adaptive learning rates ( e.g. , Duchi et al . ( 2011 ) ; Zeiler ( 2012 ) ; Kingma & Ba ( 2014 ) ; Bengio ( 2015 ) ; Loshchilov & Hutter ( 2017 ) ; Luo et al . ( 2019 ) ) . For instance , Adam Kingma & Ba ( 2014 ) adaptively adjusts the learning rate so that parameters that have changed infrequently based on historical gradients are updated more quickly than parameters that have changed frequently . Another way to interpret such methods is that they change the underlying metric on the space on which the loss function is defined to an iso-tropically scaled version of the L2 metric given by a simple diagonal matrix ; we change the metrics an-isotropically . We show that our method can be used in conjunction with such methods by simply using the stochastic gradient computed with our metrics to boost performance . As the stochastic gradient is computed based on sampling , different runs of the algorithm can result in different local optima . To reduce the variance , several methods have been been formulated , e.g. , Defazio et al . ( 2014 ) ; Johnson & Zhang ( 2013 ) . We are not motivated by variance reduction , rather , inducing correlation in the parameter tensor to improve generalization . However , as our method smooths the gradient , our experiments show reduced variance with our metrics compared to SGD . Another method motivated by variance reduction is Osher et al . ( 2018 ) ( see applications Wang et al . ( 2019 ) ; Liang et al . ( 2020 ) ; Wang et al . ( 2020 ) ) , where the stochastic gradient is pre-multiplied with an inverse Laplacian smoothing matrix . For CNNs , the gradient with respect to parameters is rasterized in row or column order of network filters before smoothing . Our work is inspired by Osher et al . ( 2018 ) , though we are motivated by correlation in the parameter tensor . Osher et al . ( 2018 ) can be interpreted as using the gradient of the loss with respect to a Sobolev metric . One insight over Osher et al . ( 2018 ) is that keeping the structure of the parameter tensor and defining the Sobolev metric with respect to the output-channel direction boosts accuracy , while other directions do not . Secondly , we introduce a re-weightedH0 metric that preferentially treats the output-channel direction , and can be implemented with a line of Pytorch code , has linear ( in parameter size ) complexity , and performs comparably ( in many cases ) to our channel-directed Sobolev metric , boosting accuracy of SGD . Third , our Sobolev gradient , a variant of the ordinary one , has linear complexity rather than quasi-linear ( not requiring FFT as Osher et al . ( 2018 ) ) . Sobolev gradients have been used in computer vision Sundaramoorthi et al . ( 2007 ) ; Charpiat et al . ( 2007 ) for their coarse-to-fine evolution Sundaramoorthi et al . ( 2008 ) ; we adapt that formulation to CNNs . We formulate Sobolev gradients by considering the space of parameter tensors as a Riemannian manifold , and choosing the Sobolev metric on the tangent space . By choosing a metric , gradients intrinsic to the manifold can be computed and gradient flows are guaranteed to decrease loss . Other Riemannian metrics have been used for optimization in neural networks , e.g. , Amari ( 1998 ) ; MarceauCaron & Ollivier ( 2016 ) ; Hoffman et al . ( 2013 ) ; Gunasekar et al . ( 2020 ) and tangentially relate to our work . These works are based on Amari ’ s Amari ( 1998 ) information geometry on probability measures , and the metric considered is the Fisher information metric . The motivation for these methods is re-parametrization invariance of optimization , whereas our motivation is imposing correlation in the parameter space . Other works Gunasekar et al . ( 2020 ) use the Hessian metric ( in the convex case ) , but these metrics are data-dependent and the gradient is challenging to compute , requiring ( a large ) inverse matrix computation . 2 CHANNEL-DIRECTED GRADIENTS . We now present the theory to define channel-directed gradients . To do this , we formulate new metrics on the space of tensors , and then derive analytic formulas for channel-directed gradients in terms of the standard L2 gradient . As we show , our channel-directed gradients effectively smooth the components of the L2 gradient across the output direction of the parameter tensors of the CNN , which induces correlation in that direction in the gradient and thus also the parameter tensor . Another interpretation is we are changing the geometry of the loss landscape ( without changing the loss ) to a more smooth one by changing the metric of the space on which the loss is defined . 2.1 BACKGROUND ON RIEMANNIAN GRADIENTS . We present the definition of gradient on a Riemannian manifold , and show the dependence of the gradient on the chosen metric on the manifold ( see Carmo ( 1992 ) ; Abraham et al . ( 2012 ) for more details ) . A manifold X is a space that is locally linear around each point X ∈ X ; this linear space is the tangent space , denoted TXX . A Riemannian manifold has a smoothly varying positive definite bilinear form 〈· , ·〉 ( called the metric ) on the tangent space . This metric allows one to define the notion of lengths of curves on the space , in addition to other operations , including gradients . Definition 1 ( Gradient of a Function ) Let X be a Riemannian manifold , and f : X → R be a function . The directional derivative of f at X ∈ X along a direction k ∈ TXX is defined as df ( X ) · k = ddε f ( X + εk ) \|ε=0 . The gradient of f at X ∈ X is the vector , ∇f ( X ) ∈ TXX , that satisfies the relation df ( X ) · k = 〈∇f ( X ) , k〉 , for all k ∈ TXX . ( 1 ) Note that “ the ” gradient will depend on the choice of the metric on the manifold . We note that any such gradient will decrease the the function f by moving infinitesimally in the tangent space in the direction of negative the gradient as df ( X ) · k = −‖∇f ( X ) ‖2 < 0 when k = −∇f ( X ) , where ‖ · ‖ is the norm induced from the metric . The gradient flow , defined by the differential equation Ẋt = −∇f ( Xt ) , will converge to a local minimum . In our application of this theory to CNN optimization , f will be the loss function , and X will be the space of parameter tensors . A consequence of this definition is that the gradient is the direction ( up to a scale factor ) in the tangent space that optimizes the following problem : arg max k∈TXX\ { 0 } \|df ( X ) · k\| ‖k‖ . ( 2 ) Thus , the gradient can be regarded as the most efficient direction as it maximizes the ratio of the change in energy by perturbing in a direction k over the cost ( defined by the metric ) of k. Thus , by constructing the metric to have small costs for perturbations ( directions ) that we prefer for gradients , the gradient flow will move in these preferential directions while minimizng the function , and thus land in favorable local minima . 2.2 CHANNEL-DIRECTED METRICS . In existing deep network gradient-based optimization schemes , the underlying metric on the loss function is assumed to be the standard Euclidean L2 metric . We will consider a re-weighted version of the L2 metric and a Sobolev metric that favor correlation in the output channel direction of the gradient and thus the parameter tensors . To formulate the methodology , we start from a continuum formulation , where we treat weight tensors in the continuum , formulate the metrics in the continuum and then in the next sub-section derive the gradients with respect to these metrics . Finally , we discretize gradient flows in the implementation to derive iterative schemes . Let X : O × I × H × W → R denote a parameter tensor of a layer of a convolutional neural network . Here O = [ 0 , O ] denotes indices to the output channel dimension of the tensor , I = [ 0 , I ] Figure 2 : Visualization of kernels applied to the H0 gradient under different metrics for λ = 1 . This illustrates the smoothing effect of the metrics . In computation , linear cost formulas are applied to compute the gradients not using the convolution interpretation . denote the indices to the input channel , and H = [ 0 , H ] , W = [ 0 , W ] denote indices to the height and width dimension of the spatial filters of the tensor . The metric is defined on the tangent space to the space of such X . An element of the tangent space will have the same form of the tensor , i.e. , k : O × I ×H×W → R. The L2 ( called H0 from now on ) metric is defined as 〈k1 , k2〉 H0 = ∫ O , I , H , W k1 ( o , i , h , w ) · k2 ( o , i , h , w ) do didhdw , ( 3 ) where k1 , k2 are in the tangent space of tensors . We now define a re-weighted version of H0 that favors tangent vectors that have global smoothness in the direction of the O dimension : 〈k1 , k2〉H0λ = ∫ I , H , W k̄1 ( i , h , w ) · k̄2 ( i , h , w ) didhdw + λ O 〈 k1 − k̄1 , k2 − k̄2 〉 H0 , ( 4 ) where λ > 0 is a hyper-parameter , and k̄ is the average value in the output channel direction , i.e. , k̄ ( i , h , w ) = 1 O ∫ O k ( o , i , h , w ) do . ( 5 ) The metric in ( 4 ) splits the tangent vector into global translations in the output channel direction and its orthogonal complement , i.e. , the deformation . The weight λ is used to control the weighting between the translation and deformation components , i.e. , larger values of λ means that deformations more heavily influence the norm of the perturbation . As shown in the next sub-section that means gradients with respect to this metric have higher weighted channel-directed translations than deformations . Next , we introduce a channel-directed version of a Sobolev metric , defined as follows : 〈k1 , k2〉H̃1 = ∫ I , H , W k̄1 ( i , h , w ) · k̄2 ( i , h , w ) di dhdw + λO 〈 ∂k1 ∂o , ∂k2 ∂o 〉 H0 , ( 6 ) where ∂∂o indicates the partial derivative with respect to the the output channel direction . The partial derivative in the o-direction implies that tensor perturbations that are smooth along the o-direction are close with respect to these metrics , which will imply that the corresponding gradients will exhibit smoothness in this direction , i.e. , convolution filters that are nearby in the output direction will exhibit correlation . The metric is a weighted combination of the H0 metric of the derivative in the output direction , and the H0 metric of the output-directed translation . Note that the traditional Sobolev metric uses the H0 metric of the perturbation rather than the translation . Our choice is motivated by computational efficiency of the corresponding gradient , to be discussed below . The scale factors of O in the expressions above are so that the metric is scale invariant with respect to different sizes of output channels . The part of the metric with the partial derivative component implies that tensors that differ in the output channel direction by a non-smooth perturbation are far away in distance . Tensors that differ by just a channel-directed translation are close .	This paper follows a very active line of research on gradient-based optimization methods for convolutional neural networks. In particular, this work proposes a method that can be applied as an extension to popular optimization methods such as SGD and Adam. The main idea is to introduce a modification of the tensor space in order to encourage correlations in the output-channel dimension. In particular, they propose two channel re-weighting strategies: H^0 and H^1. The theoretical foundation behind the paper seems to be sound and the overall writing is clear. Results indicate that, indeed, the proposed modifications help to improve the performance of the baselines: SGD and Adam. This is shown using several datasets. Furthermore, the proposed method does not produce a significant overhead with respect to the regular operation of the baselines.	SP:dfb0838d0bd978aeff5b8b6b165d86b7e28ceccc
Remembering for the Right Reasons: Explanations Reduce Catastrophic Forgetting	1 INTRODUCTION . Humans are capable of continuously learning novel tasks by leveraging their lifetime knowledge and expanding them when they encounter a new experience . They can remember the majority of their prior knowledge despite the never-ending nature of their learning process by simply keeping a running tally of the observations distributed over time or presented in summary form . The field of continual learning or lifelong learning ( Thrun & Mitchell , 1995 ; Silver et al. , 2013 ) aims at maintaining previous performance and avoiding so-called catastrophic forgetting of previous experience ( McCloskey & Cohen , 1989 ; McClelland et al. , 1995 ) when learning new skills . The goal is to develop algorithms that continually update or add parameters to accommodate an online stream of data over time . An active line of research in continual learning explores the effectiveness of using small memory budgets to store data points from the training set ( Castro et al. , 2018 ; Rajasegaran et al. , 2020 ; Rebuffi et al. , 2017 ; Wu et al. , 2019 ) , gradients ( Lopez-Paz et al. , 2017 ) , or storing an online generative model that can fake them later ( Shin et al. , 2017 ) . Memory has been also exploited in the form of accommodating space for architecture growth and storage to fully recover the old performance when needed ( Ebrahimi et al. , 2020b ; Rusu et al. , 2016 ) . Some methods store an old snapshot of the model to distill the features ( Li & Hoiem , 2016 ) or attention maps ( Dhar et al. , 2019 ) between the teacher and student models . The internal reasoning process of deep models is often treated as a black box and remains hidden from the user . However , recent work in explainable artificial intelligence ( XAI ) has developed methods to create human-interpretable explanations for model decisions ( Simonyan et al. , 2013 ; Zhang et al. , 2018 ; Petsiuk et al. , 2018 ; Zhou et al. , 2016 ; Selvaraju et al. , 2017 ) . We posit that the catastrophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was used for a previously seen observation . Therefore , we hypothesize that forgetting can be mitigated when the model is encouraged to remember the evidence for previously made decisions . In other words , a model which can remember its final decision and can reconstruct the same prior reasoning . Based on this approach , we develop a novel strategy to exploit explainable models for improving performance . Among the various explainability techniques proposed in XAI , saliency methods have emerged as a popular tool to identify the support of a model prediction in terms of relevant features in the input . These methods produce saliency maps , defined as regions of visual evidence upon which a network makes a decision . Our goal is to investigate whether augmenting experience replay with explanation replay reduces forgetting and how enforcing to remember the explanations will affect the explanations themselves . Figure 1 illustrates our proposed method . In this work , we propose RRR , a training strategy guided by model explanations generated by any white-box differentiable explanation method ; RRR adds an explanation loss to continual learning . White-box methods generate an explanation by using some internal state of the model , such as gradients , enabling their use in end-to-end training . We evaluate our approach using various popular explanation methods including vanilla backpropagation ( Zeiler & Fergus , 2014 ) , backpropagation with smoothing gradients ( Smoothgrad ) ( Smilkov et al. , 2017 ) , Guided Backpropagation ( Springenberg et al. , 2014 ) , and Gradient Class Activation Mapping ( Grad-CAM ) ( Selvaraju et al. , 2017 ) and compare their performance versus their computational feasibility . We integrate RRR into several state of the art class incremental learning ( CIL ) methods , including iTAML ( Rajasegaran et al. , 2020 ) , EEIl ( Castro et al. , 2018 ) , BiC ( Wu et al. , 2019 ) , TOPIC ( Tao et al. , 2020 ) , iCaRL ( Rebuffi et al. , 2017 ) , EWC ( Kirkpatrick et al. , 2017 ) , and LwF ( Li & Hoiem , 2016 ) . Note that RRR does not require task IDs at test time . We qualitatively and quantitatively analyze model explanations in the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence of tasks due to the requirement to focus on the the right evidence . We empirically show the effect of RRR in standard and few-shot class incremental learning ( CIL ) scenarios on popular benchmark datasets including CIFAR100 , ImageNet100 , and Caltech-UCSD Birds 200 using different network architectures where RRR improves overall accuracy and forgetting over experience replay and other memory-based method . Our contribution is threefold : we first propose our novel , simple , yet effective memory constraint , which we call Remembering for the Right Reasons ( RRR ) , and show that it reduces catastrophic forgetting by encouraging the model to look at the same explanations it initially found for its decisions . Second , we show how RRR can be readily combined with memory-based and regularization-based CL methods to improve performance . Third , we demonstrate how guiding a continual learner to remember its explanations can improve the quality of the explanations themselves ; i.e. , the model looks at the right region in an image when making correct decisions while it focuses its maximum attention on the background when it misclassifies an object . 2 BACKGROUND : WHITE-BOX EXPLANABILITY TECHNIQUES . Here we briefly review the explainability methods we have evaluated our approach with . The core idea behind RRR is to guide explanations or saliency maps during training to preserve their values . Hence , only gradient-based saliency techniques can be used which are differentiable and hence trainable during training for the mainstream task as opposed to black-box saliency methods which can be used only after training to determine important parts of an image . Vanilla Backpropagation ( Zeiler & Fergus , 2014 ) : The simplest way to understand and visualize which pixels are most salient in an image is to look at the gradients . This is typically done by making a forward pass through the model and taking the gradient of the given output class with respect to the input . Those pixel-wise derivative values can be rendered as a normalized heatmap representing the amount of change in the output probability of a class caused by perturbing that pixel . To store a saliency map for each RGB image of size 3 ×W × H , we need an equivalent memory size of storing W ×H pixel values . Backpropagation with SmoothGrad : Smilkov et al . ( 2017 ) showed that the saliency maps obtained using raw gradients are visually noisy and using them as a proxy for feature importance is sub-optimal . They proposed a simple technique for denoising the gradients that adds pixel-wise Gaussian noise to n copies of the image , and simply averages the resulting gradients . SmoothGrad requires the same amount of memory to store the saliency maps while it takes n times longer to repeat generating each saliency map . We found n = 40 to be large enough to make a noticeable change in the saliencies in our experiments . Gradient-weighted Class Activation Mapping ( Grad-CAM ) ( Selvaraju et al. , 2017 ) : is a whitebox explainability technique which uses gradients to determine the influence of specific feature map activations on a given prediction . Because later layers in a convolutional neural network are known to encode higher-level semantics , taking the gradient of a model output with respect to the activations of these feature maps discovers which high-level semantics are important for the prediction . We refer to this layer as the target layer in our analysis . For example , when using Grad-CAM to visualize explanations for image classification , taking the gradient of the correct class prediction with respect to the last convolutional layer highlights class-discriminative regions in the image ( such as the wings of a bird when identifying bird species ) . Consider the pre-softmax score yc for class c in an image classification output . In general , any differentiable activation can be used . Consider also a single convolutional layer with K feature maps , with a single feature map noted as Ak ∈ Ru×v . Grad-CAM takes the derivative of yc with respect to each feature map Ak . It then performs global average pooling over the height and width dimensions for each of these feature map gradients , getting a vector of length K. Each element in this vector is used as a weight αck , indicating the importance of feature map k for the prediction yc . Next , these importance weights are used in computing a linear combination of the feature maps . Followed by a ReLU ( Jarrett et al. , 2009 ) to zero-out any activations with a negative influence on the prediction of class c , the final Grad-CAM output ( s ) is as below with Akij defined at location ( i , j ) in feature map Ak . αck = 1 uv u∑ i=1 v∑ j=1 ∂yc ∂Akij scGrad-CAM = ReLU ( K∑ k=1 αckA k ) ( 1 ) Unlike the common saliency map techniques of Guided BackProp ( Springenberg et al. , 2014 ) , Guided GradCAM ( Selvaraju et al. , 2016 ) , Integrated Gradients ( Sundararajan et al. , 2017b ) , Gradient Input ( Shrikumar et al. , 2016 ) , Backpropagation with SmoothGrad ( Smilkov et al. , 2017 ) etc. , vanilla Backpropagation and Grad-CAM pass important “ sanity checks ” regarding their sensitivity to data and model parameters ( Adebayo et al. , 2018 ) . We will compare using vanilla Backpropagation , Backpropagation with SmoothGrad , and Grad-CAM in RRR in Section 4 . We will refer to the function that computes the output s of these saliency method as XAI . Algorithm 1 Remembering for the Right Reasons ( RRR ) for Continual Learning 1 : function TRAIN ( fθ , Dtr , Dts ) 2 : T : # of tasks , n : # of samples in task 3 : R← 0 ∈ RT×T 4 : Mrep ← { } 5 : MRRR ← { } 6 : for k = 1 to T do 7 : for i = 1 to n do 8 : Compute cross entropy on task ( Ltask ) 9 : Compute LRRR using Eq . 2 10 : θ′ ← θ − α∇θ ( Ltask+LRRR ) 11 : end for 12 : Mrep , MRRR ← UPDATE MEM ( fkθ , Dtrk , Mrep , MRRR ) 13 : Rk , { 1···k } ← EVAL ( fkθ , Dts { 1···k } ) 14 : end for 15 : return fθ , R 16 : end function function UPDATE MEM ( fkθ , Dtrk , Mrep , MRRR ) ( xi , k , yi ) ∼ Dtrk Mrep ←Mrep ∪ { ( xi , k , yi ) } ŝ← XAI ( fkθ ( xi , k ) ) MRRR ←MRRR ∪ { ŝ } returnMrep , MRRR end function function EVAL ( fkθ , Dts { 1···k } ) for i = 1 to k do Rk , i = Accuracy ( fkθ ( x , i ) , y\|∀ ( x , y ) ∈ Dtsi ) end for return R end function 3 REMEMBERING FOR THE RIGHT REASONS ( RRR ) . Memory-based methods in continual learning have achieved high performance on vision benchmarks using a small amount of memory , i.e . storing a few samples from the training data into the replay buffer to directly train with them when learning new tasks . This simple method , known as experience replay , has been explored and shown to be effective ( Rebuffi et al. , 2017 ; Wu et al. , 2019 ; Castro et al. , 2018 ; Rajasegaran et al. , 2020 ; Ebrahimi et al. , 2020b ; Hayes et al. , 2019 ; Riemer et al. , 2018 ) . In this work we aim to go one step further and investigate the role of explanations in continual learning , particularly on mitigating forgetting and change of model explanations . We consider the problem of learning a sequence of T data distributions Dtr = { Dtr1 , · · · , DtrT } , whereDtrk = { ( xki , yki ) nk i=1 } is the data distribution for task k with n sample tuples of input ( xk ⊂ X ) and set of output labels ( yk ⊂ Y ) . The goal is to sequentially learn the model fθ : X × T → Y for each task that can map each input x to its target output , y , while maintaining its performance on all prior tasks . We aim to achieve this by using memory to enhance better knowledge transfer as well as better avoidance of catastrophic forgetting . We assume two limited memory poolsMrep for raw samples andMRRR for model explanations . In particular , Mrep = { ( xji , y j i ) m i=1 ∼ Dtrj=1···k−1 } stores m samples in total from all prior tasks to k. SimilarlyMRRR stores the saliency maps generated based on fkθ by one of the explanation methods ( XAI ) discussed in Section 2 for images in Mrep where fkθ is fθ being trained for task k. We use a single-head architecture where the task ID integer t is not required at test time . Upon finishing the kth task , we randomly select m/ ( k−1 ) samples per task from its training data and update our replay buffer memoryMrep . RRR uses model explanations on memory samples to perform continual learning such that the model preserves its reasoning for previously seen observations . We explore several explanation techniques to compute saliency maps using fkθ for the stored samples in the replay buffer to populate the xai buffer memoryMxai . The stored saliency maps will serve as reference explanations during the learning of future tasks to prevent model parameters from being altered resulting in different reasoning for the same samples . We implement RRR using an L1 loss on the error in saliency maps generated after training a new task with respect to their stored reference evidence . LRRR ( fθ , Mrep , MRRR ) = E ( ( x , y ) , ŝ ) ∼ ( Mrep , MRRR ) \|\|XAI ( fkθ ( x ) ) − ŝ\|\|1 ( 2 ) where XAI ( · ) denotes the explanation method used to compute saliency maps using the model trained on the last seen example from task k , and ŝ are the reference saliency maps generated by XAI ( fkθ ) upon learning each task prior to T and stored in to the memory . We show below that combining RRR into the objective function of state-of-the-art memory and regularization-based methods results in significant performance improvements . The full algorithm for RRR is given in Alg . 1 .	This paper proposes a method for continual learning of a sequence of supervised tasks which is based on memory-replay for remembering evidence from previously made decisions. This evidence relies on explanations of those decisions rather than data. These explanations are incorporated directly into the loss function. The proposed approach is tested against several counterparts using benchmark datasets in continual learning.	SP:4d770cdb47df15333de256052add8a4639294809
Remembering for the Right Reasons: Explanations Reduce Catastrophic Forgetting	1 INTRODUCTION . Humans are capable of continuously learning novel tasks by leveraging their lifetime knowledge and expanding them when they encounter a new experience . They can remember the majority of their prior knowledge despite the never-ending nature of their learning process by simply keeping a running tally of the observations distributed over time or presented in summary form . The field of continual learning or lifelong learning ( Thrun & Mitchell , 1995 ; Silver et al. , 2013 ) aims at maintaining previous performance and avoiding so-called catastrophic forgetting of previous experience ( McCloskey & Cohen , 1989 ; McClelland et al. , 1995 ) when learning new skills . The goal is to develop algorithms that continually update or add parameters to accommodate an online stream of data over time . An active line of research in continual learning explores the effectiveness of using small memory budgets to store data points from the training set ( Castro et al. , 2018 ; Rajasegaran et al. , 2020 ; Rebuffi et al. , 2017 ; Wu et al. , 2019 ) , gradients ( Lopez-Paz et al. , 2017 ) , or storing an online generative model that can fake them later ( Shin et al. , 2017 ) . Memory has been also exploited in the form of accommodating space for architecture growth and storage to fully recover the old performance when needed ( Ebrahimi et al. , 2020b ; Rusu et al. , 2016 ) . Some methods store an old snapshot of the model to distill the features ( Li & Hoiem , 2016 ) or attention maps ( Dhar et al. , 2019 ) between the teacher and student models . The internal reasoning process of deep models is often treated as a black box and remains hidden from the user . However , recent work in explainable artificial intelligence ( XAI ) has developed methods to create human-interpretable explanations for model decisions ( Simonyan et al. , 2013 ; Zhang et al. , 2018 ; Petsiuk et al. , 2018 ; Zhou et al. , 2016 ; Selvaraju et al. , 2017 ) . We posit that the catastrophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was used for a previously seen observation . Therefore , we hypothesize that forgetting can be mitigated when the model is encouraged to remember the evidence for previously made decisions . In other words , a model which can remember its final decision and can reconstruct the same prior reasoning . Based on this approach , we develop a novel strategy to exploit explainable models for improving performance . Among the various explainability techniques proposed in XAI , saliency methods have emerged as a popular tool to identify the support of a model prediction in terms of relevant features in the input . These methods produce saliency maps , defined as regions of visual evidence upon which a network makes a decision . Our goal is to investigate whether augmenting experience replay with explanation replay reduces forgetting and how enforcing to remember the explanations will affect the explanations themselves . Figure 1 illustrates our proposed method . In this work , we propose RRR , a training strategy guided by model explanations generated by any white-box differentiable explanation method ; RRR adds an explanation loss to continual learning . White-box methods generate an explanation by using some internal state of the model , such as gradients , enabling their use in end-to-end training . We evaluate our approach using various popular explanation methods including vanilla backpropagation ( Zeiler & Fergus , 2014 ) , backpropagation with smoothing gradients ( Smoothgrad ) ( Smilkov et al. , 2017 ) , Guided Backpropagation ( Springenberg et al. , 2014 ) , and Gradient Class Activation Mapping ( Grad-CAM ) ( Selvaraju et al. , 2017 ) and compare their performance versus their computational feasibility . We integrate RRR into several state of the art class incremental learning ( CIL ) methods , including iTAML ( Rajasegaran et al. , 2020 ) , EEIl ( Castro et al. , 2018 ) , BiC ( Wu et al. , 2019 ) , TOPIC ( Tao et al. , 2020 ) , iCaRL ( Rebuffi et al. , 2017 ) , EWC ( Kirkpatrick et al. , 2017 ) , and LwF ( Li & Hoiem , 2016 ) . Note that RRR does not require task IDs at test time . We qualitatively and quantitatively analyze model explanations in the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence of tasks due to the requirement to focus on the the right evidence . We empirically show the effect of RRR in standard and few-shot class incremental learning ( CIL ) scenarios on popular benchmark datasets including CIFAR100 , ImageNet100 , and Caltech-UCSD Birds 200 using different network architectures where RRR improves overall accuracy and forgetting over experience replay and other memory-based method . Our contribution is threefold : we first propose our novel , simple , yet effective memory constraint , which we call Remembering for the Right Reasons ( RRR ) , and show that it reduces catastrophic forgetting by encouraging the model to look at the same explanations it initially found for its decisions . Second , we show how RRR can be readily combined with memory-based and regularization-based CL methods to improve performance . Third , we demonstrate how guiding a continual learner to remember its explanations can improve the quality of the explanations themselves ; i.e. , the model looks at the right region in an image when making correct decisions while it focuses its maximum attention on the background when it misclassifies an object . 2 BACKGROUND : WHITE-BOX EXPLANABILITY TECHNIQUES . Here we briefly review the explainability methods we have evaluated our approach with . The core idea behind RRR is to guide explanations or saliency maps during training to preserve their values . Hence , only gradient-based saliency techniques can be used which are differentiable and hence trainable during training for the mainstream task as opposed to black-box saliency methods which can be used only after training to determine important parts of an image . Vanilla Backpropagation ( Zeiler & Fergus , 2014 ) : The simplest way to understand and visualize which pixels are most salient in an image is to look at the gradients . This is typically done by making a forward pass through the model and taking the gradient of the given output class with respect to the input . Those pixel-wise derivative values can be rendered as a normalized heatmap representing the amount of change in the output probability of a class caused by perturbing that pixel . To store a saliency map for each RGB image of size 3 ×W × H , we need an equivalent memory size of storing W ×H pixel values . Backpropagation with SmoothGrad : Smilkov et al . ( 2017 ) showed that the saliency maps obtained using raw gradients are visually noisy and using them as a proxy for feature importance is sub-optimal . They proposed a simple technique for denoising the gradients that adds pixel-wise Gaussian noise to n copies of the image , and simply averages the resulting gradients . SmoothGrad requires the same amount of memory to store the saliency maps while it takes n times longer to repeat generating each saliency map . We found n = 40 to be large enough to make a noticeable change in the saliencies in our experiments . Gradient-weighted Class Activation Mapping ( Grad-CAM ) ( Selvaraju et al. , 2017 ) : is a whitebox explainability technique which uses gradients to determine the influence of specific feature map activations on a given prediction . Because later layers in a convolutional neural network are known to encode higher-level semantics , taking the gradient of a model output with respect to the activations of these feature maps discovers which high-level semantics are important for the prediction . We refer to this layer as the target layer in our analysis . For example , when using Grad-CAM to visualize explanations for image classification , taking the gradient of the correct class prediction with respect to the last convolutional layer highlights class-discriminative regions in the image ( such as the wings of a bird when identifying bird species ) . Consider the pre-softmax score yc for class c in an image classification output . In general , any differentiable activation can be used . Consider also a single convolutional layer with K feature maps , with a single feature map noted as Ak ∈ Ru×v . Grad-CAM takes the derivative of yc with respect to each feature map Ak . It then performs global average pooling over the height and width dimensions for each of these feature map gradients , getting a vector of length K. Each element in this vector is used as a weight αck , indicating the importance of feature map k for the prediction yc . Next , these importance weights are used in computing a linear combination of the feature maps . Followed by a ReLU ( Jarrett et al. , 2009 ) to zero-out any activations with a negative influence on the prediction of class c , the final Grad-CAM output ( s ) is as below with Akij defined at location ( i , j ) in feature map Ak . αck = 1 uv u∑ i=1 v∑ j=1 ∂yc ∂Akij scGrad-CAM = ReLU ( K∑ k=1 αckA k ) ( 1 ) Unlike the common saliency map techniques of Guided BackProp ( Springenberg et al. , 2014 ) , Guided GradCAM ( Selvaraju et al. , 2016 ) , Integrated Gradients ( Sundararajan et al. , 2017b ) , Gradient Input ( Shrikumar et al. , 2016 ) , Backpropagation with SmoothGrad ( Smilkov et al. , 2017 ) etc. , vanilla Backpropagation and Grad-CAM pass important “ sanity checks ” regarding their sensitivity to data and model parameters ( Adebayo et al. , 2018 ) . We will compare using vanilla Backpropagation , Backpropagation with SmoothGrad , and Grad-CAM in RRR in Section 4 . We will refer to the function that computes the output s of these saliency method as XAI . Algorithm 1 Remembering for the Right Reasons ( RRR ) for Continual Learning 1 : function TRAIN ( fθ , Dtr , Dts ) 2 : T : # of tasks , n : # of samples in task 3 : R← 0 ∈ RT×T 4 : Mrep ← { } 5 : MRRR ← { } 6 : for k = 1 to T do 7 : for i = 1 to n do 8 : Compute cross entropy on task ( Ltask ) 9 : Compute LRRR using Eq . 2 10 : θ′ ← θ − α∇θ ( Ltask+LRRR ) 11 : end for 12 : Mrep , MRRR ← UPDATE MEM ( fkθ , Dtrk , Mrep , MRRR ) 13 : Rk , { 1···k } ← EVAL ( fkθ , Dts { 1···k } ) 14 : end for 15 : return fθ , R 16 : end function function UPDATE MEM ( fkθ , Dtrk , Mrep , MRRR ) ( xi , k , yi ) ∼ Dtrk Mrep ←Mrep ∪ { ( xi , k , yi ) } ŝ← XAI ( fkθ ( xi , k ) ) MRRR ←MRRR ∪ { ŝ } returnMrep , MRRR end function function EVAL ( fkθ , Dts { 1···k } ) for i = 1 to k do Rk , i = Accuracy ( fkθ ( x , i ) , y\|∀ ( x , y ) ∈ Dtsi ) end for return R end function 3 REMEMBERING FOR THE RIGHT REASONS ( RRR ) . Memory-based methods in continual learning have achieved high performance on vision benchmarks using a small amount of memory , i.e . storing a few samples from the training data into the replay buffer to directly train with them when learning new tasks . This simple method , known as experience replay , has been explored and shown to be effective ( Rebuffi et al. , 2017 ; Wu et al. , 2019 ; Castro et al. , 2018 ; Rajasegaran et al. , 2020 ; Ebrahimi et al. , 2020b ; Hayes et al. , 2019 ; Riemer et al. , 2018 ) . In this work we aim to go one step further and investigate the role of explanations in continual learning , particularly on mitigating forgetting and change of model explanations . We consider the problem of learning a sequence of T data distributions Dtr = { Dtr1 , · · · , DtrT } , whereDtrk = { ( xki , yki ) nk i=1 } is the data distribution for task k with n sample tuples of input ( xk ⊂ X ) and set of output labels ( yk ⊂ Y ) . The goal is to sequentially learn the model fθ : X × T → Y for each task that can map each input x to its target output , y , while maintaining its performance on all prior tasks . We aim to achieve this by using memory to enhance better knowledge transfer as well as better avoidance of catastrophic forgetting . We assume two limited memory poolsMrep for raw samples andMRRR for model explanations . In particular , Mrep = { ( xji , y j i ) m i=1 ∼ Dtrj=1···k−1 } stores m samples in total from all prior tasks to k. SimilarlyMRRR stores the saliency maps generated based on fkθ by one of the explanation methods ( XAI ) discussed in Section 2 for images in Mrep where fkθ is fθ being trained for task k. We use a single-head architecture where the task ID integer t is not required at test time . Upon finishing the kth task , we randomly select m/ ( k−1 ) samples per task from its training data and update our replay buffer memoryMrep . RRR uses model explanations on memory samples to perform continual learning such that the model preserves its reasoning for previously seen observations . We explore several explanation techniques to compute saliency maps using fkθ for the stored samples in the replay buffer to populate the xai buffer memoryMxai . The stored saliency maps will serve as reference explanations during the learning of future tasks to prevent model parameters from being altered resulting in different reasoning for the same samples . We implement RRR using an L1 loss on the error in saliency maps generated after training a new task with respect to their stored reference evidence . LRRR ( fθ , Mrep , MRRR ) = E ( ( x , y ) , ŝ ) ∼ ( Mrep , MRRR ) \|\|XAI ( fkθ ( x ) ) − ŝ\|\|1 ( 2 ) where XAI ( · ) denotes the explanation method used to compute saliency maps using the model trained on the last seen example from task k , and ŝ are the reference saliency maps generated by XAI ( fkθ ) upon learning each task prior to T and stored in to the memory . We show below that combining RRR into the objective function of state-of-the-art memory and regularization-based methods results in significant performance improvements . The full algorithm for RRR is given in Alg . 1 .	The proposed technique is simple. It helps existing replay methods to boost their accuracy for class-incremental learning. In addition to save some training examples of previous tasks or classes, it also saves the saliency maps of these examples as explanations of classification. The size the saliency maps is small and requires very little memory. For continual learning, it includes additional term in the loss function to take care of the saved saliency maps. Experimental results show that this additional saved information can help improve existing replay methods for class-incremental learning.	SP:4d770cdb47df15333de256052add8a4639294809
Remembering for the Right Reasons: Explanations Reduce Catastrophic Forgetting	1 INTRODUCTION . Humans are capable of continuously learning novel tasks by leveraging their lifetime knowledge and expanding them when they encounter a new experience . They can remember the majority of their prior knowledge despite the never-ending nature of their learning process by simply keeping a running tally of the observations distributed over time or presented in summary form . The field of continual learning or lifelong learning ( Thrun & Mitchell , 1995 ; Silver et al. , 2013 ) aims at maintaining previous performance and avoiding so-called catastrophic forgetting of previous experience ( McCloskey & Cohen , 1989 ; McClelland et al. , 1995 ) when learning new skills . The goal is to develop algorithms that continually update or add parameters to accommodate an online stream of data over time . An active line of research in continual learning explores the effectiveness of using small memory budgets to store data points from the training set ( Castro et al. , 2018 ; Rajasegaran et al. , 2020 ; Rebuffi et al. , 2017 ; Wu et al. , 2019 ) , gradients ( Lopez-Paz et al. , 2017 ) , or storing an online generative model that can fake them later ( Shin et al. , 2017 ) . Memory has been also exploited in the form of accommodating space for architecture growth and storage to fully recover the old performance when needed ( Ebrahimi et al. , 2020b ; Rusu et al. , 2016 ) . Some methods store an old snapshot of the model to distill the features ( Li & Hoiem , 2016 ) or attention maps ( Dhar et al. , 2019 ) between the teacher and student models . The internal reasoning process of deep models is often treated as a black box and remains hidden from the user . However , recent work in explainable artificial intelligence ( XAI ) has developed methods to create human-interpretable explanations for model decisions ( Simonyan et al. , 2013 ; Zhang et al. , 2018 ; Petsiuk et al. , 2018 ; Zhou et al. , 2016 ; Selvaraju et al. , 2017 ) . We posit that the catastrophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was used for a previously seen observation . Therefore , we hypothesize that forgetting can be mitigated when the model is encouraged to remember the evidence for previously made decisions . In other words , a model which can remember its final decision and can reconstruct the same prior reasoning . Based on this approach , we develop a novel strategy to exploit explainable models for improving performance . Among the various explainability techniques proposed in XAI , saliency methods have emerged as a popular tool to identify the support of a model prediction in terms of relevant features in the input . These methods produce saliency maps , defined as regions of visual evidence upon which a network makes a decision . Our goal is to investigate whether augmenting experience replay with explanation replay reduces forgetting and how enforcing to remember the explanations will affect the explanations themselves . Figure 1 illustrates our proposed method . In this work , we propose RRR , a training strategy guided by model explanations generated by any white-box differentiable explanation method ; RRR adds an explanation loss to continual learning . White-box methods generate an explanation by using some internal state of the model , such as gradients , enabling their use in end-to-end training . We evaluate our approach using various popular explanation methods including vanilla backpropagation ( Zeiler & Fergus , 2014 ) , backpropagation with smoothing gradients ( Smoothgrad ) ( Smilkov et al. , 2017 ) , Guided Backpropagation ( Springenberg et al. , 2014 ) , and Gradient Class Activation Mapping ( Grad-CAM ) ( Selvaraju et al. , 2017 ) and compare their performance versus their computational feasibility . We integrate RRR into several state of the art class incremental learning ( CIL ) methods , including iTAML ( Rajasegaran et al. , 2020 ) , EEIl ( Castro et al. , 2018 ) , BiC ( Wu et al. , 2019 ) , TOPIC ( Tao et al. , 2020 ) , iCaRL ( Rebuffi et al. , 2017 ) , EWC ( Kirkpatrick et al. , 2017 ) , and LwF ( Li & Hoiem , 2016 ) . Note that RRR does not require task IDs at test time . We qualitatively and quantitatively analyze model explanations in the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence of tasks due to the requirement to focus on the the right evidence . We empirically show the effect of RRR in standard and few-shot class incremental learning ( CIL ) scenarios on popular benchmark datasets including CIFAR100 , ImageNet100 , and Caltech-UCSD Birds 200 using different network architectures where RRR improves overall accuracy and forgetting over experience replay and other memory-based method . Our contribution is threefold : we first propose our novel , simple , yet effective memory constraint , which we call Remembering for the Right Reasons ( RRR ) , and show that it reduces catastrophic forgetting by encouraging the model to look at the same explanations it initially found for its decisions . Second , we show how RRR can be readily combined with memory-based and regularization-based CL methods to improve performance . Third , we demonstrate how guiding a continual learner to remember its explanations can improve the quality of the explanations themselves ; i.e. , the model looks at the right region in an image when making correct decisions while it focuses its maximum attention on the background when it misclassifies an object . 2 BACKGROUND : WHITE-BOX EXPLANABILITY TECHNIQUES . Here we briefly review the explainability methods we have evaluated our approach with . The core idea behind RRR is to guide explanations or saliency maps during training to preserve their values . Hence , only gradient-based saliency techniques can be used which are differentiable and hence trainable during training for the mainstream task as opposed to black-box saliency methods which can be used only after training to determine important parts of an image . Vanilla Backpropagation ( Zeiler & Fergus , 2014 ) : The simplest way to understand and visualize which pixels are most salient in an image is to look at the gradients . This is typically done by making a forward pass through the model and taking the gradient of the given output class with respect to the input . Those pixel-wise derivative values can be rendered as a normalized heatmap representing the amount of change in the output probability of a class caused by perturbing that pixel . To store a saliency map for each RGB image of size 3 ×W × H , we need an equivalent memory size of storing W ×H pixel values . Backpropagation with SmoothGrad : Smilkov et al . ( 2017 ) showed that the saliency maps obtained using raw gradients are visually noisy and using them as a proxy for feature importance is sub-optimal . They proposed a simple technique for denoising the gradients that adds pixel-wise Gaussian noise to n copies of the image , and simply averages the resulting gradients . SmoothGrad requires the same amount of memory to store the saliency maps while it takes n times longer to repeat generating each saliency map . We found n = 40 to be large enough to make a noticeable change in the saliencies in our experiments . Gradient-weighted Class Activation Mapping ( Grad-CAM ) ( Selvaraju et al. , 2017 ) : is a whitebox explainability technique which uses gradients to determine the influence of specific feature map activations on a given prediction . Because later layers in a convolutional neural network are known to encode higher-level semantics , taking the gradient of a model output with respect to the activations of these feature maps discovers which high-level semantics are important for the prediction . We refer to this layer as the target layer in our analysis . For example , when using Grad-CAM to visualize explanations for image classification , taking the gradient of the correct class prediction with respect to the last convolutional layer highlights class-discriminative regions in the image ( such as the wings of a bird when identifying bird species ) . Consider the pre-softmax score yc for class c in an image classification output . In general , any differentiable activation can be used . Consider also a single convolutional layer with K feature maps , with a single feature map noted as Ak ∈ Ru×v . Grad-CAM takes the derivative of yc with respect to each feature map Ak . It then performs global average pooling over the height and width dimensions for each of these feature map gradients , getting a vector of length K. Each element in this vector is used as a weight αck , indicating the importance of feature map k for the prediction yc . Next , these importance weights are used in computing a linear combination of the feature maps . Followed by a ReLU ( Jarrett et al. , 2009 ) to zero-out any activations with a negative influence on the prediction of class c , the final Grad-CAM output ( s ) is as below with Akij defined at location ( i , j ) in feature map Ak . αck = 1 uv u∑ i=1 v∑ j=1 ∂yc ∂Akij scGrad-CAM = ReLU ( K∑ k=1 αckA k ) ( 1 ) Unlike the common saliency map techniques of Guided BackProp ( Springenberg et al. , 2014 ) , Guided GradCAM ( Selvaraju et al. , 2016 ) , Integrated Gradients ( Sundararajan et al. , 2017b ) , Gradient Input ( Shrikumar et al. , 2016 ) , Backpropagation with SmoothGrad ( Smilkov et al. , 2017 ) etc. , vanilla Backpropagation and Grad-CAM pass important “ sanity checks ” regarding their sensitivity to data and model parameters ( Adebayo et al. , 2018 ) . We will compare using vanilla Backpropagation , Backpropagation with SmoothGrad , and Grad-CAM in RRR in Section 4 . We will refer to the function that computes the output s of these saliency method as XAI . Algorithm 1 Remembering for the Right Reasons ( RRR ) for Continual Learning 1 : function TRAIN ( fθ , Dtr , Dts ) 2 : T : # of tasks , n : # of samples in task 3 : R← 0 ∈ RT×T 4 : Mrep ← { } 5 : MRRR ← { } 6 : for k = 1 to T do 7 : for i = 1 to n do 8 : Compute cross entropy on task ( Ltask ) 9 : Compute LRRR using Eq . 2 10 : θ′ ← θ − α∇θ ( Ltask+LRRR ) 11 : end for 12 : Mrep , MRRR ← UPDATE MEM ( fkθ , Dtrk , Mrep , MRRR ) 13 : Rk , { 1···k } ← EVAL ( fkθ , Dts { 1···k } ) 14 : end for 15 : return fθ , R 16 : end function function UPDATE MEM ( fkθ , Dtrk , Mrep , MRRR ) ( xi , k , yi ) ∼ Dtrk Mrep ←Mrep ∪ { ( xi , k , yi ) } ŝ← XAI ( fkθ ( xi , k ) ) MRRR ←MRRR ∪ { ŝ } returnMrep , MRRR end function function EVAL ( fkθ , Dts { 1···k } ) for i = 1 to k do Rk , i = Accuracy ( fkθ ( x , i ) , y\|∀ ( x , y ) ∈ Dtsi ) end for return R end function 3 REMEMBERING FOR THE RIGHT REASONS ( RRR ) . Memory-based methods in continual learning have achieved high performance on vision benchmarks using a small amount of memory , i.e . storing a few samples from the training data into the replay buffer to directly train with them when learning new tasks . This simple method , known as experience replay , has been explored and shown to be effective ( Rebuffi et al. , 2017 ; Wu et al. , 2019 ; Castro et al. , 2018 ; Rajasegaran et al. , 2020 ; Ebrahimi et al. , 2020b ; Hayes et al. , 2019 ; Riemer et al. , 2018 ) . In this work we aim to go one step further and investigate the role of explanations in continual learning , particularly on mitigating forgetting and change of model explanations . We consider the problem of learning a sequence of T data distributions Dtr = { Dtr1 , · · · , DtrT } , whereDtrk = { ( xki , yki ) nk i=1 } is the data distribution for task k with n sample tuples of input ( xk ⊂ X ) and set of output labels ( yk ⊂ Y ) . The goal is to sequentially learn the model fθ : X × T → Y for each task that can map each input x to its target output , y , while maintaining its performance on all prior tasks . We aim to achieve this by using memory to enhance better knowledge transfer as well as better avoidance of catastrophic forgetting . We assume two limited memory poolsMrep for raw samples andMRRR for model explanations . In particular , Mrep = { ( xji , y j i ) m i=1 ∼ Dtrj=1···k−1 } stores m samples in total from all prior tasks to k. SimilarlyMRRR stores the saliency maps generated based on fkθ by one of the explanation methods ( XAI ) discussed in Section 2 for images in Mrep where fkθ is fθ being trained for task k. We use a single-head architecture where the task ID integer t is not required at test time . Upon finishing the kth task , we randomly select m/ ( k−1 ) samples per task from its training data and update our replay buffer memoryMrep . RRR uses model explanations on memory samples to perform continual learning such that the model preserves its reasoning for previously seen observations . We explore several explanation techniques to compute saliency maps using fkθ for the stored samples in the replay buffer to populate the xai buffer memoryMxai . The stored saliency maps will serve as reference explanations during the learning of future tasks to prevent model parameters from being altered resulting in different reasoning for the same samples . We implement RRR using an L1 loss on the error in saliency maps generated after training a new task with respect to their stored reference evidence . LRRR ( fθ , Mrep , MRRR ) = E ( ( x , y ) , ŝ ) ∼ ( Mrep , MRRR ) \|\|XAI ( fkθ ( x ) ) − ŝ\|\|1 ( 2 ) where XAI ( · ) denotes the explanation method used to compute saliency maps using the model trained on the last seen example from task k , and ŝ are the reference saliency maps generated by XAI ( fkθ ) upon learning each task prior to T and stored in to the memory . We show below that combining RRR into the objective function of state-of-the-art memory and regularization-based methods results in significant performance improvements . The full algorithm for RRR is given in Alg . 1 .	This paper tackles the problem of catastrophic forgetting in a continual learning scenario, in which the same classifier is trained incrementally on new classification tasks, each defined on a new set of output classes, and asked to retain performance on all the previous tasks. To tackle this problem, a stream of approaches keep a minimal "replay buffer" of examples and their labels from each task, that can be accessed during sequential task learning. This paper belongs to this stream of approaches. A naive way of using the replay buffer is to mix the buffer to the current task training set. Additionally, this paper proposes to store gradient attribution maps in the replay buffer and use the stored gradient attribution maps as additional targets for the current model (RRR loss): the model should have "explanations" of previous examples that roughly don't change upon seeing new evidence from other classes. The authors show that the proposed RRR loss can be applied to multiple baselines and improves performance on a continual learning and few-shot continual learning scenario.	SP:4d770cdb47df15333de256052add8a4639294809
Grounded Compositional Generalization with Environment Interactions	1 INTRODUCTION . Compositional generalization is a key skill for flexible and efficient learning . Humans leverage compositionality to create and recognize new combinations of familiar concepts ( Chomsky , 1957 ; Minsky , 1986 ) . Though there are many progresses for machine learning and deep learning in various areas recently ( LeCun et al. , 2015 ) , current main learning algorithms are not able to perform compositional generalization , and require many samples to train models . Such efficient learning is even more important when machines interact with the environment for grounding , because interactions are usually slow . Machine learning has been mostly developed with an assumption that training and test distributions are identical . Compositional generalization , however , is a kind of out-of-distribution generalization ( Bengio , 2017 ) , where training and test distributions are different . During training , dataset does not contain the information of the difference , so it can only be given as prior knowledge . In compositional generalization , a sample is a combination of several components . Test distribution changes as test samples are new combinations of seen components in training . For example , if we can find “ large apple ” and “ small orange ” in some environments , then we can also find “ large orange ” among multiple objects in a new environment . The recombination is enabled when an output component depends only on the corresponding input components , and invariant of other components ( please see Section 4.1 for more details ) . So there are two aspects to consider . What are the components in output , and how to find the corresponding input signals . We propose to use interactions between agent and the environment to define output components . This is analogues to model-free reinforcement learning ( Sutton & Barto , 2018 ) , where an agent does not have an environment model , but leans to act at each step during the interactions with the environment . Then we use entropy regularization ( Li et al. , 2019 ; Li & Eisner , 2019 ) to learn the minimal input components for outputs . We evaluate the approach with gSCAN dataset ( Ruis et al. , 2020 ) , which is designed to study compositional generalization in grounded agent instruction learning . Please see Figure 1 for examples . The results show the proposed approach significantly outperforms baselines in most tasks , with more than 25 % absolute average accuracy increase , and the high accuracy indicates that the proposed approach addresses the designed grounded compositional generalization problems in these tasks . We also look into the impact of entropy regularization and other changes with ablation study . We hope this work will be helpful in advancing grounded compositional generalization and artificial intelligence research . The contributions of this paper can be summarized as follows . • This is the first work to enable accurate compositional generalization in grounded instruction learning problem , serving for analyzing and understanding the mechanism . • The novelty of this paper is to find that the combination of environment interaction and entropy regularization helps the generalization . 2 RELATED WORK . Compositional generalization research has a long history , and recently there are increasing focus on this area . SCAN dataset ( Lake , 2019 ) was proposed to study compositional generalization in instruction learning . It maps a command sentence to a sequence of actions . This dataset has a property that input words and output actions have direct correspondence . Though some NLP tasks , such as machine translation , have similar property , not all problems fit to the setting . Also this dataset does not contain an environment for an agent to take actions . SCAN dataset inspired multiple approaches ( Russin et al. , 2019 ; Andreas , 2019 ; Li et al. , 2019 ; Lake , 2019 ; Gordon et al. , 2020 ; Liu et al. , 2020 ) . Some of them lead to general techniques for compositional generalization . For example , entropy regularization ( Li et al. , 2019 ) is proposed to avoid redundant dependency on input , and it is a core idea of the approach in this paper . Compositional generalization has applications in various fields such as question answering ( Andreas et al. , 2016 ; Hudson & Manning , 2019 ; Keysers et al. , 2020 ) , counting ( Rodriguez & Wiles , 1998 ; Weiss et al. , 2018 ) , systematic behaviour ( Wong & Wang , 2007 ; Brakel & Frank , 2009 ) , and hierarchical structure ( Linzen et al. , 2016 ) . Another related work is independent disentangled representation ( Higgins et al. , 2017 ; Locatello et al. , 2019 ) , but they do not address compositional generalization . Compositionality is also helpful for reasoning ( Talmor et al. , 2020 ) and continual learning ( Jin et al. , 2020 ; Li et al. , 2020 ) . Grounded SCAN ( gSCAN ) dataset was proposed to introduce environment and grounding to agent instruction learning with compositional generalization ( Ruis et al. , 2020 ) . It has a command sentence as input and a sequence of actions as output . However , the input command does not tell the specific way to act , but agent needs to understand the environment and take corresponding actions . This also avoids direct mapping between input words and output actions . Different approaches have been proposed to address this problem . As compositional generalization requires prior knowledge for distribution change , these approaches correspond to different ways to provide the prior knowledge . Andreas ( 2019 ) uses linguistic knowledge to augment training data . Kuo et al . ( 2020 ) uses external syntactic parser and WordNet . In this paper , we apply the prior knowledge for the interactions of agent and the environment . 3 PROBLEM DESCRIPTION . gSCAN dataset contains episodes of commands and actions , and each episode has an input and an output . The input includes a command and an environment . A command is a sequence of words . An environment contains an agent and a set of objects . An agent has its position and initial direction . An object has its position and attributes of color , shape and size . The output contains a sequence of actions ( Figure 1 ) . We hope to extract prior knowledge from environment interactions . We first notice that an agent should know whether it is going to change position . This means we can break an episode to a sequence of steps , where each step corresponds to an action for position change . An agent should also know the change of directions between steps . So we separate the direction change and the manner of action in a step . Therefore , we have three output for each step : direction , action and manner . We further convert the environment to be agent centered , and rotate the environment to make agent facing forward . We also assume automatic collision prevention . If the agent tries to push or pull an object to collide with other objects or boarder , then it stops . This makes us focus on addressing grounded compositional generalization problem . In summary , when we consider agent interaction with the environment , we can convert the problem to a set of step-wise label prediction problems with multiple outputs . Input contains command , environment , state , and output contains direction , action and manner . 4 APPROACH . In this section , we describe the approach for grounded compositional generalization . As we use entropy regularization in different modules , we first introduce it , then move to the model architecture . 4.1 ENTROPY REGULARIZATION . The difficulty of compositional generalization is that there might be incorrect dependency between input and output components . For example , when “ red ” and “ square ” do not appear together in any training sample , a model might learn that square is not red . However , this causes errors for compositional generalization in test . To avoid such case , we hope the representation of shape not influenced by input information of color . Entropy regularization ( Li et al. , 2019 ; Li & Eisner , 2019 ) aims at reducing entropy of a representation to avoid dependency on redundant input components . Given an representation x , we compute the L2 norm and add normal noise to each element of the representation . This decreases the channel capacity , so that the entropy for the representation reduces . We then feed the noised representation to the next layer , and add the norm to loss function . L = Loriginal + λL2 ( x ) EntReg ( x ) = x+ αN ( 0 , I ) where α is a weight of noise , positive for training and zero for inference . A representation can be fed to multiple networks , requiring different regularization for each input node . So we design entropy regularization layer , where we achieve non-linear mapping by expanding each node xi to a vector hi ∈ RH with ReLU activation , and maps it back to a node yi with linear activation , then apply entropy regularization on y to get y′ . The input x and output ERL ( x ) = y′ have the same size . hi = ffAi ( xi , H ) , yi = ff B i ( hi , 1 ) , y ′ = EntReg ( y ) . We write ff ( x , K ) for feed-forward neural network with x as input and K as output size . We use EntReg for word embeddings and ERL for environment inputs . 4.2 MODEL ARCHITECTURE . Input contains one-hot representation of a command xc with sentence length n and vocabulary size V c , and a sequence of m objects . We denote object attributes xa with vector size V a , and object positions xp with vector size V p. We also has a binary state s ∈ { 0 , 1 } indicating whether it is the first step of an episode . Output has three types . Direction yd has Cd classes , action ya has Ca classes , and manner ym has Cm classes . xc = xc1 , . . . , x c n ∈ RV c×n , xa = xa1 , . . . , x a m ∈ RV a×m , xp = xp1 , . . . , x p m ∈ RV p×m . The model has the following modules , as summarized in Figure 2 and Algorithm 1 . Command module ( CM ) This module takes a command xc as input and returns a representation r with size K , r = CM ( xc , K ) . r is expected to be the embedding of certain type of keyword , e.g . action , color , etc . To enable compositional generalization , we hope to separate representations of types ( e.g . color ) and values ( e.g . red ) . We use types for attention maps and values for attended values in attention mechanism . For a new combination of values , the types are still recognizable , so that attention maps are correct and the corresponding values are extracted ( Li et al. , 2019 ) . We design that each word xci in command x c has two embeddings for type ti and value vi with corresponding embedding matrices , Et ∈ Rkt×V c , Ev ∈ Rkv×V c , where kt and kv are embedding sizes , respectively . We then apply entropy regularization on both t and v to reduce redundant dependency . t = EntReg ( Etxc ) ∈ Rkt×n , v = EntReg ( Evxc ) ∈ Rkv×n . We use attention mechanism with a query q ∈ Rkt as learnable parameters , keys t , values v , and temperature τ c ∈ R. We compute score wc and attention map ac . The attended value u is fed to a feed-forward network with output size K. wc = qt ∈ Rn , ac = Softmax ( wc/τ c ) ∈ Rn , u = vac ∈ Rkv , r = ffθ ( u , K ) ∈ RK . Grounding module ( GM ) This module finds target in the environment according to the command . We use N command modules for queries . For each query ri , we have an dedicated entropy regularization layer on attributes x′ai , because different queries correspond to different type of attributes , and other attributes would be redundant . We then compute a score wgi . For i = 1 , . . . , N , ri = CM ( xc , V a ) ∈ RV a x′ai = ERL ( x a ) ∈ RV a×m , wgi = rix ′a i ∈ Rm . Algorithm 1 The proposed approach . Input includes command xc , state s , object attributes xa and object positions xp . K is an embedding size . Please see Section 4 for more information . Command module Input : xc , K Output : r : embedding 1 : t = EntReg ( Etxc ) 2 : v = EntReg ( Evxc ) 3 : wc = qt 4 : ac = Softmax ( wc/τ c ) 5 : u = vac 6 : r = ffθ ( u , K ) Grounding module Input : xc , xa , xp Output : p : target position 1 : for i = 1 , . . . , N do 2 : ri = CM ( xc , V a ) 3 : x′ai = ERL ( x a ) 4 : wgi = rix ′a i 5 : end for 6 : wg = ∑N i=1 w g i 7 : ag = Softmax ( wg/τg ) 8 : p = xpag Prediction module Input : xc , xa , xp , s , K Output : ŷ : prediction 1 : p = GM ( xc , xa , xp ) 2 : for each output type i do 3 : ki = CM ( xc , K ) 4 : for each node j do 5 : p′i , j = ERL ( p ) 6 : zi , j = [ kTi , p ′T i , j , s T ] T 7 : li , j = ff φ i , j ( zi , j , 1 ) 8 : end for 9 : ŷi = Softmax ( li ) 10 : end for The scores are added as wg . Attention map ag is computed by Softmax with temperature τg ∈ R , and is applied to get the attended object position p. wg = N∑ i=1 wgi ∈ R m , ag = Softmax ( wg/τg ) ∈ Rm , p = xpag ∈ RV p . Prediction module ( PM ) Prediction module takes command , environment and state as input , and outputs a prediction . We have three separate prediction modules for direction , action and manner , respectively . Modules correspond to different keywords but share the same grounded target and state . So for each prediction module , we use one command module to extract a keyword k with size K for the prediction . We also have a environment module for target object position p , and all the prediction modules share it as an input . There is a input of state s , and this is also a shared input for each prediction module . We build a dedicated feed-forward neural network from input to each output node without weight sharing . In each network of output type i and node j , we use entropy regularization layer for target position p. This is because different output nodes may need to be computed from different input components , and other components are redundant , so we hope to reduce dependency of each output node to input nodes ( more discussion in Section 6.1 ) . Then all the inputs are concatenated to form a vector z with size L = K + V p + 1. ki = CM ( xc , K ) ∈ RK , p′i , j = ERL ( p ) ∈ RV p , zi , j = [ k T i , p ′T i , j , s T ] T ∈ RL . We then feed it to another feed-forward network to get a output node li , j . They are concatenated to form a logit li , and we use Softmax to output ŷi . Ci is the number of classes for the output type . li , j = ff φ i , j ( zi , j , 1 ) ∈ R , li = [ li,1 , . . . , li , Ci ] ∈ R Ci , ŷi = Softmax ( li ) ∈ RCi We use cross entropy and the norms for entropy regularization with weight λ as training objective .	This paper tries to address a very important problem, compositional generalization in grounded agent instruction learning. It proposes to use interactions between agent and the environment to define output components, and entropy regularization to reduce redundant dependency on input. It shows significant improvements in most of gSCAN tasks. The paper also has an ablation study that investigates the effectiveness of entropy regularization and other factors.	SP:0117ddd034d838c0c7e226111c52c13712cb454d
Grounded Compositional Generalization with Environment Interactions	1 INTRODUCTION . Compositional generalization is a key skill for flexible and efficient learning . Humans leverage compositionality to create and recognize new combinations of familiar concepts ( Chomsky , 1957 ; Minsky , 1986 ) . Though there are many progresses for machine learning and deep learning in various areas recently ( LeCun et al. , 2015 ) , current main learning algorithms are not able to perform compositional generalization , and require many samples to train models . Such efficient learning is even more important when machines interact with the environment for grounding , because interactions are usually slow . Machine learning has been mostly developed with an assumption that training and test distributions are identical . Compositional generalization , however , is a kind of out-of-distribution generalization ( Bengio , 2017 ) , where training and test distributions are different . During training , dataset does not contain the information of the difference , so it can only be given as prior knowledge . In compositional generalization , a sample is a combination of several components . Test distribution changes as test samples are new combinations of seen components in training . For example , if we can find “ large apple ” and “ small orange ” in some environments , then we can also find “ large orange ” among multiple objects in a new environment . The recombination is enabled when an output component depends only on the corresponding input components , and invariant of other components ( please see Section 4.1 for more details ) . So there are two aspects to consider . What are the components in output , and how to find the corresponding input signals . We propose to use interactions between agent and the environment to define output components . This is analogues to model-free reinforcement learning ( Sutton & Barto , 2018 ) , where an agent does not have an environment model , but leans to act at each step during the interactions with the environment . Then we use entropy regularization ( Li et al. , 2019 ; Li & Eisner , 2019 ) to learn the minimal input components for outputs . We evaluate the approach with gSCAN dataset ( Ruis et al. , 2020 ) , which is designed to study compositional generalization in grounded agent instruction learning . Please see Figure 1 for examples . The results show the proposed approach significantly outperforms baselines in most tasks , with more than 25 % absolute average accuracy increase , and the high accuracy indicates that the proposed approach addresses the designed grounded compositional generalization problems in these tasks . We also look into the impact of entropy regularization and other changes with ablation study . We hope this work will be helpful in advancing grounded compositional generalization and artificial intelligence research . The contributions of this paper can be summarized as follows . • This is the first work to enable accurate compositional generalization in grounded instruction learning problem , serving for analyzing and understanding the mechanism . • The novelty of this paper is to find that the combination of environment interaction and entropy regularization helps the generalization . 2 RELATED WORK . Compositional generalization research has a long history , and recently there are increasing focus on this area . SCAN dataset ( Lake , 2019 ) was proposed to study compositional generalization in instruction learning . It maps a command sentence to a sequence of actions . This dataset has a property that input words and output actions have direct correspondence . Though some NLP tasks , such as machine translation , have similar property , not all problems fit to the setting . Also this dataset does not contain an environment for an agent to take actions . SCAN dataset inspired multiple approaches ( Russin et al. , 2019 ; Andreas , 2019 ; Li et al. , 2019 ; Lake , 2019 ; Gordon et al. , 2020 ; Liu et al. , 2020 ) . Some of them lead to general techniques for compositional generalization . For example , entropy regularization ( Li et al. , 2019 ) is proposed to avoid redundant dependency on input , and it is a core idea of the approach in this paper . Compositional generalization has applications in various fields such as question answering ( Andreas et al. , 2016 ; Hudson & Manning , 2019 ; Keysers et al. , 2020 ) , counting ( Rodriguez & Wiles , 1998 ; Weiss et al. , 2018 ) , systematic behaviour ( Wong & Wang , 2007 ; Brakel & Frank , 2009 ) , and hierarchical structure ( Linzen et al. , 2016 ) . Another related work is independent disentangled representation ( Higgins et al. , 2017 ; Locatello et al. , 2019 ) , but they do not address compositional generalization . Compositionality is also helpful for reasoning ( Talmor et al. , 2020 ) and continual learning ( Jin et al. , 2020 ; Li et al. , 2020 ) . Grounded SCAN ( gSCAN ) dataset was proposed to introduce environment and grounding to agent instruction learning with compositional generalization ( Ruis et al. , 2020 ) . It has a command sentence as input and a sequence of actions as output . However , the input command does not tell the specific way to act , but agent needs to understand the environment and take corresponding actions . This also avoids direct mapping between input words and output actions . Different approaches have been proposed to address this problem . As compositional generalization requires prior knowledge for distribution change , these approaches correspond to different ways to provide the prior knowledge . Andreas ( 2019 ) uses linguistic knowledge to augment training data . Kuo et al . ( 2020 ) uses external syntactic parser and WordNet . In this paper , we apply the prior knowledge for the interactions of agent and the environment . 3 PROBLEM DESCRIPTION . gSCAN dataset contains episodes of commands and actions , and each episode has an input and an output . The input includes a command and an environment . A command is a sequence of words . An environment contains an agent and a set of objects . An agent has its position and initial direction . An object has its position and attributes of color , shape and size . The output contains a sequence of actions ( Figure 1 ) . We hope to extract prior knowledge from environment interactions . We first notice that an agent should know whether it is going to change position . This means we can break an episode to a sequence of steps , where each step corresponds to an action for position change . An agent should also know the change of directions between steps . So we separate the direction change and the manner of action in a step . Therefore , we have three output for each step : direction , action and manner . We further convert the environment to be agent centered , and rotate the environment to make agent facing forward . We also assume automatic collision prevention . If the agent tries to push or pull an object to collide with other objects or boarder , then it stops . This makes us focus on addressing grounded compositional generalization problem . In summary , when we consider agent interaction with the environment , we can convert the problem to a set of step-wise label prediction problems with multiple outputs . Input contains command , environment , state , and output contains direction , action and manner . 4 APPROACH . In this section , we describe the approach for grounded compositional generalization . As we use entropy regularization in different modules , we first introduce it , then move to the model architecture . 4.1 ENTROPY REGULARIZATION . The difficulty of compositional generalization is that there might be incorrect dependency between input and output components . For example , when “ red ” and “ square ” do not appear together in any training sample , a model might learn that square is not red . However , this causes errors for compositional generalization in test . To avoid such case , we hope the representation of shape not influenced by input information of color . Entropy regularization ( Li et al. , 2019 ; Li & Eisner , 2019 ) aims at reducing entropy of a representation to avoid dependency on redundant input components . Given an representation x , we compute the L2 norm and add normal noise to each element of the representation . This decreases the channel capacity , so that the entropy for the representation reduces . We then feed the noised representation to the next layer , and add the norm to loss function . L = Loriginal + λL2 ( x ) EntReg ( x ) = x+ αN ( 0 , I ) where α is a weight of noise , positive for training and zero for inference . A representation can be fed to multiple networks , requiring different regularization for each input node . So we design entropy regularization layer , where we achieve non-linear mapping by expanding each node xi to a vector hi ∈ RH with ReLU activation , and maps it back to a node yi with linear activation , then apply entropy regularization on y to get y′ . The input x and output ERL ( x ) = y′ have the same size . hi = ffAi ( xi , H ) , yi = ff B i ( hi , 1 ) , y ′ = EntReg ( y ) . We write ff ( x , K ) for feed-forward neural network with x as input and K as output size . We use EntReg for word embeddings and ERL for environment inputs . 4.2 MODEL ARCHITECTURE . Input contains one-hot representation of a command xc with sentence length n and vocabulary size V c , and a sequence of m objects . We denote object attributes xa with vector size V a , and object positions xp with vector size V p. We also has a binary state s ∈ { 0 , 1 } indicating whether it is the first step of an episode . Output has three types . Direction yd has Cd classes , action ya has Ca classes , and manner ym has Cm classes . xc = xc1 , . . . , x c n ∈ RV c×n , xa = xa1 , . . . , x a m ∈ RV a×m , xp = xp1 , . . . , x p m ∈ RV p×m . The model has the following modules , as summarized in Figure 2 and Algorithm 1 . Command module ( CM ) This module takes a command xc as input and returns a representation r with size K , r = CM ( xc , K ) . r is expected to be the embedding of certain type of keyword , e.g . action , color , etc . To enable compositional generalization , we hope to separate representations of types ( e.g . color ) and values ( e.g . red ) . We use types for attention maps and values for attended values in attention mechanism . For a new combination of values , the types are still recognizable , so that attention maps are correct and the corresponding values are extracted ( Li et al. , 2019 ) . We design that each word xci in command x c has two embeddings for type ti and value vi with corresponding embedding matrices , Et ∈ Rkt×V c , Ev ∈ Rkv×V c , where kt and kv are embedding sizes , respectively . We then apply entropy regularization on both t and v to reduce redundant dependency . t = EntReg ( Etxc ) ∈ Rkt×n , v = EntReg ( Evxc ) ∈ Rkv×n . We use attention mechanism with a query q ∈ Rkt as learnable parameters , keys t , values v , and temperature τ c ∈ R. We compute score wc and attention map ac . The attended value u is fed to a feed-forward network with output size K. wc = qt ∈ Rn , ac = Softmax ( wc/τ c ) ∈ Rn , u = vac ∈ Rkv , r = ffθ ( u , K ) ∈ RK . Grounding module ( GM ) This module finds target in the environment according to the command . We use N command modules for queries . For each query ri , we have an dedicated entropy regularization layer on attributes x′ai , because different queries correspond to different type of attributes , and other attributes would be redundant . We then compute a score wgi . For i = 1 , . . . , N , ri = CM ( xc , V a ) ∈ RV a x′ai = ERL ( x a ) ∈ RV a×m , wgi = rix ′a i ∈ Rm . Algorithm 1 The proposed approach . Input includes command xc , state s , object attributes xa and object positions xp . K is an embedding size . Please see Section 4 for more information . Command module Input : xc , K Output : r : embedding 1 : t = EntReg ( Etxc ) 2 : v = EntReg ( Evxc ) 3 : wc = qt 4 : ac = Softmax ( wc/τ c ) 5 : u = vac 6 : r = ffθ ( u , K ) Grounding module Input : xc , xa , xp Output : p : target position 1 : for i = 1 , . . . , N do 2 : ri = CM ( xc , V a ) 3 : x′ai = ERL ( x a ) 4 : wgi = rix ′a i 5 : end for 6 : wg = ∑N i=1 w g i 7 : ag = Softmax ( wg/τg ) 8 : p = xpag Prediction module Input : xc , xa , xp , s , K Output : ŷ : prediction 1 : p = GM ( xc , xa , xp ) 2 : for each output type i do 3 : ki = CM ( xc , K ) 4 : for each node j do 5 : p′i , j = ERL ( p ) 6 : zi , j = [ kTi , p ′T i , j , s T ] T 7 : li , j = ff φ i , j ( zi , j , 1 ) 8 : end for 9 : ŷi = Softmax ( li ) 10 : end for The scores are added as wg . Attention map ag is computed by Softmax with temperature τg ∈ R , and is applied to get the attended object position p. wg = N∑ i=1 wgi ∈ R m , ag = Softmax ( wg/τg ) ∈ Rm , p = xpag ∈ RV p . Prediction module ( PM ) Prediction module takes command , environment and state as input , and outputs a prediction . We have three separate prediction modules for direction , action and manner , respectively . Modules correspond to different keywords but share the same grounded target and state . So for each prediction module , we use one command module to extract a keyword k with size K for the prediction . We also have a environment module for target object position p , and all the prediction modules share it as an input . There is a input of state s , and this is also a shared input for each prediction module . We build a dedicated feed-forward neural network from input to each output node without weight sharing . In each network of output type i and node j , we use entropy regularization layer for target position p. This is because different output nodes may need to be computed from different input components , and other components are redundant , so we hope to reduce dependency of each output node to input nodes ( more discussion in Section 6.1 ) . Then all the inputs are concatenated to form a vector z with size L = K + V p + 1. ki = CM ( xc , K ) ∈ RK , p′i , j = ERL ( p ) ∈ RV p , zi , j = [ k T i , p ′T i , j , s T ] T ∈ RL . We then feed it to another feed-forward network to get a output node li , j . They are concatenated to form a logit li , and we use Softmax to output ŷi . Ci is the number of classes for the output type . li , j = ff φ i , j ( zi , j , 1 ) ∈ R , li = [ li,1 , . . . , li , Ci ] ∈ R Ci , ŷi = Softmax ( li ) ∈ RCi We use cross entropy and the norms for entropy regularization with weight λ as training objective .	The paper proposes a new regularization method that constrains the mapping between the inputs and output spaces for achieving compositional generalization in simple grounded environments like gSCAN. The problem is interesting and important and the paper is corroborated by good experiments with 25% accuracy increase and also generalization to longer commands. However, the paper has clarity issues with descriptions that are sometimes vague or not precise enough and quite frequent language mistakes. It also doesn't discuss almost at all existing works and mention the only very briefly, making it harder to judge the strength of the new approach as there's not enough context.	SP:0117ddd034d838c0c7e226111c52c13712cb454d
Grounded Compositional Generalization with Environment Interactions	1 INTRODUCTION . Compositional generalization is a key skill for flexible and efficient learning . Humans leverage compositionality to create and recognize new combinations of familiar concepts ( Chomsky , 1957 ; Minsky , 1986 ) . Though there are many progresses for machine learning and deep learning in various areas recently ( LeCun et al. , 2015 ) , current main learning algorithms are not able to perform compositional generalization , and require many samples to train models . Such efficient learning is even more important when machines interact with the environment for grounding , because interactions are usually slow . Machine learning has been mostly developed with an assumption that training and test distributions are identical . Compositional generalization , however , is a kind of out-of-distribution generalization ( Bengio , 2017 ) , where training and test distributions are different . During training , dataset does not contain the information of the difference , so it can only be given as prior knowledge . In compositional generalization , a sample is a combination of several components . Test distribution changes as test samples are new combinations of seen components in training . For example , if we can find “ large apple ” and “ small orange ” in some environments , then we can also find “ large orange ” among multiple objects in a new environment . The recombination is enabled when an output component depends only on the corresponding input components , and invariant of other components ( please see Section 4.1 for more details ) . So there are two aspects to consider . What are the components in output , and how to find the corresponding input signals . We propose to use interactions between agent and the environment to define output components . This is analogues to model-free reinforcement learning ( Sutton & Barto , 2018 ) , where an agent does not have an environment model , but leans to act at each step during the interactions with the environment . Then we use entropy regularization ( Li et al. , 2019 ; Li & Eisner , 2019 ) to learn the minimal input components for outputs . We evaluate the approach with gSCAN dataset ( Ruis et al. , 2020 ) , which is designed to study compositional generalization in grounded agent instruction learning . Please see Figure 1 for examples . The results show the proposed approach significantly outperforms baselines in most tasks , with more than 25 % absolute average accuracy increase , and the high accuracy indicates that the proposed approach addresses the designed grounded compositional generalization problems in these tasks . We also look into the impact of entropy regularization and other changes with ablation study . We hope this work will be helpful in advancing grounded compositional generalization and artificial intelligence research . The contributions of this paper can be summarized as follows . • This is the first work to enable accurate compositional generalization in grounded instruction learning problem , serving for analyzing and understanding the mechanism . • The novelty of this paper is to find that the combination of environment interaction and entropy regularization helps the generalization . 2 RELATED WORK . Compositional generalization research has a long history , and recently there are increasing focus on this area . SCAN dataset ( Lake , 2019 ) was proposed to study compositional generalization in instruction learning . It maps a command sentence to a sequence of actions . This dataset has a property that input words and output actions have direct correspondence . Though some NLP tasks , such as machine translation , have similar property , not all problems fit to the setting . Also this dataset does not contain an environment for an agent to take actions . SCAN dataset inspired multiple approaches ( Russin et al. , 2019 ; Andreas , 2019 ; Li et al. , 2019 ; Lake , 2019 ; Gordon et al. , 2020 ; Liu et al. , 2020 ) . Some of them lead to general techniques for compositional generalization . For example , entropy regularization ( Li et al. , 2019 ) is proposed to avoid redundant dependency on input , and it is a core idea of the approach in this paper . Compositional generalization has applications in various fields such as question answering ( Andreas et al. , 2016 ; Hudson & Manning , 2019 ; Keysers et al. , 2020 ) , counting ( Rodriguez & Wiles , 1998 ; Weiss et al. , 2018 ) , systematic behaviour ( Wong & Wang , 2007 ; Brakel & Frank , 2009 ) , and hierarchical structure ( Linzen et al. , 2016 ) . Another related work is independent disentangled representation ( Higgins et al. , 2017 ; Locatello et al. , 2019 ) , but they do not address compositional generalization . Compositionality is also helpful for reasoning ( Talmor et al. , 2020 ) and continual learning ( Jin et al. , 2020 ; Li et al. , 2020 ) . Grounded SCAN ( gSCAN ) dataset was proposed to introduce environment and grounding to agent instruction learning with compositional generalization ( Ruis et al. , 2020 ) . It has a command sentence as input and a sequence of actions as output . However , the input command does not tell the specific way to act , but agent needs to understand the environment and take corresponding actions . This also avoids direct mapping between input words and output actions . Different approaches have been proposed to address this problem . As compositional generalization requires prior knowledge for distribution change , these approaches correspond to different ways to provide the prior knowledge . Andreas ( 2019 ) uses linguistic knowledge to augment training data . Kuo et al . ( 2020 ) uses external syntactic parser and WordNet . In this paper , we apply the prior knowledge for the interactions of agent and the environment . 3 PROBLEM DESCRIPTION . gSCAN dataset contains episodes of commands and actions , and each episode has an input and an output . The input includes a command and an environment . A command is a sequence of words . An environment contains an agent and a set of objects . An agent has its position and initial direction . An object has its position and attributes of color , shape and size . The output contains a sequence of actions ( Figure 1 ) . We hope to extract prior knowledge from environment interactions . We first notice that an agent should know whether it is going to change position . This means we can break an episode to a sequence of steps , where each step corresponds to an action for position change . An agent should also know the change of directions between steps . So we separate the direction change and the manner of action in a step . Therefore , we have three output for each step : direction , action and manner . We further convert the environment to be agent centered , and rotate the environment to make agent facing forward . We also assume automatic collision prevention . If the agent tries to push or pull an object to collide with other objects or boarder , then it stops . This makes us focus on addressing grounded compositional generalization problem . In summary , when we consider agent interaction with the environment , we can convert the problem to a set of step-wise label prediction problems with multiple outputs . Input contains command , environment , state , and output contains direction , action and manner . 4 APPROACH . In this section , we describe the approach for grounded compositional generalization . As we use entropy regularization in different modules , we first introduce it , then move to the model architecture . 4.1 ENTROPY REGULARIZATION . The difficulty of compositional generalization is that there might be incorrect dependency between input and output components . For example , when “ red ” and “ square ” do not appear together in any training sample , a model might learn that square is not red . However , this causes errors for compositional generalization in test . To avoid such case , we hope the representation of shape not influenced by input information of color . Entropy regularization ( Li et al. , 2019 ; Li & Eisner , 2019 ) aims at reducing entropy of a representation to avoid dependency on redundant input components . Given an representation x , we compute the L2 norm and add normal noise to each element of the representation . This decreases the channel capacity , so that the entropy for the representation reduces . We then feed the noised representation to the next layer , and add the norm to loss function . L = Loriginal + λL2 ( x ) EntReg ( x ) = x+ αN ( 0 , I ) where α is a weight of noise , positive for training and zero for inference . A representation can be fed to multiple networks , requiring different regularization for each input node . So we design entropy regularization layer , where we achieve non-linear mapping by expanding each node xi to a vector hi ∈ RH with ReLU activation , and maps it back to a node yi with linear activation , then apply entropy regularization on y to get y′ . The input x and output ERL ( x ) = y′ have the same size . hi = ffAi ( xi , H ) , yi = ff B i ( hi , 1 ) , y ′ = EntReg ( y ) . We write ff ( x , K ) for feed-forward neural network with x as input and K as output size . We use EntReg for word embeddings and ERL for environment inputs . 4.2 MODEL ARCHITECTURE . Input contains one-hot representation of a command xc with sentence length n and vocabulary size V c , and a sequence of m objects . We denote object attributes xa with vector size V a , and object positions xp with vector size V p. We also has a binary state s ∈ { 0 , 1 } indicating whether it is the first step of an episode . Output has three types . Direction yd has Cd classes , action ya has Ca classes , and manner ym has Cm classes . xc = xc1 , . . . , x c n ∈ RV c×n , xa = xa1 , . . . , x a m ∈ RV a×m , xp = xp1 , . . . , x p m ∈ RV p×m . The model has the following modules , as summarized in Figure 2 and Algorithm 1 . Command module ( CM ) This module takes a command xc as input and returns a representation r with size K , r = CM ( xc , K ) . r is expected to be the embedding of certain type of keyword , e.g . action , color , etc . To enable compositional generalization , we hope to separate representations of types ( e.g . color ) and values ( e.g . red ) . We use types for attention maps and values for attended values in attention mechanism . For a new combination of values , the types are still recognizable , so that attention maps are correct and the corresponding values are extracted ( Li et al. , 2019 ) . We design that each word xci in command x c has two embeddings for type ti and value vi with corresponding embedding matrices , Et ∈ Rkt×V c , Ev ∈ Rkv×V c , where kt and kv are embedding sizes , respectively . We then apply entropy regularization on both t and v to reduce redundant dependency . t = EntReg ( Etxc ) ∈ Rkt×n , v = EntReg ( Evxc ) ∈ Rkv×n . We use attention mechanism with a query q ∈ Rkt as learnable parameters , keys t , values v , and temperature τ c ∈ R. We compute score wc and attention map ac . The attended value u is fed to a feed-forward network with output size K. wc = qt ∈ Rn , ac = Softmax ( wc/τ c ) ∈ Rn , u = vac ∈ Rkv , r = ffθ ( u , K ) ∈ RK . Grounding module ( GM ) This module finds target in the environment according to the command . We use N command modules for queries . For each query ri , we have an dedicated entropy regularization layer on attributes x′ai , because different queries correspond to different type of attributes , and other attributes would be redundant . We then compute a score wgi . For i = 1 , . . . , N , ri = CM ( xc , V a ) ∈ RV a x′ai = ERL ( x a ) ∈ RV a×m , wgi = rix ′a i ∈ Rm . Algorithm 1 The proposed approach . Input includes command xc , state s , object attributes xa and object positions xp . K is an embedding size . Please see Section 4 for more information . Command module Input : xc , K Output : r : embedding 1 : t = EntReg ( Etxc ) 2 : v = EntReg ( Evxc ) 3 : wc = qt 4 : ac = Softmax ( wc/τ c ) 5 : u = vac 6 : r = ffθ ( u , K ) Grounding module Input : xc , xa , xp Output : p : target position 1 : for i = 1 , . . . , N do 2 : ri = CM ( xc , V a ) 3 : x′ai = ERL ( x a ) 4 : wgi = rix ′a i 5 : end for 6 : wg = ∑N i=1 w g i 7 : ag = Softmax ( wg/τg ) 8 : p = xpag Prediction module Input : xc , xa , xp , s , K Output : ŷ : prediction 1 : p = GM ( xc , xa , xp ) 2 : for each output type i do 3 : ki = CM ( xc , K ) 4 : for each node j do 5 : p′i , j = ERL ( p ) 6 : zi , j = [ kTi , p ′T i , j , s T ] T 7 : li , j = ff φ i , j ( zi , j , 1 ) 8 : end for 9 : ŷi = Softmax ( li ) 10 : end for The scores are added as wg . Attention map ag is computed by Softmax with temperature τg ∈ R , and is applied to get the attended object position p. wg = N∑ i=1 wgi ∈ R m , ag = Softmax ( wg/τg ) ∈ Rm , p = xpag ∈ RV p . Prediction module ( PM ) Prediction module takes command , environment and state as input , and outputs a prediction . We have three separate prediction modules for direction , action and manner , respectively . Modules correspond to different keywords but share the same grounded target and state . So for each prediction module , we use one command module to extract a keyword k with size K for the prediction . We also have a environment module for target object position p , and all the prediction modules share it as an input . There is a input of state s , and this is also a shared input for each prediction module . We build a dedicated feed-forward neural network from input to each output node without weight sharing . In each network of output type i and node j , we use entropy regularization layer for target position p. This is because different output nodes may need to be computed from different input components , and other components are redundant , so we hope to reduce dependency of each output node to input nodes ( more discussion in Section 6.1 ) . Then all the inputs are concatenated to form a vector z with size L = K + V p + 1. ki = CM ( xc , K ) ∈ RK , p′i , j = ERL ( p ) ∈ RV p , zi , j = [ k T i , p ′T i , j , s T ] T ∈ RL . We then feed it to another feed-forward network to get a output node li , j . They are concatenated to form a logit li , and we use Softmax to output ŷi . Ci is the number of classes for the output type . li , j = ff φ i , j ( zi , j , 1 ) ∈ R , li = [ li,1 , . . . , li , Ci ] ∈ R Ci , ŷi = Softmax ( li ) ∈ RCi We use cross entropy and the norms for entropy regularization with weight λ as training objective .	This paper proposes a new model for the gSCAN dataset (Ruis et al. 2020) which is a synthetically-generated dataset that challenges models to generalize to new compositions of attributes and objects in an instruction. The paper proposes to use "entropy regularization" as a way to enforce that spurious correlations between input tokens and output actions are not learned. The proposed model shows promising results on the gSCAN benchmark, achieving nearly 100% accuracy on all but one task.	SP:0117ddd034d838c0c7e226111c52c13712cb454d
Semantic-Guided Representation Enhancement for Self-supervised Monocular Trained Depth Estimation	1 INTRODUCTION . Depth estimation is a long standing problem in computer vision community , which offers useful information to a wide range of tasks including robotic perception , augmented reality and autonomous driving , etc . Compared with depth estimation methods which rely on active vision or multi-view paradigms ( Schonberger & Frahm , 2016 ; Li et al. , 2019 ) , estimating depth from only a single image is highly ill-posed , thus brings greater challenges for higher quality results . In recent years , monocular depth estimation has witnessed new renaissances with the advent of deep learning ( He et al. , 2016 ; Jaderberg et al. , 2015 ; Simonyan & Zisserman , 2014 ) . By learning deep representations in a supervised manner , various of networks ( Eigen & Fergus , 2015 ; Eigen et al. , 2014 ; Laina et al. , 2016 ) are capable of producing high quality depth maps thanks to the large corpus of training data . At the mean time , consider the lack of labeled training data for network training , recent advances ( Zhou et al. , 2017 ; Godard et al. , 2019 ; 2017 ) show that monocular depth estimation can be accomplished in a self-supervised way . The network can be trained by unlabeled image sequences using two-view geometric constraints , while achieving comparable results with the supervised paradigm . The learning-based methods managed to handle the highly ill-posed monocular depth estimation problem by implicitly learning the mapping between visual appearance and its corresponding depth value . However , despite the great effectiveness for learning-based depth estimation , these methods still struggle to conduct precise depth estimation on challenging image regions such as semantic category borders or thin object areas . For example , the estimated object depth usually fails to align with the real object borders , and the depth of foreground objects which have thin structures tends to be submerged in the background . We attribute these phenomena to the limited depth representation ability that ( 1 ) the pixel-wise local depth information can not be well represented by current depth network , especially on highly ambiguous , semantic border areas , ( 2 ) current depth representations are not capable of well describing depth foreground/background relationships globally . These issues lead to the wrong depth estimation from the true scene structures , which hinders the further applications in the real-world tasks . In this paper , we address this problem via enhancing both local and global depth feature representations for self-supervised monocular depth estimation via semantic guidance . As semantic segmentation conducts explicit category-level scene understandings and produces well-aligned object boundary detection , we propose an extra semantic estimation branch inside the self-supervised paradigm . The semantic branch offers rich contextual features which is fused with depth features during multiscale learning . Under this framework , we propose to enhance the depth feature using semantic guidance in a local-to-global way . To improve the local depth representations on semantic category borders , inspired by the sampling and enhancing strategies used in semantic segmentation ( Kirillov et al. , 2020 ) , our first contribution is to propose a Semantic-guided Edge Enhancement Module ( SEEM ) which is specially designed to enhance the local point-based depth representations that locate on the semantic edges . Different from the method in ( Kirillov et al. , 2020 ) , we enhance the point features using multi-level of representations from different domains under a self-supervised framework . To be specific , we sample a set of point positions lying on the binary semantic category borders , and extract the point-wise features of the corresponding edge positions from the encoded feature , depth decoding feature as well as the semantic decoding feature . We then merge and enhance the point-wise features and feed them to the final depth decoding features to promote the edge-aware depth inference for self-supervision . For global depth representation enhancement , our second contribution is to propose a semantic-guided multi-level attention module to improve the global depth representation . Different from the conventional self-attentions ( Fu et al. , 2019 ; Vaswani et al. , 2017 ) which are implemented as single modules on the bottleneck feature block , we propose to explore the self-attentions on different level of decoding layers . In this way , both semantic and depth representations can be further promoted by exploring and leveraging the pixel-wise correlations inside of their fused features . We validate our method mainly on KITTI 2015 ( Geiger et al. , 2012 ) , and the Cityscapes ( Cordts et al. , 2016 ) benchmark is also used for evaluating the generalization ability . Experiments show that the proposed method significantly improves the depth estimation on category edges and thin scene structures . Extensive quantitative and qualitative results validate the superiority of our method that it outperforms the state-of-the-art methods for self-supervised monocular depth estimation . 2 RELATED WORK . There exist extensive researches on monocular depth estimation , including geometry-based methods ( Schonberger & Frahm , 2016 ; Enqvist et al. , 2011 ) and learning-based methods ( Eigen et al. , 2014 ; Laina et al. , 2016 ) . In this paper , however , we concentrate only on the self-supervised depth training and semantic-guided depth estimation , which is highly related to the research focus of this paper . Self-supervised depth estimation . Self-supervised methods enable the networks to learn depth representation from merely unlabeled image sequences that they reformulate the depth supervision loss into the image reconstruction loss . Godard et al . ( 2017 ) and Garg et al . ( 2016 ) first propose the self-supervised method on stereo images , then Zhou et al . ( 2017 ) propose a monocular trained approach using a separate motion estimation network . Based on these frameworks , a large corpus of works seek to promote self-supervised learning from different aspects . For more robust selfsupervision signals , Mahjourian et al . ( 2018 ) propose to use 3D information as extra supervisory signal , another kind of methods leverage additional information such as the optical flow ( Ranjan et al. , 2019 ; Wang et al. , 2019b ) to strengthen depth supervision via consistency constraints . In order to solve the loss deviation problems on non-rigid motion areas , selective masks are used to filter out the these areas when computing losses . Prior works generate the mask by the network itself ( Zhou et al. , 2017 ; Yang et al. , 2017 ; Vijayanarasimhan et al. , 2017 ) , while the succeeding methods produce the mask by leveraging geometric clues ( Bian et al. , 2019 ; Wang et al. , 2019a ; Godard et al. , 2019 ; Klingner et al. , 2020 ) , which is proved to be more effective . There also exist other methods trying to enhance the network performance with traditional SfM ( Schonberger & Frahm , 2016 ) , which offer pseudo labels for depth estimation . Guizilini et al . ( 2020a ) propose a novel network architecture to improve depth estimation . In this paper , we do not consider the performance improvements contributed by this novel architecture in experimental comparisons , and compare different methods with conventional backbones . Depth estimation using semantics . Semantic information are shown to provide positive effects toward depth estimation framework in the previous works . The methods can be categorized into two groups by their way to use the semantic information . One group of the methods leverage the semantic labels directly to guide the depth learning . The works of Ramirez et al . ( 2018 ) and Chen et al . ( 2019a ) propose depth constraints by leveraging the semantic labels of the scene . Klingner et al . ( 2020 ) and Casser et al . ( 2019 ) address the non-rigid motion issues by handling the moving object areas highlighted by the semantic map . Wang et al . ( 2020 ) use a divide-and-conquer strategy to conduct depth estimation with different semantic categories , and Zhu et al . ( 2020 ) offer a depth morphing paradigm with the help of semantic foreground edges . The other group of methods enhance the depth representation by feature manipulation . Chen et al . ( 2019a ) generated depth and semantic maps by a single latent representation . Ochs et al . ( 2019 ) proposed a segmentation-like loss item for depth estimation . Guizilini et al . ( 2020b ) leveraged a pre-trained semantic segmentation network and further conduct feature merging by PAC-based ( Su et al. , 2019 ) module . In this paper , we propose an individual semantic network for depth estimation . But different with Guizilini et al . ( 2020b ) , our semantic branch shares the same encoder and is trained together with the depth network for better scene representation . We leverage the semantic information in both ways . The semantic edges are used to guide the edge-based point feature sampling for local depth depth representation enhancement . At the mean time , we utilize the semantic feature blocks to conduct multi-scale feature fusion for global depth representation enhancement . The proposed method constructs a holistic way for feature representation enhancing in self-supervised depth estimation . 3 SELF-SUPERVISED DEPTH ESTIMATION FRAMEWORK . The proposed method builds upon the self-supervised depth estimation framework . An image triplet ( It−1 , It , It+1 ) is used as the input , where It acts as the target image and It′ ∈ { It−1 , It+1 } are the source images . During training , the target image It is fed into the depth network fD to get the estimated depth Dt = fD ( It ) , while the adjacent image pairs ( It , It′ ) are put into the ego-motion network for the 6-DoF ego-motion estimations Tt′→t . Then , the synthesized target images It′→t can be estimated using the source images It′ , depth Dt and the ego-motions Tt′→t , following the formulation from Zhou et al . ( 2017 ) . The self-supervised loss can be calculated by L ( It , It′→t ) = Lp ( It , It′→t ) + λLs ( It , It′→t ) , ( 1 ) where λ is the weighting factor between the photometric loss Lp and smoothness loss Ls . Following Godard et al . ( 2019 ) , we implement the minimum photometric loss Lp ( It , It′→t ) = min t′ ( α 2 ( 1− SSIM ( It , It′→t ) ) + ( 1− α ) ‖It − It′→t‖1 ) , ( 2 ) where SSIM is the Structural Similarity item ( Wang et al. , 2004 ) , α is the weighting factor which is set to 0.85 . The smoothness loss item ( Godard et al. , 2017 ) is implemented to smooth the depth map by the consistency between the image and depth gradient Ls ( It ) = \|∂xDt\| e−\|∂xIt\| + \|∂yDt\| e−\|∂yIt\| , ( 3 ) where ∂x , ∂x denote gradient operation on x , y-axis . In our implementation , we also conduct automasking and multi-scale depth upsampling strategy as proposed in Godard et al . ( 2019 ) to further improve depth estimation . 4 THE PROPOSED METHOD . In this paper , we promote the self-supervised depth estimation paradigm via proposing a semanticguided representation enhancing depth network . The overview of the method is shown in Figure 1 . We propose a multi-task framework which consists of the depth and semantic decoding branch with shared feature encoder . During the training stage , the semantic branch feeds contextual features to the depth branch to enhance depth comprehension . Under this multi-task framework , we propose a Semantic-guided Edge Enhancement Module ( SEEM ) to enhance the local depth feature representation on semantic category edges . Meanwhile , we enhance the global depth feature representations by proposing the semantic-guided multi-level self-attention module , which improves both the semantic and depth feature representations by digging into the pixel-wise dependencies .	This paper proposed a novel framework to improve self-supervised monocular depth estimation leveraging semantic features at local and global level. The proposed framework includes a semantic-guided edge enhancement module to extract and enhance point-based features around semantic boundaries. The proposed framework also incorporates feature fusion through a self-attention module at different feature levels for global fusion. Evaluation on KITTI datasets compared with recent state-of-the-arts are provided. The proposed method outperformed other monocular depth estimation methods including the ones that also leverage semantic information.	SP:529dd8f14cc4459b84dcb6a2eb5a35a520e10124
Semantic-Guided Representation Enhancement for Self-supervised Monocular Trained Depth Estimation	1 INTRODUCTION . Depth estimation is a long standing problem in computer vision community , which offers useful information to a wide range of tasks including robotic perception , augmented reality and autonomous driving , etc . Compared with depth estimation methods which rely on active vision or multi-view paradigms ( Schonberger & Frahm , 2016 ; Li et al. , 2019 ) , estimating depth from only a single image is highly ill-posed , thus brings greater challenges for higher quality results . In recent years , monocular depth estimation has witnessed new renaissances with the advent of deep learning ( He et al. , 2016 ; Jaderberg et al. , 2015 ; Simonyan & Zisserman , 2014 ) . By learning deep representations in a supervised manner , various of networks ( Eigen & Fergus , 2015 ; Eigen et al. , 2014 ; Laina et al. , 2016 ) are capable of producing high quality depth maps thanks to the large corpus of training data . At the mean time , consider the lack of labeled training data for network training , recent advances ( Zhou et al. , 2017 ; Godard et al. , 2019 ; 2017 ) show that monocular depth estimation can be accomplished in a self-supervised way . The network can be trained by unlabeled image sequences using two-view geometric constraints , while achieving comparable results with the supervised paradigm . The learning-based methods managed to handle the highly ill-posed monocular depth estimation problem by implicitly learning the mapping between visual appearance and its corresponding depth value . However , despite the great effectiveness for learning-based depth estimation , these methods still struggle to conduct precise depth estimation on challenging image regions such as semantic category borders or thin object areas . For example , the estimated object depth usually fails to align with the real object borders , and the depth of foreground objects which have thin structures tends to be submerged in the background . We attribute these phenomena to the limited depth representation ability that ( 1 ) the pixel-wise local depth information can not be well represented by current depth network , especially on highly ambiguous , semantic border areas , ( 2 ) current depth representations are not capable of well describing depth foreground/background relationships globally . These issues lead to the wrong depth estimation from the true scene structures , which hinders the further applications in the real-world tasks . In this paper , we address this problem via enhancing both local and global depth feature representations for self-supervised monocular depth estimation via semantic guidance . As semantic segmentation conducts explicit category-level scene understandings and produces well-aligned object boundary detection , we propose an extra semantic estimation branch inside the self-supervised paradigm . The semantic branch offers rich contextual features which is fused with depth features during multiscale learning . Under this framework , we propose to enhance the depth feature using semantic guidance in a local-to-global way . To improve the local depth representations on semantic category borders , inspired by the sampling and enhancing strategies used in semantic segmentation ( Kirillov et al. , 2020 ) , our first contribution is to propose a Semantic-guided Edge Enhancement Module ( SEEM ) which is specially designed to enhance the local point-based depth representations that locate on the semantic edges . Different from the method in ( Kirillov et al. , 2020 ) , we enhance the point features using multi-level of representations from different domains under a self-supervised framework . To be specific , we sample a set of point positions lying on the binary semantic category borders , and extract the point-wise features of the corresponding edge positions from the encoded feature , depth decoding feature as well as the semantic decoding feature . We then merge and enhance the point-wise features and feed them to the final depth decoding features to promote the edge-aware depth inference for self-supervision . For global depth representation enhancement , our second contribution is to propose a semantic-guided multi-level attention module to improve the global depth representation . Different from the conventional self-attentions ( Fu et al. , 2019 ; Vaswani et al. , 2017 ) which are implemented as single modules on the bottleneck feature block , we propose to explore the self-attentions on different level of decoding layers . In this way , both semantic and depth representations can be further promoted by exploring and leveraging the pixel-wise correlations inside of their fused features . We validate our method mainly on KITTI 2015 ( Geiger et al. , 2012 ) , and the Cityscapes ( Cordts et al. , 2016 ) benchmark is also used for evaluating the generalization ability . Experiments show that the proposed method significantly improves the depth estimation on category edges and thin scene structures . Extensive quantitative and qualitative results validate the superiority of our method that it outperforms the state-of-the-art methods for self-supervised monocular depth estimation . 2 RELATED WORK . There exist extensive researches on monocular depth estimation , including geometry-based methods ( Schonberger & Frahm , 2016 ; Enqvist et al. , 2011 ) and learning-based methods ( Eigen et al. , 2014 ; Laina et al. , 2016 ) . In this paper , however , we concentrate only on the self-supervised depth training and semantic-guided depth estimation , which is highly related to the research focus of this paper . Self-supervised depth estimation . Self-supervised methods enable the networks to learn depth representation from merely unlabeled image sequences that they reformulate the depth supervision loss into the image reconstruction loss . Godard et al . ( 2017 ) and Garg et al . ( 2016 ) first propose the self-supervised method on stereo images , then Zhou et al . ( 2017 ) propose a monocular trained approach using a separate motion estimation network . Based on these frameworks , a large corpus of works seek to promote self-supervised learning from different aspects . For more robust selfsupervision signals , Mahjourian et al . ( 2018 ) propose to use 3D information as extra supervisory signal , another kind of methods leverage additional information such as the optical flow ( Ranjan et al. , 2019 ; Wang et al. , 2019b ) to strengthen depth supervision via consistency constraints . In order to solve the loss deviation problems on non-rigid motion areas , selective masks are used to filter out the these areas when computing losses . Prior works generate the mask by the network itself ( Zhou et al. , 2017 ; Yang et al. , 2017 ; Vijayanarasimhan et al. , 2017 ) , while the succeeding methods produce the mask by leveraging geometric clues ( Bian et al. , 2019 ; Wang et al. , 2019a ; Godard et al. , 2019 ; Klingner et al. , 2020 ) , which is proved to be more effective . There also exist other methods trying to enhance the network performance with traditional SfM ( Schonberger & Frahm , 2016 ) , which offer pseudo labels for depth estimation . Guizilini et al . ( 2020a ) propose a novel network architecture to improve depth estimation . In this paper , we do not consider the performance improvements contributed by this novel architecture in experimental comparisons , and compare different methods with conventional backbones . Depth estimation using semantics . Semantic information are shown to provide positive effects toward depth estimation framework in the previous works . The methods can be categorized into two groups by their way to use the semantic information . One group of the methods leverage the semantic labels directly to guide the depth learning . The works of Ramirez et al . ( 2018 ) and Chen et al . ( 2019a ) propose depth constraints by leveraging the semantic labels of the scene . Klingner et al . ( 2020 ) and Casser et al . ( 2019 ) address the non-rigid motion issues by handling the moving object areas highlighted by the semantic map . Wang et al . ( 2020 ) use a divide-and-conquer strategy to conduct depth estimation with different semantic categories , and Zhu et al . ( 2020 ) offer a depth morphing paradigm with the help of semantic foreground edges . The other group of methods enhance the depth representation by feature manipulation . Chen et al . ( 2019a ) generated depth and semantic maps by a single latent representation . Ochs et al . ( 2019 ) proposed a segmentation-like loss item for depth estimation . Guizilini et al . ( 2020b ) leveraged a pre-trained semantic segmentation network and further conduct feature merging by PAC-based ( Su et al. , 2019 ) module . In this paper , we propose an individual semantic network for depth estimation . But different with Guizilini et al . ( 2020b ) , our semantic branch shares the same encoder and is trained together with the depth network for better scene representation . We leverage the semantic information in both ways . The semantic edges are used to guide the edge-based point feature sampling for local depth depth representation enhancement . At the mean time , we utilize the semantic feature blocks to conduct multi-scale feature fusion for global depth representation enhancement . The proposed method constructs a holistic way for feature representation enhancing in self-supervised depth estimation . 3 SELF-SUPERVISED DEPTH ESTIMATION FRAMEWORK . The proposed method builds upon the self-supervised depth estimation framework . An image triplet ( It−1 , It , It+1 ) is used as the input , where It acts as the target image and It′ ∈ { It−1 , It+1 } are the source images . During training , the target image It is fed into the depth network fD to get the estimated depth Dt = fD ( It ) , while the adjacent image pairs ( It , It′ ) are put into the ego-motion network for the 6-DoF ego-motion estimations Tt′→t . Then , the synthesized target images It′→t can be estimated using the source images It′ , depth Dt and the ego-motions Tt′→t , following the formulation from Zhou et al . ( 2017 ) . The self-supervised loss can be calculated by L ( It , It′→t ) = Lp ( It , It′→t ) + λLs ( It , It′→t ) , ( 1 ) where λ is the weighting factor between the photometric loss Lp and smoothness loss Ls . Following Godard et al . ( 2019 ) , we implement the minimum photometric loss Lp ( It , It′→t ) = min t′ ( α 2 ( 1− SSIM ( It , It′→t ) ) + ( 1− α ) ‖It − It′→t‖1 ) , ( 2 ) where SSIM is the Structural Similarity item ( Wang et al. , 2004 ) , α is the weighting factor which is set to 0.85 . The smoothness loss item ( Godard et al. , 2017 ) is implemented to smooth the depth map by the consistency between the image and depth gradient Ls ( It ) = \|∂xDt\| e−\|∂xIt\| + \|∂yDt\| e−\|∂yIt\| , ( 3 ) where ∂x , ∂x denote gradient operation on x , y-axis . In our implementation , we also conduct automasking and multi-scale depth upsampling strategy as proposed in Godard et al . ( 2019 ) to further improve depth estimation . 4 THE PROPOSED METHOD . In this paper , we promote the self-supervised depth estimation paradigm via proposing a semanticguided representation enhancing depth network . The overview of the method is shown in Figure 1 . We propose a multi-task framework which consists of the depth and semantic decoding branch with shared feature encoder . During the training stage , the semantic branch feeds contextual features to the depth branch to enhance depth comprehension . Under this multi-task framework , we propose a Semantic-guided Edge Enhancement Module ( SEEM ) to enhance the local depth feature representation on semantic category edges . Meanwhile , we enhance the global depth feature representations by proposing the semantic-guided multi-level self-attention module , which improves both the semantic and depth feature representations by digging into the pixel-wise dependencies .	The authors tackle the problem of self-supervised depth estimation and particularly address the issue of poor depth estimation on object boundaries. The introduction motivates the problem well, and the related work covers most of the relevant papers. The authors propose two main modifications allowing them to leverage an off-the-shelf semantic segmentation network: SEEM (semantic guided edge enhancement module) and a multi-level self-attention mechanism that fuses depth and semantic features at different levels. The combination of these contributions along with an off-the-shelf semantic segmentation network achieves impressive results, especially on the boundaries of objects.	SP:529dd8f14cc4459b84dcb6a2eb5a35a520e10124
Semantic-Guided Representation Enhancement for Self-supervised Monocular Trained Depth Estimation	1 INTRODUCTION . Depth estimation is a long standing problem in computer vision community , which offers useful information to a wide range of tasks including robotic perception , augmented reality and autonomous driving , etc . Compared with depth estimation methods which rely on active vision or multi-view paradigms ( Schonberger & Frahm , 2016 ; Li et al. , 2019 ) , estimating depth from only a single image is highly ill-posed , thus brings greater challenges for higher quality results . In recent years , monocular depth estimation has witnessed new renaissances with the advent of deep learning ( He et al. , 2016 ; Jaderberg et al. , 2015 ; Simonyan & Zisserman , 2014 ) . By learning deep representations in a supervised manner , various of networks ( Eigen & Fergus , 2015 ; Eigen et al. , 2014 ; Laina et al. , 2016 ) are capable of producing high quality depth maps thanks to the large corpus of training data . At the mean time , consider the lack of labeled training data for network training , recent advances ( Zhou et al. , 2017 ; Godard et al. , 2019 ; 2017 ) show that monocular depth estimation can be accomplished in a self-supervised way . The network can be trained by unlabeled image sequences using two-view geometric constraints , while achieving comparable results with the supervised paradigm . The learning-based methods managed to handle the highly ill-posed monocular depth estimation problem by implicitly learning the mapping between visual appearance and its corresponding depth value . However , despite the great effectiveness for learning-based depth estimation , these methods still struggle to conduct precise depth estimation on challenging image regions such as semantic category borders or thin object areas . For example , the estimated object depth usually fails to align with the real object borders , and the depth of foreground objects which have thin structures tends to be submerged in the background . We attribute these phenomena to the limited depth representation ability that ( 1 ) the pixel-wise local depth information can not be well represented by current depth network , especially on highly ambiguous , semantic border areas , ( 2 ) current depth representations are not capable of well describing depth foreground/background relationships globally . These issues lead to the wrong depth estimation from the true scene structures , which hinders the further applications in the real-world tasks . In this paper , we address this problem via enhancing both local and global depth feature representations for self-supervised monocular depth estimation via semantic guidance . As semantic segmentation conducts explicit category-level scene understandings and produces well-aligned object boundary detection , we propose an extra semantic estimation branch inside the self-supervised paradigm . The semantic branch offers rich contextual features which is fused with depth features during multiscale learning . Under this framework , we propose to enhance the depth feature using semantic guidance in a local-to-global way . To improve the local depth representations on semantic category borders , inspired by the sampling and enhancing strategies used in semantic segmentation ( Kirillov et al. , 2020 ) , our first contribution is to propose a Semantic-guided Edge Enhancement Module ( SEEM ) which is specially designed to enhance the local point-based depth representations that locate on the semantic edges . Different from the method in ( Kirillov et al. , 2020 ) , we enhance the point features using multi-level of representations from different domains under a self-supervised framework . To be specific , we sample a set of point positions lying on the binary semantic category borders , and extract the point-wise features of the corresponding edge positions from the encoded feature , depth decoding feature as well as the semantic decoding feature . We then merge and enhance the point-wise features and feed them to the final depth decoding features to promote the edge-aware depth inference for self-supervision . For global depth representation enhancement , our second contribution is to propose a semantic-guided multi-level attention module to improve the global depth representation . Different from the conventional self-attentions ( Fu et al. , 2019 ; Vaswani et al. , 2017 ) which are implemented as single modules on the bottleneck feature block , we propose to explore the self-attentions on different level of decoding layers . In this way , both semantic and depth representations can be further promoted by exploring and leveraging the pixel-wise correlations inside of their fused features . We validate our method mainly on KITTI 2015 ( Geiger et al. , 2012 ) , and the Cityscapes ( Cordts et al. , 2016 ) benchmark is also used for evaluating the generalization ability . Experiments show that the proposed method significantly improves the depth estimation on category edges and thin scene structures . Extensive quantitative and qualitative results validate the superiority of our method that it outperforms the state-of-the-art methods for self-supervised monocular depth estimation . 2 RELATED WORK . There exist extensive researches on monocular depth estimation , including geometry-based methods ( Schonberger & Frahm , 2016 ; Enqvist et al. , 2011 ) and learning-based methods ( Eigen et al. , 2014 ; Laina et al. , 2016 ) . In this paper , however , we concentrate only on the self-supervised depth training and semantic-guided depth estimation , which is highly related to the research focus of this paper . Self-supervised depth estimation . Self-supervised methods enable the networks to learn depth representation from merely unlabeled image sequences that they reformulate the depth supervision loss into the image reconstruction loss . Godard et al . ( 2017 ) and Garg et al . ( 2016 ) first propose the self-supervised method on stereo images , then Zhou et al . ( 2017 ) propose a monocular trained approach using a separate motion estimation network . Based on these frameworks , a large corpus of works seek to promote self-supervised learning from different aspects . For more robust selfsupervision signals , Mahjourian et al . ( 2018 ) propose to use 3D information as extra supervisory signal , another kind of methods leverage additional information such as the optical flow ( Ranjan et al. , 2019 ; Wang et al. , 2019b ) to strengthen depth supervision via consistency constraints . In order to solve the loss deviation problems on non-rigid motion areas , selective masks are used to filter out the these areas when computing losses . Prior works generate the mask by the network itself ( Zhou et al. , 2017 ; Yang et al. , 2017 ; Vijayanarasimhan et al. , 2017 ) , while the succeeding methods produce the mask by leveraging geometric clues ( Bian et al. , 2019 ; Wang et al. , 2019a ; Godard et al. , 2019 ; Klingner et al. , 2020 ) , which is proved to be more effective . There also exist other methods trying to enhance the network performance with traditional SfM ( Schonberger & Frahm , 2016 ) , which offer pseudo labels for depth estimation . Guizilini et al . ( 2020a ) propose a novel network architecture to improve depth estimation . In this paper , we do not consider the performance improvements contributed by this novel architecture in experimental comparisons , and compare different methods with conventional backbones . Depth estimation using semantics . Semantic information are shown to provide positive effects toward depth estimation framework in the previous works . The methods can be categorized into two groups by their way to use the semantic information . One group of the methods leverage the semantic labels directly to guide the depth learning . The works of Ramirez et al . ( 2018 ) and Chen et al . ( 2019a ) propose depth constraints by leveraging the semantic labels of the scene . Klingner et al . ( 2020 ) and Casser et al . ( 2019 ) address the non-rigid motion issues by handling the moving object areas highlighted by the semantic map . Wang et al . ( 2020 ) use a divide-and-conquer strategy to conduct depth estimation with different semantic categories , and Zhu et al . ( 2020 ) offer a depth morphing paradigm with the help of semantic foreground edges . The other group of methods enhance the depth representation by feature manipulation . Chen et al . ( 2019a ) generated depth and semantic maps by a single latent representation . Ochs et al . ( 2019 ) proposed a segmentation-like loss item for depth estimation . Guizilini et al . ( 2020b ) leveraged a pre-trained semantic segmentation network and further conduct feature merging by PAC-based ( Su et al. , 2019 ) module . In this paper , we propose an individual semantic network for depth estimation . But different with Guizilini et al . ( 2020b ) , our semantic branch shares the same encoder and is trained together with the depth network for better scene representation . We leverage the semantic information in both ways . The semantic edges are used to guide the edge-based point feature sampling for local depth depth representation enhancement . At the mean time , we utilize the semantic feature blocks to conduct multi-scale feature fusion for global depth representation enhancement . The proposed method constructs a holistic way for feature representation enhancing in self-supervised depth estimation . 3 SELF-SUPERVISED DEPTH ESTIMATION FRAMEWORK . The proposed method builds upon the self-supervised depth estimation framework . An image triplet ( It−1 , It , It+1 ) is used as the input , where It acts as the target image and It′ ∈ { It−1 , It+1 } are the source images . During training , the target image It is fed into the depth network fD to get the estimated depth Dt = fD ( It ) , while the adjacent image pairs ( It , It′ ) are put into the ego-motion network for the 6-DoF ego-motion estimations Tt′→t . Then , the synthesized target images It′→t can be estimated using the source images It′ , depth Dt and the ego-motions Tt′→t , following the formulation from Zhou et al . ( 2017 ) . The self-supervised loss can be calculated by L ( It , It′→t ) = Lp ( It , It′→t ) + λLs ( It , It′→t ) , ( 1 ) where λ is the weighting factor between the photometric loss Lp and smoothness loss Ls . Following Godard et al . ( 2019 ) , we implement the minimum photometric loss Lp ( It , It′→t ) = min t′ ( α 2 ( 1− SSIM ( It , It′→t ) ) + ( 1− α ) ‖It − It′→t‖1 ) , ( 2 ) where SSIM is the Structural Similarity item ( Wang et al. , 2004 ) , α is the weighting factor which is set to 0.85 . The smoothness loss item ( Godard et al. , 2017 ) is implemented to smooth the depth map by the consistency between the image and depth gradient Ls ( It ) = \|∂xDt\| e−\|∂xIt\| + \|∂yDt\| e−\|∂yIt\| , ( 3 ) where ∂x , ∂x denote gradient operation on x , y-axis . In our implementation , we also conduct automasking and multi-scale depth upsampling strategy as proposed in Godard et al . ( 2019 ) to further improve depth estimation . 4 THE PROPOSED METHOD . In this paper , we promote the self-supervised depth estimation paradigm via proposing a semanticguided representation enhancing depth network . The overview of the method is shown in Figure 1 . We propose a multi-task framework which consists of the depth and semantic decoding branch with shared feature encoder . During the training stage , the semantic branch feeds contextual features to the depth branch to enhance depth comprehension . Under this multi-task framework , we propose a Semantic-guided Edge Enhancement Module ( SEEM ) to enhance the local depth feature representation on semantic category edges . Meanwhile , we enhance the global depth feature representations by proposing the semantic-guided multi-level self-attention module , which improves both the semantic and depth feature representations by digging into the pixel-wise dependencies .	The paper presents a method for semantic-guided self-supervised depth estimation from monocular images. They propose semantic guidance to improve depth estimation performance. This is obtained by applying semantic guidance at multiple levels in the decoder via an attention layer and via feature enhancement in edges areas of the image. Experiments show the proposed method reaches state of the art performance for self-supervised monocular depth estimation in structure-from-motion setting.	SP:529dd8f14cc4459b84dcb6a2eb5a35a520e10124
Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors	1 INTRODUCTION . Imaging inverse problems seek to recover an unknown image x 2 Rn from its noisy measurements y 2 Rm . Such problems arise in many fields , ranging from low-level computer vision to biomedical imaging . Since many imaging inverse problems are ill-posed , it is common to regularize the solution by using prior information on the unknown image . Widely-adopted image priors include total variation , low-rank penalties , and transform-domain sparsity ( Rudin et al. , 1992 ; Figueiredo & Nowak , 2001 ; 2003 ; Hu et al. , 2012 ; Elad & Aharon , 2006 ) . There has been considerable recent interest in plug-and-play priors ( PnP ) ( Venkatakrishnan et al. , 2013 ; Sreehari et al. , 2016 ) and regularization by denoising ( RED ) ( Romano et al. , 2017 ) , as frameworks for exploiting image denoisers as priors for image recovery . The popularity of deep learning has led to a wide adoption of deep denoisers within PnP/RED , leading to their state-of-the-art performance in a variety of applications , including image restoration ( Mataev et al. , 2019 ) , phase retrieval ( Metzler et al. , 2018 ) , and tomographic imaging ( Wu et al. , 2020 ) . Their empirical success has also prompted a follow-up theoretical work clarifying the existence of explicit regularizers ( Reehorst & Schniter , 2019 ) , providing new interpretations based on fixed-point projections ( Cohen et al. , 2020 ) , and analyzing their coordinate/online variants ( Sun et al. , 2019a ; Wu et al. , 2020 ) . Nonetheless , current PnP/RED algorithms are inherently serial . As illustrated in Fig . 1 , this makes them suboptimal on multicore systems that are often required for processing large-scale datasets ( Recht et al. , 2011 ) , such as those involving biomedical ( Ong et al. , 2020 ) and astronomical images ( Akiyama et al. , 2019 ) We address this gap by proposing a novel asynchronous RED ( ASYNC-RED ) algorithm . The algorithm decomposes the inference problem into a sequence of partial ( block-coordinate ) updates on x executed asynchronously in parallel over a multicore system . ASYNC-RED leads to a more efficient usage of available cores by avoiding synchronization of partial updates . ASYNC-RED is also scalable in terms of the number of measurements , since it processes only a small random subset of y at every iteration . We present two new theoretical results on the convergence of ASYNC-RED based on a unified set of explicit assumptions on the data-fidelity and the denoiser . Specifically , we establish its fixed-point convergence in the batch setting and extend this analysis to the randomized minibatch scenario . Our results extend recent work on serial block-coordinate RED ( BC-RED ) ( Sun et al. , Published as a conference paper at ICLR 2021 < latexit sha1_base64= '' +7+mD4Qola2MF8vLZ024UevUrMM= '' > 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< /latexit > 2019a ) and are fully consistent with the traditional asynchronous parallel optimization methods ( Lian et al. , 2015 ; Sun et al. , 2017 ) . We numerically validate ASYNC-RED on image recovery from linear and noisy measurements using pre-trained deep denoisers as image priors . 2 BACKGROUND . Inverse problems . Inverse problems are traditionally formulated as a composite optimization problem bx = arg min x2Rn g ( x ) + h ( x ) , ( 1 ) where g is the data-fidelity term that ensures consistency of x with the measured data y and h is the regularizer that infuses the prior knowledge on x . For example , consider the smooth ` 2-norm data-fidelity term g ( x ) = ky Axk22 , which assumes a linear observation model y = Ax + e , and the nonsmooth TV regularizer h ( x ) = ⌧kDxk1 , where ⌧ > 0 is the regularization parameter and D is the image gradient ( Rudin et al. , 1992 ) . Regularization by denoising ( RED ) . RED is a recent methodology for imaging inverse problems that seeks vectors x⇤ 2 Rn satisfying G ( x⇤ ) = rg ( x⇤ ) + ⌧ ( x⇤ D ( x⇤ ) ) = 0 , x⇤ 2 zer ( G ) : = { x 2 Rn : G ( x ) = 0 } ( 2 ) where rg denotes the gradient of the data-fidelity term and D : Rn ! Rn is an image denoiser parameterized by > 0 . Under additional technical assumptions , the solutions x⇤ 2 zer ( G ) can be associated with an explicit objective function of form ( 1 ) . Specifically , when D is locally homogeneous and has a symmetric Jacobian satisfying strong passivity ( Romano et al. , 2017 ; Reehorst & Schniter , 2019 ) , H ( x ) : = x D ( x ) corresponds to the gradient of a convex regularizer h ( x ) = 1 2 xT ( x D ( x ) ) . ( 3 ) A simple strategy , known as GM-RED , for computing x⇤ 2 zer ( G ) is based on the first-order fixed-point iteration xt = xt 1 G ( xt 1 ) , with G : Rn ! Rn , ( 4 ) where > 0 denotes the stepsize . In this paper , we extend this first-order RED algorithm to design ASYNC-RED . Since many denoisers do not satisfy the assumptions necessary for having an explicit objective ( Reehorst & Schniter , 2019 ) , our theoretical analysis considers a broader setting where D does not necessarily correspond to any explicit regularizer . The benefit of our analysis is that it accommodates powerful deep denoisers ( such as DnCNN ( Zhang et al. , 2017a ) ) that have been shown to achieve the state-of-the-art performance ( Sun et al. , 2019a ; Wu et al. , 2020 ; Cohen et al. , 2020 ) . Published as a conference paper at ICLR 2021 Plug-and-play priors ( PnP ) and other related work . There are other lines of works that combine the iterative methods with advanced denoisers . One closely-related framework is known as the deep mean-shift priors ( Bigdeli et al. , 2017 ) . It develops an implicit regularizer whose gradient is specified by a denoising autoencoder . Another well-known framework is PnP , which generalizes proximal methods by replacing the proximal map with an image denoiser ( Venkatakrishnan et al. , 2013 ) . Applications and theoretical analysis of PnP are widely studied in ( Sreehari et al. , 2016 ; Zhang et al. , 2017b ; Sun et al. , 2019 ; Zhang et al. , 2019 ; Ahmad et al. , 2020 ; Wei et al. , 2020 ) and ( Chan et al. , 2017 ; Meinhardt et al. , 2017 ; Buzzard et al. , 2018 ; Sun et al. , 2019b ; Tirer & Giryes , 2019 ; Teodoro et al. , 2019 ; Ryu et al. , 2019 ; Xu et al. , 2020 ) , respectively . In particular , Buzzard et al . ( 2018 ) proposed a parallel extension of PnP called Consensus Equilibrium ( CE ) , which enables synchronous parallel updates of x . Note that while we developed ASYNC-RED as a variant of RED , our framework and analysis can be also potentially applied to PnP/CE . The plug-in strategy can be also applied to another family of algorithms known as approximate message passing ( AMP ) ( Metzler et al. , 2016a ; b ; Fletcher et al. , 2018 ) . The AMP-based algorithms are known to be nearly-optimal for random measurement matrices , but are generally unstable for general A ( Rangan et al. , 2014 ; 2015 ) . Asynchronous parallel optimization . There are two main lines of work in asynchronous parallel optimization , the one involving the asynchrony in coordinate updates ( Iutzeler et al. , 2013 ; Liu et al. , 2015 ; Peng et al. , 2016 ; Bianchi et al. , 2015 ; Sun et al. , 2017 ; Hannah & Yin , 2018 ; Hannah et al. , 2019 ) , and the other focusing on the study of various asynchronous stochastic gradient methods ( Recht et al. , 2011 ; Lian et al. , 2015 ; Liu et al. , 2018 ; Zhou et al. , 2018 ; Lian et al. , 2018 ) . Our work contributes to the area by developing a novel deep-regularized asynchronous parallel method with provable convergence guarantees .	This paper proposed for the first time the asynchronous variants of deterministic and stochastic regularization-by-denoising (RED) algorithms which have become popular recently in image recovery and reconstruction applications since they leverage the power of pretrained deep denoising neural networks into the traditional model-based schemes, and often achieve state-of-the-art recovery results. These new variants are aimed to fully utilize the multi-cores structure of computational devices and to maximize the practicality of PnP/RED methods in large-scale inverse problems. The authors provide gradient-norm convergence analysis for the proposed algorithms under standard assumptions, along with detailed numerical studies demonstrating practical advantageous of proposed methods.	SP:db54672a85a9533d2afb420b9deb50e905bd33ec
Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors	1 INTRODUCTION . Imaging inverse problems seek to recover an unknown image x 2 Rn from its noisy measurements y 2 Rm . Such problems arise in many fields , ranging from low-level computer vision to biomedical imaging . Since many imaging inverse problems are ill-posed , it is common to regularize the solution by using prior information on the unknown image . Widely-adopted image priors include total variation , low-rank penalties , and transform-domain sparsity ( Rudin et al. , 1992 ; Figueiredo & Nowak , 2001 ; 2003 ; Hu et al. , 2012 ; Elad & Aharon , 2006 ) . There has been considerable recent interest in plug-and-play priors ( PnP ) ( Venkatakrishnan et al. , 2013 ; Sreehari et al. , 2016 ) and regularization by denoising ( RED ) ( Romano et al. , 2017 ) , as frameworks for exploiting image denoisers as priors for image recovery . The popularity of deep learning has led to a wide adoption of deep denoisers within PnP/RED , leading to their state-of-the-art performance in a variety of applications , including image restoration ( Mataev et al. , 2019 ) , phase retrieval ( Metzler et al. , 2018 ) , and tomographic imaging ( Wu et al. , 2020 ) . Their empirical success has also prompted a follow-up theoretical work clarifying the existence of explicit regularizers ( Reehorst & Schniter , 2019 ) , providing new interpretations based on fixed-point projections ( Cohen et al. , 2020 ) , and analyzing their coordinate/online variants ( Sun et al. , 2019a ; Wu et al. , 2020 ) . Nonetheless , current PnP/RED algorithms are inherently serial . As illustrated in Fig . 1 , this makes them suboptimal on multicore systems that are often required for processing large-scale datasets ( Recht et al. , 2011 ) , such as those involving biomedical ( Ong et al. , 2020 ) and astronomical images ( Akiyama et al. , 2019 ) We address this gap by proposing a novel asynchronous RED ( ASYNC-RED ) algorithm . The algorithm decomposes the inference problem into a sequence of partial ( block-coordinate ) updates on x executed asynchronously in parallel over a multicore system . ASYNC-RED leads to a more efficient usage of available cores by avoiding synchronization of partial updates . ASYNC-RED is also scalable in terms of the number of measurements , since it processes only a small random subset of y at every iteration . We present two new theoretical results on the convergence of ASYNC-RED based on a unified set of explicit assumptions on the data-fidelity and the denoiser . Specifically , we establish its fixed-point convergence in the batch setting and extend this analysis to the randomized minibatch scenario . Our results extend recent work on serial block-coordinate RED ( BC-RED ) ( Sun et al. , Published as a conference paper at ICLR 2021 < latexit sha1_base64= '' +7+mD4Qola2MF8vLZ024UevUrMM= '' > 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< /latexit > 2019a ) and are fully consistent with the traditional asynchronous parallel optimization methods ( Lian et al. , 2015 ; Sun et al. , 2017 ) . We numerically validate ASYNC-RED on image recovery from linear and noisy measurements using pre-trained deep denoisers as image priors . 2 BACKGROUND . Inverse problems . Inverse problems are traditionally formulated as a composite optimization problem bx = arg min x2Rn g ( x ) + h ( x ) , ( 1 ) where g is the data-fidelity term that ensures consistency of x with the measured data y and h is the regularizer that infuses the prior knowledge on x . For example , consider the smooth ` 2-norm data-fidelity term g ( x ) = ky Axk22 , which assumes a linear observation model y = Ax + e , and the nonsmooth TV regularizer h ( x ) = ⌧kDxk1 , where ⌧ > 0 is the regularization parameter and D is the image gradient ( Rudin et al. , 1992 ) . Regularization by denoising ( RED ) . RED is a recent methodology for imaging inverse problems that seeks vectors x⇤ 2 Rn satisfying G ( x⇤ ) = rg ( x⇤ ) + ⌧ ( x⇤ D ( x⇤ ) ) = 0 , x⇤ 2 zer ( G ) : = { x 2 Rn : G ( x ) = 0 } ( 2 ) where rg denotes the gradient of the data-fidelity term and D : Rn ! Rn is an image denoiser parameterized by > 0 . Under additional technical assumptions , the solutions x⇤ 2 zer ( G ) can be associated with an explicit objective function of form ( 1 ) . Specifically , when D is locally homogeneous and has a symmetric Jacobian satisfying strong passivity ( Romano et al. , 2017 ; Reehorst & Schniter , 2019 ) , H ( x ) : = x D ( x ) corresponds to the gradient of a convex regularizer h ( x ) = 1 2 xT ( x D ( x ) ) . ( 3 ) A simple strategy , known as GM-RED , for computing x⇤ 2 zer ( G ) is based on the first-order fixed-point iteration xt = xt 1 G ( xt 1 ) , with G : Rn ! Rn , ( 4 ) where > 0 denotes the stepsize . In this paper , we extend this first-order RED algorithm to design ASYNC-RED . Since many denoisers do not satisfy the assumptions necessary for having an explicit objective ( Reehorst & Schniter , 2019 ) , our theoretical analysis considers a broader setting where D does not necessarily correspond to any explicit regularizer . The benefit of our analysis is that it accommodates powerful deep denoisers ( such as DnCNN ( Zhang et al. , 2017a ) ) that have been shown to achieve the state-of-the-art performance ( Sun et al. , 2019a ; Wu et al. , 2020 ; Cohen et al. , 2020 ) . Published as a conference paper at ICLR 2021 Plug-and-play priors ( PnP ) and other related work . There are other lines of works that combine the iterative methods with advanced denoisers . One closely-related framework is known as the deep mean-shift priors ( Bigdeli et al. , 2017 ) . It develops an implicit regularizer whose gradient is specified by a denoising autoencoder . Another well-known framework is PnP , which generalizes proximal methods by replacing the proximal map with an image denoiser ( Venkatakrishnan et al. , 2013 ) . Applications and theoretical analysis of PnP are widely studied in ( Sreehari et al. , 2016 ; Zhang et al. , 2017b ; Sun et al. , 2019 ; Zhang et al. , 2019 ; Ahmad et al. , 2020 ; Wei et al. , 2020 ) and ( Chan et al. , 2017 ; Meinhardt et al. , 2017 ; Buzzard et al. , 2018 ; Sun et al. , 2019b ; Tirer & Giryes , 2019 ; Teodoro et al. , 2019 ; Ryu et al. , 2019 ; Xu et al. , 2020 ) , respectively . In particular , Buzzard et al . ( 2018 ) proposed a parallel extension of PnP called Consensus Equilibrium ( CE ) , which enables synchronous parallel updates of x . Note that while we developed ASYNC-RED as a variant of RED , our framework and analysis can be also potentially applied to PnP/CE . The plug-in strategy can be also applied to another family of algorithms known as approximate message passing ( AMP ) ( Metzler et al. , 2016a ; b ; Fletcher et al. , 2018 ) . The AMP-based algorithms are known to be nearly-optimal for random measurement matrices , but are generally unstable for general A ( Rangan et al. , 2014 ; 2015 ) . Asynchronous parallel optimization . There are two main lines of work in asynchronous parallel optimization , the one involving the asynchrony in coordinate updates ( Iutzeler et al. , 2013 ; Liu et al. , 2015 ; Peng et al. , 2016 ; Bianchi et al. , 2015 ; Sun et al. , 2017 ; Hannah & Yin , 2018 ; Hannah et al. , 2019 ) , and the other focusing on the study of various asynchronous stochastic gradient methods ( Recht et al. , 2011 ; Lian et al. , 2015 ; Liu et al. , 2018 ; Zhou et al. , 2018 ; Lian et al. , 2018 ) . Our work contributes to the area by developing a novel deep-regularized asynchronous parallel method with provable convergence guarantees .	Due to the growth of data sets in a lot of applications, it is important to develop algorithms to achieve great performance but with significantly reduced computational cost. The paper proposes asynchronous type of parallel algorithms by combining the pre-trained deep denoisers. In particular, batch gradient and stochastic gradient are applied to the proposed algorithm framework by taking advantage of the coordinate separable structures of the problem. Convergence of the algorithms are guaranteed under the four specified assumptions. Numerical experiments on the CT image reconstruction have justified the proposed efficiency and significant improvement in terms of running time. The importance and contribution of this work in compressive sensing algorithms stand. However, the novelty of the methods look incremental. There are some other issues listed as follows.	SP:db54672a85a9533d2afb420b9deb50e905bd33ec
Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors	1 INTRODUCTION . Imaging inverse problems seek to recover an unknown image x 2 Rn from its noisy measurements y 2 Rm . Such problems arise in many fields , ranging from low-level computer vision to biomedical imaging . Since many imaging inverse problems are ill-posed , it is common to regularize the solution by using prior information on the unknown image . Widely-adopted image priors include total variation , low-rank penalties , and transform-domain sparsity ( Rudin et al. , 1992 ; Figueiredo & Nowak , 2001 ; 2003 ; Hu et al. , 2012 ; Elad & Aharon , 2006 ) . There has been considerable recent interest in plug-and-play priors ( PnP ) ( Venkatakrishnan et al. , 2013 ; Sreehari et al. , 2016 ) and regularization by denoising ( RED ) ( Romano et al. , 2017 ) , as frameworks for exploiting image denoisers as priors for image recovery . The popularity of deep learning has led to a wide adoption of deep denoisers within PnP/RED , leading to their state-of-the-art performance in a variety of applications , including image restoration ( Mataev et al. , 2019 ) , phase retrieval ( Metzler et al. , 2018 ) , and tomographic imaging ( Wu et al. , 2020 ) . Their empirical success has also prompted a follow-up theoretical work clarifying the existence of explicit regularizers ( Reehorst & Schniter , 2019 ) , providing new interpretations based on fixed-point projections ( Cohen et al. , 2020 ) , and analyzing their coordinate/online variants ( Sun et al. , 2019a ; Wu et al. , 2020 ) . Nonetheless , current PnP/RED algorithms are inherently serial . As illustrated in Fig . 1 , this makes them suboptimal on multicore systems that are often required for processing large-scale datasets ( Recht et al. , 2011 ) , such as those involving biomedical ( Ong et al. , 2020 ) and astronomical images ( Akiyama et al. , 2019 ) We address this gap by proposing a novel asynchronous RED ( ASYNC-RED ) algorithm . The algorithm decomposes the inference problem into a sequence of partial ( block-coordinate ) updates on x executed asynchronously in parallel over a multicore system . ASYNC-RED leads to a more efficient usage of available cores by avoiding synchronization of partial updates . ASYNC-RED is also scalable in terms of the number of measurements , since it processes only a small random subset of y at every iteration . We present two new theoretical results on the convergence of ASYNC-RED based on a unified set of explicit assumptions on the data-fidelity and the denoiser . Specifically , we establish its fixed-point convergence in the batch setting and extend this analysis to the randomized minibatch scenario . Our results extend recent work on serial block-coordinate RED ( BC-RED ) ( Sun et al. , Published as a conference paper at ICLR 2021 < latexit sha1_base64= '' +7+mD4Qola2MF8vLZ024UevUrMM= '' > 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< /latexit > 2019a ) and are fully consistent with the traditional asynchronous parallel optimization methods ( Lian et al. , 2015 ; Sun et al. , 2017 ) . We numerically validate ASYNC-RED on image recovery from linear and noisy measurements using pre-trained deep denoisers as image priors . 2 BACKGROUND . Inverse problems . Inverse problems are traditionally formulated as a composite optimization problem bx = arg min x2Rn g ( x ) + h ( x ) , ( 1 ) where g is the data-fidelity term that ensures consistency of x with the measured data y and h is the regularizer that infuses the prior knowledge on x . For example , consider the smooth ` 2-norm data-fidelity term g ( x ) = ky Axk22 , which assumes a linear observation model y = Ax + e , and the nonsmooth TV regularizer h ( x ) = ⌧kDxk1 , where ⌧ > 0 is the regularization parameter and D is the image gradient ( Rudin et al. , 1992 ) . Regularization by denoising ( RED ) . RED is a recent methodology for imaging inverse problems that seeks vectors x⇤ 2 Rn satisfying G ( x⇤ ) = rg ( x⇤ ) + ⌧ ( x⇤ D ( x⇤ ) ) = 0 , x⇤ 2 zer ( G ) : = { x 2 Rn : G ( x ) = 0 } ( 2 ) where rg denotes the gradient of the data-fidelity term and D : Rn ! Rn is an image denoiser parameterized by > 0 . Under additional technical assumptions , the solutions x⇤ 2 zer ( G ) can be associated with an explicit objective function of form ( 1 ) . Specifically , when D is locally homogeneous and has a symmetric Jacobian satisfying strong passivity ( Romano et al. , 2017 ; Reehorst & Schniter , 2019 ) , H ( x ) : = x D ( x ) corresponds to the gradient of a convex regularizer h ( x ) = 1 2 xT ( x D ( x ) ) . ( 3 ) A simple strategy , known as GM-RED , for computing x⇤ 2 zer ( G ) is based on the first-order fixed-point iteration xt = xt 1 G ( xt 1 ) , with G : Rn ! Rn , ( 4 ) where > 0 denotes the stepsize . In this paper , we extend this first-order RED algorithm to design ASYNC-RED . Since many denoisers do not satisfy the assumptions necessary for having an explicit objective ( Reehorst & Schniter , 2019 ) , our theoretical analysis considers a broader setting where D does not necessarily correspond to any explicit regularizer . The benefit of our analysis is that it accommodates powerful deep denoisers ( such as DnCNN ( Zhang et al. , 2017a ) ) that have been shown to achieve the state-of-the-art performance ( Sun et al. , 2019a ; Wu et al. , 2020 ; Cohen et al. , 2020 ) . Published as a conference paper at ICLR 2021 Plug-and-play priors ( PnP ) and other related work . There are other lines of works that combine the iterative methods with advanced denoisers . One closely-related framework is known as the deep mean-shift priors ( Bigdeli et al. , 2017 ) . It develops an implicit regularizer whose gradient is specified by a denoising autoencoder . Another well-known framework is PnP , which generalizes proximal methods by replacing the proximal map with an image denoiser ( Venkatakrishnan et al. , 2013 ) . Applications and theoretical analysis of PnP are widely studied in ( Sreehari et al. , 2016 ; Zhang et al. , 2017b ; Sun et al. , 2019 ; Zhang et al. , 2019 ; Ahmad et al. , 2020 ; Wei et al. , 2020 ) and ( Chan et al. , 2017 ; Meinhardt et al. , 2017 ; Buzzard et al. , 2018 ; Sun et al. , 2019b ; Tirer & Giryes , 2019 ; Teodoro et al. , 2019 ; Ryu et al. , 2019 ; Xu et al. , 2020 ) , respectively . In particular , Buzzard et al . ( 2018 ) proposed a parallel extension of PnP called Consensus Equilibrium ( CE ) , which enables synchronous parallel updates of x . Note that while we developed ASYNC-RED as a variant of RED , our framework and analysis can be also potentially applied to PnP/CE . The plug-in strategy can be also applied to another family of algorithms known as approximate message passing ( AMP ) ( Metzler et al. , 2016a ; b ; Fletcher et al. , 2018 ) . The AMP-based algorithms are known to be nearly-optimal for random measurement matrices , but are generally unstable for general A ( Rangan et al. , 2014 ; 2015 ) . Asynchronous parallel optimization . There are two main lines of work in asynchronous parallel optimization , the one involving the asynchrony in coordinate updates ( Iutzeler et al. , 2013 ; Liu et al. , 2015 ; Peng et al. , 2016 ; Bianchi et al. , 2015 ; Sun et al. , 2017 ; Hannah & Yin , 2018 ; Hannah et al. , 2019 ) , and the other focusing on the study of various asynchronous stochastic gradient methods ( Recht et al. , 2011 ; Lian et al. , 2015 ; Liu et al. , 2018 ; Zhou et al. , 2018 ; Lian et al. , 2018 ) . Our work contributes to the area by developing a novel deep-regularized asynchronous parallel method with provable convergence guarantees .	The paper describes a novel implementation of RED, regularization by denoising, which better leverages multicore architectures to achieve a significant speedup. The proposed implementation splits the gradient step into smaller components, which can each be executed independently on different cores and then used to update a shared copy. The crucial result is two sets of convergence guarantees showing that this delayed update will not cause too much error, even if the updates from different cores arrive at different times. The speedups achieved range from 6× to 8× on two tasks (compressive sensing and computed tomography reconstruction).	SP:db54672a85a9533d2afb420b9deb50e905bd33ec
Policy-Driven Attack: Learning to Query for Hard-label Black-box Adversarial Examples	1 INTRODUCTION . It is widely known that deep neural networks ( DNNs ) are vulnerable to adversarial examples , which are crafted via perturbing clean examples to cause the victim model to make incorrect predictions . In a white-box setting where the adversaries have full access to the architecture and parameters of the victim model , gradients w.r.t . network inputs can be easily calculated via back-propagation , and thus first-order optimization techniques can be directly applied to craft adversarial examples in this setting ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ; Carlini & Wagner , 2017 ; Madry et al. , 2018 ; Rony et al. , 2019 ) . However , in black-box settings , input gradients are no longer readily available since all model internals are kept secret . Over the past few years , the community has made massive efforts in developing black-box attacks . In order to gain high attack success rates , delicate queries to the victim model are normally required . Recent methods can be roughly categorized into score-based attacks ( Chen et al. , 2017 ; Ilyas et al. , 2018 ; Nitin Bhagoji et al. , 2018 ; Ilyas et al. , 2019 ; Yan et al. , 2019 ; Li et al. , 2020b ; Tu et al. , 2019 ; Du et al. , 2019 ; Li et al. , 2019 ; Bai et al. , 2020 ) and hard-label attacks ( a.k.a , decision-based attacks ) ( Brendel et al. , 2018 ; Cheng et al. , 2019 ; Dong et al. , 2019 ; Shi et al. , 2019 ; Brunner et al. , 2019 ; Chen et al. , 2020 ; Rahmati et al. , 2020 ; Li et al. , 2020a ; Shi et al. , 2020 ; Chen & Gu , 2020 ) , based on the amount of information exposed to the adversaries from the output of victim model . When the prediction probabilities of the victim model are accessible , an intelligent adversary would generally prefer score-based attacks , while in a more practical scenario where only the top-1 class prediction is available , the adversaries will have to resort to hard-label attacks . Since less information is exposed from such feedback of the victim model , hard-label attacks often bare higher query complexity than that of score-based attacks , making their attack process costly and time intensive . * The first two authors contributed equally to the work . Work was done when ZY was an intern at ByteDance AI Lab . In this paper , we aim at reducing the query complexity of hard-label black-box attacks . We cast the problem of progressively refining the candidate adversarial example ( by skillfully querying the victim model and analyzing its feedback ) into a reinforcement learning formulation . At each iteration , we search along a set of chosen directions to see whether there exists any new candidate adversarial example that is perceptually more similar to its benign counterpart , i.e. , in the sense of requiring less distortion . A reward is assigned to each of such search directions ( treated as actions ) , based on the amount of distortion reduction yielded after updating the adversarial example along that direction . Such a reinforcement learning formulation enables us to learn the non-differentiable mapping from search directions to their potential of refining the current adversarial example , directly and precisely . The policy network is expected to be capable of providing the most promising search direction for updating candidate adversarial examples to reduce the required distortion of the adversarial examples from their benign counterparts . As we will show , the proposed policy network can learn from not only the queries that had been performed following the evolving policy but also peer experience from other black-box attacks . As such , it is possible to pre-train the policy network on a small number of query-reward pairs obtained from the performance log of prior attacks ( with or without policy ) to the same victim model . Experiments show that our policy-driven attack ( PDA ) can achieve significantly lower distortions than existing state-of-the-arts under the same query budgets . 2 RELATED WORK . In this paper , we focus on the hard-label black-box setting where only the top-1 decision of the victim model is available . Since less information ( of the victim model ) is exposed after each query , attacks in this category are generally required to query the victim model more times than those in the white-box or score-based settings . For example , an initial attempt named boundary attack ( Brendel et al. , 2018 ) could require ∼million queries before convergence . It proposed to start from an image that is already adversarial , and tried to reduce the distortion by walking towards the benign image along the decision boundary . Recent methods in this category focused more on gradient estimation which could provide more promising search directions , while relying only on top-1 class predictions . Ilyas et al . ( 2018 ) advocated to use NES ( Wierstra et al. , 2014 ; Salimans et al. , 2017 ) to estimate the gradients over proxy scores , and then mounted a variant of PGD attack ( Madry et al. , 2018 ) with the estimated gradients . Towards improving the efficiency of gradient estimation , Cheng et al . ( 2019 ) and Chen et al . ( 2020 ) further introduced a continuous optimization formulation and an unbiased gradient estimation with careful error control , respectively . The gradients were estimated via issuing probe queries from a standard Gaussian distribution . To generate probes from some more powerful distributions , Dong et al . ( 2019 ) proposed to use the covariance matrix adaptation evolution strategy , while Shi et al . ( 2020 ) suggested to use customized distribution to model the sensitivity of each pixel . In contrast to these methods , our PDA proposes to use a policy network which is learned from prior intentions to advocate promising search directions to reduce the query complexity . We note that some works also proposed to exploit DNN models to generate black-box attacks . For example , Naseer et al . ( 2019 ) used DNNs to promote the transferability of black-box attacks , while several score-based black-box attacks proposed to train DNN models for assisting the generation of queries ( Li et al. , 2019 ; Du et al. , 2019 ; Bai et al. , 2020 ) . Our method is naturally different from them in problem settings ( score-based vs hard-label ) and problem formulations . In the autonomous field , Hamdi et al . ( 2020 ) proposed to formulate the generation of semantic attacks as a reinforcement learning problem to find parameters of environment ( e.g. , camera viewpoint ) that can fool the recognition system . To the best of our knowledge , our work is the first to incorporate reinforcement learning into the black-box attacking scenario for estimating perturbation directions , and we advocate the community to consider more about this principled formulation in the future . In addition to the novel reinforcement learning formulation , we also introduce a specific architecture for the policy network which enjoys superior generalization performance , while these methods adopted off-the-shelf auto-encoding architectures . 3 OUR POLICY-DRIVEN ATTACK . We study the problem of attacking an image classifier in the hard-label setting . The goal of the adversaries is to perturb an benign image x ∈ Rn to fool a k-way victim classifier f : Rn → Rk into making an incorrect decision : arg maxi f ( x ′ ) i 6= y , where x′ is the adversarial example generated by perturbing the benign image and y is the true label of x . The adversaries would generally prefer adversarial examples x′ with smaller distortions ‖x− x′‖2 achieved using less queries , since these properties make the attack less suspicious and also save the cost . In this section , we first briefly review some background information that motivate our method ( in Section 3.1 ) , and then detail our reinforcement learning formulation ( in Section 3.2 and Section 3.3 ) and the architecture of our policy network ( in Section 3.4 ) . 3.1 MOTIVATIONS . Most recent hard-label attacks followed a common pipeline of searching from a starting point which was already an adversarial image1 yet not close enough to the benign one . Unlike the white-box and score-based black-box setting in which the input gradients can be calculated and used as the most effective perturbation direction , in the concerned hard-label setting , outputs of the victim model only flip on the decision boundary while keeping constant away from the boundary , making it difficult to evaluate different directions almost everywhere . In this context , the search of promising perturbation directions was restricted into the regions near the decision boundary , since these regions are arguably more informative , and binary search was used to reach the decision boundary efficiently . Let us take a very recent attack named HopSkipJumpAttack ( Chen et al. , 2020 ) as an example . Given the current estimation x′s of the adversarial example at each iteration , HopSkipJumpAttack first performed binary search to project it onto the decision boundary of the victim model . Denote x′ as the updated example that was on the decision boundary already , HopSkipJumpAttack then sampled many probes around x′ from an isotropic Gaussian distribution , and issued these probes to the victim model as queries . The feedback of the victim model was utilized to estimate the gradient direction at x′ , and it was updated along this direction to obtain a new estimation of the adversarial example . This process was repeated many times until the query budget was exhausted . In comparison with boundary attack ( Brendel et al. , 2018 ) , HopSkipJumpAttack was in general far more query-efficient , though a large number of queries had to be consumed for probing the local geometrics of the decision boundary of the victim model . Its superiority came from using the estimated gradient directions as the search directions , which motivated us to explore even better search directions at each iteration of the attack . As will be shown in the appendices , drawing on some geometric insights , we found that the gradient directions are in fact not the optimal search directions in the framework of HopSkipJumpAttack . We also found that the task of performing hardlabel black-box attack could be naturally cast into a reinforcement learning task , thus we attempt to explore the possibility of developing a model-based method for predicting the most promising search directions for attacks . Feedbacks from the victim can provide supervision and thus the policy models in our reinforcement learning framework can be trained/fine-tuned on the fly during each attack process , such that little query is required once the model has been well-trained . 3.2 ATTACK AS REINFORCEMENT LEARNING PROBLEM . In this paper , we consider both targeted attacks and untargeted attacks . Given a benign example x , its label y , and the victim model f , an environment E ( x , y , f ) is naturally formed . The adversaries shall play the role of agent , trying to interact with the environment by issuing queries and collecting feedbacks , under a certain policy . The current example x′t on the decision boundary of the victim model ( or called the candidate adversarial example ) represents the state at each timestamp t. The agent uses a learnable policy network g which will be carefully introduced in Section 3.4 to guide its actions , and the action is to update the candidate adversarial example such that less distortions are required to fool the victim model . The action here incorporates searching along a promising direction at/‖at‖ where at ∈ Rn is sampled from an isotropic Gaussian distribution whose mean vector is given by the policy network µt = g ( x′t , y , y ′ ) ∈ Rn where y′ is the target label , and its covariance matrix is given by Σ = σI ∈ Rn×n , in which the value of σ ∈ R is set to be gradually increased as the attack on each sample progresses , and I ∈ Rn×n indicates the n × n identity matrix . With at , the agent searches along its direction at/‖at‖ to see whether any better candidate adversarial example can be found . For targeted attacks , the target label y′ is chosen by the agent from the beginning and kept unchanged during the attack process . For untargeted attacks , x′t should be on the decision boundary where one side is the ground-truth label y and the other side could be regarded as the “ target label ” y′ . As will be carefully introduced in Section 3.3 , a reward rt ∈ R 1In practice , it is performed by randomly sampling until the adversarial constraint is satisfied , i.e. , it is not classified as y by the victim model , or by directly choosing a benign sample from the adversarial class . Algorithm 1 Policy-Driven Attack Algorithm 1 : Input : the environment E ( x , y , f ) ; the target label y′ , initial adversarial image x′1 ∈ Rn which lies on the decision boundary ; the policy network g. 2 : Output : an adversarial example . 3 : Initialize the step index t← 1 . 4 : while the query count limit not reached do 5 : // Determine the baseline lt to evaluate the potential of different actions 6 : µt ← g ( x′t , y , y′ ) , z ← BS ( x′t + δ · µt‖µt‖2 , x , f ) , where BS ( · , · , · ) performs binary search 7 : Set the distortion reduction of z as baseline : lt ← max { ‖x′t − x‖2 − ‖z − x‖2 , lmin } 8 : 9 : // Collect actions and rewards , and update the policy network 10 : Sample M actions : at , i ∼ N ( µt , σtI ) , i ∈ { 1 , 2 , . . . , M } 11 : Assign rewards rt , i to each actions with our mechanism introduced in Section 3.3 12 : Update the policy network using one-step REINFORCE on M pairs : ( at , i , rt , i ) 13 : 14 : // Update adversarial image using predicted direction 15 : µt ← g ( x′t , y , y′ ) if ‖z − x‖2 ≤ ‖x′t − x‖2 , otherwise µt ← at , i∗ , i∗ = arg maxi rt , i 16 : x′t+1 ← BS ( x′t + · µt‖µt‖2 , x , f ) 17 : 18 : // Update other variables 19 : Double σt if all M rewards are zeros : σt+1 ← 2 · σt ; else keep it : σt+1 ← σt 20 : t← t+ 1 21 : end while 22 : return final adversarial image x′t based on the performance of each action and the corresponding at is given to the agent for updating the parameter of the policy network . All details of our PDA are summarized in Algorithm 1 . Powered by the reinforcement learning framework , we can use policy gradient algorithms to train the policy network g to generate promising search directions in a direct way . For simplicity , we use the one-step REINFORCE ( Williams , 1992 ) in the sequel of this paper and leave the exploration of more advanced policy gradient algorithms to future work . 3.3 REWARD AND ACTION Figure 1 illustrates how we assign the scalar reward rt given current candidate adversarial example x′t and an action at . The decision boundary is illustrated by a horizontal straight line ( denoted by B ) in the figure , the benign counterpart x is assumed to be below B , and the circle C centered at x′t with a small radius δ shows all possible locations after jumping along the directions of some actions by δ from x′t . As described earlier , the reward rt should be assigned based on the amount of potential distortion reduction brought by at . A direct evaluation can be achieved by jumping along the direction of at first and then projecting the updated example back onto the decision boundary via binary search , to see how much improvement is obtained . However , since we evaluateM actions at , i simultaneously at an iteration ( see Algorithm 1 ) and binary search needs to be performed for each of them , and the overall process would be prohibitively ( query- ) expensive . On this point , to efficiently assess the performance of an action , we instead evaluate whether the reduction of distortion by taking a particular action can exceed particular baselines . Concretely , we first evaluate µt = g ( x′t , y , y ′ ) as an action directly by using binary search in a way as just described . Suppose that it can reduce the required adversarial distortion2 by lt , then we setup two levels of baselines ‖x− x′t‖2 − β1 · lt and ‖x−x′t‖2−β2 · lt to see whether other actions can lead to adversarial examples with closer distance ( than these baselines ) from the benign example x , in which β1 = 0 and β2 = 0.25 . As shown in Figure 1 , for an action a ∈ { at , i } , we first obtain x′t + δ ·a/‖a‖2 and then move it towards x to see how much reward it can obtain . The two arcs V1 and V2 indicates where the same progress as the two baselines can be achieved , thus we can further project x′t + δ ·a/‖a‖2 onto the arcs to see if the projections ( i.e , xV1s and x V2 s ) are still adversarial . It can be seen that x V1 s is still adversarial yet x V2 s is not . We assign a reward 1 to such an action a . If both the projections are still adversarial we shall assign a reward of 2 , and if neither of them is adversarial , zero reward is assigned . Since xV1s is not adversarial could imply that xV2s is also not adversarial , such a way reduces the number of queries for assessing each action to at most 2 ( xV1s and x V2 s ) and makes our PDA more query-efficient .	This paper proposes a new hard-label black-box adversarial attack method based on reinforcement learning. The authors formulate the black-box attacking problem as a reinforcement learning problem, and design a policy network to learn the appropriate attack directions, in order to achieve more efficient attacks. The proposed policy-driven attack (PDA) algorithm is able to craft adversarial examples with lower distortions under the same query budgets. Experiment results show that with a pre-trained policy model, PDA outperforms two baseline methods, Boundary Attack and HopSkipJumpAttack.	SP:6b7fdf95219b32e4dca1b3fb084f2c11a5d29fd9
Policy-Driven Attack: Learning to Query for Hard-label Black-box Adversarial Examples	1 INTRODUCTION . It is widely known that deep neural networks ( DNNs ) are vulnerable to adversarial examples , which are crafted via perturbing clean examples to cause the victim model to make incorrect predictions . In a white-box setting where the adversaries have full access to the architecture and parameters of the victim model , gradients w.r.t . network inputs can be easily calculated via back-propagation , and thus first-order optimization techniques can be directly applied to craft adversarial examples in this setting ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ; Carlini & Wagner , 2017 ; Madry et al. , 2018 ; Rony et al. , 2019 ) . However , in black-box settings , input gradients are no longer readily available since all model internals are kept secret . Over the past few years , the community has made massive efforts in developing black-box attacks . In order to gain high attack success rates , delicate queries to the victim model are normally required . Recent methods can be roughly categorized into score-based attacks ( Chen et al. , 2017 ; Ilyas et al. , 2018 ; Nitin Bhagoji et al. , 2018 ; Ilyas et al. , 2019 ; Yan et al. , 2019 ; Li et al. , 2020b ; Tu et al. , 2019 ; Du et al. , 2019 ; Li et al. , 2019 ; Bai et al. , 2020 ) and hard-label attacks ( a.k.a , decision-based attacks ) ( Brendel et al. , 2018 ; Cheng et al. , 2019 ; Dong et al. , 2019 ; Shi et al. , 2019 ; Brunner et al. , 2019 ; Chen et al. , 2020 ; Rahmati et al. , 2020 ; Li et al. , 2020a ; Shi et al. , 2020 ; Chen & Gu , 2020 ) , based on the amount of information exposed to the adversaries from the output of victim model . When the prediction probabilities of the victim model are accessible , an intelligent adversary would generally prefer score-based attacks , while in a more practical scenario where only the top-1 class prediction is available , the adversaries will have to resort to hard-label attacks . Since less information is exposed from such feedback of the victim model , hard-label attacks often bare higher query complexity than that of score-based attacks , making their attack process costly and time intensive . * The first two authors contributed equally to the work . Work was done when ZY was an intern at ByteDance AI Lab . In this paper , we aim at reducing the query complexity of hard-label black-box attacks . We cast the problem of progressively refining the candidate adversarial example ( by skillfully querying the victim model and analyzing its feedback ) into a reinforcement learning formulation . At each iteration , we search along a set of chosen directions to see whether there exists any new candidate adversarial example that is perceptually more similar to its benign counterpart , i.e. , in the sense of requiring less distortion . A reward is assigned to each of such search directions ( treated as actions ) , based on the amount of distortion reduction yielded after updating the adversarial example along that direction . Such a reinforcement learning formulation enables us to learn the non-differentiable mapping from search directions to their potential of refining the current adversarial example , directly and precisely . The policy network is expected to be capable of providing the most promising search direction for updating candidate adversarial examples to reduce the required distortion of the adversarial examples from their benign counterparts . As we will show , the proposed policy network can learn from not only the queries that had been performed following the evolving policy but also peer experience from other black-box attacks . As such , it is possible to pre-train the policy network on a small number of query-reward pairs obtained from the performance log of prior attacks ( with or without policy ) to the same victim model . Experiments show that our policy-driven attack ( PDA ) can achieve significantly lower distortions than existing state-of-the-arts under the same query budgets . 2 RELATED WORK . In this paper , we focus on the hard-label black-box setting where only the top-1 decision of the victim model is available . Since less information ( of the victim model ) is exposed after each query , attacks in this category are generally required to query the victim model more times than those in the white-box or score-based settings . For example , an initial attempt named boundary attack ( Brendel et al. , 2018 ) could require ∼million queries before convergence . It proposed to start from an image that is already adversarial , and tried to reduce the distortion by walking towards the benign image along the decision boundary . Recent methods in this category focused more on gradient estimation which could provide more promising search directions , while relying only on top-1 class predictions . Ilyas et al . ( 2018 ) advocated to use NES ( Wierstra et al. , 2014 ; Salimans et al. , 2017 ) to estimate the gradients over proxy scores , and then mounted a variant of PGD attack ( Madry et al. , 2018 ) with the estimated gradients . Towards improving the efficiency of gradient estimation , Cheng et al . ( 2019 ) and Chen et al . ( 2020 ) further introduced a continuous optimization formulation and an unbiased gradient estimation with careful error control , respectively . The gradients were estimated via issuing probe queries from a standard Gaussian distribution . To generate probes from some more powerful distributions , Dong et al . ( 2019 ) proposed to use the covariance matrix adaptation evolution strategy , while Shi et al . ( 2020 ) suggested to use customized distribution to model the sensitivity of each pixel . In contrast to these methods , our PDA proposes to use a policy network which is learned from prior intentions to advocate promising search directions to reduce the query complexity . We note that some works also proposed to exploit DNN models to generate black-box attacks . For example , Naseer et al . ( 2019 ) used DNNs to promote the transferability of black-box attacks , while several score-based black-box attacks proposed to train DNN models for assisting the generation of queries ( Li et al. , 2019 ; Du et al. , 2019 ; Bai et al. , 2020 ) . Our method is naturally different from them in problem settings ( score-based vs hard-label ) and problem formulations . In the autonomous field , Hamdi et al . ( 2020 ) proposed to formulate the generation of semantic attacks as a reinforcement learning problem to find parameters of environment ( e.g. , camera viewpoint ) that can fool the recognition system . To the best of our knowledge , our work is the first to incorporate reinforcement learning into the black-box attacking scenario for estimating perturbation directions , and we advocate the community to consider more about this principled formulation in the future . In addition to the novel reinforcement learning formulation , we also introduce a specific architecture for the policy network which enjoys superior generalization performance , while these methods adopted off-the-shelf auto-encoding architectures . 3 OUR POLICY-DRIVEN ATTACK . We study the problem of attacking an image classifier in the hard-label setting . The goal of the adversaries is to perturb an benign image x ∈ Rn to fool a k-way victim classifier f : Rn → Rk into making an incorrect decision : arg maxi f ( x ′ ) i 6= y , where x′ is the adversarial example generated by perturbing the benign image and y is the true label of x . The adversaries would generally prefer adversarial examples x′ with smaller distortions ‖x− x′‖2 achieved using less queries , since these properties make the attack less suspicious and also save the cost . In this section , we first briefly review some background information that motivate our method ( in Section 3.1 ) , and then detail our reinforcement learning formulation ( in Section 3.2 and Section 3.3 ) and the architecture of our policy network ( in Section 3.4 ) . 3.1 MOTIVATIONS . Most recent hard-label attacks followed a common pipeline of searching from a starting point which was already an adversarial image1 yet not close enough to the benign one . Unlike the white-box and score-based black-box setting in which the input gradients can be calculated and used as the most effective perturbation direction , in the concerned hard-label setting , outputs of the victim model only flip on the decision boundary while keeping constant away from the boundary , making it difficult to evaluate different directions almost everywhere . In this context , the search of promising perturbation directions was restricted into the regions near the decision boundary , since these regions are arguably more informative , and binary search was used to reach the decision boundary efficiently . Let us take a very recent attack named HopSkipJumpAttack ( Chen et al. , 2020 ) as an example . Given the current estimation x′s of the adversarial example at each iteration , HopSkipJumpAttack first performed binary search to project it onto the decision boundary of the victim model . Denote x′ as the updated example that was on the decision boundary already , HopSkipJumpAttack then sampled many probes around x′ from an isotropic Gaussian distribution , and issued these probes to the victim model as queries . The feedback of the victim model was utilized to estimate the gradient direction at x′ , and it was updated along this direction to obtain a new estimation of the adversarial example . This process was repeated many times until the query budget was exhausted . In comparison with boundary attack ( Brendel et al. , 2018 ) , HopSkipJumpAttack was in general far more query-efficient , though a large number of queries had to be consumed for probing the local geometrics of the decision boundary of the victim model . Its superiority came from using the estimated gradient directions as the search directions , which motivated us to explore even better search directions at each iteration of the attack . As will be shown in the appendices , drawing on some geometric insights , we found that the gradient directions are in fact not the optimal search directions in the framework of HopSkipJumpAttack . We also found that the task of performing hardlabel black-box attack could be naturally cast into a reinforcement learning task , thus we attempt to explore the possibility of developing a model-based method for predicting the most promising search directions for attacks . Feedbacks from the victim can provide supervision and thus the policy models in our reinforcement learning framework can be trained/fine-tuned on the fly during each attack process , such that little query is required once the model has been well-trained . 3.2 ATTACK AS REINFORCEMENT LEARNING PROBLEM . In this paper , we consider both targeted attacks and untargeted attacks . Given a benign example x , its label y , and the victim model f , an environment E ( x , y , f ) is naturally formed . The adversaries shall play the role of agent , trying to interact with the environment by issuing queries and collecting feedbacks , under a certain policy . The current example x′t on the decision boundary of the victim model ( or called the candidate adversarial example ) represents the state at each timestamp t. The agent uses a learnable policy network g which will be carefully introduced in Section 3.4 to guide its actions , and the action is to update the candidate adversarial example such that less distortions are required to fool the victim model . The action here incorporates searching along a promising direction at/‖at‖ where at ∈ Rn is sampled from an isotropic Gaussian distribution whose mean vector is given by the policy network µt = g ( x′t , y , y ′ ) ∈ Rn where y′ is the target label , and its covariance matrix is given by Σ = σI ∈ Rn×n , in which the value of σ ∈ R is set to be gradually increased as the attack on each sample progresses , and I ∈ Rn×n indicates the n × n identity matrix . With at , the agent searches along its direction at/‖at‖ to see whether any better candidate adversarial example can be found . For targeted attacks , the target label y′ is chosen by the agent from the beginning and kept unchanged during the attack process . For untargeted attacks , x′t should be on the decision boundary where one side is the ground-truth label y and the other side could be regarded as the “ target label ” y′ . As will be carefully introduced in Section 3.3 , a reward rt ∈ R 1In practice , it is performed by randomly sampling until the adversarial constraint is satisfied , i.e. , it is not classified as y by the victim model , or by directly choosing a benign sample from the adversarial class . Algorithm 1 Policy-Driven Attack Algorithm 1 : Input : the environment E ( x , y , f ) ; the target label y′ , initial adversarial image x′1 ∈ Rn which lies on the decision boundary ; the policy network g. 2 : Output : an adversarial example . 3 : Initialize the step index t← 1 . 4 : while the query count limit not reached do 5 : // Determine the baseline lt to evaluate the potential of different actions 6 : µt ← g ( x′t , y , y′ ) , z ← BS ( x′t + δ · µt‖µt‖2 , x , f ) , where BS ( · , · , · ) performs binary search 7 : Set the distortion reduction of z as baseline : lt ← max { ‖x′t − x‖2 − ‖z − x‖2 , lmin } 8 : 9 : // Collect actions and rewards , and update the policy network 10 : Sample M actions : at , i ∼ N ( µt , σtI ) , i ∈ { 1 , 2 , . . . , M } 11 : Assign rewards rt , i to each actions with our mechanism introduced in Section 3.3 12 : Update the policy network using one-step REINFORCE on M pairs : ( at , i , rt , i ) 13 : 14 : // Update adversarial image using predicted direction 15 : µt ← g ( x′t , y , y′ ) if ‖z − x‖2 ≤ ‖x′t − x‖2 , otherwise µt ← at , i∗ , i∗ = arg maxi rt , i 16 : x′t+1 ← BS ( x′t + · µt‖µt‖2 , x , f ) 17 : 18 : // Update other variables 19 : Double σt if all M rewards are zeros : σt+1 ← 2 · σt ; else keep it : σt+1 ← σt 20 : t← t+ 1 21 : end while 22 : return final adversarial image x′t based on the performance of each action and the corresponding at is given to the agent for updating the parameter of the policy network . All details of our PDA are summarized in Algorithm 1 . Powered by the reinforcement learning framework , we can use policy gradient algorithms to train the policy network g to generate promising search directions in a direct way . For simplicity , we use the one-step REINFORCE ( Williams , 1992 ) in the sequel of this paper and leave the exploration of more advanced policy gradient algorithms to future work . 3.3 REWARD AND ACTION Figure 1 illustrates how we assign the scalar reward rt given current candidate adversarial example x′t and an action at . The decision boundary is illustrated by a horizontal straight line ( denoted by B ) in the figure , the benign counterpart x is assumed to be below B , and the circle C centered at x′t with a small radius δ shows all possible locations after jumping along the directions of some actions by δ from x′t . As described earlier , the reward rt should be assigned based on the amount of potential distortion reduction brought by at . A direct evaluation can be achieved by jumping along the direction of at first and then projecting the updated example back onto the decision boundary via binary search , to see how much improvement is obtained . However , since we evaluateM actions at , i simultaneously at an iteration ( see Algorithm 1 ) and binary search needs to be performed for each of them , and the overall process would be prohibitively ( query- ) expensive . On this point , to efficiently assess the performance of an action , we instead evaluate whether the reduction of distortion by taking a particular action can exceed particular baselines . Concretely , we first evaluate µt = g ( x′t , y , y ′ ) as an action directly by using binary search in a way as just described . Suppose that it can reduce the required adversarial distortion2 by lt , then we setup two levels of baselines ‖x− x′t‖2 − β1 · lt and ‖x−x′t‖2−β2 · lt to see whether other actions can lead to adversarial examples with closer distance ( than these baselines ) from the benign example x , in which β1 = 0 and β2 = 0.25 . As shown in Figure 1 , for an action a ∈ { at , i } , we first obtain x′t + δ ·a/‖a‖2 and then move it towards x to see how much reward it can obtain . The two arcs V1 and V2 indicates where the same progress as the two baselines can be achieved , thus we can further project x′t + δ ·a/‖a‖2 onto the arcs to see if the projections ( i.e , xV1s and x V2 s ) are still adversarial . It can be seen that x V1 s is still adversarial yet x V2 s is not . We assign a reward 1 to such an action a . If both the projections are still adversarial we shall assign a reward of 2 , and if neither of them is adversarial , zero reward is assigned . Since xV1s is not adversarial could imply that xV2s is also not adversarial , such a way reduces the number of queries for assessing each action to at most 2 ( xV1s and x V2 s ) and makes our PDA more query-efficient .	This work proposes to formulate the problem of black-box adversarial attacks as learning a policy network that predicts offsets to some initial guesses of adversarial examples. The proposed PDA method tackles the problematic situation of queries with only hard labels. A specific reward, architecture, and pretraining setups are designed for the policy network to learn useful directions in updating the adversarial examples. Experiments on MNIST, CIFAR10, and ImageNet show the proposed PDA produced less distorted adversarial examples with less sampling budget for the black box attacks than two recent baseline attacks.	SP:6b7fdf95219b32e4dca1b3fb084f2c11a5d29fd9
Policy-Driven Attack: Learning to Query for Hard-label Black-box Adversarial Examples	1 INTRODUCTION . It is widely known that deep neural networks ( DNNs ) are vulnerable to adversarial examples , which are crafted via perturbing clean examples to cause the victim model to make incorrect predictions . In a white-box setting where the adversaries have full access to the architecture and parameters of the victim model , gradients w.r.t . network inputs can be easily calculated via back-propagation , and thus first-order optimization techniques can be directly applied to craft adversarial examples in this setting ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ; Carlini & Wagner , 2017 ; Madry et al. , 2018 ; Rony et al. , 2019 ) . However , in black-box settings , input gradients are no longer readily available since all model internals are kept secret . Over the past few years , the community has made massive efforts in developing black-box attacks . In order to gain high attack success rates , delicate queries to the victim model are normally required . Recent methods can be roughly categorized into score-based attacks ( Chen et al. , 2017 ; Ilyas et al. , 2018 ; Nitin Bhagoji et al. , 2018 ; Ilyas et al. , 2019 ; Yan et al. , 2019 ; Li et al. , 2020b ; Tu et al. , 2019 ; Du et al. , 2019 ; Li et al. , 2019 ; Bai et al. , 2020 ) and hard-label attacks ( a.k.a , decision-based attacks ) ( Brendel et al. , 2018 ; Cheng et al. , 2019 ; Dong et al. , 2019 ; Shi et al. , 2019 ; Brunner et al. , 2019 ; Chen et al. , 2020 ; Rahmati et al. , 2020 ; Li et al. , 2020a ; Shi et al. , 2020 ; Chen & Gu , 2020 ) , based on the amount of information exposed to the adversaries from the output of victim model . When the prediction probabilities of the victim model are accessible , an intelligent adversary would generally prefer score-based attacks , while in a more practical scenario where only the top-1 class prediction is available , the adversaries will have to resort to hard-label attacks . Since less information is exposed from such feedback of the victim model , hard-label attacks often bare higher query complexity than that of score-based attacks , making their attack process costly and time intensive . * The first two authors contributed equally to the work . Work was done when ZY was an intern at ByteDance AI Lab . In this paper , we aim at reducing the query complexity of hard-label black-box attacks . We cast the problem of progressively refining the candidate adversarial example ( by skillfully querying the victim model and analyzing its feedback ) into a reinforcement learning formulation . At each iteration , we search along a set of chosen directions to see whether there exists any new candidate adversarial example that is perceptually more similar to its benign counterpart , i.e. , in the sense of requiring less distortion . A reward is assigned to each of such search directions ( treated as actions ) , based on the amount of distortion reduction yielded after updating the adversarial example along that direction . Such a reinforcement learning formulation enables us to learn the non-differentiable mapping from search directions to their potential of refining the current adversarial example , directly and precisely . The policy network is expected to be capable of providing the most promising search direction for updating candidate adversarial examples to reduce the required distortion of the adversarial examples from their benign counterparts . As we will show , the proposed policy network can learn from not only the queries that had been performed following the evolving policy but also peer experience from other black-box attacks . As such , it is possible to pre-train the policy network on a small number of query-reward pairs obtained from the performance log of prior attacks ( with or without policy ) to the same victim model . Experiments show that our policy-driven attack ( PDA ) can achieve significantly lower distortions than existing state-of-the-arts under the same query budgets . 2 RELATED WORK . In this paper , we focus on the hard-label black-box setting where only the top-1 decision of the victim model is available . Since less information ( of the victim model ) is exposed after each query , attacks in this category are generally required to query the victim model more times than those in the white-box or score-based settings . For example , an initial attempt named boundary attack ( Brendel et al. , 2018 ) could require ∼million queries before convergence . It proposed to start from an image that is already adversarial , and tried to reduce the distortion by walking towards the benign image along the decision boundary . Recent methods in this category focused more on gradient estimation which could provide more promising search directions , while relying only on top-1 class predictions . Ilyas et al . ( 2018 ) advocated to use NES ( Wierstra et al. , 2014 ; Salimans et al. , 2017 ) to estimate the gradients over proxy scores , and then mounted a variant of PGD attack ( Madry et al. , 2018 ) with the estimated gradients . Towards improving the efficiency of gradient estimation , Cheng et al . ( 2019 ) and Chen et al . ( 2020 ) further introduced a continuous optimization formulation and an unbiased gradient estimation with careful error control , respectively . The gradients were estimated via issuing probe queries from a standard Gaussian distribution . To generate probes from some more powerful distributions , Dong et al . ( 2019 ) proposed to use the covariance matrix adaptation evolution strategy , while Shi et al . ( 2020 ) suggested to use customized distribution to model the sensitivity of each pixel . In contrast to these methods , our PDA proposes to use a policy network which is learned from prior intentions to advocate promising search directions to reduce the query complexity . We note that some works also proposed to exploit DNN models to generate black-box attacks . For example , Naseer et al . ( 2019 ) used DNNs to promote the transferability of black-box attacks , while several score-based black-box attacks proposed to train DNN models for assisting the generation of queries ( Li et al. , 2019 ; Du et al. , 2019 ; Bai et al. , 2020 ) . Our method is naturally different from them in problem settings ( score-based vs hard-label ) and problem formulations . In the autonomous field , Hamdi et al . ( 2020 ) proposed to formulate the generation of semantic attacks as a reinforcement learning problem to find parameters of environment ( e.g. , camera viewpoint ) that can fool the recognition system . To the best of our knowledge , our work is the first to incorporate reinforcement learning into the black-box attacking scenario for estimating perturbation directions , and we advocate the community to consider more about this principled formulation in the future . In addition to the novel reinforcement learning formulation , we also introduce a specific architecture for the policy network which enjoys superior generalization performance , while these methods adopted off-the-shelf auto-encoding architectures . 3 OUR POLICY-DRIVEN ATTACK . We study the problem of attacking an image classifier in the hard-label setting . The goal of the adversaries is to perturb an benign image x ∈ Rn to fool a k-way victim classifier f : Rn → Rk into making an incorrect decision : arg maxi f ( x ′ ) i 6= y , where x′ is the adversarial example generated by perturbing the benign image and y is the true label of x . The adversaries would generally prefer adversarial examples x′ with smaller distortions ‖x− x′‖2 achieved using less queries , since these properties make the attack less suspicious and also save the cost . In this section , we first briefly review some background information that motivate our method ( in Section 3.1 ) , and then detail our reinforcement learning formulation ( in Section 3.2 and Section 3.3 ) and the architecture of our policy network ( in Section 3.4 ) . 3.1 MOTIVATIONS . Most recent hard-label attacks followed a common pipeline of searching from a starting point which was already an adversarial image1 yet not close enough to the benign one . Unlike the white-box and score-based black-box setting in which the input gradients can be calculated and used as the most effective perturbation direction , in the concerned hard-label setting , outputs of the victim model only flip on the decision boundary while keeping constant away from the boundary , making it difficult to evaluate different directions almost everywhere . In this context , the search of promising perturbation directions was restricted into the regions near the decision boundary , since these regions are arguably more informative , and binary search was used to reach the decision boundary efficiently . Let us take a very recent attack named HopSkipJumpAttack ( Chen et al. , 2020 ) as an example . Given the current estimation x′s of the adversarial example at each iteration , HopSkipJumpAttack first performed binary search to project it onto the decision boundary of the victim model . Denote x′ as the updated example that was on the decision boundary already , HopSkipJumpAttack then sampled many probes around x′ from an isotropic Gaussian distribution , and issued these probes to the victim model as queries . The feedback of the victim model was utilized to estimate the gradient direction at x′ , and it was updated along this direction to obtain a new estimation of the adversarial example . This process was repeated many times until the query budget was exhausted . In comparison with boundary attack ( Brendel et al. , 2018 ) , HopSkipJumpAttack was in general far more query-efficient , though a large number of queries had to be consumed for probing the local geometrics of the decision boundary of the victim model . Its superiority came from using the estimated gradient directions as the search directions , which motivated us to explore even better search directions at each iteration of the attack . As will be shown in the appendices , drawing on some geometric insights , we found that the gradient directions are in fact not the optimal search directions in the framework of HopSkipJumpAttack . We also found that the task of performing hardlabel black-box attack could be naturally cast into a reinforcement learning task , thus we attempt to explore the possibility of developing a model-based method for predicting the most promising search directions for attacks . Feedbacks from the victim can provide supervision and thus the policy models in our reinforcement learning framework can be trained/fine-tuned on the fly during each attack process , such that little query is required once the model has been well-trained . 3.2 ATTACK AS REINFORCEMENT LEARNING PROBLEM . In this paper , we consider both targeted attacks and untargeted attacks . Given a benign example x , its label y , and the victim model f , an environment E ( x , y , f ) is naturally formed . The adversaries shall play the role of agent , trying to interact with the environment by issuing queries and collecting feedbacks , under a certain policy . The current example x′t on the decision boundary of the victim model ( or called the candidate adversarial example ) represents the state at each timestamp t. The agent uses a learnable policy network g which will be carefully introduced in Section 3.4 to guide its actions , and the action is to update the candidate adversarial example such that less distortions are required to fool the victim model . The action here incorporates searching along a promising direction at/‖at‖ where at ∈ Rn is sampled from an isotropic Gaussian distribution whose mean vector is given by the policy network µt = g ( x′t , y , y ′ ) ∈ Rn where y′ is the target label , and its covariance matrix is given by Σ = σI ∈ Rn×n , in which the value of σ ∈ R is set to be gradually increased as the attack on each sample progresses , and I ∈ Rn×n indicates the n × n identity matrix . With at , the agent searches along its direction at/‖at‖ to see whether any better candidate adversarial example can be found . For targeted attacks , the target label y′ is chosen by the agent from the beginning and kept unchanged during the attack process . For untargeted attacks , x′t should be on the decision boundary where one side is the ground-truth label y and the other side could be regarded as the “ target label ” y′ . As will be carefully introduced in Section 3.3 , a reward rt ∈ R 1In practice , it is performed by randomly sampling until the adversarial constraint is satisfied , i.e. , it is not classified as y by the victim model , or by directly choosing a benign sample from the adversarial class . Algorithm 1 Policy-Driven Attack Algorithm 1 : Input : the environment E ( x , y , f ) ; the target label y′ , initial adversarial image x′1 ∈ Rn which lies on the decision boundary ; the policy network g. 2 : Output : an adversarial example . 3 : Initialize the step index t← 1 . 4 : while the query count limit not reached do 5 : // Determine the baseline lt to evaluate the potential of different actions 6 : µt ← g ( x′t , y , y′ ) , z ← BS ( x′t + δ · µt‖µt‖2 , x , f ) , where BS ( · , · , · ) performs binary search 7 : Set the distortion reduction of z as baseline : lt ← max { ‖x′t − x‖2 − ‖z − x‖2 , lmin } 8 : 9 : // Collect actions and rewards , and update the policy network 10 : Sample M actions : at , i ∼ N ( µt , σtI ) , i ∈ { 1 , 2 , . . . , M } 11 : Assign rewards rt , i to each actions with our mechanism introduced in Section 3.3 12 : Update the policy network using one-step REINFORCE on M pairs : ( at , i , rt , i ) 13 : 14 : // Update adversarial image using predicted direction 15 : µt ← g ( x′t , y , y′ ) if ‖z − x‖2 ≤ ‖x′t − x‖2 , otherwise µt ← at , i∗ , i∗ = arg maxi rt , i 16 : x′t+1 ← BS ( x′t + · µt‖µt‖2 , x , f ) 17 : 18 : // Update other variables 19 : Double σt if all M rewards are zeros : σt+1 ← 2 · σt ; else keep it : σt+1 ← σt 20 : t← t+ 1 21 : end while 22 : return final adversarial image x′t based on the performance of each action and the corresponding at is given to the agent for updating the parameter of the policy network . All details of our PDA are summarized in Algorithm 1 . Powered by the reinforcement learning framework , we can use policy gradient algorithms to train the policy network g to generate promising search directions in a direct way . For simplicity , we use the one-step REINFORCE ( Williams , 1992 ) in the sequel of this paper and leave the exploration of more advanced policy gradient algorithms to future work . 3.3 REWARD AND ACTION Figure 1 illustrates how we assign the scalar reward rt given current candidate adversarial example x′t and an action at . The decision boundary is illustrated by a horizontal straight line ( denoted by B ) in the figure , the benign counterpart x is assumed to be below B , and the circle C centered at x′t with a small radius δ shows all possible locations after jumping along the directions of some actions by δ from x′t . As described earlier , the reward rt should be assigned based on the amount of potential distortion reduction brought by at . A direct evaluation can be achieved by jumping along the direction of at first and then projecting the updated example back onto the decision boundary via binary search , to see how much improvement is obtained . However , since we evaluateM actions at , i simultaneously at an iteration ( see Algorithm 1 ) and binary search needs to be performed for each of them , and the overall process would be prohibitively ( query- ) expensive . On this point , to efficiently assess the performance of an action , we instead evaluate whether the reduction of distortion by taking a particular action can exceed particular baselines . Concretely , we first evaluate µt = g ( x′t , y , y ′ ) as an action directly by using binary search in a way as just described . Suppose that it can reduce the required adversarial distortion2 by lt , then we setup two levels of baselines ‖x− x′t‖2 − β1 · lt and ‖x−x′t‖2−β2 · lt to see whether other actions can lead to adversarial examples with closer distance ( than these baselines ) from the benign example x , in which β1 = 0 and β2 = 0.25 . As shown in Figure 1 , for an action a ∈ { at , i } , we first obtain x′t + δ ·a/‖a‖2 and then move it towards x to see how much reward it can obtain . The two arcs V1 and V2 indicates where the same progress as the two baselines can be achieved , thus we can further project x′t + δ ·a/‖a‖2 onto the arcs to see if the projections ( i.e , xV1s and x V2 s ) are still adversarial . It can be seen that x V1 s is still adversarial yet x V2 s is not . We assign a reward 1 to such an action a . If both the projections are still adversarial we shall assign a reward of 2 , and if neither of them is adversarial , zero reward is assigned . Since xV1s is not adversarial could imply that xV2s is also not adversarial , such a way reduces the number of queries for assessing each action to at most 2 ( xV1s and x V2 s ) and makes our PDA more query-efficient .	This paper proposes a new hard-label black-box adversarial attack method based on reinforcement learning. The general idea is to improve the adversarial noise compressing efficiency taking advantage of past queries and the policy network. Experiments are conducted on MNIST, CIFAR-10 and ImageNet and achieved superior performance compared to other hard-label black-box attacks. In addition, ablation studies are conducted to verify the effectiveness of the proposed attack.	SP:6b7fdf95219b32e4dca1b3fb084f2c11a5d29fd9
Adaptive Federated Optimization	1 INTRODUCTION . Federated learning ( FL ) is a machine learning paradigm in which multiple clients cooperate to learn a model under the orchestration of a central server ( McMahan et al. , 2017 ) . In FL , raw client data is never shared with the server or other clients . This distinguishes FL from traditional distributed optimization , and requires contending with heterogeneous data . FL has two primary settings , crosssilo ( eg . FL between large institutions ) and cross-device ( eg . FL across edge devices ) ( Kairouz et al. , 2019 , Table 1 ) . In cross-silo FL , most clients participate in every round and can maintain state between rounds . In the more challenging cross-device FL , our primary focus , only a small fraction of clients participate in each round , and clients can not maintain state across rounds . For a more in-depth discussion of FL and the challenges involved , we defer to Kairouz et al . ( 2019 ) and Li et al . ( 2019a ) . Standard optimization methods , such as distributed SGD , are often unsuitable in FL and can incur high communication costs . To remedy this , many federated optimization methods use local client updates , in which clients update their models multiple times before communicating with the server . This can greatly reduce the amount of communication required to train a model . One such method is FEDAVG ( McMahan et al. , 2017 ) , in which clients perform multiple epochs of SGD on their local datasets . The clients communicate their models to the server , which averages them to form a new global model . While FEDAVG has seen great success , recent works have highlighted its convergence issues in some settings ( Karimireddy et al. , 2019 ; Hsu et al. , 2019 ) . This is due to a variety of factors including ( 1 ) client drift ( Karimireddy et al. , 2019 ) , where local client models move away from globally optimal models , and ( 2 ) a lack of adaptivity . FEDAVG is similar in spirit to SGD , and may be unsuitable for settings with heavy-tail stochastic gradient noise distributions , which often arise when training language models ( Zhang et al. , 2019a ) . Such settings benefit from adaptive learning rates , which incorporate knowledge of past iterations to perform more informed optimization . In this paper , we focus on the second issue and present a simple framework for incorporating adaptivity in FL . In particular , we propose a general optimization framework in which ( 1 ) clients perform multiple epochs of training using a client optimizer to minimize loss on their local data and ( 2 ) server updates its global model by applying a gradient-based server optimizer to the average of the clients ’ model updates . We show that FEDAVG is the special case where SGD is used as both client and server optimizer and server learning rate is 1 . This framework can also seamlessly incorporate ∗Authors contributed equally to this work adaptivity by using adaptive optimizers as client or server optimizers . Building upon this , we develop novel adaptive optimization techniques for FL by using per-coordinate methods as server optimizers . By focusing on adaptive server optimization , we enable use of adaptive learning rates without increase in client storage or communication costs , and ensure compatibility with cross-device FL . Main contributions In light of the above , we highlight the main contributions of the paper . • We study a general framework for federated optimization using server and client optimizers . This framework generalizes many existing federated optimization methods , including FEDAVG . • We use this framework to design novel , cross-device compatible , adaptive federated optimization methods , and provide convergence analysis in general nonconvex settings . To the best of our knowledge , these are the first methods for FL using adaptive server optimization . We show an important interplay between the number of local steps and the heterogeneity among clients . • We introduce comprehensive and reproducible empirical benchmarks for comparing federated optimization methods . These benchmarks consist of seven diverse and representative FL tasks involving both image and text data , with varying amounts of heterogeneity and numbers of clients . • We demonstrate strong empirical performance of our adaptive optimizers throughout , improving upon commonly used baselines . Our results show that our methods can be easier to tune , and highlight their utility in cross-device settings . Related work FEDAVG was first introduced by McMahan et al . ( 2017 ) , who showed it can dramatically reduce communication costs . Many variants have since been proposed to tackle issues such as convergence and client drift . Examples include adding a regularization term in the client objectives towards the broadcast model ( Li et al. , 2018 ) , and server momentum ( Hsu et al. , 2019 ) . When clients are homogeneous , FEDAVG reduces to local SGD ( Zinkevich et al. , 2010 ) , which has been analyzed by many works ( Stich , 2019 ; Yu et al. , 2019 ; Wang & Joshi , 2018 ; Stich & Karimireddy , 2019 ; Basu et al. , 2019 ) . In order to analyze FEDAVG in heterogeneous settings , many works derive convergence rates depending on the amount of heterogeneity ( Li et al. , 2018 ; Wang et al. , 2019 ; Khaled et al. , 2019 ; Li et al. , 2019b ) . Typically , the convergence rate of FEDAVG gets worse with client heterogeneity . By using control variates to reduce client drift , the SCAFFOLD method ( Karimireddy et al. , 2019 ) achieves convergence rates that are independent of the amount of heterogeneity . While effective in cross-silo FL , the method is incompatible with cross-device FL as it requires clients to maintain state across rounds . For more detailed comparisons , we defer to Kairouz et al . ( 2019 ) . Adaptive methods have been the subject of significant theoretical and empirical study , in both convex ( McMahan & Streeter , 2010b ; Duchi et al. , 2011 ; Kingma & Ba , 2015 ) and non-convex settings ( Li & Orabona , 2018 ; Ward et al. , 2018 ; Wu et al. , 2019 ) . Reddi et al . ( 2019 ) ; Zaheer et al . ( 2018 ) study convergence failures of ADAM in certain non-convex settings , and develop an adaptive optimizer , YOGI , designed to improve convergence . While most work on adaptive methods focuses on non-FL settings , Xie et al . ( 2019 ) propose ADAALTER , a method for FL using adaptive client optimization . Conceptually , our approach is also related to the LOOKAHEAD optimizer ( Zhang et al. , 2019b ) , which was designed for non-FL settings . Similar to ADAALTER , an adaptive FL variant of LOOKAHEAD entails adaptive client optimization ( see Appendix B.3 for more details ) . We note that both ADAALTER and LOOKAHEAD are , in fact , special cases of our framework ( see Algorithm 1 ) and the primary novelty of our work comes in focusing on adaptive server optimization . This allows us to avoid aggregating optimizer states across clients , making our methods require at most half as much communication and client memory usage per round ( see Appendix B.3 for details ) . Notation For a , b ∈ Rd , we let √ a , a2 and a/b denote the element-wise square root , square , and division of the vectors . For θi ∈ Rd , we use both θi , j and [ θi ] j to denote its jth coordinate . 2 FEDERATED LEARNING AND FEDAVG . In federated learning , we solve an optimization problem of the form : min x∈Rd f ( x ) = 1 m m∑ i=1 Fi ( x ) , ( 1 ) where Fi ( x ) = Ez∼Di [ fi ( x , z ) ] , is the loss function of the ith client , z ∈ Z , and Di is the data distribution for the ith client . For i 6= j , Di and Dj may be very different . The functions Fi ( and therefore f ) may be nonconvex . For each i and x , we assume access to an unbiased stochastic gradient gi ( x ) of the client ’ s true gradient∇Fi ( x ) . In addition , we make the following assumptions . Assumption 1 ( Lipschitz Gradient ) . The function Fi is L-smooth for all i ∈ [ m ] i.e. , ‖∇Fi ( x ) − ∇Fi ( y ) ‖ ≤ L‖x− y‖ , for all x , y ∈ Rd . Assumption 2 ( Bounded Variance ) . The function Fi have σl-bounded ( local ) variance i.e. , E [ ‖∇ [ fi ( x , z ) ] j − [ ∇Fi ( x ) ] j‖2 ] = σ2l , j for all x ∈ Rd , j ∈ [ d ] and i ∈ [ m ] . Furthermore , we assume the ( global ) variance is bounded , ( 1/m ) ∑m i=1 ‖∇ [ Fi ( x ) ] j − [ ∇f ( x ) ] j‖2 ≤ σ2g , j for all x ∈ Rd and j ∈ [ d ] . Assumption 3 ( Bounded Gradients ) . The function fi ( x , z ) have G-bounded gradients i.e. , for any i ∈ [ m ] , x ∈ Rd and z ∈ Z we have \| [ ∇fi ( x , z ) ] j \| ≤ G for all j ∈ [ d ] . With a slight abuse of notation , we use σ2l and σ 2 g to denote ∑d j=1 σ 2 l , j and ∑d j=1 σ 2 g , j . Assumptions 1 and 3 are fairly standard in nonconvex optimization literature ( Reddi et al. , 2016 ; Ward et al. , 2018 ; Zaheer et al. , 2018 ) . We make no further assumptions regarding the similarity of clients datasets . Assumption 2 is a form of bounded variance , but between the client objective functions and the overall objective function . This assumption has been used throughout various works on federated optimization ( Li et al. , 2018 ; Wang et al. , 2019 ) . Intuitively , the parameter σg quantifies similarity of client objective functions . Note σg = 0 corresponds to the i.i.d . setting . A common approach to solving ( 1 ) in federated settings is FEDAVG ( McMahan et al. , 2017 ) . At each round of FEDAVG , a subset of clients are selected ( typically randomly ) and the server broadcasts its global model to each client . In parallel , the clients run SGD on their own loss function , and send the resulting model to the server . The server then updates its global model as the average of these local models . See Algorithm 3 in the appendix for more details . Suppose that at round t , the server has model xt and samples a set S of clients . Let xti denote the model of each client i ∈ S after local training . We rewrite FEDAVG ’ s update as xt+1 = 1 \|S\| ∑ i∈S xti = xt − 1 \|S\| ∑ i∈S ( xt − xti ) . Let ∆ti : = x t i − xt and ∆t : = ( 1/\|S\| ) ∑ i∈S ∆ t i . Then the server update in FEDAVG is equivalent to applying SGD to the “ pseudo-gradient ” −∆t with learning rate η = 1 . This formulation makes it clear that other choices of η are possible . One could also utilize optimizers other than SGD on the clients , or use an alternative update rule on the server . This family of algorithms , which we refer to collectively as FEDOPT , is formalized in Algorithm 1 . Algorithm 1 FEDOPT 1 : Input : x0 , CLIENTOPT , SERVEROPT 2 : for t = 0 , · · · , T − 1 do 3 : Sample a subset S of clients 4 : xti,0 = xt 5 : for each client i ∈ S in parallel do 6 : for k = 0 , · · · , K − 1 do 7 : Compute an unbiased estimate gti , k of∇Fi ( xti , k ) 8 : xti , k+1 = CLIENTOPT ( x t i , k , g t i , k , ηl , t ) 9 : ∆ti = x t i , K − xt 10 : ∆t = 1 \|S\| ∑ i∈S ∆ t i 11 : xt+1 = SERVEROPT ( xt , −∆t , η , t ) In Algorithm 1 , CLIENTOPT and SERVEROPT are gradient-based optimizers with learning rates ηl and η respectively . Intuitively , CLIENTOPT aims to minimize ( 1 ) based on each client ’ s local data while SERVEROPT optimizes from a global perspective . FEDOPT naturally allows the use of adaptive optimizers ( eg . ADAM , YOGI , etc . ) , as well as techniques such as server-side momentum ( leading to FEDAVGM , proposed by Hsu et al . ( 2019 ) ) . In its most general form , FEDOPT uses a CLIENTOPT whose updates can depend on globally aggregated statistics ( e.g . server updates in the previous iterations ) . We also allow η and ηl to depend on the round t in order to encompass learning rate schedules . While we focus on specific adaptive optimizers in this work , we can in principle use any adaptive optimizer ( e.g . AMSGRAD ( Reddi et al. , 2019 ) , ADABOUND ( Luo et al. , 2019 ) ) . While FEDOPT has intuitive benefits over FEDAVG , it also raises a fundamental question : Can the negative of the average model difference ∆t be used as a pseudo-gradient in general server optimizer updates ? In this paper , we provide an affirmative answer to this question by establishing a theoretical basis for FEDOPT . We will show that the use of the term SERVEROPT is justified , as we can guarantee convergence across a wide variety of server optimizers , including ADAGRAD , ADAM , and YOGI , thus developing principled adaptive optimizers for FL based on our framework .	This paper studies the convergence of well-known adaptive methods, ADAM, ADAGRAD, and YOGI, for the federated learning problem. In particular, while the nodes (clients) still use SGD for their local computations (same as Fed-Avg), the server uses one of the three adaptive methods mentioned above to update the model. The authors have provided convergence rates (based on gradient norms) for all three methods for the nonconvex settings. Moreover, various experiments have been conducted to compare the strengths of these methods against classic FedAvg. Overall, this paper is well-written, and the problem statement and the goal of the paper are clear. Moreover, the authors have shown the success of using adaptive methods in numerical settings.	SP:c95a81c7f0df69cd0a7a85b08e8ed5c610732f8a
Adaptive Federated Optimization	1 INTRODUCTION . Federated learning ( FL ) is a machine learning paradigm in which multiple clients cooperate to learn a model under the orchestration of a central server ( McMahan et al. , 2017 ) . In FL , raw client data is never shared with the server or other clients . This distinguishes FL from traditional distributed optimization , and requires contending with heterogeneous data . FL has two primary settings , crosssilo ( eg . FL between large institutions ) and cross-device ( eg . FL across edge devices ) ( Kairouz et al. , 2019 , Table 1 ) . In cross-silo FL , most clients participate in every round and can maintain state between rounds . In the more challenging cross-device FL , our primary focus , only a small fraction of clients participate in each round , and clients can not maintain state across rounds . For a more in-depth discussion of FL and the challenges involved , we defer to Kairouz et al . ( 2019 ) and Li et al . ( 2019a ) . Standard optimization methods , such as distributed SGD , are often unsuitable in FL and can incur high communication costs . To remedy this , many federated optimization methods use local client updates , in which clients update their models multiple times before communicating with the server . This can greatly reduce the amount of communication required to train a model . One such method is FEDAVG ( McMahan et al. , 2017 ) , in which clients perform multiple epochs of SGD on their local datasets . The clients communicate their models to the server , which averages them to form a new global model . While FEDAVG has seen great success , recent works have highlighted its convergence issues in some settings ( Karimireddy et al. , 2019 ; Hsu et al. , 2019 ) . This is due to a variety of factors including ( 1 ) client drift ( Karimireddy et al. , 2019 ) , where local client models move away from globally optimal models , and ( 2 ) a lack of adaptivity . FEDAVG is similar in spirit to SGD , and may be unsuitable for settings with heavy-tail stochastic gradient noise distributions , which often arise when training language models ( Zhang et al. , 2019a ) . Such settings benefit from adaptive learning rates , which incorporate knowledge of past iterations to perform more informed optimization . In this paper , we focus on the second issue and present a simple framework for incorporating adaptivity in FL . In particular , we propose a general optimization framework in which ( 1 ) clients perform multiple epochs of training using a client optimizer to minimize loss on their local data and ( 2 ) server updates its global model by applying a gradient-based server optimizer to the average of the clients ’ model updates . We show that FEDAVG is the special case where SGD is used as both client and server optimizer and server learning rate is 1 . This framework can also seamlessly incorporate ∗Authors contributed equally to this work adaptivity by using adaptive optimizers as client or server optimizers . Building upon this , we develop novel adaptive optimization techniques for FL by using per-coordinate methods as server optimizers . By focusing on adaptive server optimization , we enable use of adaptive learning rates without increase in client storage or communication costs , and ensure compatibility with cross-device FL . Main contributions In light of the above , we highlight the main contributions of the paper . • We study a general framework for federated optimization using server and client optimizers . This framework generalizes many existing federated optimization methods , including FEDAVG . • We use this framework to design novel , cross-device compatible , adaptive federated optimization methods , and provide convergence analysis in general nonconvex settings . To the best of our knowledge , these are the first methods for FL using adaptive server optimization . We show an important interplay between the number of local steps and the heterogeneity among clients . • We introduce comprehensive and reproducible empirical benchmarks for comparing federated optimization methods . These benchmarks consist of seven diverse and representative FL tasks involving both image and text data , with varying amounts of heterogeneity and numbers of clients . • We demonstrate strong empirical performance of our adaptive optimizers throughout , improving upon commonly used baselines . Our results show that our methods can be easier to tune , and highlight their utility in cross-device settings . Related work FEDAVG was first introduced by McMahan et al . ( 2017 ) , who showed it can dramatically reduce communication costs . Many variants have since been proposed to tackle issues such as convergence and client drift . Examples include adding a regularization term in the client objectives towards the broadcast model ( Li et al. , 2018 ) , and server momentum ( Hsu et al. , 2019 ) . When clients are homogeneous , FEDAVG reduces to local SGD ( Zinkevich et al. , 2010 ) , which has been analyzed by many works ( Stich , 2019 ; Yu et al. , 2019 ; Wang & Joshi , 2018 ; Stich & Karimireddy , 2019 ; Basu et al. , 2019 ) . In order to analyze FEDAVG in heterogeneous settings , many works derive convergence rates depending on the amount of heterogeneity ( Li et al. , 2018 ; Wang et al. , 2019 ; Khaled et al. , 2019 ; Li et al. , 2019b ) . Typically , the convergence rate of FEDAVG gets worse with client heterogeneity . By using control variates to reduce client drift , the SCAFFOLD method ( Karimireddy et al. , 2019 ) achieves convergence rates that are independent of the amount of heterogeneity . While effective in cross-silo FL , the method is incompatible with cross-device FL as it requires clients to maintain state across rounds . For more detailed comparisons , we defer to Kairouz et al . ( 2019 ) . Adaptive methods have been the subject of significant theoretical and empirical study , in both convex ( McMahan & Streeter , 2010b ; Duchi et al. , 2011 ; Kingma & Ba , 2015 ) and non-convex settings ( Li & Orabona , 2018 ; Ward et al. , 2018 ; Wu et al. , 2019 ) . Reddi et al . ( 2019 ) ; Zaheer et al . ( 2018 ) study convergence failures of ADAM in certain non-convex settings , and develop an adaptive optimizer , YOGI , designed to improve convergence . While most work on adaptive methods focuses on non-FL settings , Xie et al . ( 2019 ) propose ADAALTER , a method for FL using adaptive client optimization . Conceptually , our approach is also related to the LOOKAHEAD optimizer ( Zhang et al. , 2019b ) , which was designed for non-FL settings . Similar to ADAALTER , an adaptive FL variant of LOOKAHEAD entails adaptive client optimization ( see Appendix B.3 for more details ) . We note that both ADAALTER and LOOKAHEAD are , in fact , special cases of our framework ( see Algorithm 1 ) and the primary novelty of our work comes in focusing on adaptive server optimization . This allows us to avoid aggregating optimizer states across clients , making our methods require at most half as much communication and client memory usage per round ( see Appendix B.3 for details ) . Notation For a , b ∈ Rd , we let √ a , a2 and a/b denote the element-wise square root , square , and division of the vectors . For θi ∈ Rd , we use both θi , j and [ θi ] j to denote its jth coordinate . 2 FEDERATED LEARNING AND FEDAVG . In federated learning , we solve an optimization problem of the form : min x∈Rd f ( x ) = 1 m m∑ i=1 Fi ( x ) , ( 1 ) where Fi ( x ) = Ez∼Di [ fi ( x , z ) ] , is the loss function of the ith client , z ∈ Z , and Di is the data distribution for the ith client . For i 6= j , Di and Dj may be very different . The functions Fi ( and therefore f ) may be nonconvex . For each i and x , we assume access to an unbiased stochastic gradient gi ( x ) of the client ’ s true gradient∇Fi ( x ) . In addition , we make the following assumptions . Assumption 1 ( Lipschitz Gradient ) . The function Fi is L-smooth for all i ∈ [ m ] i.e. , ‖∇Fi ( x ) − ∇Fi ( y ) ‖ ≤ L‖x− y‖ , for all x , y ∈ Rd . Assumption 2 ( Bounded Variance ) . The function Fi have σl-bounded ( local ) variance i.e. , E [ ‖∇ [ fi ( x , z ) ] j − [ ∇Fi ( x ) ] j‖2 ] = σ2l , j for all x ∈ Rd , j ∈ [ d ] and i ∈ [ m ] . Furthermore , we assume the ( global ) variance is bounded , ( 1/m ) ∑m i=1 ‖∇ [ Fi ( x ) ] j − [ ∇f ( x ) ] j‖2 ≤ σ2g , j for all x ∈ Rd and j ∈ [ d ] . Assumption 3 ( Bounded Gradients ) . The function fi ( x , z ) have G-bounded gradients i.e. , for any i ∈ [ m ] , x ∈ Rd and z ∈ Z we have \| [ ∇fi ( x , z ) ] j \| ≤ G for all j ∈ [ d ] . With a slight abuse of notation , we use σ2l and σ 2 g to denote ∑d j=1 σ 2 l , j and ∑d j=1 σ 2 g , j . Assumptions 1 and 3 are fairly standard in nonconvex optimization literature ( Reddi et al. , 2016 ; Ward et al. , 2018 ; Zaheer et al. , 2018 ) . We make no further assumptions regarding the similarity of clients datasets . Assumption 2 is a form of bounded variance , but between the client objective functions and the overall objective function . This assumption has been used throughout various works on federated optimization ( Li et al. , 2018 ; Wang et al. , 2019 ) . Intuitively , the parameter σg quantifies similarity of client objective functions . Note σg = 0 corresponds to the i.i.d . setting . A common approach to solving ( 1 ) in federated settings is FEDAVG ( McMahan et al. , 2017 ) . At each round of FEDAVG , a subset of clients are selected ( typically randomly ) and the server broadcasts its global model to each client . In parallel , the clients run SGD on their own loss function , and send the resulting model to the server . The server then updates its global model as the average of these local models . See Algorithm 3 in the appendix for more details . Suppose that at round t , the server has model xt and samples a set S of clients . Let xti denote the model of each client i ∈ S after local training . We rewrite FEDAVG ’ s update as xt+1 = 1 \|S\| ∑ i∈S xti = xt − 1 \|S\| ∑ i∈S ( xt − xti ) . Let ∆ti : = x t i − xt and ∆t : = ( 1/\|S\| ) ∑ i∈S ∆ t i . Then the server update in FEDAVG is equivalent to applying SGD to the “ pseudo-gradient ” −∆t with learning rate η = 1 . This formulation makes it clear that other choices of η are possible . One could also utilize optimizers other than SGD on the clients , or use an alternative update rule on the server . This family of algorithms , which we refer to collectively as FEDOPT , is formalized in Algorithm 1 . Algorithm 1 FEDOPT 1 : Input : x0 , CLIENTOPT , SERVEROPT 2 : for t = 0 , · · · , T − 1 do 3 : Sample a subset S of clients 4 : xti,0 = xt 5 : for each client i ∈ S in parallel do 6 : for k = 0 , · · · , K − 1 do 7 : Compute an unbiased estimate gti , k of∇Fi ( xti , k ) 8 : xti , k+1 = CLIENTOPT ( x t i , k , g t i , k , ηl , t ) 9 : ∆ti = x t i , K − xt 10 : ∆t = 1 \|S\| ∑ i∈S ∆ t i 11 : xt+1 = SERVEROPT ( xt , −∆t , η , t ) In Algorithm 1 , CLIENTOPT and SERVEROPT are gradient-based optimizers with learning rates ηl and η respectively . Intuitively , CLIENTOPT aims to minimize ( 1 ) based on each client ’ s local data while SERVEROPT optimizes from a global perspective . FEDOPT naturally allows the use of adaptive optimizers ( eg . ADAM , YOGI , etc . ) , as well as techniques such as server-side momentum ( leading to FEDAVGM , proposed by Hsu et al . ( 2019 ) ) . In its most general form , FEDOPT uses a CLIENTOPT whose updates can depend on globally aggregated statistics ( e.g . server updates in the previous iterations ) . We also allow η and ηl to depend on the round t in order to encompass learning rate schedules . While we focus on specific adaptive optimizers in this work , we can in principle use any adaptive optimizer ( e.g . AMSGRAD ( Reddi et al. , 2019 ) , ADABOUND ( Luo et al. , 2019 ) ) . While FEDOPT has intuitive benefits over FEDAVG , it also raises a fundamental question : Can the negative of the average model difference ∆t be used as a pseudo-gradient in general server optimizer updates ? In this paper , we provide an affirmative answer to this question by establishing a theoretical basis for FEDOPT . We will show that the use of the term SERVEROPT is justified , as we can guarantee convergence across a wide variety of server optimizers , including ADAGRAD , ADAM , and YOGI , thus developing principled adaptive optimizers for FL based on our framework .	This paper presents an adaptive federated optimization framework that induces three different adaptive federated learning algorithms, which are proposed to address the issues of client drift due to data heterogeneity and lack of adaptivity. The authors presented thorough literature survey on the federated learning and formulated the meta-algorithm, FedOpt, introducing both server and client optimizers. Different from the popular FedAvg, the server optimizer in this work is an adaptive protocol which is originated from the client drift. The authors analyzed mathematically the convergence rates of the proposed framework and showed extensive experimental results on different benchmark datasets to validate the efficacy of the developed algorithms.	SP:c95a81c7f0df69cd0a7a85b08e8ed5c610732f8a
Adaptive Federated Optimization	1 INTRODUCTION . Federated learning ( FL ) is a machine learning paradigm in which multiple clients cooperate to learn a model under the orchestration of a central server ( McMahan et al. , 2017 ) . In FL , raw client data is never shared with the server or other clients . This distinguishes FL from traditional distributed optimization , and requires contending with heterogeneous data . FL has two primary settings , crosssilo ( eg . FL between large institutions ) and cross-device ( eg . FL across edge devices ) ( Kairouz et al. , 2019 , Table 1 ) . In cross-silo FL , most clients participate in every round and can maintain state between rounds . In the more challenging cross-device FL , our primary focus , only a small fraction of clients participate in each round , and clients can not maintain state across rounds . For a more in-depth discussion of FL and the challenges involved , we defer to Kairouz et al . ( 2019 ) and Li et al . ( 2019a ) . Standard optimization methods , such as distributed SGD , are often unsuitable in FL and can incur high communication costs . To remedy this , many federated optimization methods use local client updates , in which clients update their models multiple times before communicating with the server . This can greatly reduce the amount of communication required to train a model . One such method is FEDAVG ( McMahan et al. , 2017 ) , in which clients perform multiple epochs of SGD on their local datasets . The clients communicate their models to the server , which averages them to form a new global model . While FEDAVG has seen great success , recent works have highlighted its convergence issues in some settings ( Karimireddy et al. , 2019 ; Hsu et al. , 2019 ) . This is due to a variety of factors including ( 1 ) client drift ( Karimireddy et al. , 2019 ) , where local client models move away from globally optimal models , and ( 2 ) a lack of adaptivity . FEDAVG is similar in spirit to SGD , and may be unsuitable for settings with heavy-tail stochastic gradient noise distributions , which often arise when training language models ( Zhang et al. , 2019a ) . Such settings benefit from adaptive learning rates , which incorporate knowledge of past iterations to perform more informed optimization . In this paper , we focus on the second issue and present a simple framework for incorporating adaptivity in FL . In particular , we propose a general optimization framework in which ( 1 ) clients perform multiple epochs of training using a client optimizer to minimize loss on their local data and ( 2 ) server updates its global model by applying a gradient-based server optimizer to the average of the clients ’ model updates . We show that FEDAVG is the special case where SGD is used as both client and server optimizer and server learning rate is 1 . This framework can also seamlessly incorporate ∗Authors contributed equally to this work adaptivity by using adaptive optimizers as client or server optimizers . Building upon this , we develop novel adaptive optimization techniques for FL by using per-coordinate methods as server optimizers . By focusing on adaptive server optimization , we enable use of adaptive learning rates without increase in client storage or communication costs , and ensure compatibility with cross-device FL . Main contributions In light of the above , we highlight the main contributions of the paper . • We study a general framework for federated optimization using server and client optimizers . This framework generalizes many existing federated optimization methods , including FEDAVG . • We use this framework to design novel , cross-device compatible , adaptive federated optimization methods , and provide convergence analysis in general nonconvex settings . To the best of our knowledge , these are the first methods for FL using adaptive server optimization . We show an important interplay between the number of local steps and the heterogeneity among clients . • We introduce comprehensive and reproducible empirical benchmarks for comparing federated optimization methods . These benchmarks consist of seven diverse and representative FL tasks involving both image and text data , with varying amounts of heterogeneity and numbers of clients . • We demonstrate strong empirical performance of our adaptive optimizers throughout , improving upon commonly used baselines . Our results show that our methods can be easier to tune , and highlight their utility in cross-device settings . Related work FEDAVG was first introduced by McMahan et al . ( 2017 ) , who showed it can dramatically reduce communication costs . Many variants have since been proposed to tackle issues such as convergence and client drift . Examples include adding a regularization term in the client objectives towards the broadcast model ( Li et al. , 2018 ) , and server momentum ( Hsu et al. , 2019 ) . When clients are homogeneous , FEDAVG reduces to local SGD ( Zinkevich et al. , 2010 ) , which has been analyzed by many works ( Stich , 2019 ; Yu et al. , 2019 ; Wang & Joshi , 2018 ; Stich & Karimireddy , 2019 ; Basu et al. , 2019 ) . In order to analyze FEDAVG in heterogeneous settings , many works derive convergence rates depending on the amount of heterogeneity ( Li et al. , 2018 ; Wang et al. , 2019 ; Khaled et al. , 2019 ; Li et al. , 2019b ) . Typically , the convergence rate of FEDAVG gets worse with client heterogeneity . By using control variates to reduce client drift , the SCAFFOLD method ( Karimireddy et al. , 2019 ) achieves convergence rates that are independent of the amount of heterogeneity . While effective in cross-silo FL , the method is incompatible with cross-device FL as it requires clients to maintain state across rounds . For more detailed comparisons , we defer to Kairouz et al . ( 2019 ) . Adaptive methods have been the subject of significant theoretical and empirical study , in both convex ( McMahan & Streeter , 2010b ; Duchi et al. , 2011 ; Kingma & Ba , 2015 ) and non-convex settings ( Li & Orabona , 2018 ; Ward et al. , 2018 ; Wu et al. , 2019 ) . Reddi et al . ( 2019 ) ; Zaheer et al . ( 2018 ) study convergence failures of ADAM in certain non-convex settings , and develop an adaptive optimizer , YOGI , designed to improve convergence . While most work on adaptive methods focuses on non-FL settings , Xie et al . ( 2019 ) propose ADAALTER , a method for FL using adaptive client optimization . Conceptually , our approach is also related to the LOOKAHEAD optimizer ( Zhang et al. , 2019b ) , which was designed for non-FL settings . Similar to ADAALTER , an adaptive FL variant of LOOKAHEAD entails adaptive client optimization ( see Appendix B.3 for more details ) . We note that both ADAALTER and LOOKAHEAD are , in fact , special cases of our framework ( see Algorithm 1 ) and the primary novelty of our work comes in focusing on adaptive server optimization . This allows us to avoid aggregating optimizer states across clients , making our methods require at most half as much communication and client memory usage per round ( see Appendix B.3 for details ) . Notation For a , b ∈ Rd , we let √ a , a2 and a/b denote the element-wise square root , square , and division of the vectors . For θi ∈ Rd , we use both θi , j and [ θi ] j to denote its jth coordinate . 2 FEDERATED LEARNING AND FEDAVG . In federated learning , we solve an optimization problem of the form : min x∈Rd f ( x ) = 1 m m∑ i=1 Fi ( x ) , ( 1 ) where Fi ( x ) = Ez∼Di [ fi ( x , z ) ] , is the loss function of the ith client , z ∈ Z , and Di is the data distribution for the ith client . For i 6= j , Di and Dj may be very different . The functions Fi ( and therefore f ) may be nonconvex . For each i and x , we assume access to an unbiased stochastic gradient gi ( x ) of the client ’ s true gradient∇Fi ( x ) . In addition , we make the following assumptions . Assumption 1 ( Lipschitz Gradient ) . The function Fi is L-smooth for all i ∈ [ m ] i.e. , ‖∇Fi ( x ) − ∇Fi ( y ) ‖ ≤ L‖x− y‖ , for all x , y ∈ Rd . Assumption 2 ( Bounded Variance ) . The function Fi have σl-bounded ( local ) variance i.e. , E [ ‖∇ [ fi ( x , z ) ] j − [ ∇Fi ( x ) ] j‖2 ] = σ2l , j for all x ∈ Rd , j ∈ [ d ] and i ∈ [ m ] . Furthermore , we assume the ( global ) variance is bounded , ( 1/m ) ∑m i=1 ‖∇ [ Fi ( x ) ] j − [ ∇f ( x ) ] j‖2 ≤ σ2g , j for all x ∈ Rd and j ∈ [ d ] . Assumption 3 ( Bounded Gradients ) . The function fi ( x , z ) have G-bounded gradients i.e. , for any i ∈ [ m ] , x ∈ Rd and z ∈ Z we have \| [ ∇fi ( x , z ) ] j \| ≤ G for all j ∈ [ d ] . With a slight abuse of notation , we use σ2l and σ 2 g to denote ∑d j=1 σ 2 l , j and ∑d j=1 σ 2 g , j . Assumptions 1 and 3 are fairly standard in nonconvex optimization literature ( Reddi et al. , 2016 ; Ward et al. , 2018 ; Zaheer et al. , 2018 ) . We make no further assumptions regarding the similarity of clients datasets . Assumption 2 is a form of bounded variance , but between the client objective functions and the overall objective function . This assumption has been used throughout various works on federated optimization ( Li et al. , 2018 ; Wang et al. , 2019 ) . Intuitively , the parameter σg quantifies similarity of client objective functions . Note σg = 0 corresponds to the i.i.d . setting . A common approach to solving ( 1 ) in federated settings is FEDAVG ( McMahan et al. , 2017 ) . At each round of FEDAVG , a subset of clients are selected ( typically randomly ) and the server broadcasts its global model to each client . In parallel , the clients run SGD on their own loss function , and send the resulting model to the server . The server then updates its global model as the average of these local models . See Algorithm 3 in the appendix for more details . Suppose that at round t , the server has model xt and samples a set S of clients . Let xti denote the model of each client i ∈ S after local training . We rewrite FEDAVG ’ s update as xt+1 = 1 \|S\| ∑ i∈S xti = xt − 1 \|S\| ∑ i∈S ( xt − xti ) . Let ∆ti : = x t i − xt and ∆t : = ( 1/\|S\| ) ∑ i∈S ∆ t i . Then the server update in FEDAVG is equivalent to applying SGD to the “ pseudo-gradient ” −∆t with learning rate η = 1 . This formulation makes it clear that other choices of η are possible . One could also utilize optimizers other than SGD on the clients , or use an alternative update rule on the server . This family of algorithms , which we refer to collectively as FEDOPT , is formalized in Algorithm 1 . Algorithm 1 FEDOPT 1 : Input : x0 , CLIENTOPT , SERVEROPT 2 : for t = 0 , · · · , T − 1 do 3 : Sample a subset S of clients 4 : xti,0 = xt 5 : for each client i ∈ S in parallel do 6 : for k = 0 , · · · , K − 1 do 7 : Compute an unbiased estimate gti , k of∇Fi ( xti , k ) 8 : xti , k+1 = CLIENTOPT ( x t i , k , g t i , k , ηl , t ) 9 : ∆ti = x t i , K − xt 10 : ∆t = 1 \|S\| ∑ i∈S ∆ t i 11 : xt+1 = SERVEROPT ( xt , −∆t , η , t ) In Algorithm 1 , CLIENTOPT and SERVEROPT are gradient-based optimizers with learning rates ηl and η respectively . Intuitively , CLIENTOPT aims to minimize ( 1 ) based on each client ’ s local data while SERVEROPT optimizes from a global perspective . FEDOPT naturally allows the use of adaptive optimizers ( eg . ADAM , YOGI , etc . ) , as well as techniques such as server-side momentum ( leading to FEDAVGM , proposed by Hsu et al . ( 2019 ) ) . In its most general form , FEDOPT uses a CLIENTOPT whose updates can depend on globally aggregated statistics ( e.g . server updates in the previous iterations ) . We also allow η and ηl to depend on the round t in order to encompass learning rate schedules . While we focus on specific adaptive optimizers in this work , we can in principle use any adaptive optimizer ( e.g . AMSGRAD ( Reddi et al. , 2019 ) , ADABOUND ( Luo et al. , 2019 ) ) . While FEDOPT has intuitive benefits over FEDAVG , it also raises a fundamental question : Can the negative of the average model difference ∆t be used as a pseudo-gradient in general server optimizer updates ? In this paper , we provide an affirmative answer to this question by establishing a theoretical basis for FEDOPT . We will show that the use of the term SERVEROPT is justified , as we can guarantee convergence across a wide variety of server optimizers , including ADAGRAD , ADAM , and YOGI , thus developing principled adaptive optimizers for FL based on our framework .	This paper extends the server model averaging step in FedAvg to a more general adaptive optimization step on the global model, specifically, by writing the model averaging as a gradient descent step using a pseudo gradient. Three variants of this scheme (FedOpt) are presented, based on three adaptive optimizers, including AdaGrad, ADAM, and YOGI. While there exist works applying the server-side momentum method, the paper argues that the proposed framework is more general since any adaptive optimizer can be applied to extend FedAvg. Under three assumptions (Lipschitz gradient, bounded gradients, bounded variances), the paper provides a local convergence analysis with nonconvex objectives. The achieved bounds match the best-known convergence rate for FL under reasonable conditions (i.e., T is sufficiently large compared to K). They also provide guidance for how to decay the local learning rate to avoid client drift, and show that increasing local steps can help to reduce the communication rounds (under certain conditions). They also reflect the dependency of these bounds on client heterogeneity measured by global variance and indicates how to mitigate the problem in FedOpt. Experimental comparison to FedAvg, FedAvgM, and SCAFFOLD on seven benchmark FL tasks show that the proposed three adaptive FL methods are better or comparable on early-stage convergence and final validation-set performance. A study about tuning the local and global learning rate is also presented.	SP:c95a81c7f0df69cd0a7a85b08e8ed5c610732f8a
Counterfactual Generative Networks	1 INTRODUCTION . Deep neural networks ( DNNs ) are the main building blocks of many state-of-the-art machine learning systems that address diverse tasks such as image classification ( He et al. , 2016 ) , natural language processing ( Brown et al. , 2020 ) , and autonomous driving ( Ohn-Bar et al. , 2020 ) . Despite the considerable successes of DNNs , they still struggle in many situations , e.g. , classifying images perturbed by an adversary ( Szegedy et al. , 2013 ) , or failing to recognize known objects in unfamiliar contexts ( Rosenfeld et al. , 2018 ) or from unseen poses ( Alcorn et al. , 2019 ) . Many of these failures can be attributed to dataset biases ( Torralba & Efros , 2011 ) or shortcut learning ( Geirhos et al. , 2020 ) . The DNN learns the simplest correlations and tends to ignore more complex ones . This characteristic becomes problematic when the simple correlation is spurious , i.e. , not present during inference . The motivational example of ( Beery et al. , 2018 ) considers the setting of a DNN that is trained to recognize cows in images . A real-world dataset will typically depict cows on green pastures in most images . The most straightforward correlation a classifier can learn to predict the label ” cow ” is hence the connection to a green , grass-textured background . Generally , this is not a problem during inference as long as the test data follows the same distribution . However , if we provide the classifier an image depicting a purple cow on the moon , the classifier should still confidently assign the label ” cow. ” Thus , if we want to achieve robust generalization beyond the training data , we need to disentangle possibly spurious correlations from causal relationships . Distinguishing between spurious and causal correlations is one of the core questions in causality research ( Pearl , 2009 ; Peters et al. , 2017 ; Schölkopf , 2019 ) . One central concept in causality is the assumption of independent mechanisms ( IM ) , which states that a causal generative process is composed of autonomous modules that do not influence each other . In the context of image classification ( e.g. , on ImageNet ) , we can interpret the generation of an image as a causal process ( Kocaoglu et al. , 2018 ; Goyal et al. , 2019 ; Suter et al. , 2019 ) . We decompose this process into separate IMs , each controlling one factor of variation ( FoV ) of the image . Concretely , we consider three IMs : one generates the object ’ s shape , the second generates the object ’ s texture , and the third generates the background . With access to these IMs , we can produce counterfactual images , i.e. , images of unseen combinations of FoVs . We can then train an ensemble of invariant classifiers on the generated coun- terfactual images , such that every classifier relies on only a single one of those factors . The main idea is illustrated in Figure 1 . By exploiting concepts from causality , this paper links two previously distinct domains : disentangled generative models and robust classification . This allows us to scale our experiments beyond small toy datasets typically used in either domain . The main contributions of our work are as follows : • We present an approach for generating high-quality counterfactual images with direct control over shape , texture , and background . Supervision is only provided by the class label and certain inductive biases we impose on the learning problem . • We demonstrate the usefulness of the generated counterfactual images for the downstream task of image classification on both MNIST and ImageNet . Our model improves the classifier ’ s out-of-domain robustness while only marginally degrading its overall accuracy . • We show that our generative model demonstrates interesting emerging properties , such as generating high-quality binary object masks and unsupervised image inpainting . We release our code at https : //github.com/autonomousvision/counterfactual generative networks 2 STRUCTURAL CAUSAL MODELS FOR IMAGE GENERATION . In this section , we first introduce our ideas on a conceptual level . Concretely , we form a connection between the areas of causality , disentangled representation learning , and invariant classifiers , and highlight that domain randomization ( Tobin et al. , 2017 ) is a particular instance of these ideas . In section 3 , we will then formulate a concrete model that implements these ideas for image classification . Our goals are two-fold : ( i ) We aim at generating counterfactual images with previously unseen combinations like a cat with elephant texture or the proverbial ” bull in a china shop. ” ( ii ) We utilize these images to train a classifier invariant to chosen factors of variation . In the following , we first formalize the problem setting we address . Second , we describe how we can address this setting by structuring a generator network as a structural causal model ( SCM ) . Third , we show how to use the SCM for training robust classifiers . 2.1 PROBLEM SETTING . Consider a dataset comprised of ( high-dimensional ) observations x ( e.g . images ) , and corresponding labels y ( e.g . classes ) . A common assumption is that each x can be described by lower-dimensional , semantically meaningful factors of variation z ( e.g. , color or shape of objects in the image ) . If we can disentangle these factors , we are able to control their influence on the classifier ’ s decision . In the disentanglement literature , the factors are often assumed to be statistically independent , i.e. , z is distributed according to p ( z ) = Πni=1 ( zi ) ( Locatello et al. , 2018 ) . However , assuming independence is problematic because certain factors might be correlated in the training data , or the combination of some factors may not exist . Consider the colored MNIST dataset ( Kim et al. , 2019 ) , where both the digit ’ s color and its shape correspond to the label . The simplest decision rule a classifier can learn is to count the number of pixels of a specific color value ; no notion of the digit ’ s shape is required . This kind of correlation is not limited to constructed datasets – classifiers trained on ImageNet ( Deng et al. , 2009 ) strongly rely on texture for classification , significantly more than on the object ’ s shape ( Geirhos et al. , 2018 ) . While texture or color is a powerful classification cue , we do not want the classifier to ignore shape information completely . Therefore , we advocate a generative viewpoint . However , simply training , e.g. , a disentangled VAE ( Higgins et al. , 2017 ) on this dataset , does not allow for generating data points of unseen combinations – the VAE can not generate green zeros if all zeros in the training data are red ( see Appendix A for a visualization ) . We therefore propose a novel generative model which enables full control over several FoVs relevant for classification . We then train a classifier on these images while randomizing all factors but one . The classifier focuses on the non-randomized factor and becomes invariant wrt . the randomized ones . 2.2 STRUCTURAL CAUSAL MODELS . In representation learning , it is commonly assumed that a potentially complex function f generates images from a small set of high-level semantic variables ( e.g. , position or color of objects ) ( Bengio et al. , 2013 ) . Most previous work ( Goyal et al. , 2019 ; Suter et al. , 2019 ) imposes no restrictions on f , i.e. , a neural network is trained to map directly from a low-dimensional latent space to images . We follow the argument that rather than training a monolithic network to map from a latent space to images , the mapping should be decomposed into several functions . Each of these functions is autonomous , e.g. , we can modify the background of an image while keeping all other aspects of the image unchanged . These demands coincide with the concept of structural causal models ( SCMs ) and independent mechanisms ( IMs ) . An SCM C is defined as a collection of d ( structural ) assignments Sj : = fj ( PAj , Uj ) , j = 1 , . . . , d ( 1 ) where each random variable Sj is a function of its parents PAj ⊆ { S1 , . . . , Sd } \ { Sj } and a noise variable Uj . The noise variables U1 , . . . , Ud are jointly independent . The functions fi are independent mechanisms , intervening on one mechanism fj does not change the other mechanisms { f1 , . . . , fd } \ { fj } . The SCM C defines a unique distribution over the variables S = ( S1 , . . . , Sd ) which is referred to as the entailed distribution PCS . If one or more structural assignments are replaced , i.e. , Sk : = f̃ ( P̃Ak , Ũk ) , this is called an intervention . We consider the case of atomic interventions , when f̃ ( P̃Ak , Ũk ) puts a point mass on a real value a . The entailed distribution then changes to the intervention distribution PC ; do ( Sk : =a ) S , where the do refers to the intervention . A thorough review of these concepts can be found in ( Peters et al. , 2017 ) . Our goal is to represent the image generation process with an SCM . If we learn a sensible set of IMs , we can intervene on a subset of them and generate interventional images xIV . These images were not part of the training data x as they are generated from the intervention distribution PC ; do ( Sk : =a ) S . To generate a set of counterfactual images xCF , we fix the noise u and randomly draw a , hence answering counterfactual questions such as ” How would this image look like with a different background ? ” . In our case , a corresponds to a class label that we provide as input , denoted as yCF in the following . 2.3 TRAINING AN INVARIANT CLASSIFIER . To train an invariant classifier , we generate counterfactual images xCF , by intervening on all fj simultaneously . Towards this goal , we draw labels uniformly from the set of possible labels Y for each fj , i.e. , each IM is conditioned on a different label . We denote the domain of images generated by all possible label permutations as XCF . The task of the invariant classifier r : XCF → YCF , k is then to predict the label yCF , k that was provided to one specific IM fk – rendering r invariant wrt . all other IMs . This type of invariance is reminiscent of the idea of domain randomization ( Tobin et al. , 2017 ) . Here , the goal is to solve a robotics task while randomizing all task-irrelevant attributes . The randomization improves the performance of the learned policy in the real-world . In domain randomization , we commonly assume access to the true generative model ( the simulator ) . This assumption is not feasible if we do not have access to this model . Similar connections of causality and data augmentation have been made in ( Ilse et al. , 2020 ) . It is also possible to train on interventional images xIV , i.e. , generating a single image per sampled noise vector . Empirically , we find that counterfactual images improve performance over interventional ones . We hypothesize that counterfactuals provide a more stable signal .	The main idea of the paper, i.e., using independent causal mechanisms to generate interventional images, has already been explored by Kocaoglu et al. in Causalgan: Learning causal implicit generative models with adversarial training, ICLR'18. Same as here, the authors there also "view image generation as a causal process" and "structure a generator network as a structural causal model (SCM)" and use a conditional gan to generate the image from the labels. The generation used here based on three variables, i.e., shape, texture and background seem to be a special case. Therefore, the authors should definitely cite this work.	SP:a1c087b38201c94a7fccb11826606bbf678a8a57
Counterfactual Generative Networks	1 INTRODUCTION . Deep neural networks ( DNNs ) are the main building blocks of many state-of-the-art machine learning systems that address diverse tasks such as image classification ( He et al. , 2016 ) , natural language processing ( Brown et al. , 2020 ) , and autonomous driving ( Ohn-Bar et al. , 2020 ) . Despite the considerable successes of DNNs , they still struggle in many situations , e.g. , classifying images perturbed by an adversary ( Szegedy et al. , 2013 ) , or failing to recognize known objects in unfamiliar contexts ( Rosenfeld et al. , 2018 ) or from unseen poses ( Alcorn et al. , 2019 ) . Many of these failures can be attributed to dataset biases ( Torralba & Efros , 2011 ) or shortcut learning ( Geirhos et al. , 2020 ) . The DNN learns the simplest correlations and tends to ignore more complex ones . This characteristic becomes problematic when the simple correlation is spurious , i.e. , not present during inference . The motivational example of ( Beery et al. , 2018 ) considers the setting of a DNN that is trained to recognize cows in images . A real-world dataset will typically depict cows on green pastures in most images . The most straightforward correlation a classifier can learn to predict the label ” cow ” is hence the connection to a green , grass-textured background . Generally , this is not a problem during inference as long as the test data follows the same distribution . However , if we provide the classifier an image depicting a purple cow on the moon , the classifier should still confidently assign the label ” cow. ” Thus , if we want to achieve robust generalization beyond the training data , we need to disentangle possibly spurious correlations from causal relationships . Distinguishing between spurious and causal correlations is one of the core questions in causality research ( Pearl , 2009 ; Peters et al. , 2017 ; Schölkopf , 2019 ) . One central concept in causality is the assumption of independent mechanisms ( IM ) , which states that a causal generative process is composed of autonomous modules that do not influence each other . In the context of image classification ( e.g. , on ImageNet ) , we can interpret the generation of an image as a causal process ( Kocaoglu et al. , 2018 ; Goyal et al. , 2019 ; Suter et al. , 2019 ) . We decompose this process into separate IMs , each controlling one factor of variation ( FoV ) of the image . Concretely , we consider three IMs : one generates the object ’ s shape , the second generates the object ’ s texture , and the third generates the background . With access to these IMs , we can produce counterfactual images , i.e. , images of unseen combinations of FoVs . We can then train an ensemble of invariant classifiers on the generated coun- terfactual images , such that every classifier relies on only a single one of those factors . The main idea is illustrated in Figure 1 . By exploiting concepts from causality , this paper links two previously distinct domains : disentangled generative models and robust classification . This allows us to scale our experiments beyond small toy datasets typically used in either domain . The main contributions of our work are as follows : • We present an approach for generating high-quality counterfactual images with direct control over shape , texture , and background . Supervision is only provided by the class label and certain inductive biases we impose on the learning problem . • We demonstrate the usefulness of the generated counterfactual images for the downstream task of image classification on both MNIST and ImageNet . Our model improves the classifier ’ s out-of-domain robustness while only marginally degrading its overall accuracy . • We show that our generative model demonstrates interesting emerging properties , such as generating high-quality binary object masks and unsupervised image inpainting . We release our code at https : //github.com/autonomousvision/counterfactual generative networks 2 STRUCTURAL CAUSAL MODELS FOR IMAGE GENERATION . In this section , we first introduce our ideas on a conceptual level . Concretely , we form a connection between the areas of causality , disentangled representation learning , and invariant classifiers , and highlight that domain randomization ( Tobin et al. , 2017 ) is a particular instance of these ideas . In section 3 , we will then formulate a concrete model that implements these ideas for image classification . Our goals are two-fold : ( i ) We aim at generating counterfactual images with previously unseen combinations like a cat with elephant texture or the proverbial ” bull in a china shop. ” ( ii ) We utilize these images to train a classifier invariant to chosen factors of variation . In the following , we first formalize the problem setting we address . Second , we describe how we can address this setting by structuring a generator network as a structural causal model ( SCM ) . Third , we show how to use the SCM for training robust classifiers . 2.1 PROBLEM SETTING . Consider a dataset comprised of ( high-dimensional ) observations x ( e.g . images ) , and corresponding labels y ( e.g . classes ) . A common assumption is that each x can be described by lower-dimensional , semantically meaningful factors of variation z ( e.g. , color or shape of objects in the image ) . If we can disentangle these factors , we are able to control their influence on the classifier ’ s decision . In the disentanglement literature , the factors are often assumed to be statistically independent , i.e. , z is distributed according to p ( z ) = Πni=1 ( zi ) ( Locatello et al. , 2018 ) . However , assuming independence is problematic because certain factors might be correlated in the training data , or the combination of some factors may not exist . Consider the colored MNIST dataset ( Kim et al. , 2019 ) , where both the digit ’ s color and its shape correspond to the label . The simplest decision rule a classifier can learn is to count the number of pixels of a specific color value ; no notion of the digit ’ s shape is required . This kind of correlation is not limited to constructed datasets – classifiers trained on ImageNet ( Deng et al. , 2009 ) strongly rely on texture for classification , significantly more than on the object ’ s shape ( Geirhos et al. , 2018 ) . While texture or color is a powerful classification cue , we do not want the classifier to ignore shape information completely . Therefore , we advocate a generative viewpoint . However , simply training , e.g. , a disentangled VAE ( Higgins et al. , 2017 ) on this dataset , does not allow for generating data points of unseen combinations – the VAE can not generate green zeros if all zeros in the training data are red ( see Appendix A for a visualization ) . We therefore propose a novel generative model which enables full control over several FoVs relevant for classification . We then train a classifier on these images while randomizing all factors but one . The classifier focuses on the non-randomized factor and becomes invariant wrt . the randomized ones . 2.2 STRUCTURAL CAUSAL MODELS . In representation learning , it is commonly assumed that a potentially complex function f generates images from a small set of high-level semantic variables ( e.g. , position or color of objects ) ( Bengio et al. , 2013 ) . Most previous work ( Goyal et al. , 2019 ; Suter et al. , 2019 ) imposes no restrictions on f , i.e. , a neural network is trained to map directly from a low-dimensional latent space to images . We follow the argument that rather than training a monolithic network to map from a latent space to images , the mapping should be decomposed into several functions . Each of these functions is autonomous , e.g. , we can modify the background of an image while keeping all other aspects of the image unchanged . These demands coincide with the concept of structural causal models ( SCMs ) and independent mechanisms ( IMs ) . An SCM C is defined as a collection of d ( structural ) assignments Sj : = fj ( PAj , Uj ) , j = 1 , . . . , d ( 1 ) where each random variable Sj is a function of its parents PAj ⊆ { S1 , . . . , Sd } \ { Sj } and a noise variable Uj . The noise variables U1 , . . . , Ud are jointly independent . The functions fi are independent mechanisms , intervening on one mechanism fj does not change the other mechanisms { f1 , . . . , fd } \ { fj } . The SCM C defines a unique distribution over the variables S = ( S1 , . . . , Sd ) which is referred to as the entailed distribution PCS . If one or more structural assignments are replaced , i.e. , Sk : = f̃ ( P̃Ak , Ũk ) , this is called an intervention . We consider the case of atomic interventions , when f̃ ( P̃Ak , Ũk ) puts a point mass on a real value a . The entailed distribution then changes to the intervention distribution PC ; do ( Sk : =a ) S , where the do refers to the intervention . A thorough review of these concepts can be found in ( Peters et al. , 2017 ) . Our goal is to represent the image generation process with an SCM . If we learn a sensible set of IMs , we can intervene on a subset of them and generate interventional images xIV . These images were not part of the training data x as they are generated from the intervention distribution PC ; do ( Sk : =a ) S . To generate a set of counterfactual images xCF , we fix the noise u and randomly draw a , hence answering counterfactual questions such as ” How would this image look like with a different background ? ” . In our case , a corresponds to a class label that we provide as input , denoted as yCF in the following . 2.3 TRAINING AN INVARIANT CLASSIFIER . To train an invariant classifier , we generate counterfactual images xCF , by intervening on all fj simultaneously . Towards this goal , we draw labels uniformly from the set of possible labels Y for each fj , i.e. , each IM is conditioned on a different label . We denote the domain of images generated by all possible label permutations as XCF . The task of the invariant classifier r : XCF → YCF , k is then to predict the label yCF , k that was provided to one specific IM fk – rendering r invariant wrt . all other IMs . This type of invariance is reminiscent of the idea of domain randomization ( Tobin et al. , 2017 ) . Here , the goal is to solve a robotics task while randomizing all task-irrelevant attributes . The randomization improves the performance of the learned policy in the real-world . In domain randomization , we commonly assume access to the true generative model ( the simulator ) . This assumption is not feasible if we do not have access to this model . Similar connections of causality and data augmentation have been made in ( Ilse et al. , 2020 ) . It is also possible to train on interventional images xIV , i.e. , generating a single image per sampled noise vector . Empirically , we find that counterfactual images improve performance over interventional ones . We hypothesize that counterfactuals provide a more stable signal .	Deep neural network brittleness can be attributed to their tendency to latch on to spurious correlations in the training dataset. The proposal in the paper is to learn to generate samples where these correlations can be eliminated. To this end, the authors, distill trained conditional big gan into a transformation with explicit modules to capture the shape, texture of the foreground object, and the background. The distilled network is called Counterfactual Generator Network (CGN). Thus, an image can be generated with a background of one class, the shape of another class, and the foreground texture of a different class. Then a classifier with multiple heads is learned where each head predicts a class based on only one of the factors among shape, texture, and background.	SP:a1c087b38201c94a7fccb11826606bbf678a8a57
Counterfactual Generative Networks	1 INTRODUCTION . Deep neural networks ( DNNs ) are the main building blocks of many state-of-the-art machine learning systems that address diverse tasks such as image classification ( He et al. , 2016 ) , natural language processing ( Brown et al. , 2020 ) , and autonomous driving ( Ohn-Bar et al. , 2020 ) . Despite the considerable successes of DNNs , they still struggle in many situations , e.g. , classifying images perturbed by an adversary ( Szegedy et al. , 2013 ) , or failing to recognize known objects in unfamiliar contexts ( Rosenfeld et al. , 2018 ) or from unseen poses ( Alcorn et al. , 2019 ) . Many of these failures can be attributed to dataset biases ( Torralba & Efros , 2011 ) or shortcut learning ( Geirhos et al. , 2020 ) . The DNN learns the simplest correlations and tends to ignore more complex ones . This characteristic becomes problematic when the simple correlation is spurious , i.e. , not present during inference . The motivational example of ( Beery et al. , 2018 ) considers the setting of a DNN that is trained to recognize cows in images . A real-world dataset will typically depict cows on green pastures in most images . The most straightforward correlation a classifier can learn to predict the label ” cow ” is hence the connection to a green , grass-textured background . Generally , this is not a problem during inference as long as the test data follows the same distribution . However , if we provide the classifier an image depicting a purple cow on the moon , the classifier should still confidently assign the label ” cow. ” Thus , if we want to achieve robust generalization beyond the training data , we need to disentangle possibly spurious correlations from causal relationships . Distinguishing between spurious and causal correlations is one of the core questions in causality research ( Pearl , 2009 ; Peters et al. , 2017 ; Schölkopf , 2019 ) . One central concept in causality is the assumption of independent mechanisms ( IM ) , which states that a causal generative process is composed of autonomous modules that do not influence each other . In the context of image classification ( e.g. , on ImageNet ) , we can interpret the generation of an image as a causal process ( Kocaoglu et al. , 2018 ; Goyal et al. , 2019 ; Suter et al. , 2019 ) . We decompose this process into separate IMs , each controlling one factor of variation ( FoV ) of the image . Concretely , we consider three IMs : one generates the object ’ s shape , the second generates the object ’ s texture , and the third generates the background . With access to these IMs , we can produce counterfactual images , i.e. , images of unseen combinations of FoVs . We can then train an ensemble of invariant classifiers on the generated coun- terfactual images , such that every classifier relies on only a single one of those factors . The main idea is illustrated in Figure 1 . By exploiting concepts from causality , this paper links two previously distinct domains : disentangled generative models and robust classification . This allows us to scale our experiments beyond small toy datasets typically used in either domain . The main contributions of our work are as follows : • We present an approach for generating high-quality counterfactual images with direct control over shape , texture , and background . Supervision is only provided by the class label and certain inductive biases we impose on the learning problem . • We demonstrate the usefulness of the generated counterfactual images for the downstream task of image classification on both MNIST and ImageNet . Our model improves the classifier ’ s out-of-domain robustness while only marginally degrading its overall accuracy . • We show that our generative model demonstrates interesting emerging properties , such as generating high-quality binary object masks and unsupervised image inpainting . We release our code at https : //github.com/autonomousvision/counterfactual generative networks 2 STRUCTURAL CAUSAL MODELS FOR IMAGE GENERATION . In this section , we first introduce our ideas on a conceptual level . Concretely , we form a connection between the areas of causality , disentangled representation learning , and invariant classifiers , and highlight that domain randomization ( Tobin et al. , 2017 ) is a particular instance of these ideas . In section 3 , we will then formulate a concrete model that implements these ideas for image classification . Our goals are two-fold : ( i ) We aim at generating counterfactual images with previously unseen combinations like a cat with elephant texture or the proverbial ” bull in a china shop. ” ( ii ) We utilize these images to train a classifier invariant to chosen factors of variation . In the following , we first formalize the problem setting we address . Second , we describe how we can address this setting by structuring a generator network as a structural causal model ( SCM ) . Third , we show how to use the SCM for training robust classifiers . 2.1 PROBLEM SETTING . Consider a dataset comprised of ( high-dimensional ) observations x ( e.g . images ) , and corresponding labels y ( e.g . classes ) . A common assumption is that each x can be described by lower-dimensional , semantically meaningful factors of variation z ( e.g. , color or shape of objects in the image ) . If we can disentangle these factors , we are able to control their influence on the classifier ’ s decision . In the disentanglement literature , the factors are often assumed to be statistically independent , i.e. , z is distributed according to p ( z ) = Πni=1 ( zi ) ( Locatello et al. , 2018 ) . However , assuming independence is problematic because certain factors might be correlated in the training data , or the combination of some factors may not exist . Consider the colored MNIST dataset ( Kim et al. , 2019 ) , where both the digit ’ s color and its shape correspond to the label . The simplest decision rule a classifier can learn is to count the number of pixels of a specific color value ; no notion of the digit ’ s shape is required . This kind of correlation is not limited to constructed datasets – classifiers trained on ImageNet ( Deng et al. , 2009 ) strongly rely on texture for classification , significantly more than on the object ’ s shape ( Geirhos et al. , 2018 ) . While texture or color is a powerful classification cue , we do not want the classifier to ignore shape information completely . Therefore , we advocate a generative viewpoint . However , simply training , e.g. , a disentangled VAE ( Higgins et al. , 2017 ) on this dataset , does not allow for generating data points of unseen combinations – the VAE can not generate green zeros if all zeros in the training data are red ( see Appendix A for a visualization ) . We therefore propose a novel generative model which enables full control over several FoVs relevant for classification . We then train a classifier on these images while randomizing all factors but one . The classifier focuses on the non-randomized factor and becomes invariant wrt . the randomized ones . 2.2 STRUCTURAL CAUSAL MODELS . In representation learning , it is commonly assumed that a potentially complex function f generates images from a small set of high-level semantic variables ( e.g. , position or color of objects ) ( Bengio et al. , 2013 ) . Most previous work ( Goyal et al. , 2019 ; Suter et al. , 2019 ) imposes no restrictions on f , i.e. , a neural network is trained to map directly from a low-dimensional latent space to images . We follow the argument that rather than training a monolithic network to map from a latent space to images , the mapping should be decomposed into several functions . Each of these functions is autonomous , e.g. , we can modify the background of an image while keeping all other aspects of the image unchanged . These demands coincide with the concept of structural causal models ( SCMs ) and independent mechanisms ( IMs ) . An SCM C is defined as a collection of d ( structural ) assignments Sj : = fj ( PAj , Uj ) , j = 1 , . . . , d ( 1 ) where each random variable Sj is a function of its parents PAj ⊆ { S1 , . . . , Sd } \ { Sj } and a noise variable Uj . The noise variables U1 , . . . , Ud are jointly independent . The functions fi are independent mechanisms , intervening on one mechanism fj does not change the other mechanisms { f1 , . . . , fd } \ { fj } . The SCM C defines a unique distribution over the variables S = ( S1 , . . . , Sd ) which is referred to as the entailed distribution PCS . If one or more structural assignments are replaced , i.e. , Sk : = f̃ ( P̃Ak , Ũk ) , this is called an intervention . We consider the case of atomic interventions , when f̃ ( P̃Ak , Ũk ) puts a point mass on a real value a . The entailed distribution then changes to the intervention distribution PC ; do ( Sk : =a ) S , where the do refers to the intervention . A thorough review of these concepts can be found in ( Peters et al. , 2017 ) . Our goal is to represent the image generation process with an SCM . If we learn a sensible set of IMs , we can intervene on a subset of them and generate interventional images xIV . These images were not part of the training data x as they are generated from the intervention distribution PC ; do ( Sk : =a ) S . To generate a set of counterfactual images xCF , we fix the noise u and randomly draw a , hence answering counterfactual questions such as ” How would this image look like with a different background ? ” . In our case , a corresponds to a class label that we provide as input , denoted as yCF in the following . 2.3 TRAINING AN INVARIANT CLASSIFIER . To train an invariant classifier , we generate counterfactual images xCF , by intervening on all fj simultaneously . Towards this goal , we draw labels uniformly from the set of possible labels Y for each fj , i.e. , each IM is conditioned on a different label . We denote the domain of images generated by all possible label permutations as XCF . The task of the invariant classifier r : XCF → YCF , k is then to predict the label yCF , k that was provided to one specific IM fk – rendering r invariant wrt . all other IMs . This type of invariance is reminiscent of the idea of domain randomization ( Tobin et al. , 2017 ) . Here , the goal is to solve a robotics task while randomizing all task-irrelevant attributes . The randomization improves the performance of the learned policy in the real-world . In domain randomization , we commonly assume access to the true generative model ( the simulator ) . This assumption is not feasible if we do not have access to this model . Similar connections of causality and data augmentation have been made in ( Ilse et al. , 2020 ) . It is also possible to train on interventional images xIV , i.e. , generating a single image per sampled noise vector . Empirically , we find that counterfactual images improve performance over interventional ones . We hypothesize that counterfactuals provide a more stable signal .	This paper proposes a new generative model that generate images from 3 seperate aspects: foreground masks (shapes), forground texture, and backgrounds. Then they convexly mix these 3 aspects into one image. By doing so, they can vary each aspect individually without changing other aspects, enabling the model to generate counterfactual images. For example, we can generate a cat shape with telephene texture and sea background, which would not exist in natural images. They show that in several colorful MNISTs datasets their methods can generate new combinations of images. In ImageNet, by using the pre-trained BigNet GAN as backbones and pretrained U-Net for foreground object masks, they can distill the knowledge in BigNet into these 3 seperate aspects and generate counterfactual natural images.	SP:a1c087b38201c94a7fccb11826606bbf678a8a57
Federated Learning With Quantized Global Model Updates	1 Introduction . Federated learning ( FL ) enables wireless devices to collaboratively train a global model by utilizing locally available data and computational capabilities under the coordination of a parameter server ( PS ) while the data never leaves the devices McMahan & Ramage ( 2017 ) . In FL with M devices the goal is to minimize a loss function F ( θ ) = ∑M m=1 Bm B Fm ( θ ) with respect to the global model θ ∈ Rd , where Fm ( θ ) = 1Bm ∑ u∈Bm f ( θ , u ) is the loss function at device m , with Bm representing device m ’ s local dataset of size Bm , B , ∑M m=1 Bm , and f ( · , · ) is an empirical loss function . Having access to the global model θ , device m utilizes its local dataset and performs multiple iterations of stochastic gradient descent ( SGD ) in order to minimize the local loss function Fm ( θ ) . It then sends the local model update to the server , which aggregates the local updates from all the devices to update the global model . FL mainly targets mobile applications at the network edge , and the wireless communication links connecting these devices to the network are typically limited in bandwidth and power , and suffer from various channel impairments such as fading , shadowing , or interference ; hence the need to develop an FL framework with limited communication requirements becomes more vital . While communication-efficient FL has been widely studied , prior works mainly focused on the devices-to-PS links , assuming perfect broadcasting of the global model to the devices at each iteration . In this paper , we design an FL algorithm aiming to reduce the cost of both PS-to-device and devices-to-PS communications . To address the importance of quantization at the PS-to-device direction , we highlight that some devices simply may not have the sufficient bandwidth to receive the global model update when the model size is relatively large , particularly in the wireless setting , where the devices are away from the base station . This would result in consistent exclusion of these devices , resulting in significant performance loss . Moreover , the impact of quantization in the device-to-PS direction is less severe due to the impact of averaging local updates at the PS . Related work There is a fast-growing body of literature on the communication efficiency of FL targeting restricted bandwidth devices . Several studies address this issue by considering communications with rate limitations , and propose different compression and quantization techniques Konecny et al . ( 2016 ) ; McMahan et al . ( 2017 ) ; Konecny & Richtarik ( 2018 ) ; Dowlin et al . ( 2016 ) ; Konecny et al . ( 2015 ) ; Lin et al . ( 2018b ) ; He et al . ( 2018 ) ; M. M. Amiri & Gündüz ( 2020 ) , as well as performing local updates to reduce the frequency of communications from the devices to the PS Lin et al . ( 2018a ) ; Stich ( 2019 ) . Statistical challenges arise in FL since the data samples may not be independent and identically distributed ( iid ) across devices . The common sources of the dependence or bias in data distribution are the participating devices being located in a particular geographic region , and/or at a particular time window P. Kairouz et al . ( 2019 ) . Different approaches have been studied to mitigate the effect of non-iid data in FL McMahan et al . ( 2017 ) ; Hsieh et al . ( 2019 ) ; Li et al . ( 2020a ) ; Wang et al . ( 2020 ) ; Eichner et al . ( 2019 ) ; Zhao et al . ( 2018 ) . Also , FL suffers from a significant variability in the system , which is mainly due to the hardware , network connectivity , and available power associated with different devices Li et al . ( 2019 ) . Active device selection schemes have been introduced to alleviate significant variability in FL systems , where a subset of devices share the resources and participate at each iteration of training Kang et al . ( 2019 ) ; Nishio & Yonetani ( 2019 ) ; Amiri et al . ( 2020b ) ; Yang et al . ( 2020 ; 2019 ) . There have also been efforts in developing convergence guarantees for FL under various scenarios , considering iid data across the devices Stich ( 2019 ) ; Wang & Joshi ( 2019 ) ; Woodworth et al . ( 2019 ) ; Zhou & Cong ( 2018 ) ; Koloskova et al . ( 2020 ) , non-iid data Koloskova et al . ( 2020 ) ; Li et al . ( 2020a ) ; Haddadpour & Mahdavi ( 2019 ) ; Li et al . ( 2020c ) , participation of all the devices Khaled et al . ( 2020 ) ; Wang et al . ( 2019 ) ; Yu et al . ( 2018 ) ; Huo et al . ( 2020 ) , or only a subset of devices at each iteration Li et al . ( 2020b ) ; Karimireddy et al . ( 2020 ) ; Rizk et al . ( 2020 ) ; Li et al . ( 2020c ) ; Amiri et al . ( 2020a ) , and FL under limited communication constraints Amiri et al . ( 2020a ) ; Recht et al . ( 2011 ) ; Alistarh et al . ( 2018 ) . FL with compressed global model transmission has been studied recently in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) aiming to alleviate the communication footprint from the PS to the devices . The global model parameters are relatively skewed/diverse and the efficiency of quantization diminishes significantly when the peak-to-average ratio of the parameters is large . To overcome this , in Caldas et al . ( 2019 ) the PS first employs a linear transform in order to spread the information of the global model vector more evenly among its dimensions , and broadcasts a quantized version of the resultant vector , and the devices apply the inverse linear transform to estimate the global model . We highlight that this approach requires a relatively high computational overhead due to employing the linear transform at the PS and its inverse at the devices , where this overhead grows with the size of the model parameters . Furthermore , the performance evaluation in Caldas et al . ( 2019 ) is limited to the experimental results On the other hand , in Tang et al . ( 2019 ) the PS broadcasts quantized global model with error accumulation to compensate the quantization error . Our contributions With the exception of Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) , the literature on FL considers perfect broadcasting of the global model from the PS to the devices . With this assumption , no matter what type of local update or device-to-PS communication strategy is used , all the devices are synchronized with the same global model at each iteration . In this paper , we instead consider broadcasting a quantized version of the global model update by the PS , which provides the devices with a lossy estimate of the global model ( rather than its accurate estimate ) with which to perform local training . This further reduces the communication cost of FL , which can be particularly limited for transmission over a wireless medium while serving a massive number of devices . Also , it is interesting to investigate the impact of various hyperparameters on the performance of FL with lossy broadcasting of the global model since FL involves transmission over wireless networks with limited bandwidth . We introduce a lossy FL ( LFL ) algorithm , where at each iteration the PS broadcasts a compressed version of the global model update to all the devices through quantization . To be precise , the PS exploits the knowledge of the last global model estimate available at the devices as side information to quantize the global model update . The devices recover an estimate of the current global model by combining the received quantized global model update with their previous estimate , and perform local training using their estimate , and return the local model updates , again employing quantization . The PS updates the global model after receiving the quantized local model updates from the devices . We provide convergence analysis of the LFL algorithm investigating the impact of lossy broadcasting on the performance of FL . Numerical experiments on the MNIST and CIFAR-10 datasets illustrate the efficiency of the proposed LFL algorithm . We observe that the proposed LFL scheme , which leads to a significant communication cost saving , provides a promising performance with no visible gap to the performance of the fully lossless scenario where the communication from both PS-to-device and device-to-PS directions is assumed to be perfect . Also , it is illustrated that the proposed LFL scheme significantly outperforms the schemes introduced in Caldas et al . ( 2019 ) and Tang et al . ( 2019 ) considering compression from the PS to devices . The proposed LFL algorithm differs from the approaches in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) , since we propose broadcasting the global model update , with respect to the previous estimate at the devices , rather than the global model itself . We remark that the global model update has less variability/variance and peak-to-average ratio than the global model ( see Figure 2 ) , and hence , for the same communication load , the devices can have a more accurate estimate of the global model . However , this would require all the devices to track the global model at each iteration , even if they do not participate in the learning process by sending their local update . We argue that broadcasting the global model update to the whole set of devices , rather than a randomly chosen subset , would introduce limited additional communication cost as broadcasting is typically more efficient than sending independent information to devices . Moreover , in practice , the subset of participating devices remain the same for a number of iterations , until a device leaves or joins . Our algorithm can easily be adopted to such scenarios by sending the global model , rather than the model update , every time the subset of devices changes . Also , compared to the approach in Caldas et al . ( 2019 ) , the LFL algorithm requires a significantly smaller computational overhead . Furthermore , unlike Caldas et al . ( 2019 ) , we provide an in-depth convergence analysis of the proposed LFL algorithm . The advantage of the proposed LFL algorithm over the approaches introduced in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) is shown numerically , where , despite its significantly smaller communication load , it provides considerably higher accuracy . Notation The set of real numbers is denoted by R. For x ∈ R , \|x\| returns the absolute value of x . For a vector of real numbers x , the largest and the smallest absolute values among all the entries of x are represented by max { \|x\| } and min { \|x\| } , respectively . For an integer i , we let [ i ] , { 1 , 2 , . . . , i } . The l2-norm of vector x is denoted by ‖x‖2 .	This paper studies federated learning with quantization. The problem setting is very standard, including both iid and non-iid cases. This work proposes a new algorithm, called lossy FL, to save the communication costs, especially from the broadcasting direction. To my understanding, the algorithm is new but still very similar to double-squeeze (Tang 2019). The presentation of the paper is generally good. However, there are several major issues regarding the quality and significance of this work.	SP:4b3c0127fcd5cc73226d2b8f58df658f5599791e
Federated Learning With Quantized Global Model Updates	1 Introduction . Federated learning ( FL ) enables wireless devices to collaboratively train a global model by utilizing locally available data and computational capabilities under the coordination of a parameter server ( PS ) while the data never leaves the devices McMahan & Ramage ( 2017 ) . In FL with M devices the goal is to minimize a loss function F ( θ ) = ∑M m=1 Bm B Fm ( θ ) with respect to the global model θ ∈ Rd , where Fm ( θ ) = 1Bm ∑ u∈Bm f ( θ , u ) is the loss function at device m , with Bm representing device m ’ s local dataset of size Bm , B , ∑M m=1 Bm , and f ( · , · ) is an empirical loss function . Having access to the global model θ , device m utilizes its local dataset and performs multiple iterations of stochastic gradient descent ( SGD ) in order to minimize the local loss function Fm ( θ ) . It then sends the local model update to the server , which aggregates the local updates from all the devices to update the global model . FL mainly targets mobile applications at the network edge , and the wireless communication links connecting these devices to the network are typically limited in bandwidth and power , and suffer from various channel impairments such as fading , shadowing , or interference ; hence the need to develop an FL framework with limited communication requirements becomes more vital . While communication-efficient FL has been widely studied , prior works mainly focused on the devices-to-PS links , assuming perfect broadcasting of the global model to the devices at each iteration . In this paper , we design an FL algorithm aiming to reduce the cost of both PS-to-device and devices-to-PS communications . To address the importance of quantization at the PS-to-device direction , we highlight that some devices simply may not have the sufficient bandwidth to receive the global model update when the model size is relatively large , particularly in the wireless setting , where the devices are away from the base station . This would result in consistent exclusion of these devices , resulting in significant performance loss . Moreover , the impact of quantization in the device-to-PS direction is less severe due to the impact of averaging local updates at the PS . Related work There is a fast-growing body of literature on the communication efficiency of FL targeting restricted bandwidth devices . Several studies address this issue by considering communications with rate limitations , and propose different compression and quantization techniques Konecny et al . ( 2016 ) ; McMahan et al . ( 2017 ) ; Konecny & Richtarik ( 2018 ) ; Dowlin et al . ( 2016 ) ; Konecny et al . ( 2015 ) ; Lin et al . ( 2018b ) ; He et al . ( 2018 ) ; M. M. Amiri & Gündüz ( 2020 ) , as well as performing local updates to reduce the frequency of communications from the devices to the PS Lin et al . ( 2018a ) ; Stich ( 2019 ) . Statistical challenges arise in FL since the data samples may not be independent and identically distributed ( iid ) across devices . The common sources of the dependence or bias in data distribution are the participating devices being located in a particular geographic region , and/or at a particular time window P. Kairouz et al . ( 2019 ) . Different approaches have been studied to mitigate the effect of non-iid data in FL McMahan et al . ( 2017 ) ; Hsieh et al . ( 2019 ) ; Li et al . ( 2020a ) ; Wang et al . ( 2020 ) ; Eichner et al . ( 2019 ) ; Zhao et al . ( 2018 ) . Also , FL suffers from a significant variability in the system , which is mainly due to the hardware , network connectivity , and available power associated with different devices Li et al . ( 2019 ) . Active device selection schemes have been introduced to alleviate significant variability in FL systems , where a subset of devices share the resources and participate at each iteration of training Kang et al . ( 2019 ) ; Nishio & Yonetani ( 2019 ) ; Amiri et al . ( 2020b ) ; Yang et al . ( 2020 ; 2019 ) . There have also been efforts in developing convergence guarantees for FL under various scenarios , considering iid data across the devices Stich ( 2019 ) ; Wang & Joshi ( 2019 ) ; Woodworth et al . ( 2019 ) ; Zhou & Cong ( 2018 ) ; Koloskova et al . ( 2020 ) , non-iid data Koloskova et al . ( 2020 ) ; Li et al . ( 2020a ) ; Haddadpour & Mahdavi ( 2019 ) ; Li et al . ( 2020c ) , participation of all the devices Khaled et al . ( 2020 ) ; Wang et al . ( 2019 ) ; Yu et al . ( 2018 ) ; Huo et al . ( 2020 ) , or only a subset of devices at each iteration Li et al . ( 2020b ) ; Karimireddy et al . ( 2020 ) ; Rizk et al . ( 2020 ) ; Li et al . ( 2020c ) ; Amiri et al . ( 2020a ) , and FL under limited communication constraints Amiri et al . ( 2020a ) ; Recht et al . ( 2011 ) ; Alistarh et al . ( 2018 ) . FL with compressed global model transmission has been studied recently in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) aiming to alleviate the communication footprint from the PS to the devices . The global model parameters are relatively skewed/diverse and the efficiency of quantization diminishes significantly when the peak-to-average ratio of the parameters is large . To overcome this , in Caldas et al . ( 2019 ) the PS first employs a linear transform in order to spread the information of the global model vector more evenly among its dimensions , and broadcasts a quantized version of the resultant vector , and the devices apply the inverse linear transform to estimate the global model . We highlight that this approach requires a relatively high computational overhead due to employing the linear transform at the PS and its inverse at the devices , where this overhead grows with the size of the model parameters . Furthermore , the performance evaluation in Caldas et al . ( 2019 ) is limited to the experimental results On the other hand , in Tang et al . ( 2019 ) the PS broadcasts quantized global model with error accumulation to compensate the quantization error . Our contributions With the exception of Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) , the literature on FL considers perfect broadcasting of the global model from the PS to the devices . With this assumption , no matter what type of local update or device-to-PS communication strategy is used , all the devices are synchronized with the same global model at each iteration . In this paper , we instead consider broadcasting a quantized version of the global model update by the PS , which provides the devices with a lossy estimate of the global model ( rather than its accurate estimate ) with which to perform local training . This further reduces the communication cost of FL , which can be particularly limited for transmission over a wireless medium while serving a massive number of devices . Also , it is interesting to investigate the impact of various hyperparameters on the performance of FL with lossy broadcasting of the global model since FL involves transmission over wireless networks with limited bandwidth . We introduce a lossy FL ( LFL ) algorithm , where at each iteration the PS broadcasts a compressed version of the global model update to all the devices through quantization . To be precise , the PS exploits the knowledge of the last global model estimate available at the devices as side information to quantize the global model update . The devices recover an estimate of the current global model by combining the received quantized global model update with their previous estimate , and perform local training using their estimate , and return the local model updates , again employing quantization . The PS updates the global model after receiving the quantized local model updates from the devices . We provide convergence analysis of the LFL algorithm investigating the impact of lossy broadcasting on the performance of FL . Numerical experiments on the MNIST and CIFAR-10 datasets illustrate the efficiency of the proposed LFL algorithm . We observe that the proposed LFL scheme , which leads to a significant communication cost saving , provides a promising performance with no visible gap to the performance of the fully lossless scenario where the communication from both PS-to-device and device-to-PS directions is assumed to be perfect . Also , it is illustrated that the proposed LFL scheme significantly outperforms the schemes introduced in Caldas et al . ( 2019 ) and Tang et al . ( 2019 ) considering compression from the PS to devices . The proposed LFL algorithm differs from the approaches in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) , since we propose broadcasting the global model update , with respect to the previous estimate at the devices , rather than the global model itself . We remark that the global model update has less variability/variance and peak-to-average ratio than the global model ( see Figure 2 ) , and hence , for the same communication load , the devices can have a more accurate estimate of the global model . However , this would require all the devices to track the global model at each iteration , even if they do not participate in the learning process by sending their local update . We argue that broadcasting the global model update to the whole set of devices , rather than a randomly chosen subset , would introduce limited additional communication cost as broadcasting is typically more efficient than sending independent information to devices . Moreover , in practice , the subset of participating devices remain the same for a number of iterations , until a device leaves or joins . Our algorithm can easily be adopted to such scenarios by sending the global model , rather than the model update , every time the subset of devices changes . Also , compared to the approach in Caldas et al . ( 2019 ) , the LFL algorithm requires a significantly smaller computational overhead . Furthermore , unlike Caldas et al . ( 2019 ) , we provide an in-depth convergence analysis of the proposed LFL algorithm . The advantage of the proposed LFL algorithm over the approaches introduced in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) is shown numerically , where , despite its significantly smaller communication load , it provides considerably higher accuracy . Notation The set of real numbers is denoted by R. For x ∈ R , \|x\| returns the absolute value of x . For a vector of real numbers x , the largest and the smallest absolute values among all the entries of x are represented by max { \|x\| } and min { \|x\| } , respectively . For an integer i , we let [ i ] , { 1 , 2 , . . . , i } . The l2-norm of vector x is denoted by ‖x‖2 .	This paper suggests a lossy federated learning (LFL) algorithm where the PS broadcasts a quantized version of the global model to devices. This approach helps reduce the communication cost of federated learning and is in particular useful when the communication bandwidth is limited. Associated convergence analysis is given to present the effect of lossy broadcasting on the performance of federated learning. From experimental results, it is shown that the proposed algorithm with (q_1, q_2) = (2, 2) shows almost the same performance as the lossless schemes while enjoying a considerably reduced amount of communication bits.	SP:4b3c0127fcd5cc73226d2b8f58df658f5599791e
Federated Learning With Quantized Global Model Updates	1 Introduction . Federated learning ( FL ) enables wireless devices to collaboratively train a global model by utilizing locally available data and computational capabilities under the coordination of a parameter server ( PS ) while the data never leaves the devices McMahan & Ramage ( 2017 ) . In FL with M devices the goal is to minimize a loss function F ( θ ) = ∑M m=1 Bm B Fm ( θ ) with respect to the global model θ ∈ Rd , where Fm ( θ ) = 1Bm ∑ u∈Bm f ( θ , u ) is the loss function at device m , with Bm representing device m ’ s local dataset of size Bm , B , ∑M m=1 Bm , and f ( · , · ) is an empirical loss function . Having access to the global model θ , device m utilizes its local dataset and performs multiple iterations of stochastic gradient descent ( SGD ) in order to minimize the local loss function Fm ( θ ) . It then sends the local model update to the server , which aggregates the local updates from all the devices to update the global model . FL mainly targets mobile applications at the network edge , and the wireless communication links connecting these devices to the network are typically limited in bandwidth and power , and suffer from various channel impairments such as fading , shadowing , or interference ; hence the need to develop an FL framework with limited communication requirements becomes more vital . While communication-efficient FL has been widely studied , prior works mainly focused on the devices-to-PS links , assuming perfect broadcasting of the global model to the devices at each iteration . In this paper , we design an FL algorithm aiming to reduce the cost of both PS-to-device and devices-to-PS communications . To address the importance of quantization at the PS-to-device direction , we highlight that some devices simply may not have the sufficient bandwidth to receive the global model update when the model size is relatively large , particularly in the wireless setting , where the devices are away from the base station . This would result in consistent exclusion of these devices , resulting in significant performance loss . Moreover , the impact of quantization in the device-to-PS direction is less severe due to the impact of averaging local updates at the PS . Related work There is a fast-growing body of literature on the communication efficiency of FL targeting restricted bandwidth devices . Several studies address this issue by considering communications with rate limitations , and propose different compression and quantization techniques Konecny et al . ( 2016 ) ; McMahan et al . ( 2017 ) ; Konecny & Richtarik ( 2018 ) ; Dowlin et al . ( 2016 ) ; Konecny et al . ( 2015 ) ; Lin et al . ( 2018b ) ; He et al . ( 2018 ) ; M. M. Amiri & Gündüz ( 2020 ) , as well as performing local updates to reduce the frequency of communications from the devices to the PS Lin et al . ( 2018a ) ; Stich ( 2019 ) . Statistical challenges arise in FL since the data samples may not be independent and identically distributed ( iid ) across devices . The common sources of the dependence or bias in data distribution are the participating devices being located in a particular geographic region , and/or at a particular time window P. Kairouz et al . ( 2019 ) . Different approaches have been studied to mitigate the effect of non-iid data in FL McMahan et al . ( 2017 ) ; Hsieh et al . ( 2019 ) ; Li et al . ( 2020a ) ; Wang et al . ( 2020 ) ; Eichner et al . ( 2019 ) ; Zhao et al . ( 2018 ) . Also , FL suffers from a significant variability in the system , which is mainly due to the hardware , network connectivity , and available power associated with different devices Li et al . ( 2019 ) . Active device selection schemes have been introduced to alleviate significant variability in FL systems , where a subset of devices share the resources and participate at each iteration of training Kang et al . ( 2019 ) ; Nishio & Yonetani ( 2019 ) ; Amiri et al . ( 2020b ) ; Yang et al . ( 2020 ; 2019 ) . There have also been efforts in developing convergence guarantees for FL under various scenarios , considering iid data across the devices Stich ( 2019 ) ; Wang & Joshi ( 2019 ) ; Woodworth et al . ( 2019 ) ; Zhou & Cong ( 2018 ) ; Koloskova et al . ( 2020 ) , non-iid data Koloskova et al . ( 2020 ) ; Li et al . ( 2020a ) ; Haddadpour & Mahdavi ( 2019 ) ; Li et al . ( 2020c ) , participation of all the devices Khaled et al . ( 2020 ) ; Wang et al . ( 2019 ) ; Yu et al . ( 2018 ) ; Huo et al . ( 2020 ) , or only a subset of devices at each iteration Li et al . ( 2020b ) ; Karimireddy et al . ( 2020 ) ; Rizk et al . ( 2020 ) ; Li et al . ( 2020c ) ; Amiri et al . ( 2020a ) , and FL under limited communication constraints Amiri et al . ( 2020a ) ; Recht et al . ( 2011 ) ; Alistarh et al . ( 2018 ) . FL with compressed global model transmission has been studied recently in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) aiming to alleviate the communication footprint from the PS to the devices . The global model parameters are relatively skewed/diverse and the efficiency of quantization diminishes significantly when the peak-to-average ratio of the parameters is large . To overcome this , in Caldas et al . ( 2019 ) the PS first employs a linear transform in order to spread the information of the global model vector more evenly among its dimensions , and broadcasts a quantized version of the resultant vector , and the devices apply the inverse linear transform to estimate the global model . We highlight that this approach requires a relatively high computational overhead due to employing the linear transform at the PS and its inverse at the devices , where this overhead grows with the size of the model parameters . Furthermore , the performance evaluation in Caldas et al . ( 2019 ) is limited to the experimental results On the other hand , in Tang et al . ( 2019 ) the PS broadcasts quantized global model with error accumulation to compensate the quantization error . Our contributions With the exception of Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) , the literature on FL considers perfect broadcasting of the global model from the PS to the devices . With this assumption , no matter what type of local update or device-to-PS communication strategy is used , all the devices are synchronized with the same global model at each iteration . In this paper , we instead consider broadcasting a quantized version of the global model update by the PS , which provides the devices with a lossy estimate of the global model ( rather than its accurate estimate ) with which to perform local training . This further reduces the communication cost of FL , which can be particularly limited for transmission over a wireless medium while serving a massive number of devices . Also , it is interesting to investigate the impact of various hyperparameters on the performance of FL with lossy broadcasting of the global model since FL involves transmission over wireless networks with limited bandwidth . We introduce a lossy FL ( LFL ) algorithm , where at each iteration the PS broadcasts a compressed version of the global model update to all the devices through quantization . To be precise , the PS exploits the knowledge of the last global model estimate available at the devices as side information to quantize the global model update . The devices recover an estimate of the current global model by combining the received quantized global model update with their previous estimate , and perform local training using their estimate , and return the local model updates , again employing quantization . The PS updates the global model after receiving the quantized local model updates from the devices . We provide convergence analysis of the LFL algorithm investigating the impact of lossy broadcasting on the performance of FL . Numerical experiments on the MNIST and CIFAR-10 datasets illustrate the efficiency of the proposed LFL algorithm . We observe that the proposed LFL scheme , which leads to a significant communication cost saving , provides a promising performance with no visible gap to the performance of the fully lossless scenario where the communication from both PS-to-device and device-to-PS directions is assumed to be perfect . Also , it is illustrated that the proposed LFL scheme significantly outperforms the schemes introduced in Caldas et al . ( 2019 ) and Tang et al . ( 2019 ) considering compression from the PS to devices . The proposed LFL algorithm differs from the approaches in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) , since we propose broadcasting the global model update , with respect to the previous estimate at the devices , rather than the global model itself . We remark that the global model update has less variability/variance and peak-to-average ratio than the global model ( see Figure 2 ) , and hence , for the same communication load , the devices can have a more accurate estimate of the global model . However , this would require all the devices to track the global model at each iteration , even if they do not participate in the learning process by sending their local update . We argue that broadcasting the global model update to the whole set of devices , rather than a randomly chosen subset , would introduce limited additional communication cost as broadcasting is typically more efficient than sending independent information to devices . Moreover , in practice , the subset of participating devices remain the same for a number of iterations , until a device leaves or joins . Our algorithm can easily be adopted to such scenarios by sending the global model , rather than the model update , every time the subset of devices changes . Also , compared to the approach in Caldas et al . ( 2019 ) , the LFL algorithm requires a significantly smaller computational overhead . Furthermore , unlike Caldas et al . ( 2019 ) , we provide an in-depth convergence analysis of the proposed LFL algorithm . The advantage of the proposed LFL algorithm over the approaches introduced in Caldas et al . ( 2019 ) ; Tang et al . ( 2019 ) is shown numerically , where , despite its significantly smaller communication load , it provides considerably higher accuracy . Notation The set of real numbers is denoted by R. For x ∈ R , \|x\| returns the absolute value of x . For a vector of real numbers x , the largest and the smallest absolute values among all the entries of x are represented by max { \|x\| } and min { \|x\| } , respectively . For an integer i , we let [ i ] , { 1 , 2 , . . . , i } . The l2-norm of vector x is denoted by ‖x‖2 .	In the setting of Federated Learning, the authors propose to quantize both (1) the model send from PS to devices, and (2) the update from device to PS. Although the idea of model-broadcast-compression has appeared in previous work, the authors improve upon previous works in that (1) the authors propose to compress-and-send model updates instead of the model itself, which have smaller variance and peak-to-average ratio and therefore have more effective quantization compression, and (2) do not require pre-processing like linear transformation. The authors further show the effectiveness of their proposed algorithm, LFL, by offering a convergence analysis under appropriate conditions, and offering thorough experimental evaluations.	SP:4b3c0127fcd5cc73226d2b8f58df658f5599791e
Measuring and mitigating interference in reinforcement learning	1 INTRODUCTION . Generalization is a key property of reinforcement learning ( RL ) algorithms with function approximation . An agent must correctly generalize its recent experience to both states it has not yet encountered and other states it encountered in the past . Generalization has been extensively studied in supervised learning , inputs are sampled iid from a fixed input distribution and the targets are sampled from a fixed conditional distribution . The distribution of training data is often not iid . When learning from a stream of temporally correlated data , as in RL , the learner might fit the learned function to recent data and potentially overwrite previous learning—for example , the estimated values . This phenomenon is commonly called interference or forgetting in RL ( Bengio et al. , 2020 ; Goodrich , 2015 ; Liu et al. , 2019 ; Kirkpatrick et al. , 2017 ; Riemer et al. , 2018 ) . The conventional wisdom is that interference is particularly problematic in RL , even single-task RL , because ( a ) when an agent explores , it processes a sequence of observations , which are likely to be temporally correlated ; ( b ) the agent continually changes its policy , changing the distribution of samples over time ; and ( c ) most algorithms use bootstrap targets ( as in temporal difference learning ) , making the update targets non-stationary . It is difficult to verify this conventional wisdom , as there is no established online measure of interference for RL . There has been significant progress quantifying interference in supervised learning ( Chaudhry et al. , 2018 ; Fort et al. , 2019 ; Kemker et al. , 2018 ; Riemer et al. , 2018 ) , with some empirical work even correlating interference and properties of task sequences ( Nguyen et al. , 2019 ) , and investigations into ( un ) forgettable examples in classification ( Toneva et al. , 2019 ) . In RL , recent efforts have focused on generalization and transfer , rather than characterizing or measuring interference . Learning on new environments often results in drops in performance on previously learned environments ( Farebrother et al. , 2018 ; Packer et al. , 2018 ; Rajeswaran et al. , 2017 ; Cobbe et al. , 2018 ) . DQN-based agents can hit performance plateaus in Atari , presumably due to interference . In fact , if the learning process is segmented in the right way , the interference can be more precisely characterized with TD errors across different game contexts ( Fedus et al. , 2020 ) . Unfortunately this analysis can not be done online as learning progresses . Finally , recent work investigated several different possible measures of interference , but did not land on a clear measure ( Bengio et al. , 2020 ) . In this paper we advocate for a simpler approach to charactering interference in RL . In most systems the value estimates and actions change on every time-step conflating many different sources of non-stationarity , stochasticity , and error . If an update to the value function interferes , the result of that updated might not manifest in the policy ’ s performance for several time steps , if at all . Interference classically refers to an update negatively impacting the agent ’ s previous learning—eroding the agent ’ s knowledge stored in the value function . Therefore it makes sense to first characterize interference in the value function updates , instead of the policy or return . We define interference in terms of prediction error for two common approximate dynamic programming algorithms , approximate policy iteration and fitted Q iteration . Most value-based deep RL algorithms are based on these two algorithms . The interference is defined as the change in prediction errors , which is similar to previous definitions of interference in supervised learning . Additionally , our approach yields an online estimate of interference , which can even be directly optimized . In this work , we provide a clear justification for the use of differences in squared TD errors as the definition of interference . We highlight the definitions of interference at different granularities , and the utility of considering different statistics to summarize interference within iterations versus over time . We evaluate our interference measure by computing the correlation to a forgetting metric , which reflects instability in control performance . We show that high interference correlates with forgetting , and simultaneously show interference and forgetting properties across a variety of architectures and optimization choices . We then use our measure to highlight that updates to internal layers of the network—the representation—contribute much less to interference than updates on the last layer . This motivates the design of a new algorithm that learns representation online that explicitly minimize interference . We conclude with a demonstration that this algorithm does indeed significantly improve stability and reduce interference . 2 PROBLEM FORMULATION AND LEARNING ALGORITHMS . In reinforcement learning ( RL ) , an agent interacts with its environment , receiving observations and selecting actions to maximize a reward signal . We assume the environment can be formalized as a Markov decision process ( MDP ) . An MDP is a tuple ( S , A , Pr , R , γ , d0 ) where S is a set of states , A is an set of actions , Pr : S × A × S → [ 0 , 1 ] is the transition probability , R : S × A × S → R is the reward function , γ ∈ [ 0 , 1 ] a discount factor , and d0 is the initial distribution . The goal of the agent is to find a policy π : S ×A → [ 0 , 1 ] to maximize the expected discounted sum of rewards . Given a fixed policy π , the action-value function Qπ : S × A → R is defined as Qπ ( s , a ) : = E [ ∑∞ k=0 γ kRt+k+1\|St = s , At = a ] , where Rt+1 = R ( St , At , St+1 ) , St+1 ∼ Pr ( ·\|St , At ) , and actions are taken according to policy π : At ∼ π ( ·\|St ) . Given a policy π , the value function can be obtained using the Bellman operator for action values T π : R\|S\|×\|A\| → R\|S\|×\|A\| : ( T πQ ) ( s , a ) : = ∑ s′∈S Pr ( s ′\|s , a ) [ R ( s , a , s′ ) + γ ∑ a′∈A π ( a ′\|s′ ) Q ( s′ , a′ ) ] . Qπ is the unique solution of the Bellman equation T πQ = Q . The optimal value function Q∗ is defined as Q∗ ( s , a ) : = supπ Q ( s , a ) , with π ∗ the policy that is greedy w.r.t . Q∗ . Similarly , the optimal value function can be obtained using the Bellman optimality operator for action values T : R\|S\|×\|A\| → R\|S\|×\|A\| : ( T Q ) ( s , a ) : =∑ s′∈S Pr ( s ′\|s , a ) [ R ( s , a , s′ ) + γmaxa′∈AQ ( s′ , a′ ) ] . Q∗ is the unique solution of the Bellman equation T Q = Q . We can use neural networks to learn an approximation Qθ to the optimal action-value , with parameters θ . In this work , we restrict our attention to Iterative Value Estimation algorithms . These are algorithms where there is an explicit evaluation phase with a fixed policy , where the agent has several steps Teval to improve its value estimates . Two examples of such algorithms are Approximate Policy Iteration ( API ) and Fitted Q-Iteration ( FQI ) ( Ernst et al. , 2005 ) . In API , for the current policy πk , the agent updates its estimate of Qπk by taking Teval steps in the environment and performing a mini-batch update from a replay buffer on each step using the Sarsa update : θt+1 ← θt + αδt∇θtQθt ( St , At ) where δt : = Rt+1 + γQθt ( St+1 , πk ( St+1 ) ) −Qθt ( St , At ) . In FQI , the policy and targets Qk are held fixed for Teval steps , with these fixed targets used as a regression target in the update . Again , a mini-batch update update from a replay buffer is used on each step as above , but with a different δ = Rt+1 + γmaxa′∈AQk ( St+1 , a′ ) − Qθt ( St , At ) . The procedure for both algorithms is summarized in Algorithm 1 . Algorithm 1 Iterative Value Estimation : A General Framework for API and FQI Initialize weights θ0 . Initialize an empty buffer of size B. for t← 0 , 1 , 2 , . . . do If t mod Teval = 0 then Qk ← Qθt , update πk to be greedy w.r.t Qk , bk to be -greedy Choose at ∼ bk ( st ) , observe ( st+1 , rt+1 ) , and add the transition to the buffer Sample a mini-batch of transitions Bt from the buffer and update the weights : θt+1 ← θt + α 1\|Bt\| ∑ ( s , a , r , s′ ) ∈Bt δ ( θt ; s , a , r , s ′ ) ∇θQθt ( s , a ) where for API : δ ( θt ; s , a , r , s′ ) = r + γQθt ( s ′ , πk ( s ′ ) ) ) −Qθt ( s , a ) and for FQI : δ ( θt ; s , a , r , s′ ) = r + γmaxa′ Qk ( s′ , a′ ) −Qθt ( s , a ) 3 DEFINING INTERFERENCE IN VALUE ESTIMATION . In this section , we define interference for Iterative Value Estimation algorithms . Because these methods have an evaluation phase which corresponds to one iteration , we can more clearly define interference within one iteration . We discuss interference at four different levels of granularity . Within each iteration—in each evaluation phase—we can ask : did the agent ’ s knowledge about its value estimates improve or degrade ? The evaluation phase is more similar to a standard prediction problem , where the goal is simply to improve the estimates of the action-values towards a clear target . Let f∗ be the target function , which is either f∗ ( s , a ) = Qπk ( s , a ) for API or f∗ ( s , a ) = E [ R+ maxa′ Qk ( S′ , a′ ) \|S = s , A = a ] for FQI . Pointwise Interference At the most fine-grained , we can ask if an update , going from θt to θt+1 , resulted in interference for a specific point ( s , a ) . The change in accuracy at s , a after an update is Accuracy Change ( ( s , a ) , θt , θt+1 ) : = ( f∗ ( s , a ) −Qθt+1 ( s , a ) ) 2 − ( f∗ ( s , a ) −Qθt ( s , a ) ) 2 where if this number is negative it reflects that accuracy improved . This change resulted in interference if it is positive , and zero interference if it is negative , and so we have Pointwise Interference ( ( s , a ) , θt , θt+1 ) : = max ( Accuracy Change ( ( s , a ) , θt , θt+1 ) , 0 ) . Update Interference At a less fine-grained level , we can ask if the update generally improved our accuracy—our knowledge in our value estimates—across points . Update Interference ( θt , θt+1 ) : = max ( E ( S , A ) ∼d [ Accuracy Change ( ( S , A ) , θt , θt+1 ) ] , 0 ) where ( s , a ) are sampled according to distribution d , such as from a buffer of collected experience . Notice that this differs from the expected Pointwise Interference . There are settings where they could produce notably different values . For example , an agent could have high positive and negative Accuracy Change that cancel . The Update Interference reflects that , on average , the agent ’ s knowledge has not changed : it improved in some places , and degraded in others . The expected Pointwise Interference , on the other hand , would be high , because for some points interference was high . We focus first on Update Interference , since it is the coarser measure ; future work is to look in a more fine-grained way at Pointwise Interference . Both Pointwise Interference and Update Interference are about one step . At an even higher level , we can then ask how much interference we have across multiple steps , both within an iteration and across multiple iterations . At this higher level , it becomes more sensible to consider upper percentiles , to ask if there was significant interference within an iteration and across iterations . For this we take expectations over only the top α percentage of values . In finance , this is typically called the expected tail loss or conditional value at risk . Previous work in RL ( Chan et al. , 2020 ) has used conditional value at risk to measure the long-term risk of RL algorithms . Iteration Interference reflects if there was significant interference in updating during the evaluation phase ( an iteration ) . Even a few update steps having significant interference within an iteration could cause significant instability ; an average over the steps might wash out those few significant steps . We therefore define Iteration Interference for iteration k using expectation over the top 10 % of values Iteration Interference ( k ) : = E [ X\|X ≥ Percentile0.9 ( X ) ] for X = Update Interference ( θT , k , θT+1 , k ) where T is the time step in the iteration k , uniformly distributed and Percentile0.9 ( X ) is the 0.9- percentile of the distribution of X . Other percentiles could be considered , where smaller percentiles average over more values and a percentile of 0.5 gives the median . Interference Across Iterations reflects if an agent has many iterations with significant interference . Once again , even a few iterations with significant interference could destabilize learning ; expectations over tails are more suitable than over all iterations . For iteration K a random variable , Interference Across Iterations : = E [ X\|X ≥ Percentile0.9 ( X ) ] for X = Iteration Interference ( K ) These definitions are quite generic , assuming only that we have well-defined targets for the evaluation phase and an algorithm that proceeds in iterations . Though we have only discussed API and FQI , the algorithm DQN also fits well into this class—and so could be analyzed—because the use of target networks mimics FQI . The primary difference is that the policy changes within an iteration . This is not logistically an issue , as updates are off-policy from a replay buffer ; but , it does add an additional changing variable . We therefore focus our experiments on API and FQI .	This paper studies the reason for interference, aka catastrophic forgetting, when using parametric models for Reinforcement Learning. The authors draw the connection with previous methods and introduce some reasonable measure of interference. Then, they introduce a method to explicitly address the problem of interference, showing some good empirical results w.r.t. selected baselines.	SP:1df24e8b97ffad63accec0ed8a3477c211601fd5
Measuring and mitigating interference in reinforcement learning	1 INTRODUCTION . Generalization is a key property of reinforcement learning ( RL ) algorithms with function approximation . An agent must correctly generalize its recent experience to both states it has not yet encountered and other states it encountered in the past . Generalization has been extensively studied in supervised learning , inputs are sampled iid from a fixed input distribution and the targets are sampled from a fixed conditional distribution . The distribution of training data is often not iid . When learning from a stream of temporally correlated data , as in RL , the learner might fit the learned function to recent data and potentially overwrite previous learning—for example , the estimated values . This phenomenon is commonly called interference or forgetting in RL ( Bengio et al. , 2020 ; Goodrich , 2015 ; Liu et al. , 2019 ; Kirkpatrick et al. , 2017 ; Riemer et al. , 2018 ) . The conventional wisdom is that interference is particularly problematic in RL , even single-task RL , because ( a ) when an agent explores , it processes a sequence of observations , which are likely to be temporally correlated ; ( b ) the agent continually changes its policy , changing the distribution of samples over time ; and ( c ) most algorithms use bootstrap targets ( as in temporal difference learning ) , making the update targets non-stationary . It is difficult to verify this conventional wisdom , as there is no established online measure of interference for RL . There has been significant progress quantifying interference in supervised learning ( Chaudhry et al. , 2018 ; Fort et al. , 2019 ; Kemker et al. , 2018 ; Riemer et al. , 2018 ) , with some empirical work even correlating interference and properties of task sequences ( Nguyen et al. , 2019 ) , and investigations into ( un ) forgettable examples in classification ( Toneva et al. , 2019 ) . In RL , recent efforts have focused on generalization and transfer , rather than characterizing or measuring interference . Learning on new environments often results in drops in performance on previously learned environments ( Farebrother et al. , 2018 ; Packer et al. , 2018 ; Rajeswaran et al. , 2017 ; Cobbe et al. , 2018 ) . DQN-based agents can hit performance plateaus in Atari , presumably due to interference . In fact , if the learning process is segmented in the right way , the interference can be more precisely characterized with TD errors across different game contexts ( Fedus et al. , 2020 ) . Unfortunately this analysis can not be done online as learning progresses . Finally , recent work investigated several different possible measures of interference , but did not land on a clear measure ( Bengio et al. , 2020 ) . In this paper we advocate for a simpler approach to charactering interference in RL . In most systems the value estimates and actions change on every time-step conflating many different sources of non-stationarity , stochasticity , and error . If an update to the value function interferes , the result of that updated might not manifest in the policy ’ s performance for several time steps , if at all . Interference classically refers to an update negatively impacting the agent ’ s previous learning—eroding the agent ’ s knowledge stored in the value function . Therefore it makes sense to first characterize interference in the value function updates , instead of the policy or return . We define interference in terms of prediction error for two common approximate dynamic programming algorithms , approximate policy iteration and fitted Q iteration . Most value-based deep RL algorithms are based on these two algorithms . The interference is defined as the change in prediction errors , which is similar to previous definitions of interference in supervised learning . Additionally , our approach yields an online estimate of interference , which can even be directly optimized . In this work , we provide a clear justification for the use of differences in squared TD errors as the definition of interference . We highlight the definitions of interference at different granularities , and the utility of considering different statistics to summarize interference within iterations versus over time . We evaluate our interference measure by computing the correlation to a forgetting metric , which reflects instability in control performance . We show that high interference correlates with forgetting , and simultaneously show interference and forgetting properties across a variety of architectures and optimization choices . We then use our measure to highlight that updates to internal layers of the network—the representation—contribute much less to interference than updates on the last layer . This motivates the design of a new algorithm that learns representation online that explicitly minimize interference . We conclude with a demonstration that this algorithm does indeed significantly improve stability and reduce interference . 2 PROBLEM FORMULATION AND LEARNING ALGORITHMS . In reinforcement learning ( RL ) , an agent interacts with its environment , receiving observations and selecting actions to maximize a reward signal . We assume the environment can be formalized as a Markov decision process ( MDP ) . An MDP is a tuple ( S , A , Pr , R , γ , d0 ) where S is a set of states , A is an set of actions , Pr : S × A × S → [ 0 , 1 ] is the transition probability , R : S × A × S → R is the reward function , γ ∈ [ 0 , 1 ] a discount factor , and d0 is the initial distribution . The goal of the agent is to find a policy π : S ×A → [ 0 , 1 ] to maximize the expected discounted sum of rewards . Given a fixed policy π , the action-value function Qπ : S × A → R is defined as Qπ ( s , a ) : = E [ ∑∞ k=0 γ kRt+k+1\|St = s , At = a ] , where Rt+1 = R ( St , At , St+1 ) , St+1 ∼ Pr ( ·\|St , At ) , and actions are taken according to policy π : At ∼ π ( ·\|St ) . Given a policy π , the value function can be obtained using the Bellman operator for action values T π : R\|S\|×\|A\| → R\|S\|×\|A\| : ( T πQ ) ( s , a ) : = ∑ s′∈S Pr ( s ′\|s , a ) [ R ( s , a , s′ ) + γ ∑ a′∈A π ( a ′\|s′ ) Q ( s′ , a′ ) ] . Qπ is the unique solution of the Bellman equation T πQ = Q . The optimal value function Q∗ is defined as Q∗ ( s , a ) : = supπ Q ( s , a ) , with π ∗ the policy that is greedy w.r.t . Q∗ . Similarly , the optimal value function can be obtained using the Bellman optimality operator for action values T : R\|S\|×\|A\| → R\|S\|×\|A\| : ( T Q ) ( s , a ) : =∑ s′∈S Pr ( s ′\|s , a ) [ R ( s , a , s′ ) + γmaxa′∈AQ ( s′ , a′ ) ] . Q∗ is the unique solution of the Bellman equation T Q = Q . We can use neural networks to learn an approximation Qθ to the optimal action-value , with parameters θ . In this work , we restrict our attention to Iterative Value Estimation algorithms . These are algorithms where there is an explicit evaluation phase with a fixed policy , where the agent has several steps Teval to improve its value estimates . Two examples of such algorithms are Approximate Policy Iteration ( API ) and Fitted Q-Iteration ( FQI ) ( Ernst et al. , 2005 ) . In API , for the current policy πk , the agent updates its estimate of Qπk by taking Teval steps in the environment and performing a mini-batch update from a replay buffer on each step using the Sarsa update : θt+1 ← θt + αδt∇θtQθt ( St , At ) where δt : = Rt+1 + γQθt ( St+1 , πk ( St+1 ) ) −Qθt ( St , At ) . In FQI , the policy and targets Qk are held fixed for Teval steps , with these fixed targets used as a regression target in the update . Again , a mini-batch update update from a replay buffer is used on each step as above , but with a different δ = Rt+1 + γmaxa′∈AQk ( St+1 , a′ ) − Qθt ( St , At ) . The procedure for both algorithms is summarized in Algorithm 1 . Algorithm 1 Iterative Value Estimation : A General Framework for API and FQI Initialize weights θ0 . Initialize an empty buffer of size B. for t← 0 , 1 , 2 , . . . do If t mod Teval = 0 then Qk ← Qθt , update πk to be greedy w.r.t Qk , bk to be -greedy Choose at ∼ bk ( st ) , observe ( st+1 , rt+1 ) , and add the transition to the buffer Sample a mini-batch of transitions Bt from the buffer and update the weights : θt+1 ← θt + α 1\|Bt\| ∑ ( s , a , r , s′ ) ∈Bt δ ( θt ; s , a , r , s ′ ) ∇θQθt ( s , a ) where for API : δ ( θt ; s , a , r , s′ ) = r + γQθt ( s ′ , πk ( s ′ ) ) ) −Qθt ( s , a ) and for FQI : δ ( θt ; s , a , r , s′ ) = r + γmaxa′ Qk ( s′ , a′ ) −Qθt ( s , a ) 3 DEFINING INTERFERENCE IN VALUE ESTIMATION . In this section , we define interference for Iterative Value Estimation algorithms . Because these methods have an evaluation phase which corresponds to one iteration , we can more clearly define interference within one iteration . We discuss interference at four different levels of granularity . Within each iteration—in each evaluation phase—we can ask : did the agent ’ s knowledge about its value estimates improve or degrade ? The evaluation phase is more similar to a standard prediction problem , where the goal is simply to improve the estimates of the action-values towards a clear target . Let f∗ be the target function , which is either f∗ ( s , a ) = Qπk ( s , a ) for API or f∗ ( s , a ) = E [ R+ maxa′ Qk ( S′ , a′ ) \|S = s , A = a ] for FQI . Pointwise Interference At the most fine-grained , we can ask if an update , going from θt to θt+1 , resulted in interference for a specific point ( s , a ) . The change in accuracy at s , a after an update is Accuracy Change ( ( s , a ) , θt , θt+1 ) : = ( f∗ ( s , a ) −Qθt+1 ( s , a ) ) 2 − ( f∗ ( s , a ) −Qθt ( s , a ) ) 2 where if this number is negative it reflects that accuracy improved . This change resulted in interference if it is positive , and zero interference if it is negative , and so we have Pointwise Interference ( ( s , a ) , θt , θt+1 ) : = max ( Accuracy Change ( ( s , a ) , θt , θt+1 ) , 0 ) . Update Interference At a less fine-grained level , we can ask if the update generally improved our accuracy—our knowledge in our value estimates—across points . Update Interference ( θt , θt+1 ) : = max ( E ( S , A ) ∼d [ Accuracy Change ( ( S , A ) , θt , θt+1 ) ] , 0 ) where ( s , a ) are sampled according to distribution d , such as from a buffer of collected experience . Notice that this differs from the expected Pointwise Interference . There are settings where they could produce notably different values . For example , an agent could have high positive and negative Accuracy Change that cancel . The Update Interference reflects that , on average , the agent ’ s knowledge has not changed : it improved in some places , and degraded in others . The expected Pointwise Interference , on the other hand , would be high , because for some points interference was high . We focus first on Update Interference , since it is the coarser measure ; future work is to look in a more fine-grained way at Pointwise Interference . Both Pointwise Interference and Update Interference are about one step . At an even higher level , we can then ask how much interference we have across multiple steps , both within an iteration and across multiple iterations . At this higher level , it becomes more sensible to consider upper percentiles , to ask if there was significant interference within an iteration and across iterations . For this we take expectations over only the top α percentage of values . In finance , this is typically called the expected tail loss or conditional value at risk . Previous work in RL ( Chan et al. , 2020 ) has used conditional value at risk to measure the long-term risk of RL algorithms . Iteration Interference reflects if there was significant interference in updating during the evaluation phase ( an iteration ) . Even a few update steps having significant interference within an iteration could cause significant instability ; an average over the steps might wash out those few significant steps . We therefore define Iteration Interference for iteration k using expectation over the top 10 % of values Iteration Interference ( k ) : = E [ X\|X ≥ Percentile0.9 ( X ) ] for X = Update Interference ( θT , k , θT+1 , k ) where T is the time step in the iteration k , uniformly distributed and Percentile0.9 ( X ) is the 0.9- percentile of the distribution of X . Other percentiles could be considered , where smaller percentiles average over more values and a percentile of 0.5 gives the median . Interference Across Iterations reflects if an agent has many iterations with significant interference . Once again , even a few iterations with significant interference could destabilize learning ; expectations over tails are more suitable than over all iterations . For iteration K a random variable , Interference Across Iterations : = E [ X\|X ≥ Percentile0.9 ( X ) ] for X = Iteration Interference ( K ) These definitions are quite generic , assuming only that we have well-defined targets for the evaluation phase and an algorithm that proceeds in iterations . Though we have only discussed API and FQI , the algorithm DQN also fits well into this class—and so could be analyzed—because the use of target networks mimics FQI . The primary difference is that the policy changes within an iteration . This is not logistically an issue , as updates are off-policy from a replay buffer ; but , it does add an additional changing variable . We therefore focus our experiments on API and FQI .	The paper studies interference and forgetting in the context of reinforcement learning (RL). On the example of the Iterative Value Estimation family of algorithms, the authors define interference as the increase of the true Q target prediction error after updating Q function parameters. Since the true Q target is usually unknown, the authors propose to use the difference of squared TD-errors between updates as a proxy for interference. The paper further defines forgetting as the difference between the current agent performance and the best performance across all previous updates. On CartPole and Acrobot environments, the authors show a positive correlation between the proposed measures of interference and forgetting. They further qualitatively demonstrate that updates of the last layer weights result in higher interference compared to updates of intermediate layers. Finally, the authors propose an algorithm based on meta-learning for learning representations that minimize interference resulting in more stable return plots on Acrobot.	SP:1df24e8b97ffad63accec0ed8a3477c211601fd5
Measuring and mitigating interference in reinforcement learning	1 INTRODUCTION . Generalization is a key property of reinforcement learning ( RL ) algorithms with function approximation . An agent must correctly generalize its recent experience to both states it has not yet encountered and other states it encountered in the past . Generalization has been extensively studied in supervised learning , inputs are sampled iid from a fixed input distribution and the targets are sampled from a fixed conditional distribution . The distribution of training data is often not iid . When learning from a stream of temporally correlated data , as in RL , the learner might fit the learned function to recent data and potentially overwrite previous learning—for example , the estimated values . This phenomenon is commonly called interference or forgetting in RL ( Bengio et al. , 2020 ; Goodrich , 2015 ; Liu et al. , 2019 ; Kirkpatrick et al. , 2017 ; Riemer et al. , 2018 ) . The conventional wisdom is that interference is particularly problematic in RL , even single-task RL , because ( a ) when an agent explores , it processes a sequence of observations , which are likely to be temporally correlated ; ( b ) the agent continually changes its policy , changing the distribution of samples over time ; and ( c ) most algorithms use bootstrap targets ( as in temporal difference learning ) , making the update targets non-stationary . It is difficult to verify this conventional wisdom , as there is no established online measure of interference for RL . There has been significant progress quantifying interference in supervised learning ( Chaudhry et al. , 2018 ; Fort et al. , 2019 ; Kemker et al. , 2018 ; Riemer et al. , 2018 ) , with some empirical work even correlating interference and properties of task sequences ( Nguyen et al. , 2019 ) , and investigations into ( un ) forgettable examples in classification ( Toneva et al. , 2019 ) . In RL , recent efforts have focused on generalization and transfer , rather than characterizing or measuring interference . Learning on new environments often results in drops in performance on previously learned environments ( Farebrother et al. , 2018 ; Packer et al. , 2018 ; Rajeswaran et al. , 2017 ; Cobbe et al. , 2018 ) . DQN-based agents can hit performance plateaus in Atari , presumably due to interference . In fact , if the learning process is segmented in the right way , the interference can be more precisely characterized with TD errors across different game contexts ( Fedus et al. , 2020 ) . Unfortunately this analysis can not be done online as learning progresses . Finally , recent work investigated several different possible measures of interference , but did not land on a clear measure ( Bengio et al. , 2020 ) . In this paper we advocate for a simpler approach to charactering interference in RL . In most systems the value estimates and actions change on every time-step conflating many different sources of non-stationarity , stochasticity , and error . If an update to the value function interferes , the result of that updated might not manifest in the policy ’ s performance for several time steps , if at all . Interference classically refers to an update negatively impacting the agent ’ s previous learning—eroding the agent ’ s knowledge stored in the value function . Therefore it makes sense to first characterize interference in the value function updates , instead of the policy or return . We define interference in terms of prediction error for two common approximate dynamic programming algorithms , approximate policy iteration and fitted Q iteration . Most value-based deep RL algorithms are based on these two algorithms . The interference is defined as the change in prediction errors , which is similar to previous definitions of interference in supervised learning . Additionally , our approach yields an online estimate of interference , which can even be directly optimized . In this work , we provide a clear justification for the use of differences in squared TD errors as the definition of interference . We highlight the definitions of interference at different granularities , and the utility of considering different statistics to summarize interference within iterations versus over time . We evaluate our interference measure by computing the correlation to a forgetting metric , which reflects instability in control performance . We show that high interference correlates with forgetting , and simultaneously show interference and forgetting properties across a variety of architectures and optimization choices . We then use our measure to highlight that updates to internal layers of the network—the representation—contribute much less to interference than updates on the last layer . This motivates the design of a new algorithm that learns representation online that explicitly minimize interference . We conclude with a demonstration that this algorithm does indeed significantly improve stability and reduce interference . 2 PROBLEM FORMULATION AND LEARNING ALGORITHMS . In reinforcement learning ( RL ) , an agent interacts with its environment , receiving observations and selecting actions to maximize a reward signal . We assume the environment can be formalized as a Markov decision process ( MDP ) . An MDP is a tuple ( S , A , Pr , R , γ , d0 ) where S is a set of states , A is an set of actions , Pr : S × A × S → [ 0 , 1 ] is the transition probability , R : S × A × S → R is the reward function , γ ∈ [ 0 , 1 ] a discount factor , and d0 is the initial distribution . The goal of the agent is to find a policy π : S ×A → [ 0 , 1 ] to maximize the expected discounted sum of rewards . Given a fixed policy π , the action-value function Qπ : S × A → R is defined as Qπ ( s , a ) : = E [ ∑∞ k=0 γ kRt+k+1\|St = s , At = a ] , where Rt+1 = R ( St , At , St+1 ) , St+1 ∼ Pr ( ·\|St , At ) , and actions are taken according to policy π : At ∼ π ( ·\|St ) . Given a policy π , the value function can be obtained using the Bellman operator for action values T π : R\|S\|×\|A\| → R\|S\|×\|A\| : ( T πQ ) ( s , a ) : = ∑ s′∈S Pr ( s ′\|s , a ) [ R ( s , a , s′ ) + γ ∑ a′∈A π ( a ′\|s′ ) Q ( s′ , a′ ) ] . Qπ is the unique solution of the Bellman equation T πQ = Q . The optimal value function Q∗ is defined as Q∗ ( s , a ) : = supπ Q ( s , a ) , with π ∗ the policy that is greedy w.r.t . Q∗ . Similarly , the optimal value function can be obtained using the Bellman optimality operator for action values T : R\|S\|×\|A\| → R\|S\|×\|A\| : ( T Q ) ( s , a ) : =∑ s′∈S Pr ( s ′\|s , a ) [ R ( s , a , s′ ) + γmaxa′∈AQ ( s′ , a′ ) ] . Q∗ is the unique solution of the Bellman equation T Q = Q . We can use neural networks to learn an approximation Qθ to the optimal action-value , with parameters θ . In this work , we restrict our attention to Iterative Value Estimation algorithms . These are algorithms where there is an explicit evaluation phase with a fixed policy , where the agent has several steps Teval to improve its value estimates . Two examples of such algorithms are Approximate Policy Iteration ( API ) and Fitted Q-Iteration ( FQI ) ( Ernst et al. , 2005 ) . In API , for the current policy πk , the agent updates its estimate of Qπk by taking Teval steps in the environment and performing a mini-batch update from a replay buffer on each step using the Sarsa update : θt+1 ← θt + αδt∇θtQθt ( St , At ) where δt : = Rt+1 + γQθt ( St+1 , πk ( St+1 ) ) −Qθt ( St , At ) . In FQI , the policy and targets Qk are held fixed for Teval steps , with these fixed targets used as a regression target in the update . Again , a mini-batch update update from a replay buffer is used on each step as above , but with a different δ = Rt+1 + γmaxa′∈AQk ( St+1 , a′ ) − Qθt ( St , At ) . The procedure for both algorithms is summarized in Algorithm 1 . Algorithm 1 Iterative Value Estimation : A General Framework for API and FQI Initialize weights θ0 . Initialize an empty buffer of size B. for t← 0 , 1 , 2 , . . . do If t mod Teval = 0 then Qk ← Qθt , update πk to be greedy w.r.t Qk , bk to be -greedy Choose at ∼ bk ( st ) , observe ( st+1 , rt+1 ) , and add the transition to the buffer Sample a mini-batch of transitions Bt from the buffer and update the weights : θt+1 ← θt + α 1\|Bt\| ∑ ( s , a , r , s′ ) ∈Bt δ ( θt ; s , a , r , s ′ ) ∇θQθt ( s , a ) where for API : δ ( θt ; s , a , r , s′ ) = r + γQθt ( s ′ , πk ( s ′ ) ) ) −Qθt ( s , a ) and for FQI : δ ( θt ; s , a , r , s′ ) = r + γmaxa′ Qk ( s′ , a′ ) −Qθt ( s , a ) 3 DEFINING INTERFERENCE IN VALUE ESTIMATION . In this section , we define interference for Iterative Value Estimation algorithms . Because these methods have an evaluation phase which corresponds to one iteration , we can more clearly define interference within one iteration . We discuss interference at four different levels of granularity . Within each iteration—in each evaluation phase—we can ask : did the agent ’ s knowledge about its value estimates improve or degrade ? The evaluation phase is more similar to a standard prediction problem , where the goal is simply to improve the estimates of the action-values towards a clear target . Let f∗ be the target function , which is either f∗ ( s , a ) = Qπk ( s , a ) for API or f∗ ( s , a ) = E [ R+ maxa′ Qk ( S′ , a′ ) \|S = s , A = a ] for FQI . Pointwise Interference At the most fine-grained , we can ask if an update , going from θt to θt+1 , resulted in interference for a specific point ( s , a ) . The change in accuracy at s , a after an update is Accuracy Change ( ( s , a ) , θt , θt+1 ) : = ( f∗ ( s , a ) −Qθt+1 ( s , a ) ) 2 − ( f∗ ( s , a ) −Qθt ( s , a ) ) 2 where if this number is negative it reflects that accuracy improved . This change resulted in interference if it is positive , and zero interference if it is negative , and so we have Pointwise Interference ( ( s , a ) , θt , θt+1 ) : = max ( Accuracy Change ( ( s , a ) , θt , θt+1 ) , 0 ) . Update Interference At a less fine-grained level , we can ask if the update generally improved our accuracy—our knowledge in our value estimates—across points . Update Interference ( θt , θt+1 ) : = max ( E ( S , A ) ∼d [ Accuracy Change ( ( S , A ) , θt , θt+1 ) ] , 0 ) where ( s , a ) are sampled according to distribution d , such as from a buffer of collected experience . Notice that this differs from the expected Pointwise Interference . There are settings where they could produce notably different values . For example , an agent could have high positive and negative Accuracy Change that cancel . The Update Interference reflects that , on average , the agent ’ s knowledge has not changed : it improved in some places , and degraded in others . The expected Pointwise Interference , on the other hand , would be high , because for some points interference was high . We focus first on Update Interference , since it is the coarser measure ; future work is to look in a more fine-grained way at Pointwise Interference . Both Pointwise Interference and Update Interference are about one step . At an even higher level , we can then ask how much interference we have across multiple steps , both within an iteration and across multiple iterations . At this higher level , it becomes more sensible to consider upper percentiles , to ask if there was significant interference within an iteration and across iterations . For this we take expectations over only the top α percentage of values . In finance , this is typically called the expected tail loss or conditional value at risk . Previous work in RL ( Chan et al. , 2020 ) has used conditional value at risk to measure the long-term risk of RL algorithms . Iteration Interference reflects if there was significant interference in updating during the evaluation phase ( an iteration ) . Even a few update steps having significant interference within an iteration could cause significant instability ; an average over the steps might wash out those few significant steps . We therefore define Iteration Interference for iteration k using expectation over the top 10 % of values Iteration Interference ( k ) : = E [ X\|X ≥ Percentile0.9 ( X ) ] for X = Update Interference ( θT , k , θT+1 , k ) where T is the time step in the iteration k , uniformly distributed and Percentile0.9 ( X ) is the 0.9- percentile of the distribution of X . Other percentiles could be considered , where smaller percentiles average over more values and a percentile of 0.5 gives the median . Interference Across Iterations reflects if an agent has many iterations with significant interference . Once again , even a few iterations with significant interference could destabilize learning ; expectations over tails are more suitable than over all iterations . For iteration K a random variable , Interference Across Iterations : = E [ X\|X ≥ Percentile0.9 ( X ) ] for X = Iteration Interference ( K ) These definitions are quite generic , assuming only that we have well-defined targets for the evaluation phase and an algorithm that proceeds in iterations . Though we have only discussed API and FQI , the algorithm DQN also fits well into this class—and so could be analyzed—because the use of target networks mimics FQI . The primary difference is that the policy changes within an iteration . This is not logistically an issue , as updates are off-policy from a replay buffer ; but , it does add an additional changing variable . We therefore focus our experiments on API and FQI .	This paper studies the the interference problem under the API and FPI setting. It designs new measure for interference, and shows that the interference measure is correlated with forgetting. With the help of the interference measure, the paper studies the importance of the final layer of the neural network and proposes a new algorithm to mitigate interference.	SP:1df24e8b97ffad63accec0ed8a3477c211601fd5
Learning and Generalization in Univariate Overparameterized Normalizing Flows	1 INTRODUCTION . Neural network models trained using simple first-order iterative algorithms have been very effective in both supervised and unsupervised learning . Theoretical reasoning of this phenomenon requires one to consider simple but quintessential formulations , where this can be demonstrated by mathematical proof , along with experimental evidence for the underlying intuition . First , the minimization of training loss is typically a non-smooth and non-convex optimization over the parameters of neural networks , so it is surprising that neural networks can be trained efficiently by first-order iterative algorithms . Second , even large neural networks whose number parameters are more than the size of training data often generalize well with a small loss on the unseen test data , instead of overfitting the seen training data . Recent work in supervised learning attempts to provide theoretical justification for why overparameterized neural networks can train and generalize efficiently in the above sense . In supervised learning , the empirical risk minimization with quadratic loss is a non-convex optimization problem even for a fully connected neural network with one hidden layer of neurons with ReLU activations . Around 2018 , it was realized that when the hidden layer size is large compared to the dataset size or compared to some measure of complexity of the data , one can provably show efficient training and generalization for these networks , e.g . Jacot et al . ( 2018 ) ; Li & Liang ( 2018 ) ; Du et al . ( 2018 ) ; Allen-Zhu et al . ( 2019 ) ; Arora et al . ( 2019 ) . Of these , Allen-Zhu et al . ( 2019 ) is directly relevant to our paper and will be discussed later . The role of overparameterization , and provable training and generalization guarantees for neural networks are less well understood in unsupervised learning . Generative models or learning a data distribution from given samples is an important problem in unsupervised learning . Popular generative models based on neural networks include Generative Adversarial Networks ( GANs ) ( e.g. , Goodfellow et al . ( 2014 ) ) , Variational AutoEncoders ( VAEs ) ( e.g. , Kingma & Welling ( 2014 ) ) , and Normalizing Flows ( e.g. , Rezende & Mohamed ( 2015 ) ) . GANs and VAEs have shown impressive capability to generate samples of photo-realistic images but they can not give probability density estimates for new data points . Training of GANs and VAEs has various additional challenges such as mode collapse , posterior collapse , vanishing gradients , training instability , etc . as shown in e.g . Bowman et al . ( 2016 ) ; Salimans et al . ( 2016 ) ; Arora et al . ( 2018 ) ; Lucic et al . ( 2018 ) . In contrast to the generative models such as GANs and VAEs , when normalizing flows learn distributions , they can do both sampling and density estimation , leading to wide-ranging applications as mentioned in the surveys by Kobyzev et al . ( 2020 ) and Papamakarios et al . ( 2019 ) . Theoretical understanding of learning and generalization in normalizing flows ( more generally , generative models and unsupervised learning ) is a natural and important open question , and our main technical contribution is to extend known techniques from supervised learning to make progress towards answering this question . In this paper , we study learning and generalization in the case of univariate overparameterized normalizing flows . Restriction to the univariate case is technically non-trivial and interesting in its own right : univariate ReLU networks have been studied in recent supervised learning literature ( e.g. , Savarese et al . ( 2019 ) , Williams et al . ( 2019 ) , Sahs et al . ( 2020 ) and Daubechies et al . ( 2019 ) ) . Multidimensional flows are qualitatively more complex and our 1D analysis sheds some light on them ( see Sec . 4 ) . Before stating our contributions , we briefly introduce normalizing flows ; details appear in Section 2 . Normalizing Flows . We work with one-dimensional probability distributions with continuous density . The general idea behind normalizing flows ( NFs ) , restricted to 1D can be summarized as follows : Let X ∈ R be a random variable denoting the data distribution . We also fix a base distribution with associated random variable Z which is typically standard Gaussian , though in this paper we will work with the exponential distribution as well . Given i.i.d . samples of X , the goal is to learn a continuous strictly monotone increasing map fX : R→ R that transports the distribution of X to the distribution of Z : in other words , the distribution of f−1X ( Z ) is that of X . The learning of fX is done by representing it by a neural network and setting up an appropriate loss function . The monotonicity requirement on f which makes f invertible , while not essential , greatly simplifies the problem and is present in all the works we are aware of . It is not clear how to set up a tractable optimization problem without this requirement . Since the function represented by standard neural networks are not necessarily monotone , the design of the neural net is altered to make it monotone . For our 1D situation , one-hidden layer networks are of the form N ( x ) = ∑m i=1 aiσ ( wix+ bi ) , where m is the size of the hidden layer and the ai , wi , bi are the parameters of the network . We will assume that the activation functions used are monotone . Here we distinguish between two such alterations : ( 1 ) Changing the parametrization of the neural network . This can be done in multiple ways : instead of ai , wi we use a2i , w 2 i ( or other functions , such as the exponential function , of ai , wi that take on only positive values ) ( Huang et al. , 2018 ; Cao et al. , 2019 ) . This approach appears to be the most popular . In this paper , we also suggest another related alteration : we simply restrict the parameters ai , wi to be positive . This is achieved by enforcing this constraint during training . ( 2 ) Instead of using N ( x ) for f ( x ) we use φ ( N ( x ) ) for f ′ ( x ) = dfdx , where φ : R→ R + takes on only positive values . Positivity of f ′ implies monotonicity of f . Note that no restrictions on the parameters are required ; however , because we parametrize f ′ , the function f needs to be reconstructed using numerical quadrature . This approach is used by Wehenkel & Louppe ( 2019 ) . We will refer to the models in the first class as constrained normalizing flows ( CNFs ) and those in the second class as unconstrained normalizing flows ( UNFs ) . Our Contributions . In this paper , we study both constrained and unconstrained univariate NFs theoretically as well as empirically . The existing analyses for overparametrized neural networks in the supervised setting work with a linear approximation of the neural network , termed pseudo network in Allen-Zhu et al . ( 2019 ) . They show that ( 1 ) there is a pseudo network with weights close to the initial ones approximating the target function , ( 2 ) the loss surfaces of the neural network and the pseudo network are close and moreover the latter is convex for convex loss functions . This allows for proof of the convergence of the training of neural network to global optima . One can try to adapt the approach of using a linear approximation of the neural network to analyze training of NFs . However , one immediately encounters some new roadblocks : the loss surface of the pseudo networks is non-convex in both CNFs and UNFs . In both cases , we identify novel variations that make the optimization problem for associated pseudo network convex : For CNFs , instead of using a2i , w 2 i as parameters , we simply impose the constraints ai ≥ and wi ≥ for some small constant . The optimization algorithm now is projected SGD , which in this case incurs essentially no extra cost over SGD due to the simplicity of the positivity constraints . Apart from making the optimization problem convex , in experiments this variation slightly improves the training of NFs compared to the reparametrization approaches , and may be useful in practical settings . Similarly , for UNFs we identify two changes from the model of Wehenkel & Louppe ( 2019 ) that make the associated optimization problem convex , while still retaining empirical effectiveness : ( 1 ) Instead of Clenshaw–Curtis quadrature employed in Wehenkel & Louppe ( 2019 ) which uses positive and negative coefficients , we use the simple rectangle quadrature which uses only positive coefficients . This change makes the model somewhat slow ( it uses twice as many samples and time to get similar performance on the examples we tried ) . ( 2 ) Instead of the standard Gaussian distribution as the base distribution , we use the exponential distribution . In experiments , this does not cause much change . Our results point to a dichotomy between these two classes of NFs : our variant of UNFs can be theoretically analyzed when the networks are overparametrized to prove that the UNF indeed learns the data distribution . To our knowledge , this is the first “ end-to-end ” analysis of an NF model , and a neural generative model using gradient-based algorithms used in practice . This proof , while following the high-level scheme of Allen-Zhu et al . ( 2019 ) proof , has a number of differences , conceptual as well as technical , due to different settings . E.g. , our loss function involves a function and its integral estimated by quadrature . On the other hand , for CNFs , our empirical and theoretical findings provide evidence that overparametrization makes training slower to the extent that models of similar size which learn the data distribution well for UNFs , fail to do so for CNFs . We also analyze CNFs theoretically in the overparametrized setting and point to potential sources of the difficulty . The case of moderatesized networks , where training and generalization do take place empirically , is likely to be difficult to analyze theoretically as presently this setting is open for the simpler supervised learning case . We hope that our results will pave the way for further progress . We make some remarks on the multidimensional case in Sec . 4 . In summary , our contributions include : • To our knowledge , first efficient training and generalization proof for NFs ( in 1D ) . • Identification of architectural variants of UNFs that admit analysis via overparametrization . • Identification of “ barriers ” to the analysis of CNFs . Related Work . Previous work on normalizing flows has studied different variants such as planar and radial flows in Rezende & Mohamed ( 2015 ) , Sylvester flow in van den Berg et al . ( 2018 ) , Householder flow in Tomczak & Welling ( 2016 ) , masked autoregressive flow in Papamakarios et al . ( 2017 ) . Most variants of normalizing flows are specific to certain applications , and the expressive power ( i.e. , which base and data distributions they can map between ) and complexity of normalizing flow models have been studied recently , e.g . Kong & Chaudhuri ( 2020 ) and Teshima et al . ( 2020 ) . Invertible transformations defined by monotonic neural networks can be combined into autoregressive flows that are universal density approximators of continuous probability distributions ; see Masked Autoregressive Flows ( MAF ) Papamakarios et al . ( 2017 ) , UNMM-MAF by Wehenkel & Louppe ( 2019 ) , Neural Autoregressive Flows ( NAF ) by Huang et al . ( 2018 ) , Block Neural Autoregressive Flow ( B-NAF ) by Cao et al . ( 2019 ) . Unconstrained Monotonic Neural Network ( UMNN ) models proposed by Wehenkel & Louppe ( 2019 ) are particularly relevant to the technical part of our paper . Lei et al . ( 2020 ) show that when the generator is a two-layer tanh , sigmoid or leaky ReLU network , Wasserstein GAN trained with stochastic gradient descent-ascent converges to a global solution with polynomial time and sample complexity . Using the moments method and a learning algorithm motivated by tensor decomposition , Li & Dou ( 2020 ) show that GANs can efficiently learn a large class of distributions including those generated by two-layer networks . Nguyen et al . ( 2019b ) show that two-layer autoencoders with ReLU or threshold activations can be trained with normalized gradient descent over the reconstruction loss to provably learn the parameters of any generative bilinear model ( e.g. , mixture of Gaussians , sparse coding model ) . Nguyen et al . ( 2019a ) extend the work of Du et al . ( 2018 ) on supervised learning mentioned earlier to study weakly-trained ( i.e. , only encoder is trained ) and jointly-trained ( i.e. , both encoder and decoder are trained ) two-layer autoencoders , and show joint training requires less overparameterization and converges to a global optimum . The effect of overparameterization in unsupervised learning has also been of recent interest . Buhai et al . ( 2020 ) do an empirical study to show that across a variety of latent variable models and training algorithms , overparameterization can significantly increase the number of recovered ground truth latent variables . Radhakrishnan et al . ( 2020 ) show that overparameterized autoencoders and sequence encoders essentially implement associative memory by storing training samples as attractors in a dynamical system . Outline . A brief outline of our paper is as follows . Section 2 contains preliminaries and an overview of our results about constrained and unconstrained normalizing flows . Appendix B shows the existence of a pseudo network whose loss closely approximates the loss of the target function . Appendix C shows the coupling or closeness of their gradients over random initialization . Appendices D and E contain complete proofs of our optimization and generalization results , respectively . Section 3 and Appendix G contain our empirical studies towards validating our theoretical results .	This paper proved that for a certain modified version of sufficiently-overparametrized univariate normalizing flows where the underlying neural network has only one hidden layer, with high probability it can learn a distribution that is close enough to the target distribution where the distance can be measured in, e.g., KL divergence. The width of the network, number of samples, and the number of quadrature points are required to be at least polynomial in inverse of error rate and complexity measure of the target distribution. The authors also provided theoretical evidence and did experiments on synthetic Gaussian mixture datasets to show that another variation of the normalizing flow model does not benefit from overparametrization under this one-hidden-layer univariate setting.	SP:8180451eca79f017df91f0066ee46d8b149eaa48
Learning and Generalization in Univariate Overparameterized Normalizing Flows	1 INTRODUCTION . Neural network models trained using simple first-order iterative algorithms have been very effective in both supervised and unsupervised learning . Theoretical reasoning of this phenomenon requires one to consider simple but quintessential formulations , where this can be demonstrated by mathematical proof , along with experimental evidence for the underlying intuition . First , the minimization of training loss is typically a non-smooth and non-convex optimization over the parameters of neural networks , so it is surprising that neural networks can be trained efficiently by first-order iterative algorithms . Second , even large neural networks whose number parameters are more than the size of training data often generalize well with a small loss on the unseen test data , instead of overfitting the seen training data . Recent work in supervised learning attempts to provide theoretical justification for why overparameterized neural networks can train and generalize efficiently in the above sense . In supervised learning , the empirical risk minimization with quadratic loss is a non-convex optimization problem even for a fully connected neural network with one hidden layer of neurons with ReLU activations . Around 2018 , it was realized that when the hidden layer size is large compared to the dataset size or compared to some measure of complexity of the data , one can provably show efficient training and generalization for these networks , e.g . Jacot et al . ( 2018 ) ; Li & Liang ( 2018 ) ; Du et al . ( 2018 ) ; Allen-Zhu et al . ( 2019 ) ; Arora et al . ( 2019 ) . Of these , Allen-Zhu et al . ( 2019 ) is directly relevant to our paper and will be discussed later . The role of overparameterization , and provable training and generalization guarantees for neural networks are less well understood in unsupervised learning . Generative models or learning a data distribution from given samples is an important problem in unsupervised learning . Popular generative models based on neural networks include Generative Adversarial Networks ( GANs ) ( e.g. , Goodfellow et al . ( 2014 ) ) , Variational AutoEncoders ( VAEs ) ( e.g. , Kingma & Welling ( 2014 ) ) , and Normalizing Flows ( e.g. , Rezende & Mohamed ( 2015 ) ) . GANs and VAEs have shown impressive capability to generate samples of photo-realistic images but they can not give probability density estimates for new data points . Training of GANs and VAEs has various additional challenges such as mode collapse , posterior collapse , vanishing gradients , training instability , etc . as shown in e.g . Bowman et al . ( 2016 ) ; Salimans et al . ( 2016 ) ; Arora et al . ( 2018 ) ; Lucic et al . ( 2018 ) . In contrast to the generative models such as GANs and VAEs , when normalizing flows learn distributions , they can do both sampling and density estimation , leading to wide-ranging applications as mentioned in the surveys by Kobyzev et al . ( 2020 ) and Papamakarios et al . ( 2019 ) . Theoretical understanding of learning and generalization in normalizing flows ( more generally , generative models and unsupervised learning ) is a natural and important open question , and our main technical contribution is to extend known techniques from supervised learning to make progress towards answering this question . In this paper , we study learning and generalization in the case of univariate overparameterized normalizing flows . Restriction to the univariate case is technically non-trivial and interesting in its own right : univariate ReLU networks have been studied in recent supervised learning literature ( e.g. , Savarese et al . ( 2019 ) , Williams et al . ( 2019 ) , Sahs et al . ( 2020 ) and Daubechies et al . ( 2019 ) ) . Multidimensional flows are qualitatively more complex and our 1D analysis sheds some light on them ( see Sec . 4 ) . Before stating our contributions , we briefly introduce normalizing flows ; details appear in Section 2 . Normalizing Flows . We work with one-dimensional probability distributions with continuous density . The general idea behind normalizing flows ( NFs ) , restricted to 1D can be summarized as follows : Let X ∈ R be a random variable denoting the data distribution . We also fix a base distribution with associated random variable Z which is typically standard Gaussian , though in this paper we will work with the exponential distribution as well . Given i.i.d . samples of X , the goal is to learn a continuous strictly monotone increasing map fX : R→ R that transports the distribution of X to the distribution of Z : in other words , the distribution of f−1X ( Z ) is that of X . The learning of fX is done by representing it by a neural network and setting up an appropriate loss function . The monotonicity requirement on f which makes f invertible , while not essential , greatly simplifies the problem and is present in all the works we are aware of . It is not clear how to set up a tractable optimization problem without this requirement . Since the function represented by standard neural networks are not necessarily monotone , the design of the neural net is altered to make it monotone . For our 1D situation , one-hidden layer networks are of the form N ( x ) = ∑m i=1 aiσ ( wix+ bi ) , where m is the size of the hidden layer and the ai , wi , bi are the parameters of the network . We will assume that the activation functions used are monotone . Here we distinguish between two such alterations : ( 1 ) Changing the parametrization of the neural network . This can be done in multiple ways : instead of ai , wi we use a2i , w 2 i ( or other functions , such as the exponential function , of ai , wi that take on only positive values ) ( Huang et al. , 2018 ; Cao et al. , 2019 ) . This approach appears to be the most popular . In this paper , we also suggest another related alteration : we simply restrict the parameters ai , wi to be positive . This is achieved by enforcing this constraint during training . ( 2 ) Instead of using N ( x ) for f ( x ) we use φ ( N ( x ) ) for f ′ ( x ) = dfdx , where φ : R→ R + takes on only positive values . Positivity of f ′ implies monotonicity of f . Note that no restrictions on the parameters are required ; however , because we parametrize f ′ , the function f needs to be reconstructed using numerical quadrature . This approach is used by Wehenkel & Louppe ( 2019 ) . We will refer to the models in the first class as constrained normalizing flows ( CNFs ) and those in the second class as unconstrained normalizing flows ( UNFs ) . Our Contributions . In this paper , we study both constrained and unconstrained univariate NFs theoretically as well as empirically . The existing analyses for overparametrized neural networks in the supervised setting work with a linear approximation of the neural network , termed pseudo network in Allen-Zhu et al . ( 2019 ) . They show that ( 1 ) there is a pseudo network with weights close to the initial ones approximating the target function , ( 2 ) the loss surfaces of the neural network and the pseudo network are close and moreover the latter is convex for convex loss functions . This allows for proof of the convergence of the training of neural network to global optima . One can try to adapt the approach of using a linear approximation of the neural network to analyze training of NFs . However , one immediately encounters some new roadblocks : the loss surface of the pseudo networks is non-convex in both CNFs and UNFs . In both cases , we identify novel variations that make the optimization problem for associated pseudo network convex : For CNFs , instead of using a2i , w 2 i as parameters , we simply impose the constraints ai ≥ and wi ≥ for some small constant . The optimization algorithm now is projected SGD , which in this case incurs essentially no extra cost over SGD due to the simplicity of the positivity constraints . Apart from making the optimization problem convex , in experiments this variation slightly improves the training of NFs compared to the reparametrization approaches , and may be useful in practical settings . Similarly , for UNFs we identify two changes from the model of Wehenkel & Louppe ( 2019 ) that make the associated optimization problem convex , while still retaining empirical effectiveness : ( 1 ) Instead of Clenshaw–Curtis quadrature employed in Wehenkel & Louppe ( 2019 ) which uses positive and negative coefficients , we use the simple rectangle quadrature which uses only positive coefficients . This change makes the model somewhat slow ( it uses twice as many samples and time to get similar performance on the examples we tried ) . ( 2 ) Instead of the standard Gaussian distribution as the base distribution , we use the exponential distribution . In experiments , this does not cause much change . Our results point to a dichotomy between these two classes of NFs : our variant of UNFs can be theoretically analyzed when the networks are overparametrized to prove that the UNF indeed learns the data distribution . To our knowledge , this is the first “ end-to-end ” analysis of an NF model , and a neural generative model using gradient-based algorithms used in practice . This proof , while following the high-level scheme of Allen-Zhu et al . ( 2019 ) proof , has a number of differences , conceptual as well as technical , due to different settings . E.g. , our loss function involves a function and its integral estimated by quadrature . On the other hand , for CNFs , our empirical and theoretical findings provide evidence that overparametrization makes training slower to the extent that models of similar size which learn the data distribution well for UNFs , fail to do so for CNFs . We also analyze CNFs theoretically in the overparametrized setting and point to potential sources of the difficulty . The case of moderatesized networks , where training and generalization do take place empirically , is likely to be difficult to analyze theoretically as presently this setting is open for the simpler supervised learning case . We hope that our results will pave the way for further progress . We make some remarks on the multidimensional case in Sec . 4 . In summary , our contributions include : • To our knowledge , first efficient training and generalization proof for NFs ( in 1D ) . • Identification of architectural variants of UNFs that admit analysis via overparametrization . • Identification of “ barriers ” to the analysis of CNFs . Related Work . Previous work on normalizing flows has studied different variants such as planar and radial flows in Rezende & Mohamed ( 2015 ) , Sylvester flow in van den Berg et al . ( 2018 ) , Householder flow in Tomczak & Welling ( 2016 ) , masked autoregressive flow in Papamakarios et al . ( 2017 ) . Most variants of normalizing flows are specific to certain applications , and the expressive power ( i.e. , which base and data distributions they can map between ) and complexity of normalizing flow models have been studied recently , e.g . Kong & Chaudhuri ( 2020 ) and Teshima et al . ( 2020 ) . Invertible transformations defined by monotonic neural networks can be combined into autoregressive flows that are universal density approximators of continuous probability distributions ; see Masked Autoregressive Flows ( MAF ) Papamakarios et al . ( 2017 ) , UNMM-MAF by Wehenkel & Louppe ( 2019 ) , Neural Autoregressive Flows ( NAF ) by Huang et al . ( 2018 ) , Block Neural Autoregressive Flow ( B-NAF ) by Cao et al . ( 2019 ) . Unconstrained Monotonic Neural Network ( UMNN ) models proposed by Wehenkel & Louppe ( 2019 ) are particularly relevant to the technical part of our paper . Lei et al . ( 2020 ) show that when the generator is a two-layer tanh , sigmoid or leaky ReLU network , Wasserstein GAN trained with stochastic gradient descent-ascent converges to a global solution with polynomial time and sample complexity . Using the moments method and a learning algorithm motivated by tensor decomposition , Li & Dou ( 2020 ) show that GANs can efficiently learn a large class of distributions including those generated by two-layer networks . Nguyen et al . ( 2019b ) show that two-layer autoencoders with ReLU or threshold activations can be trained with normalized gradient descent over the reconstruction loss to provably learn the parameters of any generative bilinear model ( e.g. , mixture of Gaussians , sparse coding model ) . Nguyen et al . ( 2019a ) extend the work of Du et al . ( 2018 ) on supervised learning mentioned earlier to study weakly-trained ( i.e. , only encoder is trained ) and jointly-trained ( i.e. , both encoder and decoder are trained ) two-layer autoencoders , and show joint training requires less overparameterization and converges to a global optimum . The effect of overparameterization in unsupervised learning has also been of recent interest . Buhai et al . ( 2020 ) do an empirical study to show that across a variety of latent variable models and training algorithms , overparameterization can significantly increase the number of recovered ground truth latent variables . Radhakrishnan et al . ( 2020 ) show that overparameterized autoencoders and sequence encoders essentially implement associative memory by storing training samples as attractors in a dynamical system . Outline . A brief outline of our paper is as follows . Section 2 contains preliminaries and an overview of our results about constrained and unconstrained normalizing flows . Appendix B shows the existence of a pseudo network whose loss closely approximates the loss of the target function . Appendix C shows the coupling or closeness of their gradients over random initialization . Appendices D and E contain complete proofs of our optimization and generalization results , respectively . Section 3 and Appendix G contain our empirical studies towards validating our theoretical results .	This paper studies overparameterization over unsupervised learning. In detail, it uses constrained normalizing flows (CNF) and unconstrained normalizing flows (UNF) to learn the underlying unknown one-dimensional distribution, which can be parameterized by a two-layer neural network. The authors propose theoretical results for UNF and suggest that by selecting wide enough neural networks, a great number of random samples and number of quadrature points, a two-layer neural network is able to learn the true UNF up to small error. Experiment results are presented for both CNF and UNF, which back up their claim.	SP:8180451eca79f017df91f0066ee46d8b149eaa48
Learning and Generalization in Univariate Overparameterized Normalizing Flows	1 INTRODUCTION . Neural network models trained using simple first-order iterative algorithms have been very effective in both supervised and unsupervised learning . Theoretical reasoning of this phenomenon requires one to consider simple but quintessential formulations , where this can be demonstrated by mathematical proof , along with experimental evidence for the underlying intuition . First , the minimization of training loss is typically a non-smooth and non-convex optimization over the parameters of neural networks , so it is surprising that neural networks can be trained efficiently by first-order iterative algorithms . Second , even large neural networks whose number parameters are more than the size of training data often generalize well with a small loss on the unseen test data , instead of overfitting the seen training data . Recent work in supervised learning attempts to provide theoretical justification for why overparameterized neural networks can train and generalize efficiently in the above sense . In supervised learning , the empirical risk minimization with quadratic loss is a non-convex optimization problem even for a fully connected neural network with one hidden layer of neurons with ReLU activations . Around 2018 , it was realized that when the hidden layer size is large compared to the dataset size or compared to some measure of complexity of the data , one can provably show efficient training and generalization for these networks , e.g . Jacot et al . ( 2018 ) ; Li & Liang ( 2018 ) ; Du et al . ( 2018 ) ; Allen-Zhu et al . ( 2019 ) ; Arora et al . ( 2019 ) . Of these , Allen-Zhu et al . ( 2019 ) is directly relevant to our paper and will be discussed later . The role of overparameterization , and provable training and generalization guarantees for neural networks are less well understood in unsupervised learning . Generative models or learning a data distribution from given samples is an important problem in unsupervised learning . Popular generative models based on neural networks include Generative Adversarial Networks ( GANs ) ( e.g. , Goodfellow et al . ( 2014 ) ) , Variational AutoEncoders ( VAEs ) ( e.g. , Kingma & Welling ( 2014 ) ) , and Normalizing Flows ( e.g. , Rezende & Mohamed ( 2015 ) ) . GANs and VAEs have shown impressive capability to generate samples of photo-realistic images but they can not give probability density estimates for new data points . Training of GANs and VAEs has various additional challenges such as mode collapse , posterior collapse , vanishing gradients , training instability , etc . as shown in e.g . Bowman et al . ( 2016 ) ; Salimans et al . ( 2016 ) ; Arora et al . ( 2018 ) ; Lucic et al . ( 2018 ) . In contrast to the generative models such as GANs and VAEs , when normalizing flows learn distributions , they can do both sampling and density estimation , leading to wide-ranging applications as mentioned in the surveys by Kobyzev et al . ( 2020 ) and Papamakarios et al . ( 2019 ) . Theoretical understanding of learning and generalization in normalizing flows ( more generally , generative models and unsupervised learning ) is a natural and important open question , and our main technical contribution is to extend known techniques from supervised learning to make progress towards answering this question . In this paper , we study learning and generalization in the case of univariate overparameterized normalizing flows . Restriction to the univariate case is technically non-trivial and interesting in its own right : univariate ReLU networks have been studied in recent supervised learning literature ( e.g. , Savarese et al . ( 2019 ) , Williams et al . ( 2019 ) , Sahs et al . ( 2020 ) and Daubechies et al . ( 2019 ) ) . Multidimensional flows are qualitatively more complex and our 1D analysis sheds some light on them ( see Sec . 4 ) . Before stating our contributions , we briefly introduce normalizing flows ; details appear in Section 2 . Normalizing Flows . We work with one-dimensional probability distributions with continuous density . The general idea behind normalizing flows ( NFs ) , restricted to 1D can be summarized as follows : Let X ∈ R be a random variable denoting the data distribution . We also fix a base distribution with associated random variable Z which is typically standard Gaussian , though in this paper we will work with the exponential distribution as well . Given i.i.d . samples of X , the goal is to learn a continuous strictly monotone increasing map fX : R→ R that transports the distribution of X to the distribution of Z : in other words , the distribution of f−1X ( Z ) is that of X . The learning of fX is done by representing it by a neural network and setting up an appropriate loss function . The monotonicity requirement on f which makes f invertible , while not essential , greatly simplifies the problem and is present in all the works we are aware of . It is not clear how to set up a tractable optimization problem without this requirement . Since the function represented by standard neural networks are not necessarily monotone , the design of the neural net is altered to make it monotone . For our 1D situation , one-hidden layer networks are of the form N ( x ) = ∑m i=1 aiσ ( wix+ bi ) , where m is the size of the hidden layer and the ai , wi , bi are the parameters of the network . We will assume that the activation functions used are monotone . Here we distinguish between two such alterations : ( 1 ) Changing the parametrization of the neural network . This can be done in multiple ways : instead of ai , wi we use a2i , w 2 i ( or other functions , such as the exponential function , of ai , wi that take on only positive values ) ( Huang et al. , 2018 ; Cao et al. , 2019 ) . This approach appears to be the most popular . In this paper , we also suggest another related alteration : we simply restrict the parameters ai , wi to be positive . This is achieved by enforcing this constraint during training . ( 2 ) Instead of using N ( x ) for f ( x ) we use φ ( N ( x ) ) for f ′ ( x ) = dfdx , where φ : R→ R + takes on only positive values . Positivity of f ′ implies monotonicity of f . Note that no restrictions on the parameters are required ; however , because we parametrize f ′ , the function f needs to be reconstructed using numerical quadrature . This approach is used by Wehenkel & Louppe ( 2019 ) . We will refer to the models in the first class as constrained normalizing flows ( CNFs ) and those in the second class as unconstrained normalizing flows ( UNFs ) . Our Contributions . In this paper , we study both constrained and unconstrained univariate NFs theoretically as well as empirically . The existing analyses for overparametrized neural networks in the supervised setting work with a linear approximation of the neural network , termed pseudo network in Allen-Zhu et al . ( 2019 ) . They show that ( 1 ) there is a pseudo network with weights close to the initial ones approximating the target function , ( 2 ) the loss surfaces of the neural network and the pseudo network are close and moreover the latter is convex for convex loss functions . This allows for proof of the convergence of the training of neural network to global optima . One can try to adapt the approach of using a linear approximation of the neural network to analyze training of NFs . However , one immediately encounters some new roadblocks : the loss surface of the pseudo networks is non-convex in both CNFs and UNFs . In both cases , we identify novel variations that make the optimization problem for associated pseudo network convex : For CNFs , instead of using a2i , w 2 i as parameters , we simply impose the constraints ai ≥ and wi ≥ for some small constant . The optimization algorithm now is projected SGD , which in this case incurs essentially no extra cost over SGD due to the simplicity of the positivity constraints . Apart from making the optimization problem convex , in experiments this variation slightly improves the training of NFs compared to the reparametrization approaches , and may be useful in practical settings . Similarly , for UNFs we identify two changes from the model of Wehenkel & Louppe ( 2019 ) that make the associated optimization problem convex , while still retaining empirical effectiveness : ( 1 ) Instead of Clenshaw–Curtis quadrature employed in Wehenkel & Louppe ( 2019 ) which uses positive and negative coefficients , we use the simple rectangle quadrature which uses only positive coefficients . This change makes the model somewhat slow ( it uses twice as many samples and time to get similar performance on the examples we tried ) . ( 2 ) Instead of the standard Gaussian distribution as the base distribution , we use the exponential distribution . In experiments , this does not cause much change . Our results point to a dichotomy between these two classes of NFs : our variant of UNFs can be theoretically analyzed when the networks are overparametrized to prove that the UNF indeed learns the data distribution . To our knowledge , this is the first “ end-to-end ” analysis of an NF model , and a neural generative model using gradient-based algorithms used in practice . This proof , while following the high-level scheme of Allen-Zhu et al . ( 2019 ) proof , has a number of differences , conceptual as well as technical , due to different settings . E.g. , our loss function involves a function and its integral estimated by quadrature . On the other hand , for CNFs , our empirical and theoretical findings provide evidence that overparametrization makes training slower to the extent that models of similar size which learn the data distribution well for UNFs , fail to do so for CNFs . We also analyze CNFs theoretically in the overparametrized setting and point to potential sources of the difficulty . The case of moderatesized networks , where training and generalization do take place empirically , is likely to be difficult to analyze theoretically as presently this setting is open for the simpler supervised learning case . We hope that our results will pave the way for further progress . We make some remarks on the multidimensional case in Sec . 4 . In summary , our contributions include : • To our knowledge , first efficient training and generalization proof for NFs ( in 1D ) . • Identification of architectural variants of UNFs that admit analysis via overparametrization . • Identification of “ barriers ” to the analysis of CNFs . Related Work . Previous work on normalizing flows has studied different variants such as planar and radial flows in Rezende & Mohamed ( 2015 ) , Sylvester flow in van den Berg et al . ( 2018 ) , Householder flow in Tomczak & Welling ( 2016 ) , masked autoregressive flow in Papamakarios et al . ( 2017 ) . Most variants of normalizing flows are specific to certain applications , and the expressive power ( i.e. , which base and data distributions they can map between ) and complexity of normalizing flow models have been studied recently , e.g . Kong & Chaudhuri ( 2020 ) and Teshima et al . ( 2020 ) . Invertible transformations defined by monotonic neural networks can be combined into autoregressive flows that are universal density approximators of continuous probability distributions ; see Masked Autoregressive Flows ( MAF ) Papamakarios et al . ( 2017 ) , UNMM-MAF by Wehenkel & Louppe ( 2019 ) , Neural Autoregressive Flows ( NAF ) by Huang et al . ( 2018 ) , Block Neural Autoregressive Flow ( B-NAF ) by Cao et al . ( 2019 ) . Unconstrained Monotonic Neural Network ( UMNN ) models proposed by Wehenkel & Louppe ( 2019 ) are particularly relevant to the technical part of our paper . Lei et al . ( 2020 ) show that when the generator is a two-layer tanh , sigmoid or leaky ReLU network , Wasserstein GAN trained with stochastic gradient descent-ascent converges to a global solution with polynomial time and sample complexity . Using the moments method and a learning algorithm motivated by tensor decomposition , Li & Dou ( 2020 ) show that GANs can efficiently learn a large class of distributions including those generated by two-layer networks . Nguyen et al . ( 2019b ) show that two-layer autoencoders with ReLU or threshold activations can be trained with normalized gradient descent over the reconstruction loss to provably learn the parameters of any generative bilinear model ( e.g. , mixture of Gaussians , sparse coding model ) . Nguyen et al . ( 2019a ) extend the work of Du et al . ( 2018 ) on supervised learning mentioned earlier to study weakly-trained ( i.e. , only encoder is trained ) and jointly-trained ( i.e. , both encoder and decoder are trained ) two-layer autoencoders , and show joint training requires less overparameterization and converges to a global optimum . The effect of overparameterization in unsupervised learning has also been of recent interest . Buhai et al . ( 2020 ) do an empirical study to show that across a variety of latent variable models and training algorithms , overparameterization can significantly increase the number of recovered ground truth latent variables . Radhakrishnan et al . ( 2020 ) show that overparameterized autoencoders and sequence encoders essentially implement associative memory by storing training samples as attractors in a dynamical system . Outline . A brief outline of our paper is as follows . Section 2 contains preliminaries and an overview of our results about constrained and unconstrained normalizing flows . Appendix B shows the existence of a pseudo network whose loss closely approximates the loss of the target function . Appendix C shows the coupling or closeness of their gradients over random initialization . Appendices D and E contain complete proofs of our optimization and generalization results , respectively . Section 3 and Appendix G contain our empirical studies towards validating our theoretical results .	The paper studies the role of overparameterization in learning normalizing flow models. More specifically, the authors analyze the optimization and generalization of such a model when the transport map f is parameterized by a two-layer neural network with potentially many hidden units (or highly over-parameterized). Importantly, the focus is on univariate data distributions.	SP:8180451eca79f017df91f0066ee46d8b149eaa48
Learning Generalizable Visual Representations via Interactive Gameplay	1 INTRODUCTION . We are interested in studying what facets of their environment artificial agents learn to represent through interaction and gameplay . We study this question within the context of hide-and-seek , for which proficiency requires an ability to navigate around in an environment and manipulate objects as well as an understanding of visual relationships , object affordances , and perspective . Inspired by behavior observed in juvenile ravens ( Burghardt , 2005 ) , we focus on a variant of hide-and-seek called cache in which agents hide objects instead of themselves . Advances in deep reinforcement learning have shown that , in abstract games ( e.g . Go and Chess ) and visually simplistic environments ( e.g . Atari and grid-worlds ) with limited interaction , artificial agents exhibit surprising emergent behaviours that enable proficient gameplay ( Mnih et al. , 2015 ; Silver et al. , 2017 ) ; indeed , recent work ( Chen et al. , 2019 ; Baker et al. , 2020 ) has shown this in the context of hiding games . Our interest , however , is in understanding how agents learn to represent their visual environment , through gameplay that requires varied interaction , in a high-fidelity environment grounded in the real world . This requires a fundamental shift away from existing popular environments and a rethinking of how the capabilities of artificial agents are evaluated . Our agents must first be embodied within an environment allowing for diverse interaction and providing rich visual output . For this we leverage AI2-THOR ( Kolve et al. , 2017 ) , a near photo-realistic , interactive , simulated , 3D environment of indoor living spaces , see Fig . 1a . Our agents are parameterized using deep neural networks , and trained adversarially using the paradigms of reinforce- ment ( Mnih et al. , 2016 ) and self-supervised learning . After our agents are trained to play cache , we then probe how they have learned to represent their environment . To this end we distinguish two distinct categories of representations generated by our agents . The first , static image representations ( SIRs ) , correspond to the output of a CNN applied to the agent ’ s egocentric visual input . The second , dynamic image representations ( DIRs ) , correspond to the output of the agents ’ RNN . While SIRs are timeless , operating only on single images , DIRs have the capacity to incorporate the agent ’ s previous actions and observations . Representation learning within the computer vision community is largely focused on developing SIRs whose quality is measured by their utility in downstream tasks ( e.g . classification , depthprediction , etc ) ( Zamir et al. , 2018 ) . Our first set of experiments show that our agents develop lowlevel visual understanding of individual images measured by their capacity to perform a collection of standard tasks from the computer vision literature , these tasks include pixel-to-pixel depth ( Saxena et al. , 2006 ) and surface normal ( Fouhey et al. , 2013 ) prediction , from a single image . While SIRs are clearly an important facet of representation learning , they are also definitionally unable to represent an environment as a whole : without the ability to integrate observations through time , a representation can only ever capture a single snapshot of space and time . To represent an environment holistically , we require DIRs . Unlike for SIRs , we are unaware of any well-established benchmarks for DIRs . In order to investigate what has been learned by our agent ’ s DIRs we develop a suite of experiments loosely inspired by experiments performed on infants and young children . These experiments then demonstrate our agents ’ ability to integrate observations through time and understand spatial relationships between objects ( Casasola et al. , 2003 ) , occlusion ( Hespos et al. , 2009 ) , object permanence ( Piaget , 1954 ) , and seriation ( Piaget , 1954 ) of free space . It is important to stress that this work focuses on studying how play and interaction contribute to representation learning in artificial agents and not on developing a new , state-of-the-art , methodology for representation learning . Nevertheless , to better situate our results in context of existing work , we provide strong baselines in our experiments , e.g . in our low-level vision experiments we compare against a fully supervised model trained on ImageNet ( Deng et al. , 2009 ) . Our results provide compelling evidence that : ( a ) on a suite of low level computer vision tasks within AI2-THOR , static representations learned by playing cache perform very competitively ( and often outperform ) strong unsupervised and fully supervised methods , ( b ) these static representations , trained using only synthetic images , obtain non-trivial transfer to downstream tasks using real-world images , ( c ) unlike representations learned from datasets of single images , agents trained via embodied gameplay learn to integrate visual information through time , demonstrating an elementary understanding of free space , objects and their relationships , and ( d ) embodied gameplay provides a natural means by which to generate rich experiences for representation learning beyond random sampling and relatively simpler tasks like visual navigation . In summary , we highlight the contributions : ( 1 ) Cache – we introduce cache within the AI2-THOR environment , an adversarial game which permits the study of representation learning in the context of interactive , visual , gameplay . ( 2 ) Cache agent – training agents to play Cache is non-trivial . We produce a strong Cache agent which integrates several methodological novelties ( see , e.g. , perspective simulation and visual dynamics replay in Sec . 4 ) and even outperforms humans at hiding on training scenes ( see Sec . 5 ) . ( 3 ) Static and dynamic representation study – we provide comprehensive evaluations of how our Cache agent has learned to represent its environment providing insight into the advantages and current limitations of interactive-gameplay-based representation learning . 2 RELATED WORK . Deep reinforcement learning for games . As games provide an interactive environment that enable agents to receive observations , take actions , and receive rewards , they are a popular testbed for RL algorithms . Reinforcement learning has been studied in the context of numerous single , and multi , agent games such as Atari Breakout ( Mnih et al. , 2015 ) , VizDoom ( Lample & Chaplot , 2017 ) , Go ( Silver et al. , 2017 ) , StarCraft ( Vinyals et al. , 2019 ) , Dota ( Berner et al. , 2019 ) and Hide-and-Seek ( Baker et al. , 2020 ) . The goal of these works is proficiency : to create an agent which can achieve ( super- ) human performance with respect to the game ’ s success criteria . In contrast , our goal is to understand how an agent has learned to represent its environment through gameplay and to show that such an agent ’ s representations can be employed for downstream tasks beyond gameplay . Passive representation learning . There is a large body of recent works that address the problem of representation learning from static images or videos . Image colorization ( Zhang et al. , 2016 ) , egomotion estimation ( Agrawal et al. , 2015 ) , predicting image rotation ( Gidaris et al. , 2018 ) , context prediction ( Doersch et al. , 2015 ) , future frame prediction ( Vondrick et al. , 2016 ) and more recent contrastive learning based approaches ( He et al. , 2020 ; Chen et al. , 2020b ) are among the successful examples of passive representation learning . We refer to them as passive since the representations are learned from a fixed set of images or videos . Our approach in contrast is interactive in that the images observed during learning are decided by the actions of the agent . Interactive representation learning . Learning representations in dynamic and interactive environments has been addressed in the literature as well . In the following we mention a few examples . Burda et al . ( 2019 ) explore curiosity-driven representation learning from a suite of games . Anand et al . ( 2019 ) address representation learning of the latent factors used for generating the interactive environment . Zhan et al . ( 2018 ) learn to improve exploration in video games by predicting the memory state . Ghosh et al . ( 2019 ) learn a functional representation for decision making as opposed to a representation for the observation space only . Whitney et al . ( 2020 ) simultaneously learn embeddings of state and action sequences . Jonschkowski & Brock ( 2015 ) learn a representation by measuring inconsistencies with a set of pre-defined priors . Pinto et al . ( 2016 ) learn visual representations by pushing and grasping objects using a robotics arm . The representations learned using these approaches are typically tested on tasks akin to tasks they were trained with ( e.g. , a different level of the same game ) . In contrast , we investigate whether cognitive primitives such as depth estimation and object permanence maybe be learned via gameplay in a visual dynamic environment . 3 PLAYING CACHE IN SIMULATION . We situate our agents within AI2-THOR , a simulated 3D environment of indoor living spaces within which multiple agents can navigate around and interact with objects ( e.g . by picking up , placing , opening , closing , cutting , switching on , etc. ) . In past works , AI2-THOR has been leveraged to teach agents to interact with their world ( Zhu et al. , 2017 ; Hu et al. , 2018 ; Huang et al. , 2019 ; Gan et al. , 2020b ; Wortsman et al. , 2019 ; Gordon et al. , 2017 ) , interact with each other ( Jain et al. , 2019 ; 2020 ) as well as learn from these interactions ( Nagarajan & Grauman , 2020 ; Lohmann et al. , 2020 ) . AI2-THOR contains a total of 150 unique scenes equally distributed into five scene types : kitchens , living rooms , bedrooms , bathrooms , and foyers . We train our cache agents on a subset of the kitchen Published as a conference paper at ICLR 2021 Inp t L A & R c NN A . E D C . D c D A M Me c Ma Inp t L A & R c NN A . E D C . D c D A M Me c Ma Inp t L A & R c NN A . E D C . 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The trainable components are shown in yellow , the inputs and outputs in pink , and the intermediate outputs in blue . Refer to text for details . and living room scenes as these scenes are relatively large and often include many opportunities for interaction ; but we use all scenes types across our suite of experiments . Details regarding scene splits across train , validation , and test for the different experiments can be found in Sec . A.1 . In a game of cache , two agents ( a hider and a seeker ) compete , with the hiding agent attempting to place a given object in the environment so that the seeking agent can not find it . This game is zero-sum with the hiding agent winning if and only if the seeking agent can not find the object . We partition the game of cache into five conceptually-distinct stages : exploration and mapping ( E & M ) , perspective simulation ( PS ) , object hiding ( OH ) , object manipulation ( OM ) , and seeking ( S ) ; see Figures 1b to 1f . A game of cache begins with the hiding agent exploring its environment and building an internal map corresponding to the locations it has visited ( E & M ) . The agent then chooses globally , among the many locations it has visited , a location where it believes it can hide the object so that the seeker will not be able to find it ( PS ) . After moving to this location the agent makes a local decision about where the object should be placed , e.g . behind a TV or inside a drawer ( OH ) . The agent then performs the low-level manipulation task of moving the object in its hand to the desired location ( OM ) , if this manipulation fails then the agent tries another local hiding location ( OH ) . Finally the seeking agent searches for the hidden object ( S ) . Pragmatically , these stages are distinguished by the actions available to the agent ( see Table G.1 for a description of all 219 actions ) and their reward structures ( see Sec . C.5 ) . For more details regarding these stages , see Sec . B .	This paper examines the representations learned during adversarial gameplay, specifically a hide-and-seek game called Cache. The hiding agent must place an object in a room such that the seeker agent cannot find it. The authors argue that the adversarial nature of the game shapes the representations. Inspired by psychology experiments performed on children, the authors examine both static and dynamic representations to probe whether they contain information about properties of the environment such as object permanence.	SP:6230795b336182834de7e33ddc67aa69cba3efa4
Learning Generalizable Visual Representations via Interactive Gameplay	1 INTRODUCTION . We are interested in studying what facets of their environment artificial agents learn to represent through interaction and gameplay . We study this question within the context of hide-and-seek , for which proficiency requires an ability to navigate around in an environment and manipulate objects as well as an understanding of visual relationships , object affordances , and perspective . Inspired by behavior observed in juvenile ravens ( Burghardt , 2005 ) , we focus on a variant of hide-and-seek called cache in which agents hide objects instead of themselves . Advances in deep reinforcement learning have shown that , in abstract games ( e.g . Go and Chess ) and visually simplistic environments ( e.g . Atari and grid-worlds ) with limited interaction , artificial agents exhibit surprising emergent behaviours that enable proficient gameplay ( Mnih et al. , 2015 ; Silver et al. , 2017 ) ; indeed , recent work ( Chen et al. , 2019 ; Baker et al. , 2020 ) has shown this in the context of hiding games . Our interest , however , is in understanding how agents learn to represent their visual environment , through gameplay that requires varied interaction , in a high-fidelity environment grounded in the real world . This requires a fundamental shift away from existing popular environments and a rethinking of how the capabilities of artificial agents are evaluated . Our agents must first be embodied within an environment allowing for diverse interaction and providing rich visual output . For this we leverage AI2-THOR ( Kolve et al. , 2017 ) , a near photo-realistic , interactive , simulated , 3D environment of indoor living spaces , see Fig . 1a . Our agents are parameterized using deep neural networks , and trained adversarially using the paradigms of reinforce- ment ( Mnih et al. , 2016 ) and self-supervised learning . After our agents are trained to play cache , we then probe how they have learned to represent their environment . To this end we distinguish two distinct categories of representations generated by our agents . The first , static image representations ( SIRs ) , correspond to the output of a CNN applied to the agent ’ s egocentric visual input . The second , dynamic image representations ( DIRs ) , correspond to the output of the agents ’ RNN . While SIRs are timeless , operating only on single images , DIRs have the capacity to incorporate the agent ’ s previous actions and observations . Representation learning within the computer vision community is largely focused on developing SIRs whose quality is measured by their utility in downstream tasks ( e.g . classification , depthprediction , etc ) ( Zamir et al. , 2018 ) . Our first set of experiments show that our agents develop lowlevel visual understanding of individual images measured by their capacity to perform a collection of standard tasks from the computer vision literature , these tasks include pixel-to-pixel depth ( Saxena et al. , 2006 ) and surface normal ( Fouhey et al. , 2013 ) prediction , from a single image . While SIRs are clearly an important facet of representation learning , they are also definitionally unable to represent an environment as a whole : without the ability to integrate observations through time , a representation can only ever capture a single snapshot of space and time . To represent an environment holistically , we require DIRs . Unlike for SIRs , we are unaware of any well-established benchmarks for DIRs . In order to investigate what has been learned by our agent ’ s DIRs we develop a suite of experiments loosely inspired by experiments performed on infants and young children . These experiments then demonstrate our agents ’ ability to integrate observations through time and understand spatial relationships between objects ( Casasola et al. , 2003 ) , occlusion ( Hespos et al. , 2009 ) , object permanence ( Piaget , 1954 ) , and seriation ( Piaget , 1954 ) of free space . It is important to stress that this work focuses on studying how play and interaction contribute to representation learning in artificial agents and not on developing a new , state-of-the-art , methodology for representation learning . Nevertheless , to better situate our results in context of existing work , we provide strong baselines in our experiments , e.g . in our low-level vision experiments we compare against a fully supervised model trained on ImageNet ( Deng et al. , 2009 ) . Our results provide compelling evidence that : ( a ) on a suite of low level computer vision tasks within AI2-THOR , static representations learned by playing cache perform very competitively ( and often outperform ) strong unsupervised and fully supervised methods , ( b ) these static representations , trained using only synthetic images , obtain non-trivial transfer to downstream tasks using real-world images , ( c ) unlike representations learned from datasets of single images , agents trained via embodied gameplay learn to integrate visual information through time , demonstrating an elementary understanding of free space , objects and their relationships , and ( d ) embodied gameplay provides a natural means by which to generate rich experiences for representation learning beyond random sampling and relatively simpler tasks like visual navigation . In summary , we highlight the contributions : ( 1 ) Cache – we introduce cache within the AI2-THOR environment , an adversarial game which permits the study of representation learning in the context of interactive , visual , gameplay . ( 2 ) Cache agent – training agents to play Cache is non-trivial . We produce a strong Cache agent which integrates several methodological novelties ( see , e.g. , perspective simulation and visual dynamics replay in Sec . 4 ) and even outperforms humans at hiding on training scenes ( see Sec . 5 ) . ( 3 ) Static and dynamic representation study – we provide comprehensive evaluations of how our Cache agent has learned to represent its environment providing insight into the advantages and current limitations of interactive-gameplay-based representation learning . 2 RELATED WORK . Deep reinforcement learning for games . As games provide an interactive environment that enable agents to receive observations , take actions , and receive rewards , they are a popular testbed for RL algorithms . Reinforcement learning has been studied in the context of numerous single , and multi , agent games such as Atari Breakout ( Mnih et al. , 2015 ) , VizDoom ( Lample & Chaplot , 2017 ) , Go ( Silver et al. , 2017 ) , StarCraft ( Vinyals et al. , 2019 ) , Dota ( Berner et al. , 2019 ) and Hide-and-Seek ( Baker et al. , 2020 ) . The goal of these works is proficiency : to create an agent which can achieve ( super- ) human performance with respect to the game ’ s success criteria . In contrast , our goal is to understand how an agent has learned to represent its environment through gameplay and to show that such an agent ’ s representations can be employed for downstream tasks beyond gameplay . Passive representation learning . There is a large body of recent works that address the problem of representation learning from static images or videos . Image colorization ( Zhang et al. , 2016 ) , egomotion estimation ( Agrawal et al. , 2015 ) , predicting image rotation ( Gidaris et al. , 2018 ) , context prediction ( Doersch et al. , 2015 ) , future frame prediction ( Vondrick et al. , 2016 ) and more recent contrastive learning based approaches ( He et al. , 2020 ; Chen et al. , 2020b ) are among the successful examples of passive representation learning . We refer to them as passive since the representations are learned from a fixed set of images or videos . Our approach in contrast is interactive in that the images observed during learning are decided by the actions of the agent . Interactive representation learning . Learning representations in dynamic and interactive environments has been addressed in the literature as well . In the following we mention a few examples . Burda et al . ( 2019 ) explore curiosity-driven representation learning from a suite of games . Anand et al . ( 2019 ) address representation learning of the latent factors used for generating the interactive environment . Zhan et al . ( 2018 ) learn to improve exploration in video games by predicting the memory state . Ghosh et al . ( 2019 ) learn a functional representation for decision making as opposed to a representation for the observation space only . Whitney et al . ( 2020 ) simultaneously learn embeddings of state and action sequences . Jonschkowski & Brock ( 2015 ) learn a representation by measuring inconsistencies with a set of pre-defined priors . Pinto et al . ( 2016 ) learn visual representations by pushing and grasping objects using a robotics arm . The representations learned using these approaches are typically tested on tasks akin to tasks they were trained with ( e.g. , a different level of the same game ) . In contrast , we investigate whether cognitive primitives such as depth estimation and object permanence maybe be learned via gameplay in a visual dynamic environment . 3 PLAYING CACHE IN SIMULATION . We situate our agents within AI2-THOR , a simulated 3D environment of indoor living spaces within which multiple agents can navigate around and interact with objects ( e.g . by picking up , placing , opening , closing , cutting , switching on , etc. ) . In past works , AI2-THOR has been leveraged to teach agents to interact with their world ( Zhu et al. , 2017 ; Hu et al. , 2018 ; Huang et al. , 2019 ; Gan et al. , 2020b ; Wortsman et al. , 2019 ; Gordon et al. , 2017 ) , interact with each other ( Jain et al. , 2019 ; 2020 ) as well as learn from these interactions ( Nagarajan & Grauman , 2020 ; Lohmann et al. , 2020 ) . AI2-THOR contains a total of 150 unique scenes equally distributed into five scene types : kitchens , living rooms , bedrooms , bathrooms , and foyers . We train our cache agents on a subset of the kitchen Published as a conference paper at ICLR 2021 Inp t L A & R c NN A . E D C . D c D A M Me c Ma Inp t L A & R c NN A . E D C . D c D A M Me c Ma Inp t L A & R c NN A . E D C . 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The trainable components are shown in yellow , the inputs and outputs in pink , and the intermediate outputs in blue . Refer to text for details . and living room scenes as these scenes are relatively large and often include many opportunities for interaction ; but we use all scenes types across our suite of experiments . Details regarding scene splits across train , validation , and test for the different experiments can be found in Sec . A.1 . In a game of cache , two agents ( a hider and a seeker ) compete , with the hiding agent attempting to place a given object in the environment so that the seeking agent can not find it . This game is zero-sum with the hiding agent winning if and only if the seeking agent can not find the object . We partition the game of cache into five conceptually-distinct stages : exploration and mapping ( E & M ) , perspective simulation ( PS ) , object hiding ( OH ) , object manipulation ( OM ) , and seeking ( S ) ; see Figures 1b to 1f . A game of cache begins with the hiding agent exploring its environment and building an internal map corresponding to the locations it has visited ( E & M ) . The agent then chooses globally , among the many locations it has visited , a location where it believes it can hide the object so that the seeker will not be able to find it ( PS ) . After moving to this location the agent makes a local decision about where the object should be placed , e.g . behind a TV or inside a drawer ( OH ) . The agent then performs the low-level manipulation task of moving the object in its hand to the desired location ( OM ) , if this manipulation fails then the agent tries another local hiding location ( OH ) . Finally the seeking agent searches for the hidden object ( S ) . Pragmatically , these stages are distinguished by the actions available to the agent ( see Table G.1 for a description of all 219 actions ) and their reward structures ( see Sec . C.5 ) . For more details regarding these stages , see Sec . B .	This paper proposes embodied game-playing with artificial agents as a method to learn better representations of their environment. They describe a game, cache, which is a variant of hide-and-seek played in a virtual environment and a method for training an agent to play the game. They present results which demonstrate that the static representations learned through game-playing perform better than other pre-training tasks within the same virtual environment, on both virtual vision and real world vision applications. They also show that the dynamic representations are useful for completing object permanence tests inspired by developmental psychology research.	SP:6230795b336182834de7e33ddc67aa69cba3efa4
Learning Generalizable Visual Representations via Interactive Gameplay	1 INTRODUCTION . We are interested in studying what facets of their environment artificial agents learn to represent through interaction and gameplay . We study this question within the context of hide-and-seek , for which proficiency requires an ability to navigate around in an environment and manipulate objects as well as an understanding of visual relationships , object affordances , and perspective . Inspired by behavior observed in juvenile ravens ( Burghardt , 2005 ) , we focus on a variant of hide-and-seek called cache in which agents hide objects instead of themselves . Advances in deep reinforcement learning have shown that , in abstract games ( e.g . Go and Chess ) and visually simplistic environments ( e.g . Atari and grid-worlds ) with limited interaction , artificial agents exhibit surprising emergent behaviours that enable proficient gameplay ( Mnih et al. , 2015 ; Silver et al. , 2017 ) ; indeed , recent work ( Chen et al. , 2019 ; Baker et al. , 2020 ) has shown this in the context of hiding games . Our interest , however , is in understanding how agents learn to represent their visual environment , through gameplay that requires varied interaction , in a high-fidelity environment grounded in the real world . This requires a fundamental shift away from existing popular environments and a rethinking of how the capabilities of artificial agents are evaluated . Our agents must first be embodied within an environment allowing for diverse interaction and providing rich visual output . For this we leverage AI2-THOR ( Kolve et al. , 2017 ) , a near photo-realistic , interactive , simulated , 3D environment of indoor living spaces , see Fig . 1a . Our agents are parameterized using deep neural networks , and trained adversarially using the paradigms of reinforce- ment ( Mnih et al. , 2016 ) and self-supervised learning . After our agents are trained to play cache , we then probe how they have learned to represent their environment . To this end we distinguish two distinct categories of representations generated by our agents . The first , static image representations ( SIRs ) , correspond to the output of a CNN applied to the agent ’ s egocentric visual input . The second , dynamic image representations ( DIRs ) , correspond to the output of the agents ’ RNN . While SIRs are timeless , operating only on single images , DIRs have the capacity to incorporate the agent ’ s previous actions and observations . Representation learning within the computer vision community is largely focused on developing SIRs whose quality is measured by their utility in downstream tasks ( e.g . classification , depthprediction , etc ) ( Zamir et al. , 2018 ) . Our first set of experiments show that our agents develop lowlevel visual understanding of individual images measured by their capacity to perform a collection of standard tasks from the computer vision literature , these tasks include pixel-to-pixel depth ( Saxena et al. , 2006 ) and surface normal ( Fouhey et al. , 2013 ) prediction , from a single image . While SIRs are clearly an important facet of representation learning , they are also definitionally unable to represent an environment as a whole : without the ability to integrate observations through time , a representation can only ever capture a single snapshot of space and time . To represent an environment holistically , we require DIRs . Unlike for SIRs , we are unaware of any well-established benchmarks for DIRs . In order to investigate what has been learned by our agent ’ s DIRs we develop a suite of experiments loosely inspired by experiments performed on infants and young children . These experiments then demonstrate our agents ’ ability to integrate observations through time and understand spatial relationships between objects ( Casasola et al. , 2003 ) , occlusion ( Hespos et al. , 2009 ) , object permanence ( Piaget , 1954 ) , and seriation ( Piaget , 1954 ) of free space . It is important to stress that this work focuses on studying how play and interaction contribute to representation learning in artificial agents and not on developing a new , state-of-the-art , methodology for representation learning . Nevertheless , to better situate our results in context of existing work , we provide strong baselines in our experiments , e.g . in our low-level vision experiments we compare against a fully supervised model trained on ImageNet ( Deng et al. , 2009 ) . Our results provide compelling evidence that : ( a ) on a suite of low level computer vision tasks within AI2-THOR , static representations learned by playing cache perform very competitively ( and often outperform ) strong unsupervised and fully supervised methods , ( b ) these static representations , trained using only synthetic images , obtain non-trivial transfer to downstream tasks using real-world images , ( c ) unlike representations learned from datasets of single images , agents trained via embodied gameplay learn to integrate visual information through time , demonstrating an elementary understanding of free space , objects and their relationships , and ( d ) embodied gameplay provides a natural means by which to generate rich experiences for representation learning beyond random sampling and relatively simpler tasks like visual navigation . In summary , we highlight the contributions : ( 1 ) Cache – we introduce cache within the AI2-THOR environment , an adversarial game which permits the study of representation learning in the context of interactive , visual , gameplay . ( 2 ) Cache agent – training agents to play Cache is non-trivial . We produce a strong Cache agent which integrates several methodological novelties ( see , e.g. , perspective simulation and visual dynamics replay in Sec . 4 ) and even outperforms humans at hiding on training scenes ( see Sec . 5 ) . ( 3 ) Static and dynamic representation study – we provide comprehensive evaluations of how our Cache agent has learned to represent its environment providing insight into the advantages and current limitations of interactive-gameplay-based representation learning . 2 RELATED WORK . Deep reinforcement learning for games . As games provide an interactive environment that enable agents to receive observations , take actions , and receive rewards , they are a popular testbed for RL algorithms . Reinforcement learning has been studied in the context of numerous single , and multi , agent games such as Atari Breakout ( Mnih et al. , 2015 ) , VizDoom ( Lample & Chaplot , 2017 ) , Go ( Silver et al. , 2017 ) , StarCraft ( Vinyals et al. , 2019 ) , Dota ( Berner et al. , 2019 ) and Hide-and-Seek ( Baker et al. , 2020 ) . The goal of these works is proficiency : to create an agent which can achieve ( super- ) human performance with respect to the game ’ s success criteria . In contrast , our goal is to understand how an agent has learned to represent its environment through gameplay and to show that such an agent ’ s representations can be employed for downstream tasks beyond gameplay . Passive representation learning . There is a large body of recent works that address the problem of representation learning from static images or videos . Image colorization ( Zhang et al. , 2016 ) , egomotion estimation ( Agrawal et al. , 2015 ) , predicting image rotation ( Gidaris et al. , 2018 ) , context prediction ( Doersch et al. , 2015 ) , future frame prediction ( Vondrick et al. , 2016 ) and more recent contrastive learning based approaches ( He et al. , 2020 ; Chen et al. , 2020b ) are among the successful examples of passive representation learning . We refer to them as passive since the representations are learned from a fixed set of images or videos . Our approach in contrast is interactive in that the images observed during learning are decided by the actions of the agent . Interactive representation learning . Learning representations in dynamic and interactive environments has been addressed in the literature as well . In the following we mention a few examples . Burda et al . ( 2019 ) explore curiosity-driven representation learning from a suite of games . Anand et al . ( 2019 ) address representation learning of the latent factors used for generating the interactive environment . Zhan et al . ( 2018 ) learn to improve exploration in video games by predicting the memory state . Ghosh et al . ( 2019 ) learn a functional representation for decision making as opposed to a representation for the observation space only . Whitney et al . ( 2020 ) simultaneously learn embeddings of state and action sequences . Jonschkowski & Brock ( 2015 ) learn a representation by measuring inconsistencies with a set of pre-defined priors . Pinto et al . ( 2016 ) learn visual representations by pushing and grasping objects using a robotics arm . The representations learned using these approaches are typically tested on tasks akin to tasks they were trained with ( e.g. , a different level of the same game ) . In contrast , we investigate whether cognitive primitives such as depth estimation and object permanence maybe be learned via gameplay in a visual dynamic environment . 3 PLAYING CACHE IN SIMULATION . We situate our agents within AI2-THOR , a simulated 3D environment of indoor living spaces within which multiple agents can navigate around and interact with objects ( e.g . by picking up , placing , opening , closing , cutting , switching on , etc. ) . In past works , AI2-THOR has been leveraged to teach agents to interact with their world ( Zhu et al. , 2017 ; Hu et al. , 2018 ; Huang et al. , 2019 ; Gan et al. , 2020b ; Wortsman et al. , 2019 ; Gordon et al. , 2017 ) , interact with each other ( Jain et al. , 2019 ; 2020 ) as well as learn from these interactions ( Nagarajan & Grauman , 2020 ; Lohmann et al. , 2020 ) . AI2-THOR contains a total of 150 unique scenes equally distributed into five scene types : kitchens , living rooms , bedrooms , bathrooms , and foyers . We train our cache agents on a subset of the kitchen Published as a conference paper at ICLR 2021 Inp t L A & R c NN A . E D C . D c D A M Me c Ma Inp t L A & R c NN A . E D C . D c D A M Me c Ma Inp t L A & R c NN A . E D C . 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The trainable components are shown in yellow , the inputs and outputs in pink , and the intermediate outputs in blue . Refer to text for details . and living room scenes as these scenes are relatively large and often include many opportunities for interaction ; but we use all scenes types across our suite of experiments . Details regarding scene splits across train , validation , and test for the different experiments can be found in Sec . A.1 . In a game of cache , two agents ( a hider and a seeker ) compete , with the hiding agent attempting to place a given object in the environment so that the seeking agent can not find it . This game is zero-sum with the hiding agent winning if and only if the seeking agent can not find the object . We partition the game of cache into five conceptually-distinct stages : exploration and mapping ( E & M ) , perspective simulation ( PS ) , object hiding ( OH ) , object manipulation ( OM ) , and seeking ( S ) ; see Figures 1b to 1f . A game of cache begins with the hiding agent exploring its environment and building an internal map corresponding to the locations it has visited ( E & M ) . The agent then chooses globally , among the many locations it has visited , a location where it believes it can hide the object so that the seeker will not be able to find it ( PS ) . After moving to this location the agent makes a local decision about where the object should be placed , e.g . behind a TV or inside a drawer ( OH ) . The agent then performs the low-level manipulation task of moving the object in its hand to the desired location ( OM ) , if this manipulation fails then the agent tries another local hiding location ( OH ) . Finally the seeking agent searches for the hidden object ( S ) . Pragmatically , these stages are distinguished by the actions available to the agent ( see Table G.1 for a description of all 219 actions ) and their reward structures ( see Sec . C.5 ) . For more details regarding these stages , see Sec . B .	In this paper, the author's propose an embodied adversarial reinforcement learning agent that can play a variation of hide-and-seek called Cache. This environment is a high fidelity interactive world. The authors argue that the agents are able to learn flexible representations of their observations which encode information such as object permanence, free space and containment.	SP:6230795b336182834de7e33ddc67aa69cba3efa4
Improved Estimation of Concentration Under $\ell_p$-Norm Distance Metrics Using Half Spaces	1 INTRODUCTION . Despite achieving exceptional performance in benign settings , modern machine learning models have been shown to be highly vulnerable to inputs , known as adversarial examples , crafted with targeted but imperceptible perturbations ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) . This discovery has prompted a wave of research studies to propose defense mechanisms , including heuristic approaches ( Papernot et al. , 2016 ; Mądry et al. , 2018 ; Zhang et al. , 2019 ) and certifiable methods ( Wong & Kolter , 2018 ; Gowal et al. , 2019 ; Cohen et al. , 2019 ) . Unfortunately , none of these methods can successfully produce adversarially-robust models , even for classification tasks on toy datasets such as CIFAR-10 . To explain the prevalence of adversarial examples , a line of theoretical works ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Shafahi et al. , 2019 ; Dohmatob , 2019 ; Bhagoji et al. , 2019 ) have proven upper bounds on the maximum achievable adversarial robustness by imposing different assumptions on the underlying metric probability space . In particular , Mahloujifar et al . ( 2019a ) generalized the previous results showing that adversarial examples are inevitable as long as the input distributions are concentrated with respect to the perturbation metric . Thus , the question of whether or not natural image distributions are concentrated is highly relevant , as if they are it would rule out any possibility of there being adversarially robust image classifiers . Recently , Mahloujifar et al . ( 2019b ) proposed an empirical method to measure the concentration of an arbitrary distribution using data samples , then employed it to estimate a lower bound on intrinsic robustness ( see Definition 2.2 for its formal definition ) for several image benchmarks . By demonstrating the gap between the estimated bounds of intrinsic robustness and the robustness performance achieved by the best current models , they further concluded concentration of measure is not the sole reason behind the adversarial vulnerability of existing classifiers for benchmark image distributions . However , due to the heuristic nature of the proposed algorithm , it remains elusive whether the estimates it produces can serve as useful approximations of the underlying intrinsic robustness limits , thus hindering understanding of how much of the actual adversarial risk can be explained by the concentration of measure phenomenon . In this work , we address this issue by first characterizing the optimum of the actual concentration problem for general Gaussian spaces , then using our theoretical insights to develop an alternative algorithm for measuring concentration empirically that significantly improves both the accuracy and efficiency of estimates of intrinsic robustness . While we do not demonstrate a specific classifier which achieves this robustness upper bound , our results rule out inherent image distribution concentration as the reason for our current inability to find adversarially robust models . Contributions . We generalize the Gaussian Isoperimetric Inequality to non-spherical Gaussian distributions and ` p-norm distance metrics with p ≥ 2 ( including ` ∞ ) ( Theorem 3.3 ) . Motivated by the optimal concentration results for special Gaussian spaces ( Remark 3.4 ) , we develop a samplebased algorithm to estimate the concentration of measure using half spaces that works for arbitrary distribution and any ` p-norm distance ( Section 4 ) . Compared with prior approaches , we empirically demonstrate the significant increase in efficacy of our method under ` ∞-norm distance metric ( Section 6 ) . Not only does the proposed method converge to its limit with an order of magnitude fewer data ( Section 6.2 ) , it also finds a much tighter lower bound of intrinsic robustness for both simulated datasets whose underlying concentration function is analytically derivable and various benchmark image datasets ( Section 6.1 ) . In particular , we improve the best current estimated lower bound of intrinsic robustness from approximately 82 % to above 93 % for CIFAR-10 under ` ∞-norm bounded perturbations with = 8/255 . These tighter concentration estimates produced by our algorithm provide strong evidence that concentration of measure should not be considered as the main cause of adversarial vulnerability , at least for the image benchmarks evaluated in our experiments . Related Work . Several prior works have sought to empirically estimate lower bounds on intrinsic robustness using data samples . The pioneering work of Gilmer et al . ( 2018 ) introduced the connection between adversarial examples and the concentration phenomenon for uniform n-spheres , then proposed a simple heuristic to find a half space that expands slowly under Euclidean distance for the MNIST dataset . Our work can be seen as a strict generalization of Gilmer et al . ( 2018 ) ’ s , which applies to arbitrary ` p-norm distance metrics ( including ` ∞ ) . By characterizing the optimal transport cost between conditional distributions , Bhagoji et al . ( 2019 ) estimated a lower bound on the best possible adversarial robustness for several image datasets . However , when applied to adversaries beyond ` 2 , such as ` ∞ , the lower bound produced by their method is not informative ( that is , it is close to zero ) . The most relevant previous work is Mahloujifar et al . ( 2019b ) , which proposed a general method for measuring concentration using special collections of subsets . Although the optimal value of the considered empirical concentration problem is proven to asymptotically converge to the actual concentration , there is no guarantee that the proposed searching algorithm for solving the empirical problem finds the optimum . Our approach follows the framework introduced by Mahloujifar et al . ( 2019b ) ’ s , but considers a different collection of subsets for the empirical concentration problem . This not only results in optimality for theoretical Gaussian distributions , but also significantly improves the estimation performance for typical image benchmarks . Another line of work attempts to provide estimates of intrinsic robustness upper bounds based on generative assumptions . In order to justify the theoretically-derived impossibility results , Fawzi et al . ( 2018 ) estimated the smoothness parameters of the state-of-the-art generative models on CIFAR-10 and SVHN datasets , which yield approximated upper bounds on adversarial robustness for any classifiers . Zhang et al . ( 2020 ) generalized their results to non-smoothed data manifolds , such as datasets that can be captured by a conditional generative model . However , these methods only work for simulated generative distributions , which may deviate from the actual distributions they are intended to understand . Notation . For any n ∈ Z+ , denote by [ n ] the set { 1 , 2 , . . . , n } . Lowercase boldface letters denote vectors and uppercase boldface letters represent matrices . For any vector x and p ∈ [ 1 , ∞ ) , let xj , ‖x‖p and ‖x‖∞ be the j-th element , the ` p-norm and the ` ∞-norm of x . For any matrix A , B is said to be a square root of A if A = BB , and the induced matrix p-norm of A is defined as ‖A‖p = supx6=0 { ‖Ax‖p/‖x‖p } . Denote by N ( θ , Σ ) the Gaussian distribution with mean θ and covariance matrix Σ . Let γn be the probability measure of N ( 0 , In ) , where In denotes the identity matrix . Let Φ ( · ) be the cumulative distribution function of N ( 0 , 1 ) and Φ−1 ( · ) be its inverse . For any set A , let pow ( A ) and 1A ( · ) be all measurable subsets and the indicator function of A . Let ( X , µ , ∆ ) be a metric probability space , where ∆ : X × X → R≥0 denotes a distance metric on X . Define the empirical measure with respect to a sample set { xi } i∈ [ m ] as µ̂m ( A ) = 1m ∑ i∈ [ m ] 1A ( xi ) , ∀A ∈ pow ( X ) . Let B ( x , , ∆ ) = { x′ ∈ X : ∆ ( x′ , x ) ≤ } be the ball around x with radius . Define the -expansion of A as A ( ∆ ) = { x ∈ X : ∃ x′ ∈ B ( x , , ∆ ) ∩ A } . 2 PRELIMINARIES . In this section , we introduce the problem of measuring concentration and its connection to adversarial robustness . Consider a metric probability space of instances ( X , µ , ∆ ) . Given parameters ≥ 0 and α > 0 , the concentration of measure problem1 can be cast as the following optimization problem : minimize E∈Pow ( X ) µ ( E ( ∆ ) ) subject to µ ( E ) ≥ α . ( 2.1 ) We focus on the case where ∆ is some ` p-norm distance metric ( including ` ∞ ) in this work . Concentration of measure has been shown to be closely related to adversarial examples ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Mahloujifar et al. , 2019a ) . In particular , one can prove that for a given robust learning problem , if the input distribution is concentrated with respect to the perturbation metric , no adversarially robust model exists . The concentration parameter ( which corresponds to the optimal value of optimization problem ( 2.1 ) ) determines an inherent upper bound on the maximum adversarial robustness that any model can achieve for the given problem . To explain the connection between concentration of measure and robust learning in a more formal way , we lay out the definition of adversarial risk that we work with . We draw this definition from several previous works , including Gilmer et al . ( 2018 ) ; Bubeck et al . ( 2019 ) ; Mahloujifar et al . ( 2019a ; b ) .2 Definition 2.1 ( Adversarial Risk ) . Let ( X , µ , ∆ ) be the input metric probability space . Assume f∗ is the underlying ground-truth classifier that gives labels to any input . Given classifier f and ≥ 0 , the adversarial risk of f with respect to -perturbations measured by ∆ is defined as : AdvRisk ( f ) = Pr x∼µ [ ∃ x′ ∈ B ( x , , ∆ ) s.t . f ( x′ ) 6= f∗ ( x′ ) ] . Correspondingly , we define the adversarial robustness of f as AdvRob ( f ) = 1−AdvRisk ( f ) . When = 0 , adversarial risk degenerates to standard risk . In other words , it holds for any f that AdvRisk0 ( f ) = Risk ( f ) : = Prx∼µ [ f ( x ) 6= f∗ ( x ) ] . We remark that this definition assumes the existence of an underlying ground-truth labeling function , which does not apply to the agnostic setting where inputs can have non-deterministic labels . Initially introduced in Mahloujifar et al . ( 2019b ) , intrinsic robustness captures the maximum adversarial robustness that can be achieved by any imperfect classifier for a robust classification problem . Definition 2.2 ( Intrinsic Robustness ) . Consider the same setting as in Definition 2.1 . For any α > 0 , let Fα = { f : Risk ( f ) ≥ α } be the set of imperfect classifiers whose risk is at least α . Then the intrinsic robustness of the given robust classification problem with respect to Fα is defined as : AdvRob ( Fα ) = 1− inf f∈Fα { AdvRisk ( f ) } = sup f∈Fα { AdvRob ( f ) } . It is worth noting that the value of AdvRob ( Fα ) is only determined by the underlying input data distribution , the perturbation set and the risk threshold parameter α , which is independent of the model class one would choose for learning . By relating the robustness of a classifier to the -expansion of its induced error region , the following lemma , proved in Mahloujifar et al . ( 2019a ) , establishes a fundamental connection between the concentration of measure and the intrinsic robustness one can hope for a robust classification problem . Lemma 2.3 . Consider the same setting as in Definition 2.2 . Let h ( ∆ ) µ ( α , ) be the concentration function that captures the optimal value of the concentration of measure problem ( 2.1 ) : h ( ∆ ) µ ( α , ) = inf { µ ( E ( ∆ ) ) : E ∈ pow ( X ) and µ ( E ) ≥ α } . Then , AdvRob ( Fα ) = 1− h ( ∆ ) µ ( α , ) holds for any α > 0 and ≥ 0 . 1The standard notion of concentration of measure ( Talagrand , 1995 ) corresponds to the case where α = 0.5 . 2Other related definitions , such as the one used in most empirical works for robustness evaluation , are equivalent to this , as long as small perturbations preserve the ground truth . See Diochnos et al . ( 2018 ) for a detailed comparison of these and other definitions of adversarial robustness . Lemma 2.3 suggests that one can characterize the intrinsic robustness limit for a robust classification problem by measuring the concentration of the input data with respect to the perturbation metric . In this paper , we aim to understand and empirically estimate the intrinsic robustness limit for typical robust classification tasks by measuring concentration . It is worth noting that solving the concentration problem ( 2.1 ) itself only shows the existence of an error region E whose -expansion has certain ( small ) measure . This further implies the possibility of existing an optimally robust classifier ( with risk at least α ) , whose robustness matches the intrinsic robustness limit AdvRob ( Fα ) . However , actually finding such optimal classifier using a learning algorithm might be a much more challenging task , which is beyond the scope of this work .	The authors generalized the Gaussian Isoperimetric Inequality to non-spherical Gaussian measures with $\ell_p$ metric structures $p\geq 2$. Building on the generalized inequality, they propose a sample-based algorithm to estimate the concentration of measure using half-spaces. The main contribution is Theorem 3.3 followed by the empirical sample based algorithm for half-spaces that requires $\Omega(n\log(n) /\delta^2)$ samples.	SP:4ec13fd58b0cea4ef01b329c45b6c0042bc9f951
Improved Estimation of Concentration Under $\ell_p$-Norm Distance Metrics Using Half Spaces	1 INTRODUCTION . Despite achieving exceptional performance in benign settings , modern machine learning models have been shown to be highly vulnerable to inputs , known as adversarial examples , crafted with targeted but imperceptible perturbations ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) . This discovery has prompted a wave of research studies to propose defense mechanisms , including heuristic approaches ( Papernot et al. , 2016 ; Mądry et al. , 2018 ; Zhang et al. , 2019 ) and certifiable methods ( Wong & Kolter , 2018 ; Gowal et al. , 2019 ; Cohen et al. , 2019 ) . Unfortunately , none of these methods can successfully produce adversarially-robust models , even for classification tasks on toy datasets such as CIFAR-10 . To explain the prevalence of adversarial examples , a line of theoretical works ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Shafahi et al. , 2019 ; Dohmatob , 2019 ; Bhagoji et al. , 2019 ) have proven upper bounds on the maximum achievable adversarial robustness by imposing different assumptions on the underlying metric probability space . In particular , Mahloujifar et al . ( 2019a ) generalized the previous results showing that adversarial examples are inevitable as long as the input distributions are concentrated with respect to the perturbation metric . Thus , the question of whether or not natural image distributions are concentrated is highly relevant , as if they are it would rule out any possibility of there being adversarially robust image classifiers . Recently , Mahloujifar et al . ( 2019b ) proposed an empirical method to measure the concentration of an arbitrary distribution using data samples , then employed it to estimate a lower bound on intrinsic robustness ( see Definition 2.2 for its formal definition ) for several image benchmarks . By demonstrating the gap between the estimated bounds of intrinsic robustness and the robustness performance achieved by the best current models , they further concluded concentration of measure is not the sole reason behind the adversarial vulnerability of existing classifiers for benchmark image distributions . However , due to the heuristic nature of the proposed algorithm , it remains elusive whether the estimates it produces can serve as useful approximations of the underlying intrinsic robustness limits , thus hindering understanding of how much of the actual adversarial risk can be explained by the concentration of measure phenomenon . In this work , we address this issue by first characterizing the optimum of the actual concentration problem for general Gaussian spaces , then using our theoretical insights to develop an alternative algorithm for measuring concentration empirically that significantly improves both the accuracy and efficiency of estimates of intrinsic robustness . While we do not demonstrate a specific classifier which achieves this robustness upper bound , our results rule out inherent image distribution concentration as the reason for our current inability to find adversarially robust models . Contributions . We generalize the Gaussian Isoperimetric Inequality to non-spherical Gaussian distributions and ` p-norm distance metrics with p ≥ 2 ( including ` ∞ ) ( Theorem 3.3 ) . Motivated by the optimal concentration results for special Gaussian spaces ( Remark 3.4 ) , we develop a samplebased algorithm to estimate the concentration of measure using half spaces that works for arbitrary distribution and any ` p-norm distance ( Section 4 ) . Compared with prior approaches , we empirically demonstrate the significant increase in efficacy of our method under ` ∞-norm distance metric ( Section 6 ) . Not only does the proposed method converge to its limit with an order of magnitude fewer data ( Section 6.2 ) , it also finds a much tighter lower bound of intrinsic robustness for both simulated datasets whose underlying concentration function is analytically derivable and various benchmark image datasets ( Section 6.1 ) . In particular , we improve the best current estimated lower bound of intrinsic robustness from approximately 82 % to above 93 % for CIFAR-10 under ` ∞-norm bounded perturbations with = 8/255 . These tighter concentration estimates produced by our algorithm provide strong evidence that concentration of measure should not be considered as the main cause of adversarial vulnerability , at least for the image benchmarks evaluated in our experiments . Related Work . Several prior works have sought to empirically estimate lower bounds on intrinsic robustness using data samples . The pioneering work of Gilmer et al . ( 2018 ) introduced the connection between adversarial examples and the concentration phenomenon for uniform n-spheres , then proposed a simple heuristic to find a half space that expands slowly under Euclidean distance for the MNIST dataset . Our work can be seen as a strict generalization of Gilmer et al . ( 2018 ) ’ s , which applies to arbitrary ` p-norm distance metrics ( including ` ∞ ) . By characterizing the optimal transport cost between conditional distributions , Bhagoji et al . ( 2019 ) estimated a lower bound on the best possible adversarial robustness for several image datasets . However , when applied to adversaries beyond ` 2 , such as ` ∞ , the lower bound produced by their method is not informative ( that is , it is close to zero ) . The most relevant previous work is Mahloujifar et al . ( 2019b ) , which proposed a general method for measuring concentration using special collections of subsets . Although the optimal value of the considered empirical concentration problem is proven to asymptotically converge to the actual concentration , there is no guarantee that the proposed searching algorithm for solving the empirical problem finds the optimum . Our approach follows the framework introduced by Mahloujifar et al . ( 2019b ) ’ s , but considers a different collection of subsets for the empirical concentration problem . This not only results in optimality for theoretical Gaussian distributions , but also significantly improves the estimation performance for typical image benchmarks . Another line of work attempts to provide estimates of intrinsic robustness upper bounds based on generative assumptions . In order to justify the theoretically-derived impossibility results , Fawzi et al . ( 2018 ) estimated the smoothness parameters of the state-of-the-art generative models on CIFAR-10 and SVHN datasets , which yield approximated upper bounds on adversarial robustness for any classifiers . Zhang et al . ( 2020 ) generalized their results to non-smoothed data manifolds , such as datasets that can be captured by a conditional generative model . However , these methods only work for simulated generative distributions , which may deviate from the actual distributions they are intended to understand . Notation . For any n ∈ Z+ , denote by [ n ] the set { 1 , 2 , . . . , n } . Lowercase boldface letters denote vectors and uppercase boldface letters represent matrices . For any vector x and p ∈ [ 1 , ∞ ) , let xj , ‖x‖p and ‖x‖∞ be the j-th element , the ` p-norm and the ` ∞-norm of x . For any matrix A , B is said to be a square root of A if A = BB , and the induced matrix p-norm of A is defined as ‖A‖p = supx6=0 { ‖Ax‖p/‖x‖p } . Denote by N ( θ , Σ ) the Gaussian distribution with mean θ and covariance matrix Σ . Let γn be the probability measure of N ( 0 , In ) , where In denotes the identity matrix . Let Φ ( · ) be the cumulative distribution function of N ( 0 , 1 ) and Φ−1 ( · ) be its inverse . For any set A , let pow ( A ) and 1A ( · ) be all measurable subsets and the indicator function of A . Let ( X , µ , ∆ ) be a metric probability space , where ∆ : X × X → R≥0 denotes a distance metric on X . Define the empirical measure with respect to a sample set { xi } i∈ [ m ] as µ̂m ( A ) = 1m ∑ i∈ [ m ] 1A ( xi ) , ∀A ∈ pow ( X ) . Let B ( x , , ∆ ) = { x′ ∈ X : ∆ ( x′ , x ) ≤ } be the ball around x with radius . Define the -expansion of A as A ( ∆ ) = { x ∈ X : ∃ x′ ∈ B ( x , , ∆ ) ∩ A } . 2 PRELIMINARIES . In this section , we introduce the problem of measuring concentration and its connection to adversarial robustness . Consider a metric probability space of instances ( X , µ , ∆ ) . Given parameters ≥ 0 and α > 0 , the concentration of measure problem1 can be cast as the following optimization problem : minimize E∈Pow ( X ) µ ( E ( ∆ ) ) subject to µ ( E ) ≥ α . ( 2.1 ) We focus on the case where ∆ is some ` p-norm distance metric ( including ` ∞ ) in this work . Concentration of measure has been shown to be closely related to adversarial examples ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Mahloujifar et al. , 2019a ) . In particular , one can prove that for a given robust learning problem , if the input distribution is concentrated with respect to the perturbation metric , no adversarially robust model exists . The concentration parameter ( which corresponds to the optimal value of optimization problem ( 2.1 ) ) determines an inherent upper bound on the maximum adversarial robustness that any model can achieve for the given problem . To explain the connection between concentration of measure and robust learning in a more formal way , we lay out the definition of adversarial risk that we work with . We draw this definition from several previous works , including Gilmer et al . ( 2018 ) ; Bubeck et al . ( 2019 ) ; Mahloujifar et al . ( 2019a ; b ) .2 Definition 2.1 ( Adversarial Risk ) . Let ( X , µ , ∆ ) be the input metric probability space . Assume f∗ is the underlying ground-truth classifier that gives labels to any input . Given classifier f and ≥ 0 , the adversarial risk of f with respect to -perturbations measured by ∆ is defined as : AdvRisk ( f ) = Pr x∼µ [ ∃ x′ ∈ B ( x , , ∆ ) s.t . f ( x′ ) 6= f∗ ( x′ ) ] . Correspondingly , we define the adversarial robustness of f as AdvRob ( f ) = 1−AdvRisk ( f ) . When = 0 , adversarial risk degenerates to standard risk . In other words , it holds for any f that AdvRisk0 ( f ) = Risk ( f ) : = Prx∼µ [ f ( x ) 6= f∗ ( x ) ] . We remark that this definition assumes the existence of an underlying ground-truth labeling function , which does not apply to the agnostic setting where inputs can have non-deterministic labels . Initially introduced in Mahloujifar et al . ( 2019b ) , intrinsic robustness captures the maximum adversarial robustness that can be achieved by any imperfect classifier for a robust classification problem . Definition 2.2 ( Intrinsic Robustness ) . Consider the same setting as in Definition 2.1 . For any α > 0 , let Fα = { f : Risk ( f ) ≥ α } be the set of imperfect classifiers whose risk is at least α . Then the intrinsic robustness of the given robust classification problem with respect to Fα is defined as : AdvRob ( Fα ) = 1− inf f∈Fα { AdvRisk ( f ) } = sup f∈Fα { AdvRob ( f ) } . It is worth noting that the value of AdvRob ( Fα ) is only determined by the underlying input data distribution , the perturbation set and the risk threshold parameter α , which is independent of the model class one would choose for learning . By relating the robustness of a classifier to the -expansion of its induced error region , the following lemma , proved in Mahloujifar et al . ( 2019a ) , establishes a fundamental connection between the concentration of measure and the intrinsic robustness one can hope for a robust classification problem . Lemma 2.3 . Consider the same setting as in Definition 2.2 . Let h ( ∆ ) µ ( α , ) be the concentration function that captures the optimal value of the concentration of measure problem ( 2.1 ) : h ( ∆ ) µ ( α , ) = inf { µ ( E ( ∆ ) ) : E ∈ pow ( X ) and µ ( E ) ≥ α } . Then , AdvRob ( Fα ) = 1− h ( ∆ ) µ ( α , ) holds for any α > 0 and ≥ 0 . 1The standard notion of concentration of measure ( Talagrand , 1995 ) corresponds to the case where α = 0.5 . 2Other related definitions , such as the one used in most empirical works for robustness evaluation , are equivalent to this , as long as small perturbations preserve the ground truth . See Diochnos et al . ( 2018 ) for a detailed comparison of these and other definitions of adversarial robustness . Lemma 2.3 suggests that one can characterize the intrinsic robustness limit for a robust classification problem by measuring the concentration of the input data with respect to the perturbation metric . In this paper , we aim to understand and empirically estimate the intrinsic robustness limit for typical robust classification tasks by measuring concentration . It is worth noting that solving the concentration problem ( 2.1 ) itself only shows the existence of an error region E whose -expansion has certain ( small ) measure . This further implies the possibility of existing an optimally robust classifier ( with risk at least α ) , whose robustness matches the intrinsic robustness limit AdvRob ( Fα ) . However , actually finding such optimal classifier using a learning algorithm might be a much more challenging task , which is beyond the scope of this work .	The authors consider the problem of estimating intrinsic robustness using data samples. At a high level, intrinsic robustness is a measure that indicates the probability that a noisy version of a covariate would not be mislabeled. Mahloujifar et al. (2019a) have shown that estimating intrinsic robustness is closely related to the problem of estimating the concentration of measure. This problem focuses on finding a region such that it has the least likely epsilon neighborhood. In this work, the authors provide an efficient algorithm for measuring concentration empirically, which yields a more accurate estimation of the intrinsic robustness compared to the previous results.	SP:4ec13fd58b0cea4ef01b329c45b6c0042bc9f951
Improved Estimation of Concentration Under $\ell_p$-Norm Distance Metrics Using Half Spaces	1 INTRODUCTION . Despite achieving exceptional performance in benign settings , modern machine learning models have been shown to be highly vulnerable to inputs , known as adversarial examples , crafted with targeted but imperceptible perturbations ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) . This discovery has prompted a wave of research studies to propose defense mechanisms , including heuristic approaches ( Papernot et al. , 2016 ; Mądry et al. , 2018 ; Zhang et al. , 2019 ) and certifiable methods ( Wong & Kolter , 2018 ; Gowal et al. , 2019 ; Cohen et al. , 2019 ) . Unfortunately , none of these methods can successfully produce adversarially-robust models , even for classification tasks on toy datasets such as CIFAR-10 . To explain the prevalence of adversarial examples , a line of theoretical works ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Shafahi et al. , 2019 ; Dohmatob , 2019 ; Bhagoji et al. , 2019 ) have proven upper bounds on the maximum achievable adversarial robustness by imposing different assumptions on the underlying metric probability space . In particular , Mahloujifar et al . ( 2019a ) generalized the previous results showing that adversarial examples are inevitable as long as the input distributions are concentrated with respect to the perturbation metric . Thus , the question of whether or not natural image distributions are concentrated is highly relevant , as if they are it would rule out any possibility of there being adversarially robust image classifiers . Recently , Mahloujifar et al . ( 2019b ) proposed an empirical method to measure the concentration of an arbitrary distribution using data samples , then employed it to estimate a lower bound on intrinsic robustness ( see Definition 2.2 for its formal definition ) for several image benchmarks . By demonstrating the gap between the estimated bounds of intrinsic robustness and the robustness performance achieved by the best current models , they further concluded concentration of measure is not the sole reason behind the adversarial vulnerability of existing classifiers for benchmark image distributions . However , due to the heuristic nature of the proposed algorithm , it remains elusive whether the estimates it produces can serve as useful approximations of the underlying intrinsic robustness limits , thus hindering understanding of how much of the actual adversarial risk can be explained by the concentration of measure phenomenon . In this work , we address this issue by first characterizing the optimum of the actual concentration problem for general Gaussian spaces , then using our theoretical insights to develop an alternative algorithm for measuring concentration empirically that significantly improves both the accuracy and efficiency of estimates of intrinsic robustness . While we do not demonstrate a specific classifier which achieves this robustness upper bound , our results rule out inherent image distribution concentration as the reason for our current inability to find adversarially robust models . Contributions . We generalize the Gaussian Isoperimetric Inequality to non-spherical Gaussian distributions and ` p-norm distance metrics with p ≥ 2 ( including ` ∞ ) ( Theorem 3.3 ) . Motivated by the optimal concentration results for special Gaussian spaces ( Remark 3.4 ) , we develop a samplebased algorithm to estimate the concentration of measure using half spaces that works for arbitrary distribution and any ` p-norm distance ( Section 4 ) . Compared with prior approaches , we empirically demonstrate the significant increase in efficacy of our method under ` ∞-norm distance metric ( Section 6 ) . Not only does the proposed method converge to its limit with an order of magnitude fewer data ( Section 6.2 ) , it also finds a much tighter lower bound of intrinsic robustness for both simulated datasets whose underlying concentration function is analytically derivable and various benchmark image datasets ( Section 6.1 ) . In particular , we improve the best current estimated lower bound of intrinsic robustness from approximately 82 % to above 93 % for CIFAR-10 under ` ∞-norm bounded perturbations with = 8/255 . These tighter concentration estimates produced by our algorithm provide strong evidence that concentration of measure should not be considered as the main cause of adversarial vulnerability , at least for the image benchmarks evaluated in our experiments . Related Work . Several prior works have sought to empirically estimate lower bounds on intrinsic robustness using data samples . The pioneering work of Gilmer et al . ( 2018 ) introduced the connection between adversarial examples and the concentration phenomenon for uniform n-spheres , then proposed a simple heuristic to find a half space that expands slowly under Euclidean distance for the MNIST dataset . Our work can be seen as a strict generalization of Gilmer et al . ( 2018 ) ’ s , which applies to arbitrary ` p-norm distance metrics ( including ` ∞ ) . By characterizing the optimal transport cost between conditional distributions , Bhagoji et al . ( 2019 ) estimated a lower bound on the best possible adversarial robustness for several image datasets . However , when applied to adversaries beyond ` 2 , such as ` ∞ , the lower bound produced by their method is not informative ( that is , it is close to zero ) . The most relevant previous work is Mahloujifar et al . ( 2019b ) , which proposed a general method for measuring concentration using special collections of subsets . Although the optimal value of the considered empirical concentration problem is proven to asymptotically converge to the actual concentration , there is no guarantee that the proposed searching algorithm for solving the empirical problem finds the optimum . Our approach follows the framework introduced by Mahloujifar et al . ( 2019b ) ’ s , but considers a different collection of subsets for the empirical concentration problem . This not only results in optimality for theoretical Gaussian distributions , but also significantly improves the estimation performance for typical image benchmarks . Another line of work attempts to provide estimates of intrinsic robustness upper bounds based on generative assumptions . In order to justify the theoretically-derived impossibility results , Fawzi et al . ( 2018 ) estimated the smoothness parameters of the state-of-the-art generative models on CIFAR-10 and SVHN datasets , which yield approximated upper bounds on adversarial robustness for any classifiers . Zhang et al . ( 2020 ) generalized their results to non-smoothed data manifolds , such as datasets that can be captured by a conditional generative model . However , these methods only work for simulated generative distributions , which may deviate from the actual distributions they are intended to understand . Notation . For any n ∈ Z+ , denote by [ n ] the set { 1 , 2 , . . . , n } . Lowercase boldface letters denote vectors and uppercase boldface letters represent matrices . For any vector x and p ∈ [ 1 , ∞ ) , let xj , ‖x‖p and ‖x‖∞ be the j-th element , the ` p-norm and the ` ∞-norm of x . For any matrix A , B is said to be a square root of A if A = BB , and the induced matrix p-norm of A is defined as ‖A‖p = supx6=0 { ‖Ax‖p/‖x‖p } . Denote by N ( θ , Σ ) the Gaussian distribution with mean θ and covariance matrix Σ . Let γn be the probability measure of N ( 0 , In ) , where In denotes the identity matrix . Let Φ ( · ) be the cumulative distribution function of N ( 0 , 1 ) and Φ−1 ( · ) be its inverse . For any set A , let pow ( A ) and 1A ( · ) be all measurable subsets and the indicator function of A . Let ( X , µ , ∆ ) be a metric probability space , where ∆ : X × X → R≥0 denotes a distance metric on X . Define the empirical measure with respect to a sample set { xi } i∈ [ m ] as µ̂m ( A ) = 1m ∑ i∈ [ m ] 1A ( xi ) , ∀A ∈ pow ( X ) . Let B ( x , , ∆ ) = { x′ ∈ X : ∆ ( x′ , x ) ≤ } be the ball around x with radius . Define the -expansion of A as A ( ∆ ) = { x ∈ X : ∃ x′ ∈ B ( x , , ∆ ) ∩ A } . 2 PRELIMINARIES . In this section , we introduce the problem of measuring concentration and its connection to adversarial robustness . Consider a metric probability space of instances ( X , µ , ∆ ) . Given parameters ≥ 0 and α > 0 , the concentration of measure problem1 can be cast as the following optimization problem : minimize E∈Pow ( X ) µ ( E ( ∆ ) ) subject to µ ( E ) ≥ α . ( 2.1 ) We focus on the case where ∆ is some ` p-norm distance metric ( including ` ∞ ) in this work . Concentration of measure has been shown to be closely related to adversarial examples ( Gilmer et al. , 2018 ; Fawzi et al. , 2018 ; Mahloujifar et al. , 2019a ) . In particular , one can prove that for a given robust learning problem , if the input distribution is concentrated with respect to the perturbation metric , no adversarially robust model exists . The concentration parameter ( which corresponds to the optimal value of optimization problem ( 2.1 ) ) determines an inherent upper bound on the maximum adversarial robustness that any model can achieve for the given problem . To explain the connection between concentration of measure and robust learning in a more formal way , we lay out the definition of adversarial risk that we work with . We draw this definition from several previous works , including Gilmer et al . ( 2018 ) ; Bubeck et al . ( 2019 ) ; Mahloujifar et al . ( 2019a ; b ) .2 Definition 2.1 ( Adversarial Risk ) . Let ( X , µ , ∆ ) be the input metric probability space . Assume f∗ is the underlying ground-truth classifier that gives labels to any input . Given classifier f and ≥ 0 , the adversarial risk of f with respect to -perturbations measured by ∆ is defined as : AdvRisk ( f ) = Pr x∼µ [ ∃ x′ ∈ B ( x , , ∆ ) s.t . f ( x′ ) 6= f∗ ( x′ ) ] . Correspondingly , we define the adversarial robustness of f as AdvRob ( f ) = 1−AdvRisk ( f ) . When = 0 , adversarial risk degenerates to standard risk . In other words , it holds for any f that AdvRisk0 ( f ) = Risk ( f ) : = Prx∼µ [ f ( x ) 6= f∗ ( x ) ] . We remark that this definition assumes the existence of an underlying ground-truth labeling function , which does not apply to the agnostic setting where inputs can have non-deterministic labels . Initially introduced in Mahloujifar et al . ( 2019b ) , intrinsic robustness captures the maximum adversarial robustness that can be achieved by any imperfect classifier for a robust classification problem . Definition 2.2 ( Intrinsic Robustness ) . Consider the same setting as in Definition 2.1 . For any α > 0 , let Fα = { f : Risk ( f ) ≥ α } be the set of imperfect classifiers whose risk is at least α . Then the intrinsic robustness of the given robust classification problem with respect to Fα is defined as : AdvRob ( Fα ) = 1− inf f∈Fα { AdvRisk ( f ) } = sup f∈Fα { AdvRob ( f ) } . It is worth noting that the value of AdvRob ( Fα ) is only determined by the underlying input data distribution , the perturbation set and the risk threshold parameter α , which is independent of the model class one would choose for learning . By relating the robustness of a classifier to the -expansion of its induced error region , the following lemma , proved in Mahloujifar et al . ( 2019a ) , establishes a fundamental connection between the concentration of measure and the intrinsic robustness one can hope for a robust classification problem . Lemma 2.3 . Consider the same setting as in Definition 2.2 . Let h ( ∆ ) µ ( α , ) be the concentration function that captures the optimal value of the concentration of measure problem ( 2.1 ) : h ( ∆ ) µ ( α , ) = inf { µ ( E ( ∆ ) ) : E ∈ pow ( X ) and µ ( E ) ≥ α } . Then , AdvRob ( Fα ) = 1− h ( ∆ ) µ ( α , ) holds for any α > 0 and ≥ 0 . 1The standard notion of concentration of measure ( Talagrand , 1995 ) corresponds to the case where α = 0.5 . 2Other related definitions , such as the one used in most empirical works for robustness evaluation , are equivalent to this , as long as small perturbations preserve the ground truth . See Diochnos et al . ( 2018 ) for a detailed comparison of these and other definitions of adversarial robustness . Lemma 2.3 suggests that one can characterize the intrinsic robustness limit for a robust classification problem by measuring the concentration of the input data with respect to the perturbation metric . In this paper , we aim to understand and empirically estimate the intrinsic robustness limit for typical robust classification tasks by measuring concentration . It is worth noting that solving the concentration problem ( 2.1 ) itself only shows the existence of an error region E whose -expansion has certain ( small ) measure . This further implies the possibility of existing an optimally robust classifier ( with risk at least α ) , whose robustness matches the intrinsic robustness limit AdvRob ( Fα ) . However , actually finding such optimal classifier using a learning algorithm might be a much more challenging task , which is beyond the scope of this work .	This paper considers the estimation of the concentration of measures, which possibly causes of the vulnerability of machine learning models to adversarial attacks. Towards such a goal, the authors first extend the Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary l_p norm. An algorithm that uses half-spaces as feasible set in the concentration of measure problem is then proposed. Empirical studies are conducted to show the efficiency as well as the efficacy of the proposed method when compared to [Mahloujifar et al. (2019b)].	SP:4ec13fd58b0cea4ef01b329c45b6c0042bc9f951
CausalWorld: A Robotic Manipulation Benchmark for Causal Structure and Transfer Learning	1 INTRODUCTION Benchmarks have played a crucial role in advancing entire research fields , for instance computer vision with the introduction of CIFAR-10 and ImageNet ( Krizhevsky et al. , 2009 ; 2012 ) . When it comes to the field of reinforcement learning ( RL ) , similar breakthroughs have been achieved in domains such as game playing ( Mnih et al. , 2013 ; Silver et al. , 2017 ) , learning motor control for high-dimensional simulated robots ( Akkaya et al. , 2019 ) , multi-agent settings ( Baker et al. , 2019 ; Berner et al. , 2019 ) and for studying transfer in the context of meta-learning ( Yu et al. , 2019 ) . Nevertheless , trained agents often fail to transfer the knowledge about the learned skills from a training environment to a different but related environment sharing part of the underlying structure . This can be attributed to the fact that it is quite common to evaluate an agent on the training environments themselves , which leads to over- fitting on these narrowly defined environments ( Whiteson et al. , 2011 ) , or that algorithms are com∗Equal contribution . Correspondence to : < ossama.ahmed @ mail.mcgill.ca > , < frederik.traeuble @ tuebingen.mpg.de > †Equal Advising 1 https : //sites.google.com/view/causal-world/home pared using highly engineered and biased reward functions which may result in learning suboptimal policies with respect to the desired behaviour ; this is particularly evident in robotics . In existing benchmarks ( Yu et al. , 2019 ; Goyal et al. , 2019a ; Cobbe et al. , 2018 ; Bellemare et al. , 2013 ; James et al. , 2020 ) the amount of shared causal structure between the different environments is mostly unknown . For instance , in the Atari Arcade Learning environments , it is unclear how to quantify the underlying similarities between different Atari games and we generally do not know to which degree an agent can be expected to generalize . To overcome these limitations , we introduce a novel benchmark in a robotic manipulation environment that we call CausalWorld . It features a diverse set of environments that , in contrast to previous designs , share a large set of parameters and parts of the causal structure . Being able to intervene on these parameters ( individually or collectively ) permits the experimenter to evaluate agents ’ generalization abilities with respect to different types and magnitudes of changes in the environment . These parameters can be varied gradually , which yields a continuum of similar environments . This allows for fine-grained control of training and test distributions and the design of learning curricula . A remarkable skill that humans learn to master early on in their life is building complex structures using spatial-reasoning and dexterous manipulation abilities ( Casey et al. , 2008 ; Caldera et al. , 1999 ; Kamii et al. , 2004 ) . Playing with toy blocks constitutes a natural environment for children to develop important visual-spatial skills , helping them ‘ generalize ’ in building complex composition designs from presented or imagined goal structures ( Verdine et al. , 2017 ; Nath & Szücs , 2014 ; Dewar , 2018 ; Richardson et al. , 2014 ) . Inspired by this , CausalWorld is designed to aid in learning and investigating these skills in a simulated robotic manipulation environment corresponding to the open-source TriFinger robot platform from Wüthrich et al . ( 2020 ) , which can be built in the real world . Tasks are formulated as building 3D goal shapes using a set of available blocks by manipulating them - as seen in Fig . 1 . This yields a diverse familiy of tasks , ranging from relatively simple ( e.g . pushing a single object ) to extremely hard ( e.g . building a complex structure from a large number of objects ) . CausalWorld improves upon previous benchmarks by exposing a large set of parameters in the causal generative model of the environments , such as weight , shape and appearance of the building blocks and the robot itself . The possibility of intervening on any of these properties at any point in time allows one to set up training curricula or to evaluate an agent ’ s generalization capability with respect to different parameters . Furthermore , in contrast to previous benchmarks ( ChevalierBoisvert et al. , 2018 ; Cobbe et al. , 2018 ) , researchers may build their own real-world platform of this simulator at low cost , as detailed in Wüthrich et al . ( 2020 ) , and transfer their trained policies to the real world . Finally , by releasing this benchmark we hope to facilitate research in causal structure learning , i.e . learning the causal graph ( or certain aspects of it ) as we operate in a complex real-world environment whose dynamics follow the laws of physics , which induce causal relations between the variables . Changes to the variables we expose can be considered do-interventions on the underlying structural causal model ( SCM ) . Consequently , we believe that this benchmark offers an exciting opportunity to investigate causality and its connection to RL and robotics . Our main contributions can be summarized as follows : • We propose CausalWorld , a new benchmark comprising a parametrized family of robotic manipulation environments for advancing out-of-distribution generalization and causal structure learning in RL . • We provide a systematic way of defining curricula and disentangling generalization abilities of RL agents with respect to different changes in the environment , since we allow for dointerventions to be performed on different environment variables ( parameters and states ) individually . • We establish baseline results for some of the available tasks under different learning algorithms , thus verifying the feasibility of the tasks . • We show how different learning curricula affect generalization across different axes by reporting some of the in-distribution and out-of-distribution generalization capabilities of the trained agents . 2 CAUSALWORLD BENCHMARK . Here we make the desiderata outlined in the introduction more precise : 1 . The set of environments should be sufficiently diverse to allow for the design of challenging transfer tasks . 2 . We need to be able to intervene on different properties ( e.g . masses , colors ) individually , such that we can investigate different types of generalization . 3 . It should be possible to convert any environment to any other environment by gradually changing its properties through interventions ; this requirement is important for evaluating different levels of transfer and for defining curricula . 4 . The environments should share some causal structure to allow algorithms to transfer the learned causal knowledge from one environment to another . 5 . There should be a unified measure of success , such that an objective comparison can be made between different learning algorithms . 6 . The benchmark should make it easy for users to define meaningful distributions of environments for training and evaluation . In particular , it should facilitate evaluation of indistribution and out-of-distribution performance . 7 . The simulated benchmark should have a real-world counterpart to allow for sim2real . In light of these desiderata , we propose a setup in which a robot must build goal shapes using a set of available objects . It is worth noting that similar setups were proposed previously in a less realistic setting , e.g . in ( Janner et al. , 2018 ; Bapst et al. , 2019 ; McCarthy et al . ; Akkaya et al. , 2019 ; Fahlman , 1974 ; Winston , 1970 ; Winograd , 1972 ) . Specifically , a task is formulated as follows : given a set of available objects , the agent needs to build a specific goal structure , see Fig . 1 for an example . The vast amount of possible target shapes and environment properties ( e.g . mass , shape and appearance of objects and the robot itself ) makes this a diverse and challenging setting to evaluate different generalization aspects . CausalWorld is a simulated version ( using the Bullet physics engine ( Coumans et al. , 2013 ) ) of the open-source TriFinger robot platform from Wüthrich et al . ( 2020 ) . Each environment is defined by a set of variables such , as gravity , floor friction , stage color , floor color , joint positions , various block parameters ( e.g . size , color , mass , position , orientation ) , link colors , link masses and the goal shape . See Table 3 in the Appendix for a more extensive list of these variables . Desideratum 1 is satisfied since different environment properties and goal shapes give rise to very different tasks , ranging from relatively easy ( e.g . re-positioning a single cube ) to extremely hard ( e.g . building a complex structure ) . Desideratum 2 is satisfied because we allow for arbitrary interventions on these properties , hence users or agents may change parameters individually or jointly . Desideratum 3 is satisfied because the parameters can be changed gradually . Desideratum 4 is satisfied because all the environments share the causal structure of the robot , and one may also use subsets of environments which share even more causal structure . We satisfy desideratum 5 by defining the measure of success for all environments as the volumetric overlap of the goal shape with available objects . Further , by splitting the set of parameters into a set A , intended for training and in-distribution evaluation , and a set B , intended for out-of-distribution evaluation , we satisfy desideratum 6 . Finally , since the TriFinger robot ( Wüthrich et al. , 2020 ) can be built in the real-world , we satisfy desideratum 7 . Desideratum 7 and 2 are in partial conflict since sim2real is only possible for the tasks which are constrained to the variables on which the robot can physically act upon . Task generators : To generate meaningful families of similar goal shapes , CausalWorld allows for defining task generators which can generate a variety of different goal shapes in an environment . For instance , one task generator may generate pushing tasks , while another one may generate towerbuilding tasks ( see Fig . 2 ) . Each task generator is initialized with a default goal shape from its corresponding family and comes with a sampler to sample new goal shapes from the same family . Additionally , upon construction , one can specify the environments ’ initial state and initial goal shape structure when deviating from the default . The maximum episode time to build a given shape is number of blocks×10 seconds . CausalWorld comes with eight pre-defined task generators ( see Fig . 2 ) . • Three generators create goal shapes with a single block : Pushing with the goal shape on the floor , Picking having the goal shape defined above the floor and Pick and Place where a fixed obstacle is placed between the initial block and goal pose . • Stacking2 involves a goal shape of two stacked blocks , which can also be considered one instance of the Towers generator . • The remaining generators use a variable number of blocks to generate much more complex and challenging target shapes , as detailed in the appendix : Towers , Stacked Blocks , Creative Stacked Blocks and General . Given that building new environments using current physics simulators is often tedious , we provide a simple API for users who wish to create task generators for new challenging shape families , which may be added to CausalWorld ’ s task generators repository . Action and Observation Spaces : The robot can be chosen to operate in either joint position control mode , joint torque control mode , end-effector position control mode , or the delta of each . In any of these cases , the action is 9-dimensional ( one per joint ) . We provide two observation modes : structured and pixel . In the structured mode , the observation vector is constructed using a rule for the ordering of the relevant variables , such as joint positions , joint velocities , block positions , etc . Thus , the size of the observation space depends on the number of blocks , which could potentially change with every new goal sampled , e.g . in Towers , ( Creative ) Stacked Blocks and General . In contrast , in the pixel mode , the agent receives six RGB images ( hence the dimension of the observation is 6×3×128×128 ) , the first three images are rendered from the three cameras mounted around the TriFinger robot , and the last three images specify the goal image of the target shape rendered from the same cameras . Additionally , CausalWorld allows users to set up a fully customized observation space . Rewards : The reward function r is defined as the fractional volumetric overlap of the blocks with the goal shape , which ranges between 0 ( no overlap ) and 1 ( complete overlap ) . Since this reward function is shared across all tasks , an agent that learned r from some training tasks could in principle use it to solve unseen goal structures . There is also the possibility of modifying the reward function to 1 ) sparsify the reward further by returning a binary reward signal instead , or 2 ) add a dense reward function in order to introduce inductive biases via domain knowledge and solution guidance . We hope that the considerable complexity and diversity of goal shapes motivate and accelerate the development of algorithms that are not dependent on hand-tuned reward functions . ` Training and evaluation spaces : In this benchmark , a learning setting consists of an allowed training space ( ATS ) and an evaluation space ( ES ) , both of which are subspaces of the full parameter space . During training , in the simplest setting , parameters are sampled iid from the ATS . However , unlike existing benchmarks , CausalWorld allows in addition for curricula within the ATS as well as settings where the agent itself intervenes on the parameters within an episode ( see Fig . 3 ) . Similarly , during evaluation , parameters may be sampled iid from the evaluation space at each episode reset , or there can be interventions within an episode . Moreover , in order to retrieve the setting considered in most RL benchmarks , we could set the ATS and the ES to be identical and intervene only on object and robot states ( and keep other environment properties constant ) at each episode reset . However , to evaluate out-of-distribution generalization , one should set the two spaces ( ATS and ES ) to be different ; possibly even disjoint . Additionally , to evaluate robustness with respect to a specific parameter ( e.g . object mass ) , one may define the training and evaluation spaces which only differ in that particular parameter . In order to facilitate the definition of appropriate training and evaluation settings , we pre-define two disjoint sets , Ai and Bi , for each parameter i . Through this , one can for instance define the training space to be A1 ×A2 × ... and the evaluation space to be B1 ×B2 × ... to assess generalization with respect to all parameters simultaneously . Alternatively , the evaluation space could be defined as A1 ×A2 × ... ×Bi ×Ai+1 × ... to assess generalization with respect to parameter i only . Lastly , users may also define their own spaces which could then be integrated into the benchmark to give rise to new learning settings . Intervention actors : To provide a convenient way of specifying learning curricula , we introduce intervention actors . At each time step , such an actor takes all the exposed variables of the environment as inputs and may intervene on them . To encourage modularity , one may combine multiple actors in a learning curriculum . This actor is defined by the episode number to start intervening , the episode number to stop intervening , the timestep within the episode it should intervene and the episode periodicity of interventions . We provide a set of predefined intervention actors , including an actor which samples parameters randomly at each episode reset , which corresponds to domainrandomization . It is also easy to define custom intervention actors , we hope that this facilitates investigation into optimal learning curricula ( see Fig . 3 ) . Probing the Causal Structure in RL : The problem setting in RL is usually formulated using the language of Markov Decision Processes ( MDPs ) or Partially Observable Markov Decision Processes ( POMDPs ) ( Sutton & Barto , 1999 ) , but can be also represented by Structural Causal Models ( SCMs ) , as shown in ( Buesing et al. , 2018 ) , refer to section E in the Appendix for a detailed explanation . This is achieved by formulating all conditional probability distributions as deterministic functions that take independent noise variables as inputs . These independent noise variables can specify different scenarios while the deterministic functions reflect the causal mechanisms of the system ( Schölkopf et al. , 2021 ) . Changes in the environment can stem from two different sources : 1 . The agent may alter the state of the environment ( e.g . the position of a block ) indirectly , through its actions ( e.g . pushing the block by applying appropriate torques at the motors ) . 2 . During the execution of a learning curriculum or an evaluation protocol , we may directly intervene on any variable of the SCM , including all the latent variables of the causal model that are not accessible to the RL agent ( such as gravity , object mass or color ) . ( 1 ) is the default type of admissible interventions in RL benchmarks , whereas CausalWorld allows for interventions of type ( 2 ) in addition . The idea is that interventions on these latent variables , e.g . during a learning curriculum , will allow the agent to distinguish between spurious correlations that are only present in a particular setting and true causal relations that will hold across all settings ( i.e . interventions ) . If the agent is able to learn such a representation of the underlying SCM structure , we would expect it to perform well even in out-of-distribution scenarios ( Schölkopf et al. , 2021 ; Dittadi et al. , 2021 ) because the causal structure remains the same , even when the functional form of certain relations may vary ( e.g . when transferring to the real robot ) . Moreover , we hope that by having access to a broad range of interventions in CausalWorld it will aid the inference of the underlying SCM structure through the different causal discovery methods ( see Figure 4 for a subset of the expected SCM to be learned ) , which in turn addresses the lack of causal discovery benchmarks for real world challenges . 𝑓1 𝑓2 𝑓3 𝑓4	This paper proposed a new benchmark for studying reinforcement learning and its generalization in the context of the robotic manipulation problem. To study the generalization of a learned policy, the proposed benchmark is equipped with an interface that makes intervention easy. This interface helps to define a training space and an evaluation space so that one can systemically study both in-distribution and out-of-distribution generalization of a learned policy. At the same time, the proposed benchmark simulates an open-source robot platform, which makes sim2real transfer experiments easier.	SP:e37da841052cbdd81b629bdb5c126aa1a375d7e7
CausalWorld: A Robotic Manipulation Benchmark for Causal Structure and Transfer Learning	1 INTRODUCTION Benchmarks have played a crucial role in advancing entire research fields , for instance computer vision with the introduction of CIFAR-10 and ImageNet ( Krizhevsky et al. , 2009 ; 2012 ) . When it comes to the field of reinforcement learning ( RL ) , similar breakthroughs have been achieved in domains such as game playing ( Mnih et al. , 2013 ; Silver et al. , 2017 ) , learning motor control for high-dimensional simulated robots ( Akkaya et al. , 2019 ) , multi-agent settings ( Baker et al. , 2019 ; Berner et al. , 2019 ) and for studying transfer in the context of meta-learning ( Yu et al. , 2019 ) . Nevertheless , trained agents often fail to transfer the knowledge about the learned skills from a training environment to a different but related environment sharing part of the underlying structure . This can be attributed to the fact that it is quite common to evaluate an agent on the training environments themselves , which leads to over- fitting on these narrowly defined environments ( Whiteson et al. , 2011 ) , or that algorithms are com∗Equal contribution . Correspondence to : < ossama.ahmed @ mail.mcgill.ca > , < frederik.traeuble @ tuebingen.mpg.de > †Equal Advising 1 https : //sites.google.com/view/causal-world/home pared using highly engineered and biased reward functions which may result in learning suboptimal policies with respect to the desired behaviour ; this is particularly evident in robotics . In existing benchmarks ( Yu et al. , 2019 ; Goyal et al. , 2019a ; Cobbe et al. , 2018 ; Bellemare et al. , 2013 ; James et al. , 2020 ) the amount of shared causal structure between the different environments is mostly unknown . For instance , in the Atari Arcade Learning environments , it is unclear how to quantify the underlying similarities between different Atari games and we generally do not know to which degree an agent can be expected to generalize . To overcome these limitations , we introduce a novel benchmark in a robotic manipulation environment that we call CausalWorld . It features a diverse set of environments that , in contrast to previous designs , share a large set of parameters and parts of the causal structure . Being able to intervene on these parameters ( individually or collectively ) permits the experimenter to evaluate agents ’ generalization abilities with respect to different types and magnitudes of changes in the environment . These parameters can be varied gradually , which yields a continuum of similar environments . This allows for fine-grained control of training and test distributions and the design of learning curricula . A remarkable skill that humans learn to master early on in their life is building complex structures using spatial-reasoning and dexterous manipulation abilities ( Casey et al. , 2008 ; Caldera et al. , 1999 ; Kamii et al. , 2004 ) . Playing with toy blocks constitutes a natural environment for children to develop important visual-spatial skills , helping them ‘ generalize ’ in building complex composition designs from presented or imagined goal structures ( Verdine et al. , 2017 ; Nath & Szücs , 2014 ; Dewar , 2018 ; Richardson et al. , 2014 ) . Inspired by this , CausalWorld is designed to aid in learning and investigating these skills in a simulated robotic manipulation environment corresponding to the open-source TriFinger robot platform from Wüthrich et al . ( 2020 ) , which can be built in the real world . Tasks are formulated as building 3D goal shapes using a set of available blocks by manipulating them - as seen in Fig . 1 . This yields a diverse familiy of tasks , ranging from relatively simple ( e.g . pushing a single object ) to extremely hard ( e.g . building a complex structure from a large number of objects ) . CausalWorld improves upon previous benchmarks by exposing a large set of parameters in the causal generative model of the environments , such as weight , shape and appearance of the building blocks and the robot itself . The possibility of intervening on any of these properties at any point in time allows one to set up training curricula or to evaluate an agent ’ s generalization capability with respect to different parameters . Furthermore , in contrast to previous benchmarks ( ChevalierBoisvert et al. , 2018 ; Cobbe et al. , 2018 ) , researchers may build their own real-world platform of this simulator at low cost , as detailed in Wüthrich et al . ( 2020 ) , and transfer their trained policies to the real world . Finally , by releasing this benchmark we hope to facilitate research in causal structure learning , i.e . learning the causal graph ( or certain aspects of it ) as we operate in a complex real-world environment whose dynamics follow the laws of physics , which induce causal relations between the variables . Changes to the variables we expose can be considered do-interventions on the underlying structural causal model ( SCM ) . Consequently , we believe that this benchmark offers an exciting opportunity to investigate causality and its connection to RL and robotics . Our main contributions can be summarized as follows : • We propose CausalWorld , a new benchmark comprising a parametrized family of robotic manipulation environments for advancing out-of-distribution generalization and causal structure learning in RL . • We provide a systematic way of defining curricula and disentangling generalization abilities of RL agents with respect to different changes in the environment , since we allow for dointerventions to be performed on different environment variables ( parameters and states ) individually . • We establish baseline results for some of the available tasks under different learning algorithms , thus verifying the feasibility of the tasks . • We show how different learning curricula affect generalization across different axes by reporting some of the in-distribution and out-of-distribution generalization capabilities of the trained agents . 2 CAUSALWORLD BENCHMARK . Here we make the desiderata outlined in the introduction more precise : 1 . The set of environments should be sufficiently diverse to allow for the design of challenging transfer tasks . 2 . We need to be able to intervene on different properties ( e.g . masses , colors ) individually , such that we can investigate different types of generalization . 3 . It should be possible to convert any environment to any other environment by gradually changing its properties through interventions ; this requirement is important for evaluating different levels of transfer and for defining curricula . 4 . The environments should share some causal structure to allow algorithms to transfer the learned causal knowledge from one environment to another . 5 . There should be a unified measure of success , such that an objective comparison can be made between different learning algorithms . 6 . The benchmark should make it easy for users to define meaningful distributions of environments for training and evaluation . In particular , it should facilitate evaluation of indistribution and out-of-distribution performance . 7 . The simulated benchmark should have a real-world counterpart to allow for sim2real . In light of these desiderata , we propose a setup in which a robot must build goal shapes using a set of available objects . It is worth noting that similar setups were proposed previously in a less realistic setting , e.g . in ( Janner et al. , 2018 ; Bapst et al. , 2019 ; McCarthy et al . ; Akkaya et al. , 2019 ; Fahlman , 1974 ; Winston , 1970 ; Winograd , 1972 ) . Specifically , a task is formulated as follows : given a set of available objects , the agent needs to build a specific goal structure , see Fig . 1 for an example . The vast amount of possible target shapes and environment properties ( e.g . mass , shape and appearance of objects and the robot itself ) makes this a diverse and challenging setting to evaluate different generalization aspects . CausalWorld is a simulated version ( using the Bullet physics engine ( Coumans et al. , 2013 ) ) of the open-source TriFinger robot platform from Wüthrich et al . ( 2020 ) . Each environment is defined by a set of variables such , as gravity , floor friction , stage color , floor color , joint positions , various block parameters ( e.g . size , color , mass , position , orientation ) , link colors , link masses and the goal shape . See Table 3 in the Appendix for a more extensive list of these variables . Desideratum 1 is satisfied since different environment properties and goal shapes give rise to very different tasks , ranging from relatively easy ( e.g . re-positioning a single cube ) to extremely hard ( e.g . building a complex structure ) . Desideratum 2 is satisfied because we allow for arbitrary interventions on these properties , hence users or agents may change parameters individually or jointly . Desideratum 3 is satisfied because the parameters can be changed gradually . Desideratum 4 is satisfied because all the environments share the causal structure of the robot , and one may also use subsets of environments which share even more causal structure . We satisfy desideratum 5 by defining the measure of success for all environments as the volumetric overlap of the goal shape with available objects . Further , by splitting the set of parameters into a set A , intended for training and in-distribution evaluation , and a set B , intended for out-of-distribution evaluation , we satisfy desideratum 6 . Finally , since the TriFinger robot ( Wüthrich et al. , 2020 ) can be built in the real-world , we satisfy desideratum 7 . Desideratum 7 and 2 are in partial conflict since sim2real is only possible for the tasks which are constrained to the variables on which the robot can physically act upon . Task generators : To generate meaningful families of similar goal shapes , CausalWorld allows for defining task generators which can generate a variety of different goal shapes in an environment . For instance , one task generator may generate pushing tasks , while another one may generate towerbuilding tasks ( see Fig . 2 ) . Each task generator is initialized with a default goal shape from its corresponding family and comes with a sampler to sample new goal shapes from the same family . Additionally , upon construction , one can specify the environments ’ initial state and initial goal shape structure when deviating from the default . The maximum episode time to build a given shape is number of blocks×10 seconds . CausalWorld comes with eight pre-defined task generators ( see Fig . 2 ) . • Three generators create goal shapes with a single block : Pushing with the goal shape on the floor , Picking having the goal shape defined above the floor and Pick and Place where a fixed obstacle is placed between the initial block and goal pose . • Stacking2 involves a goal shape of two stacked blocks , which can also be considered one instance of the Towers generator . • The remaining generators use a variable number of blocks to generate much more complex and challenging target shapes , as detailed in the appendix : Towers , Stacked Blocks , Creative Stacked Blocks and General . Given that building new environments using current physics simulators is often tedious , we provide a simple API for users who wish to create task generators for new challenging shape families , which may be added to CausalWorld ’ s task generators repository . Action and Observation Spaces : The robot can be chosen to operate in either joint position control mode , joint torque control mode , end-effector position control mode , or the delta of each . In any of these cases , the action is 9-dimensional ( one per joint ) . We provide two observation modes : structured and pixel . In the structured mode , the observation vector is constructed using a rule for the ordering of the relevant variables , such as joint positions , joint velocities , block positions , etc . Thus , the size of the observation space depends on the number of blocks , which could potentially change with every new goal sampled , e.g . in Towers , ( Creative ) Stacked Blocks and General . In contrast , in the pixel mode , the agent receives six RGB images ( hence the dimension of the observation is 6×3×128×128 ) , the first three images are rendered from the three cameras mounted around the TriFinger robot , and the last three images specify the goal image of the target shape rendered from the same cameras . Additionally , CausalWorld allows users to set up a fully customized observation space . Rewards : The reward function r is defined as the fractional volumetric overlap of the blocks with the goal shape , which ranges between 0 ( no overlap ) and 1 ( complete overlap ) . Since this reward function is shared across all tasks , an agent that learned r from some training tasks could in principle use it to solve unseen goal structures . There is also the possibility of modifying the reward function to 1 ) sparsify the reward further by returning a binary reward signal instead , or 2 ) add a dense reward function in order to introduce inductive biases via domain knowledge and solution guidance . We hope that the considerable complexity and diversity of goal shapes motivate and accelerate the development of algorithms that are not dependent on hand-tuned reward functions . ` Training and evaluation spaces : In this benchmark , a learning setting consists of an allowed training space ( ATS ) and an evaluation space ( ES ) , both of which are subspaces of the full parameter space . During training , in the simplest setting , parameters are sampled iid from the ATS . However , unlike existing benchmarks , CausalWorld allows in addition for curricula within the ATS as well as settings where the agent itself intervenes on the parameters within an episode ( see Fig . 3 ) . Similarly , during evaluation , parameters may be sampled iid from the evaluation space at each episode reset , or there can be interventions within an episode . Moreover , in order to retrieve the setting considered in most RL benchmarks , we could set the ATS and the ES to be identical and intervene only on object and robot states ( and keep other environment properties constant ) at each episode reset . However , to evaluate out-of-distribution generalization , one should set the two spaces ( ATS and ES ) to be different ; possibly even disjoint . Additionally , to evaluate robustness with respect to a specific parameter ( e.g . object mass ) , one may define the training and evaluation spaces which only differ in that particular parameter . In order to facilitate the definition of appropriate training and evaluation settings , we pre-define two disjoint sets , Ai and Bi , for each parameter i . Through this , one can for instance define the training space to be A1 ×A2 × ... and the evaluation space to be B1 ×B2 × ... to assess generalization with respect to all parameters simultaneously . Alternatively , the evaluation space could be defined as A1 ×A2 × ... ×Bi ×Ai+1 × ... to assess generalization with respect to parameter i only . Lastly , users may also define their own spaces which could then be integrated into the benchmark to give rise to new learning settings . Intervention actors : To provide a convenient way of specifying learning curricula , we introduce intervention actors . At each time step , such an actor takes all the exposed variables of the environment as inputs and may intervene on them . To encourage modularity , one may combine multiple actors in a learning curriculum . This actor is defined by the episode number to start intervening , the episode number to stop intervening , the timestep within the episode it should intervene and the episode periodicity of interventions . We provide a set of predefined intervention actors , including an actor which samples parameters randomly at each episode reset , which corresponds to domainrandomization . It is also easy to define custom intervention actors , we hope that this facilitates investigation into optimal learning curricula ( see Fig . 3 ) . Probing the Causal Structure in RL : The problem setting in RL is usually formulated using the language of Markov Decision Processes ( MDPs ) or Partially Observable Markov Decision Processes ( POMDPs ) ( Sutton & Barto , 1999 ) , but can be also represented by Structural Causal Models ( SCMs ) , as shown in ( Buesing et al. , 2018 ) , refer to section E in the Appendix for a detailed explanation . This is achieved by formulating all conditional probability distributions as deterministic functions that take independent noise variables as inputs . These independent noise variables can specify different scenarios while the deterministic functions reflect the causal mechanisms of the system ( Schölkopf et al. , 2021 ) . Changes in the environment can stem from two different sources : 1 . The agent may alter the state of the environment ( e.g . the position of a block ) indirectly , through its actions ( e.g . pushing the block by applying appropriate torques at the motors ) . 2 . During the execution of a learning curriculum or an evaluation protocol , we may directly intervene on any variable of the SCM , including all the latent variables of the causal model that are not accessible to the RL agent ( such as gravity , object mass or color ) . ( 1 ) is the default type of admissible interventions in RL benchmarks , whereas CausalWorld allows for interventions of type ( 2 ) in addition . The idea is that interventions on these latent variables , e.g . during a learning curriculum , will allow the agent to distinguish between spurious correlations that are only present in a particular setting and true causal relations that will hold across all settings ( i.e . interventions ) . If the agent is able to learn such a representation of the underlying SCM structure , we would expect it to perform well even in out-of-distribution scenarios ( Schölkopf et al. , 2021 ; Dittadi et al. , 2021 ) because the causal structure remains the same , even when the functional form of certain relations may vary ( e.g . when transferring to the real robot ) . Moreover , we hope that by having access to a broad range of interventions in CausalWorld it will aid the inference of the underlying SCM structure through the different causal discovery methods ( see Figure 4 for a subset of the expected SCM to be learned ) , which in turn addresses the lack of causal discovery benchmarks for real world challenges . 𝑓1 𝑓2 𝑓3 𝑓4	Motivated by the difficulty of evaluating RL’s ability to transfer behaviors across environments, the authors propose the CausalWorld benchmark. Unlike prior benchmarks, CausalWorld exposes well-defined casual variables, in the form of task factors, and focuses on robotic manipulation of an open-source robot platform. The authors make CausalWorld easily usable for both training (in defining a learning curriculum) and evaluation (in targeting specific expected generalizations), and make it easy to extend. In their original release, the authors include eight concrete tasks to test generalization, and present baseline results on these tasks.	SP:e37da841052cbdd81b629bdb5c126aa1a375d7e7
CausalWorld: A Robotic Manipulation Benchmark for Causal Structure and Transfer Learning	1 INTRODUCTION Benchmarks have played a crucial role in advancing entire research fields , for instance computer vision with the introduction of CIFAR-10 and ImageNet ( Krizhevsky et al. , 2009 ; 2012 ) . When it comes to the field of reinforcement learning ( RL ) , similar breakthroughs have been achieved in domains such as game playing ( Mnih et al. , 2013 ; Silver et al. , 2017 ) , learning motor control for high-dimensional simulated robots ( Akkaya et al. , 2019 ) , multi-agent settings ( Baker et al. , 2019 ; Berner et al. , 2019 ) and for studying transfer in the context of meta-learning ( Yu et al. , 2019 ) . Nevertheless , trained agents often fail to transfer the knowledge about the learned skills from a training environment to a different but related environment sharing part of the underlying structure . This can be attributed to the fact that it is quite common to evaluate an agent on the training environments themselves , which leads to over- fitting on these narrowly defined environments ( Whiteson et al. , 2011 ) , or that algorithms are com∗Equal contribution . Correspondence to : < ossama.ahmed @ mail.mcgill.ca > , < frederik.traeuble @ tuebingen.mpg.de > †Equal Advising 1 https : //sites.google.com/view/causal-world/home pared using highly engineered and biased reward functions which may result in learning suboptimal policies with respect to the desired behaviour ; this is particularly evident in robotics . In existing benchmarks ( Yu et al. , 2019 ; Goyal et al. , 2019a ; Cobbe et al. , 2018 ; Bellemare et al. , 2013 ; James et al. , 2020 ) the amount of shared causal structure between the different environments is mostly unknown . For instance , in the Atari Arcade Learning environments , it is unclear how to quantify the underlying similarities between different Atari games and we generally do not know to which degree an agent can be expected to generalize . To overcome these limitations , we introduce a novel benchmark in a robotic manipulation environment that we call CausalWorld . It features a diverse set of environments that , in contrast to previous designs , share a large set of parameters and parts of the causal structure . Being able to intervene on these parameters ( individually or collectively ) permits the experimenter to evaluate agents ’ generalization abilities with respect to different types and magnitudes of changes in the environment . These parameters can be varied gradually , which yields a continuum of similar environments . This allows for fine-grained control of training and test distributions and the design of learning curricula . A remarkable skill that humans learn to master early on in their life is building complex structures using spatial-reasoning and dexterous manipulation abilities ( Casey et al. , 2008 ; Caldera et al. , 1999 ; Kamii et al. , 2004 ) . Playing with toy blocks constitutes a natural environment for children to develop important visual-spatial skills , helping them ‘ generalize ’ in building complex composition designs from presented or imagined goal structures ( Verdine et al. , 2017 ; Nath & Szücs , 2014 ; Dewar , 2018 ; Richardson et al. , 2014 ) . Inspired by this , CausalWorld is designed to aid in learning and investigating these skills in a simulated robotic manipulation environment corresponding to the open-source TriFinger robot platform from Wüthrich et al . ( 2020 ) , which can be built in the real world . Tasks are formulated as building 3D goal shapes using a set of available blocks by manipulating them - as seen in Fig . 1 . This yields a diverse familiy of tasks , ranging from relatively simple ( e.g . pushing a single object ) to extremely hard ( e.g . building a complex structure from a large number of objects ) . CausalWorld improves upon previous benchmarks by exposing a large set of parameters in the causal generative model of the environments , such as weight , shape and appearance of the building blocks and the robot itself . The possibility of intervening on any of these properties at any point in time allows one to set up training curricula or to evaluate an agent ’ s generalization capability with respect to different parameters . Furthermore , in contrast to previous benchmarks ( ChevalierBoisvert et al. , 2018 ; Cobbe et al. , 2018 ) , researchers may build their own real-world platform of this simulator at low cost , as detailed in Wüthrich et al . ( 2020 ) , and transfer their trained policies to the real world . Finally , by releasing this benchmark we hope to facilitate research in causal structure learning , i.e . learning the causal graph ( or certain aspects of it ) as we operate in a complex real-world environment whose dynamics follow the laws of physics , which induce causal relations between the variables . Changes to the variables we expose can be considered do-interventions on the underlying structural causal model ( SCM ) . Consequently , we believe that this benchmark offers an exciting opportunity to investigate causality and its connection to RL and robotics . Our main contributions can be summarized as follows : • We propose CausalWorld , a new benchmark comprising a parametrized family of robotic manipulation environments for advancing out-of-distribution generalization and causal structure learning in RL . • We provide a systematic way of defining curricula and disentangling generalization abilities of RL agents with respect to different changes in the environment , since we allow for dointerventions to be performed on different environment variables ( parameters and states ) individually . • We establish baseline results for some of the available tasks under different learning algorithms , thus verifying the feasibility of the tasks . • We show how different learning curricula affect generalization across different axes by reporting some of the in-distribution and out-of-distribution generalization capabilities of the trained agents . 2 CAUSALWORLD BENCHMARK . Here we make the desiderata outlined in the introduction more precise : 1 . The set of environments should be sufficiently diverse to allow for the design of challenging transfer tasks . 2 . We need to be able to intervene on different properties ( e.g . masses , colors ) individually , such that we can investigate different types of generalization . 3 . It should be possible to convert any environment to any other environment by gradually changing its properties through interventions ; this requirement is important for evaluating different levels of transfer and for defining curricula . 4 . The environments should share some causal structure to allow algorithms to transfer the learned causal knowledge from one environment to another . 5 . There should be a unified measure of success , such that an objective comparison can be made between different learning algorithms . 6 . The benchmark should make it easy for users to define meaningful distributions of environments for training and evaluation . In particular , it should facilitate evaluation of indistribution and out-of-distribution performance . 7 . The simulated benchmark should have a real-world counterpart to allow for sim2real . In light of these desiderata , we propose a setup in which a robot must build goal shapes using a set of available objects . It is worth noting that similar setups were proposed previously in a less realistic setting , e.g . in ( Janner et al. , 2018 ; Bapst et al. , 2019 ; McCarthy et al . ; Akkaya et al. , 2019 ; Fahlman , 1974 ; Winston , 1970 ; Winograd , 1972 ) . Specifically , a task is formulated as follows : given a set of available objects , the agent needs to build a specific goal structure , see Fig . 1 for an example . The vast amount of possible target shapes and environment properties ( e.g . mass , shape and appearance of objects and the robot itself ) makes this a diverse and challenging setting to evaluate different generalization aspects . CausalWorld is a simulated version ( using the Bullet physics engine ( Coumans et al. , 2013 ) ) of the open-source TriFinger robot platform from Wüthrich et al . ( 2020 ) . Each environment is defined by a set of variables such , as gravity , floor friction , stage color , floor color , joint positions , various block parameters ( e.g . size , color , mass , position , orientation ) , link colors , link masses and the goal shape . See Table 3 in the Appendix for a more extensive list of these variables . Desideratum 1 is satisfied since different environment properties and goal shapes give rise to very different tasks , ranging from relatively easy ( e.g . re-positioning a single cube ) to extremely hard ( e.g . building a complex structure ) . Desideratum 2 is satisfied because we allow for arbitrary interventions on these properties , hence users or agents may change parameters individually or jointly . Desideratum 3 is satisfied because the parameters can be changed gradually . Desideratum 4 is satisfied because all the environments share the causal structure of the robot , and one may also use subsets of environments which share even more causal structure . We satisfy desideratum 5 by defining the measure of success for all environments as the volumetric overlap of the goal shape with available objects . Further , by splitting the set of parameters into a set A , intended for training and in-distribution evaluation , and a set B , intended for out-of-distribution evaluation , we satisfy desideratum 6 . Finally , since the TriFinger robot ( Wüthrich et al. , 2020 ) can be built in the real-world , we satisfy desideratum 7 . Desideratum 7 and 2 are in partial conflict since sim2real is only possible for the tasks which are constrained to the variables on which the robot can physically act upon . Task generators : To generate meaningful families of similar goal shapes , CausalWorld allows for defining task generators which can generate a variety of different goal shapes in an environment . For instance , one task generator may generate pushing tasks , while another one may generate towerbuilding tasks ( see Fig . 2 ) . Each task generator is initialized with a default goal shape from its corresponding family and comes with a sampler to sample new goal shapes from the same family . Additionally , upon construction , one can specify the environments ’ initial state and initial goal shape structure when deviating from the default . The maximum episode time to build a given shape is number of blocks×10 seconds . CausalWorld comes with eight pre-defined task generators ( see Fig . 2 ) . • Three generators create goal shapes with a single block : Pushing with the goal shape on the floor , Picking having the goal shape defined above the floor and Pick and Place where a fixed obstacle is placed between the initial block and goal pose . • Stacking2 involves a goal shape of two stacked blocks , which can also be considered one instance of the Towers generator . • The remaining generators use a variable number of blocks to generate much more complex and challenging target shapes , as detailed in the appendix : Towers , Stacked Blocks , Creative Stacked Blocks and General . Given that building new environments using current physics simulators is often tedious , we provide a simple API for users who wish to create task generators for new challenging shape families , which may be added to CausalWorld ’ s task generators repository . Action and Observation Spaces : The robot can be chosen to operate in either joint position control mode , joint torque control mode , end-effector position control mode , or the delta of each . In any of these cases , the action is 9-dimensional ( one per joint ) . We provide two observation modes : structured and pixel . In the structured mode , the observation vector is constructed using a rule for the ordering of the relevant variables , such as joint positions , joint velocities , block positions , etc . Thus , the size of the observation space depends on the number of blocks , which could potentially change with every new goal sampled , e.g . in Towers , ( Creative ) Stacked Blocks and General . In contrast , in the pixel mode , the agent receives six RGB images ( hence the dimension of the observation is 6×3×128×128 ) , the first three images are rendered from the three cameras mounted around the TriFinger robot , and the last three images specify the goal image of the target shape rendered from the same cameras . Additionally , CausalWorld allows users to set up a fully customized observation space . Rewards : The reward function r is defined as the fractional volumetric overlap of the blocks with the goal shape , which ranges between 0 ( no overlap ) and 1 ( complete overlap ) . Since this reward function is shared across all tasks , an agent that learned r from some training tasks could in principle use it to solve unseen goal structures . There is also the possibility of modifying the reward function to 1 ) sparsify the reward further by returning a binary reward signal instead , or 2 ) add a dense reward function in order to introduce inductive biases via domain knowledge and solution guidance . We hope that the considerable complexity and diversity of goal shapes motivate and accelerate the development of algorithms that are not dependent on hand-tuned reward functions . ` Training and evaluation spaces : In this benchmark , a learning setting consists of an allowed training space ( ATS ) and an evaluation space ( ES ) , both of which are subspaces of the full parameter space . During training , in the simplest setting , parameters are sampled iid from the ATS . However , unlike existing benchmarks , CausalWorld allows in addition for curricula within the ATS as well as settings where the agent itself intervenes on the parameters within an episode ( see Fig . 3 ) . Similarly , during evaluation , parameters may be sampled iid from the evaluation space at each episode reset , or there can be interventions within an episode . Moreover , in order to retrieve the setting considered in most RL benchmarks , we could set the ATS and the ES to be identical and intervene only on object and robot states ( and keep other environment properties constant ) at each episode reset . However , to evaluate out-of-distribution generalization , one should set the two spaces ( ATS and ES ) to be different ; possibly even disjoint . Additionally , to evaluate robustness with respect to a specific parameter ( e.g . object mass ) , one may define the training and evaluation spaces which only differ in that particular parameter . In order to facilitate the definition of appropriate training and evaluation settings , we pre-define two disjoint sets , Ai and Bi , for each parameter i . Through this , one can for instance define the training space to be A1 ×A2 × ... and the evaluation space to be B1 ×B2 × ... to assess generalization with respect to all parameters simultaneously . Alternatively , the evaluation space could be defined as A1 ×A2 × ... ×Bi ×Ai+1 × ... to assess generalization with respect to parameter i only . Lastly , users may also define their own spaces which could then be integrated into the benchmark to give rise to new learning settings . Intervention actors : To provide a convenient way of specifying learning curricula , we introduce intervention actors . At each time step , such an actor takes all the exposed variables of the environment as inputs and may intervene on them . To encourage modularity , one may combine multiple actors in a learning curriculum . This actor is defined by the episode number to start intervening , the episode number to stop intervening , the timestep within the episode it should intervene and the episode periodicity of interventions . We provide a set of predefined intervention actors , including an actor which samples parameters randomly at each episode reset , which corresponds to domainrandomization . It is also easy to define custom intervention actors , we hope that this facilitates investigation into optimal learning curricula ( see Fig . 3 ) . Probing the Causal Structure in RL : The problem setting in RL is usually formulated using the language of Markov Decision Processes ( MDPs ) or Partially Observable Markov Decision Processes ( POMDPs ) ( Sutton & Barto , 1999 ) , but can be also represented by Structural Causal Models ( SCMs ) , as shown in ( Buesing et al. , 2018 ) , refer to section E in the Appendix for a detailed explanation . This is achieved by formulating all conditional probability distributions as deterministic functions that take independent noise variables as inputs . These independent noise variables can specify different scenarios while the deterministic functions reflect the causal mechanisms of the system ( Schölkopf et al. , 2021 ) . Changes in the environment can stem from two different sources : 1 . The agent may alter the state of the environment ( e.g . the position of a block ) indirectly , through its actions ( e.g . pushing the block by applying appropriate torques at the motors ) . 2 . During the execution of a learning curriculum or an evaluation protocol , we may directly intervene on any variable of the SCM , including all the latent variables of the causal model that are not accessible to the RL agent ( such as gravity , object mass or color ) . ( 1 ) is the default type of admissible interventions in RL benchmarks , whereas CausalWorld allows for interventions of type ( 2 ) in addition . The idea is that interventions on these latent variables , e.g . during a learning curriculum , will allow the agent to distinguish between spurious correlations that are only present in a particular setting and true causal relations that will hold across all settings ( i.e . interventions ) . If the agent is able to learn such a representation of the underlying SCM structure , we would expect it to perform well even in out-of-distribution scenarios ( Schölkopf et al. , 2021 ; Dittadi et al. , 2021 ) because the causal structure remains the same , even when the functional form of certain relations may vary ( e.g . when transferring to the real robot ) . Moreover , we hope that by having access to a broad range of interventions in CausalWorld it will aid the inference of the underlying SCM structure through the different causal discovery methods ( see Figure 4 for a subset of the expected SCM to be learned ) , which in turn addresses the lack of causal discovery benchmarks for real world challenges . 𝑓1 𝑓2 𝑓3 𝑓4	This paper proposes a a robotic manipulation benchmark for causal structure and transfer learning in a simulation environment considering 3D shape construction tasks given a set of blocks. Baseline results using model-free algorithms are provided for chosen tasks, e.g. pushing, picking, pick&place, stacking. It is also stated that a real version of the robot can be built (as it is open-sourced) for sim2real research. The paper is clearly written, nicely structured and, presents interesting and important ideas. It exposes a large set of parameters, e.g. properties of blocks (size, mass, pose), friction, goals for generalisation evaluations. Having a real-world counterpart makes it very valuable for sim2real research. Authors provide and discuss the relevant previous work detailing how their work connects to the existing literature.  A minor comment: The particular choice of the robot can be motivated, as it is a special design.	SP:e37da841052cbdd81b629bdb5c126aa1a375d7e7
Monte-Carlo Planning and Learning with Language Action Value Estimates	1 INTRODUCTION . Building an intelligent goal-oriented agent that can perceive and react via natural language is one of the grand challenges of artificial intelligence . In pursuit of this goal , we consider Interactive Fiction ( IF ) games ( Nelson , 2001 ; Montfort , 2005 ) , which are text-based simulation environments where the agent interacts with the environment only through natural language . They serve as a useful testbed for developing language-based goal-oriented agents , posing important challenges such as natural language understanding , commonsense reasoning , and non-myopic planning in the combinatorial search space of language actions . IF games naturally have a large branching factor , with at least hundreds of natural language actions that can affect the simulation of game states . This renders naive exhaustive search infeasible and raises the strong need for language-grounded planning ability , i.e . effective search space is too large to choose an optimal action , without inferring the future impact of language actions by understanding the environment state described in natural language . Still , standard planning methods such as Monte-Carlo tree search ( MCTS ) are language-agnostic and rely only on uncertainty-driven exploration , encouraging more search on less-visited states and actions . This simple uncertainty-based strategy is not sufficient to find an optimal language action under limited search time , especially when each language action is treated as a discrete token . On the other hand , recent reinforcement learning agents for IF games have started to leverage pre-trained word embeddings for language understanding ( He et al. , 2016 ; Fulda et al. , 2017 ; Hausknecht et al. , 2020 ) or knowledge graphs for commonsense reasoning ( Ammanabrolu & Hausknecht , 2020 ) , but their exploration strategies are still limited to the -greedy or the softmax policies , lacking more structured and non-myopic planning ability . As a consequence , current state-of-the-art agents for IF games still have not yet been up to the human-level play . In this paper , we introduce Monte-Carlo planning with Language Action Value Estimates ( MCLAVE ) , a planning algorithm for the environments with text-based interactions . MC-LAVE combines Monte-Carlo tree search with language-driven exploration , addressing the search inefficiency attributed to the lack of language understanding . It starts with credit assignment to language actions via Q-learning of the experiences collected from the past searches . Then , MC-LAVE assigns nonuniform search priorities to each language action based on the optimistically aggregated Q-estimates of the past actions that share similar meanings with the candidate action , so as to focus more on the semantically promising actions . This is in contrast to the previous methods that involve language understanding in the form of a knowledge graph , where the insignificant language actions are uniformly filtered out by the graph mask ( Ammanabrolu & Hausknecht , 2020 ; Ammanabrolu et al. , 2020 ) . We show that the non-uniform search empowered by language understanding in MC-LAVE yields better search efficiency while not hurting the asymptotic guarantee of MCTS . We then present our reinforcement learning approach that uses MC-LAVE as a strong policy improvement operator . Since MCTS explores the combinatorial space of action sequences , its search results can be far better than the simple greedy improvement , as demonstrated in the game of Go ( Silver et al. , 2017 ) . This final algorithm , MC-LAVE-RL , alternates between planning via MC-LAVE and supervised learning of self-generated language actions . Experimental results demonstrate that MC-LAVE-RL achieves new high scores in various IF games provided in the Jericho framework ( Hausknecht et al. , 2020 ) , showing the effectiveness of language-grounded MC-LAVE planning . 2 BACKGROUND . 2.1 INTERACTIVE FICTION GAME . Interactive Fiction ( IF ) games are fully text-based environments where the observation and the action spaces are defined as natural language . The game-playing agent observes textual descriptions of the world , selects a language-based action , and receives the associated reward . IF games can be modeled as a special case of partially observable Markov decision processes ( POMDPs ) defined by tuple 〈S , A , Ω , T , O , R , γ〉 , where S is the set of environment states s , A is the set of language actions a , Ω is the set of text observations o , T ( s′\|s , a ) = Pr ( st+1 = s′\|st = s , at = a ) is the transition function , R ( s , a ) is the reward function for taking action a in state s , O ( s ) = o is the deterministic observation function in state s , and γ ∈ ( 0 , 1 ) is the discount factor . The history at time step t , ht = { o0 , a0 , . . . , ot−1 , at−1 , ot } , is a sequence of observations and actions . The goal is to find an optimal policy π∗ that maximizes the expected cumulative rewards , i.e . π∗ = arg maxπ Eπ [ ∑∞ t=0 γ tR ( st , at ) ] . We use the same definition of observation and action space as Hausknecht et al . ( 2020 ) ; Ammanabrolu & Hausknecht ( 2020 ) ; Côté et al . ( 2018 ) , i.e . An observation is defined by ot = ( otdesc , otgame , otinv , at−1 ) where otdesc is the textual description of the current location of the agent , otgame is the simulator response to the previous action taken by the agent , otinv is the information of agent ’ s inventory , and at−1 is the previous action taken by the agent . An action is denoted by a sequence of words at = ( a1t , a 2 t , . . . , a \|at\| t ) . Finally , we denote Avalid ( ot ) ⊆ A as the set of valid actions for the observation ot , which is provided by the Jericho environment interface . Figure 1 shows an example of observation and action in ZORK1 , one of the representative IF games . 2.2 CHALLENGES IN INTERACTIVE FICTION GAME . IF games pose important challenges for reinforcement learning agents , requiring natural language understanding , commonsense reasoning , and non-myopic language-grounded planning ability in combinatorial search space of language actions ( Hausknecht et al. , 2020 ) . More concretely , consider the particular game state of ZORK1 described in Figure 1 . In this example situation , a human player would be naturally capable of performing strategic planning via language understanding and commonsense reasoning : ( 1 ) the ‘ closed trap door ’ will have to be opened and be explored to proceed with the game , ( 2 ) however , acquiring the ‘ lantern ’ should precede entering the trap door since the cellar , which is expected to exist below the trap door , could be likely pitch-dark . Without such language-grounded reasoning and planning , the agent may need to try out every actions uniformly , most of them making it vulnerable to being eaten by a monster in the cellar who always appears when there is no light source . As a result , any agent that lacks the ability of long-term planning with language reasoning is prone to be stuck at a suboptimal policy , which enters the cellar to obtain an immediate reward and does nothing further to avoid encountering the monster that kills the agent immediately . To overcome the bottleneck , the agent should be able to infer that ‘ take lantern ’ is worthy enough to be chosen in preference to other actions even though it gives no immediate reward . In this work , we will precisely address such failures of myopic planning that arise from the lack of language-grounded exploration strategy . To this end , we first propose MCTS-based long-term planning for nonmyopic-planning . Then we further improve the search efficiency of MCTS-based planning by incorporating our novel language-grounded exploration strategy based on Language Action Value Estimates , which will be detailed in the subsequent section . 2.3 MONTE-CARLO TREE SEARCH . Monte-Carlo Tree Search ( MCTS ) ( Kocsis & Szepesvári , 2006 ; Coulom , 2006 ; Browne et al. , 2012 ) is a generic online planning algorithm that combines random sampling and tree search , which has become the de-facto standard method for large sequential decision-making problems . Starting from an empty tree , MCTS repeatedly performs the phases of selection , expansion , rollout , and backpropagation to evaluate the nodes of the search tree with increased accuracy . UCT ( Kocsis & Szepesvári , 2006 ) is the standard MCTS method , which adopts UCB ( Auer et al. , 2002 ) as an action selection rule at each internal node of the search tree : arg max a [ Q ( h , a ) + c √ logN ( h ) N ( h , a ) ] ( 1 ) where h is the history of past observations and actions , Q ( h , a ) is the average value of sampled returns with taking action a in h , N ( h ) is the number of simulations performed in h , N ( h , a ) is the number of times action a is selected in h , and c is a constant that balances exploration and exploitation . However , UCT suffers from severe search inefficiency in the problems with a large action space such as IF games , since it requires to take every action at least once and relies only on uncertainty-driven ( or visit-count-based ) exploration . PUCT ( Silver et al. , 2017 ; 2018 ) partially addresses the challenges of IF games by adopting PUCB ( Rosin , 2011 ) , which involves a prior action distribution π ( ·\|h ) and eliminates the need for choosing every action at least once . The action selection rule is given by : arg max a [ Q ( h , a ) + cPUCTπ ( a\|h ) √ N ( h ) N ( h , a ) + 1 ] ( 2 ) The prior distribution π ( ·\|h ) in Eq . ( 2 ) can be trained by behavioral cloning of ( hroot , a∗root ) samples obtained by the result of tree search in previous time steps ( Anthony et al. , 2017 ; Silver et al. , 2018 ) . However , this procedure not only discards other search information such as abundant ( ht , at , rt , ht+1 ) samples obtained during the search but also hardly encourages information sharing across the semantically similar actions . Therefore , PUCT is still limited in search efficiency for the tasks with language action space , raising the need for a language-driven exploration strategy . 3 MONTE-CARLO PLANNING WITH LANGUAGE ACTION VALUE ESTIMATES . Language-grounded planning is essential to address the challenges of IF games . To this end , we need a mechanism to incorporate language understanding and commonsense reasoning into the ex- ploration strategy of search . In this section , we introduce Monte-Carlo planning with Language Action Value Estimates ( MC-LAVE ) , a novel planning algorithm that combines MCTS with languagedriven exploration . MC-LAVE does not rely on any handcrafted heuristic evaluation but starts tabula rasa , based only upon pre-trained word embeddings that can well define the semantic similarities between two language actions . In addition , it only requires a black-box simulator ( r , s′ , o′ ) ∼ G ( s , a ) , which yields ( o , a , r , o′ ) samples at each simulator query .	This paper presents a novel MCTS-based policy improvement operator called MC-LAVE designed specifically for environments with text-based action spaces. MC-LAVE adds an additional term to PUCT that shares information across semantically similar actions. This additional term for a given action $a$ is set to the soft maximum over the Q function evaluated at all $(o, \bar{a})$ pairs of transitions in the replay buffers whose transition action $\bar{a}$ is within some cosine distance $\delta$ of the action $a$. As shown in Appendix C, addition of this term preserves the expected regret bounds of PUCB. The paper reports empirical improvements over existing methods on various interactive fiction (IF) games in the Jericho suite.	SP:c8ac9e83702c206ede5bdf55988a5fb5fe73f39b
Monte-Carlo Planning and Learning with Language Action Value Estimates	1 INTRODUCTION . Building an intelligent goal-oriented agent that can perceive and react via natural language is one of the grand challenges of artificial intelligence . In pursuit of this goal , we consider Interactive Fiction ( IF ) games ( Nelson , 2001 ; Montfort , 2005 ) , which are text-based simulation environments where the agent interacts with the environment only through natural language . They serve as a useful testbed for developing language-based goal-oriented agents , posing important challenges such as natural language understanding , commonsense reasoning , and non-myopic planning in the combinatorial search space of language actions . IF games naturally have a large branching factor , with at least hundreds of natural language actions that can affect the simulation of game states . This renders naive exhaustive search infeasible and raises the strong need for language-grounded planning ability , i.e . effective search space is too large to choose an optimal action , without inferring the future impact of language actions by understanding the environment state described in natural language . Still , standard planning methods such as Monte-Carlo tree search ( MCTS ) are language-agnostic and rely only on uncertainty-driven exploration , encouraging more search on less-visited states and actions . This simple uncertainty-based strategy is not sufficient to find an optimal language action under limited search time , especially when each language action is treated as a discrete token . On the other hand , recent reinforcement learning agents for IF games have started to leverage pre-trained word embeddings for language understanding ( He et al. , 2016 ; Fulda et al. , 2017 ; Hausknecht et al. , 2020 ) or knowledge graphs for commonsense reasoning ( Ammanabrolu & Hausknecht , 2020 ) , but their exploration strategies are still limited to the -greedy or the softmax policies , lacking more structured and non-myopic planning ability . As a consequence , current state-of-the-art agents for IF games still have not yet been up to the human-level play . In this paper , we introduce Monte-Carlo planning with Language Action Value Estimates ( MCLAVE ) , a planning algorithm for the environments with text-based interactions . MC-LAVE combines Monte-Carlo tree search with language-driven exploration , addressing the search inefficiency attributed to the lack of language understanding . It starts with credit assignment to language actions via Q-learning of the experiences collected from the past searches . Then , MC-LAVE assigns nonuniform search priorities to each language action based on the optimistically aggregated Q-estimates of the past actions that share similar meanings with the candidate action , so as to focus more on the semantically promising actions . This is in contrast to the previous methods that involve language understanding in the form of a knowledge graph , where the insignificant language actions are uniformly filtered out by the graph mask ( Ammanabrolu & Hausknecht , 2020 ; Ammanabrolu et al. , 2020 ) . We show that the non-uniform search empowered by language understanding in MC-LAVE yields better search efficiency while not hurting the asymptotic guarantee of MCTS . We then present our reinforcement learning approach that uses MC-LAVE as a strong policy improvement operator . Since MCTS explores the combinatorial space of action sequences , its search results can be far better than the simple greedy improvement , as demonstrated in the game of Go ( Silver et al. , 2017 ) . This final algorithm , MC-LAVE-RL , alternates between planning via MC-LAVE and supervised learning of self-generated language actions . Experimental results demonstrate that MC-LAVE-RL achieves new high scores in various IF games provided in the Jericho framework ( Hausknecht et al. , 2020 ) , showing the effectiveness of language-grounded MC-LAVE planning . 2 BACKGROUND . 2.1 INTERACTIVE FICTION GAME . Interactive Fiction ( IF ) games are fully text-based environments where the observation and the action spaces are defined as natural language . The game-playing agent observes textual descriptions of the world , selects a language-based action , and receives the associated reward . IF games can be modeled as a special case of partially observable Markov decision processes ( POMDPs ) defined by tuple 〈S , A , Ω , T , O , R , γ〉 , where S is the set of environment states s , A is the set of language actions a , Ω is the set of text observations o , T ( s′\|s , a ) = Pr ( st+1 = s′\|st = s , at = a ) is the transition function , R ( s , a ) is the reward function for taking action a in state s , O ( s ) = o is the deterministic observation function in state s , and γ ∈ ( 0 , 1 ) is the discount factor . The history at time step t , ht = { o0 , a0 , . . . , ot−1 , at−1 , ot } , is a sequence of observations and actions . The goal is to find an optimal policy π∗ that maximizes the expected cumulative rewards , i.e . π∗ = arg maxπ Eπ [ ∑∞ t=0 γ tR ( st , at ) ] . We use the same definition of observation and action space as Hausknecht et al . ( 2020 ) ; Ammanabrolu & Hausknecht ( 2020 ) ; Côté et al . ( 2018 ) , i.e . An observation is defined by ot = ( otdesc , otgame , otinv , at−1 ) where otdesc is the textual description of the current location of the agent , otgame is the simulator response to the previous action taken by the agent , otinv is the information of agent ’ s inventory , and at−1 is the previous action taken by the agent . An action is denoted by a sequence of words at = ( a1t , a 2 t , . . . , a \|at\| t ) . Finally , we denote Avalid ( ot ) ⊆ A as the set of valid actions for the observation ot , which is provided by the Jericho environment interface . Figure 1 shows an example of observation and action in ZORK1 , one of the representative IF games . 2.2 CHALLENGES IN INTERACTIVE FICTION GAME . IF games pose important challenges for reinforcement learning agents , requiring natural language understanding , commonsense reasoning , and non-myopic language-grounded planning ability in combinatorial search space of language actions ( Hausknecht et al. , 2020 ) . More concretely , consider the particular game state of ZORK1 described in Figure 1 . In this example situation , a human player would be naturally capable of performing strategic planning via language understanding and commonsense reasoning : ( 1 ) the ‘ closed trap door ’ will have to be opened and be explored to proceed with the game , ( 2 ) however , acquiring the ‘ lantern ’ should precede entering the trap door since the cellar , which is expected to exist below the trap door , could be likely pitch-dark . Without such language-grounded reasoning and planning , the agent may need to try out every actions uniformly , most of them making it vulnerable to being eaten by a monster in the cellar who always appears when there is no light source . As a result , any agent that lacks the ability of long-term planning with language reasoning is prone to be stuck at a suboptimal policy , which enters the cellar to obtain an immediate reward and does nothing further to avoid encountering the monster that kills the agent immediately . To overcome the bottleneck , the agent should be able to infer that ‘ take lantern ’ is worthy enough to be chosen in preference to other actions even though it gives no immediate reward . In this work , we will precisely address such failures of myopic planning that arise from the lack of language-grounded exploration strategy . To this end , we first propose MCTS-based long-term planning for nonmyopic-planning . Then we further improve the search efficiency of MCTS-based planning by incorporating our novel language-grounded exploration strategy based on Language Action Value Estimates , which will be detailed in the subsequent section . 2.3 MONTE-CARLO TREE SEARCH . Monte-Carlo Tree Search ( MCTS ) ( Kocsis & Szepesvári , 2006 ; Coulom , 2006 ; Browne et al. , 2012 ) is a generic online planning algorithm that combines random sampling and tree search , which has become the de-facto standard method for large sequential decision-making problems . Starting from an empty tree , MCTS repeatedly performs the phases of selection , expansion , rollout , and backpropagation to evaluate the nodes of the search tree with increased accuracy . UCT ( Kocsis & Szepesvári , 2006 ) is the standard MCTS method , which adopts UCB ( Auer et al. , 2002 ) as an action selection rule at each internal node of the search tree : arg max a [ Q ( h , a ) + c √ logN ( h ) N ( h , a ) ] ( 1 ) where h is the history of past observations and actions , Q ( h , a ) is the average value of sampled returns with taking action a in h , N ( h ) is the number of simulations performed in h , N ( h , a ) is the number of times action a is selected in h , and c is a constant that balances exploration and exploitation . However , UCT suffers from severe search inefficiency in the problems with a large action space such as IF games , since it requires to take every action at least once and relies only on uncertainty-driven ( or visit-count-based ) exploration . PUCT ( Silver et al. , 2017 ; 2018 ) partially addresses the challenges of IF games by adopting PUCB ( Rosin , 2011 ) , which involves a prior action distribution π ( ·\|h ) and eliminates the need for choosing every action at least once . The action selection rule is given by : arg max a [ Q ( h , a ) + cPUCTπ ( a\|h ) √ N ( h ) N ( h , a ) + 1 ] ( 2 ) The prior distribution π ( ·\|h ) in Eq . ( 2 ) can be trained by behavioral cloning of ( hroot , a∗root ) samples obtained by the result of tree search in previous time steps ( Anthony et al. , 2017 ; Silver et al. , 2018 ) . However , this procedure not only discards other search information such as abundant ( ht , at , rt , ht+1 ) samples obtained during the search but also hardly encourages information sharing across the semantically similar actions . Therefore , PUCT is still limited in search efficiency for the tasks with language action space , raising the need for a language-driven exploration strategy . 3 MONTE-CARLO PLANNING WITH LANGUAGE ACTION VALUE ESTIMATES . Language-grounded planning is essential to address the challenges of IF games . To this end , we need a mechanism to incorporate language understanding and commonsense reasoning into the ex- ploration strategy of search . In this section , we introduce Monte-Carlo planning with Language Action Value Estimates ( MC-LAVE ) , a novel planning algorithm that combines MCTS with languagedriven exploration . MC-LAVE does not rely on any handcrafted heuristic evaluation but starts tabula rasa , based only upon pre-trained word embeddings that can well define the semantic similarities between two language actions . In addition , it only requires a black-box simulator ( r , s′ , o′ ) ∼ G ( s , a ) , which yields ( o , a , r , o′ ) samples at each simulator query .	This paper presents a method for combining planning an learning in text-based games. In particular it augments Monte-Carlo Tree Search to include a language-similarity bonus to encourage exploration of similar actions. This bonus works by computing a Language Action Value Estimate - which is based on increasing the score of an action by an amount corresponding to the Q-Values of similar actions the agent has experienced. Similarity here is defined by cosine distance in action-embedding space. Using this augmented MCTS algorithm the authors introduce their MC-LAVE agent which alternates between MCTS planning and policy training via supervised learning from the planned trajectories. Experiments across nine IF games show consistent improvement relative to prior RL and planning-based agents. Additional analysis is performed to show how MC-LAVE uses the language action value estimates to learn how to overcome a notable bottleneck in the game of Zork.	SP:c8ac9e83702c206ede5bdf55988a5fb5fe73f39b
Monte-Carlo Planning and Learning with Language Action Value Estimates	1 INTRODUCTION . Building an intelligent goal-oriented agent that can perceive and react via natural language is one of the grand challenges of artificial intelligence . In pursuit of this goal , we consider Interactive Fiction ( IF ) games ( Nelson , 2001 ; Montfort , 2005 ) , which are text-based simulation environments where the agent interacts with the environment only through natural language . They serve as a useful testbed for developing language-based goal-oriented agents , posing important challenges such as natural language understanding , commonsense reasoning , and non-myopic planning in the combinatorial search space of language actions . IF games naturally have a large branching factor , with at least hundreds of natural language actions that can affect the simulation of game states . This renders naive exhaustive search infeasible and raises the strong need for language-grounded planning ability , i.e . effective search space is too large to choose an optimal action , without inferring the future impact of language actions by understanding the environment state described in natural language . Still , standard planning methods such as Monte-Carlo tree search ( MCTS ) are language-agnostic and rely only on uncertainty-driven exploration , encouraging more search on less-visited states and actions . This simple uncertainty-based strategy is not sufficient to find an optimal language action under limited search time , especially when each language action is treated as a discrete token . On the other hand , recent reinforcement learning agents for IF games have started to leverage pre-trained word embeddings for language understanding ( He et al. , 2016 ; Fulda et al. , 2017 ; Hausknecht et al. , 2020 ) or knowledge graphs for commonsense reasoning ( Ammanabrolu & Hausknecht , 2020 ) , but their exploration strategies are still limited to the -greedy or the softmax policies , lacking more structured and non-myopic planning ability . As a consequence , current state-of-the-art agents for IF games still have not yet been up to the human-level play . In this paper , we introduce Monte-Carlo planning with Language Action Value Estimates ( MCLAVE ) , a planning algorithm for the environments with text-based interactions . MC-LAVE combines Monte-Carlo tree search with language-driven exploration , addressing the search inefficiency attributed to the lack of language understanding . It starts with credit assignment to language actions via Q-learning of the experiences collected from the past searches . Then , MC-LAVE assigns nonuniform search priorities to each language action based on the optimistically aggregated Q-estimates of the past actions that share similar meanings with the candidate action , so as to focus more on the semantically promising actions . This is in contrast to the previous methods that involve language understanding in the form of a knowledge graph , where the insignificant language actions are uniformly filtered out by the graph mask ( Ammanabrolu & Hausknecht , 2020 ; Ammanabrolu et al. , 2020 ) . We show that the non-uniform search empowered by language understanding in MC-LAVE yields better search efficiency while not hurting the asymptotic guarantee of MCTS . We then present our reinforcement learning approach that uses MC-LAVE as a strong policy improvement operator . Since MCTS explores the combinatorial space of action sequences , its search results can be far better than the simple greedy improvement , as demonstrated in the game of Go ( Silver et al. , 2017 ) . This final algorithm , MC-LAVE-RL , alternates between planning via MC-LAVE and supervised learning of self-generated language actions . Experimental results demonstrate that MC-LAVE-RL achieves new high scores in various IF games provided in the Jericho framework ( Hausknecht et al. , 2020 ) , showing the effectiveness of language-grounded MC-LAVE planning . 2 BACKGROUND . 2.1 INTERACTIVE FICTION GAME . Interactive Fiction ( IF ) games are fully text-based environments where the observation and the action spaces are defined as natural language . The game-playing agent observes textual descriptions of the world , selects a language-based action , and receives the associated reward . IF games can be modeled as a special case of partially observable Markov decision processes ( POMDPs ) defined by tuple 〈S , A , Ω , T , O , R , γ〉 , where S is the set of environment states s , A is the set of language actions a , Ω is the set of text observations o , T ( s′\|s , a ) = Pr ( st+1 = s′\|st = s , at = a ) is the transition function , R ( s , a ) is the reward function for taking action a in state s , O ( s ) = o is the deterministic observation function in state s , and γ ∈ ( 0 , 1 ) is the discount factor . The history at time step t , ht = { o0 , a0 , . . . , ot−1 , at−1 , ot } , is a sequence of observations and actions . The goal is to find an optimal policy π∗ that maximizes the expected cumulative rewards , i.e . π∗ = arg maxπ Eπ [ ∑∞ t=0 γ tR ( st , at ) ] . We use the same definition of observation and action space as Hausknecht et al . ( 2020 ) ; Ammanabrolu & Hausknecht ( 2020 ) ; Côté et al . ( 2018 ) , i.e . An observation is defined by ot = ( otdesc , otgame , otinv , at−1 ) where otdesc is the textual description of the current location of the agent , otgame is the simulator response to the previous action taken by the agent , otinv is the information of agent ’ s inventory , and at−1 is the previous action taken by the agent . An action is denoted by a sequence of words at = ( a1t , a 2 t , . . . , a \|at\| t ) . Finally , we denote Avalid ( ot ) ⊆ A as the set of valid actions for the observation ot , which is provided by the Jericho environment interface . Figure 1 shows an example of observation and action in ZORK1 , one of the representative IF games . 2.2 CHALLENGES IN INTERACTIVE FICTION GAME . IF games pose important challenges for reinforcement learning agents , requiring natural language understanding , commonsense reasoning , and non-myopic language-grounded planning ability in combinatorial search space of language actions ( Hausknecht et al. , 2020 ) . More concretely , consider the particular game state of ZORK1 described in Figure 1 . In this example situation , a human player would be naturally capable of performing strategic planning via language understanding and commonsense reasoning : ( 1 ) the ‘ closed trap door ’ will have to be opened and be explored to proceed with the game , ( 2 ) however , acquiring the ‘ lantern ’ should precede entering the trap door since the cellar , which is expected to exist below the trap door , could be likely pitch-dark . Without such language-grounded reasoning and planning , the agent may need to try out every actions uniformly , most of them making it vulnerable to being eaten by a monster in the cellar who always appears when there is no light source . As a result , any agent that lacks the ability of long-term planning with language reasoning is prone to be stuck at a suboptimal policy , which enters the cellar to obtain an immediate reward and does nothing further to avoid encountering the monster that kills the agent immediately . To overcome the bottleneck , the agent should be able to infer that ‘ take lantern ’ is worthy enough to be chosen in preference to other actions even though it gives no immediate reward . In this work , we will precisely address such failures of myopic planning that arise from the lack of language-grounded exploration strategy . To this end , we first propose MCTS-based long-term planning for nonmyopic-planning . Then we further improve the search efficiency of MCTS-based planning by incorporating our novel language-grounded exploration strategy based on Language Action Value Estimates , which will be detailed in the subsequent section . 2.3 MONTE-CARLO TREE SEARCH . Monte-Carlo Tree Search ( MCTS ) ( Kocsis & Szepesvári , 2006 ; Coulom , 2006 ; Browne et al. , 2012 ) is a generic online planning algorithm that combines random sampling and tree search , which has become the de-facto standard method for large sequential decision-making problems . Starting from an empty tree , MCTS repeatedly performs the phases of selection , expansion , rollout , and backpropagation to evaluate the nodes of the search tree with increased accuracy . UCT ( Kocsis & Szepesvári , 2006 ) is the standard MCTS method , which adopts UCB ( Auer et al. , 2002 ) as an action selection rule at each internal node of the search tree : arg max a [ Q ( h , a ) + c √ logN ( h ) N ( h , a ) ] ( 1 ) where h is the history of past observations and actions , Q ( h , a ) is the average value of sampled returns with taking action a in h , N ( h ) is the number of simulations performed in h , N ( h , a ) is the number of times action a is selected in h , and c is a constant that balances exploration and exploitation . However , UCT suffers from severe search inefficiency in the problems with a large action space such as IF games , since it requires to take every action at least once and relies only on uncertainty-driven ( or visit-count-based ) exploration . PUCT ( Silver et al. , 2017 ; 2018 ) partially addresses the challenges of IF games by adopting PUCB ( Rosin , 2011 ) , which involves a prior action distribution π ( ·\|h ) and eliminates the need for choosing every action at least once . The action selection rule is given by : arg max a [ Q ( h , a ) + cPUCTπ ( a\|h ) √ N ( h ) N ( h , a ) + 1 ] ( 2 ) The prior distribution π ( ·\|h ) in Eq . ( 2 ) can be trained by behavioral cloning of ( hroot , a∗root ) samples obtained by the result of tree search in previous time steps ( Anthony et al. , 2017 ; Silver et al. , 2018 ) . However , this procedure not only discards other search information such as abundant ( ht , at , rt , ht+1 ) samples obtained during the search but also hardly encourages information sharing across the semantically similar actions . Therefore , PUCT is still limited in search efficiency for the tasks with language action space , raising the need for a language-driven exploration strategy . 3 MONTE-CARLO PLANNING WITH LANGUAGE ACTION VALUE ESTIMATES . Language-grounded planning is essential to address the challenges of IF games . To this end , we need a mechanism to incorporate language understanding and commonsense reasoning into the ex- ploration strategy of search . In this section , we introduce Monte-Carlo planning with Language Action Value Estimates ( MC-LAVE ) , a novel planning algorithm that combines MCTS with languagedriven exploration . MC-LAVE does not rely on any handcrafted heuristic evaluation but starts tabula rasa , based only upon pre-trained word embeddings that can well define the semantic similarities between two language actions . In addition , it only requires a black-box simulator ( r , s′ , o′ ) ∼ G ( s , a ) , which yields ( o , a , r , o′ ) samples at each simulator query .	This paper introduces Monte-Carlo planning with language action value estimates to guide exploration. The method builds on top of MCTS w/ PUCT, where a policy distribution over actions is introduced to estimate Q for actions not seen during sampling. The modification proposed here is an additional term to the Q estimate, which is a weighted average of Q values of similar actions, where similarity is computed using word embeddings of words that make up an action. The authors show gains over MCTS w/ PUCT on 8/9 real text games.	SP:c8ac9e83702c206ede5bdf55988a5fb5fe73f39b
Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding	1 INTRODUCTION . Natural scene understanding can be achieved by decomposing the signal into its underlying factors of variation . An intuitive approach for this problem assumes that a visual representation of the world can be constructed via a generative process that receives factors as input and produces natural signals as output ( Bengio et al. , 2013 ) . This analogy is justified by the fact that our world is composed of distinct entities that can vary independently , but with regularity imposed by physics . What makes the approach appealing is that it formalizes representation learning by directly comparing representations to underlying ground-truth states , as opposed to the indirect evaluation of benchmarking against heuristic downstream tasks ( e.g . object recognition ) . However , the core issue with this approach is non-identifiability , which means a set of possible solutions may all appear equally valid to the model , while only one identifies the true generative factors . Our work is motivated by the question of whether the statistics of natural data will allow for the formulation of an identifiable model . Our core observation that enables us to make progress in ∗‡Equal contribution . Code : https : //github.com/bethgelab/slow_disentanglement addressing this question is that generative factors of natural data have sparse transitions . To estimate these generative factors , we compute statistics on measured transitions of area and position for object masks from large-scale , natural , unstructured videos . Specifically , we extracted over 300,000 object segmentation mask transitions from YouTube-VOS ( Xu et al. , 2018 ; Yang et al. , 2019 ) and KITTI-MOTS ( Voigtlaender et al. , 2019 ; Geiger et al. , 2012 ; Milan et al. , 2016 ) ( discussed in detail in Appendix D ) . We fit generalized Laplace distributions to the collected data ( Eq . 2 ) , which we indicate with orange lines in Fig . 1 . We see empirically that all marginal distributions of temporal transitions are highly sparse and that there exist complex dependencies between natural factors ( e.g . motion typically affects both position and apparent size ) . In this study , we focus on the sparse marginals , which we believe constitutes an important advance that sets the stage for solving further issues and eventually applying the technology to real-world problems . With this information at hand , we are able to provide a stronger proof for capturing the underlying generative factors of the data up to permutations and sign flips that is not covered by previous work ( Hyvärinen and Morioka , 2016 ; 2017 ; Khemakhem et al. , 2020a ) . Thus , we present the first work , to the best of our knowledge , which proposes a theoretically grounded solution that covers the statistics observed in real videos . Our contributions are : With measurements from unstructured natural video annotations we provide evidence that natural generative factors undergo sparse changes across time . We provide a proof of identifiability that relies on the observed sparse innovations to identify nonlinearly mixed sources up to a permutation and sign-flips , which we then validate with practical estimation methods for empirical comparisons . We leverage the natural scene information to create novel datasets where the latent transitions between frames follow natural statistics . These datasets provide a benchmark to evaluate how well models can uncover the true latent generative factors in the presence of realistic dynamics . We demonstrate improved disentanglement over previous models on existing datasets and our contributed ones with quantitative metrics from both the disentanglement ( Locatello et al. , 2018 ) and the nonlinear ICA community ( Hyvärinen and Morioka , 2016 ) . We show via numerous visualization techniques that the learned representations for competing models have important differences , even when quantitative metrics suggest that they are performing equally well . 2 RELATED WORK – DISENTANGLEMENT AND NONLINEAR ICA . Disentangled representation learning has its roots in blind source separation ( Cardoso , 1989 ; Jutten and Herault , 1991 ) and shares goals with fields such as inverse graphics ( Kulkarni et al. , 2015 ; Yildirim et al. , 2020 ; Barron and Malik , 2012 ) and developing models of invariant neural computation ( Hyvärinen and Hoyer , 2000 ; Wiskott and Sejnowski , 2002 ; Sohl-Dickstein et al. , 2010 ) ( see Bengio et al. , 2013 , for a review ) . A disentangled representation would be valuable for a wide variety of machine learning applications , including sample efficiency for downstream tasks ( Locatello et al. , 2018 ; Gao et al. , 2019 ) , fairness ( Locatello et al. , 2019 ; Creager et al. , 2019 ) and interpretability ( Bengio et al. , 2013 ; Higgins et al. , 2017 ; Adel et al. , 2018 ) . Since there is no agreed upon definition of disentanglement in the literature , we adopt two common measurable criteria : i ) each encoding element represents a single generative factor and ii ) the values of generative factors are trivially decodable from the encoding ( Ridgeway and Mozer , 2018 ; Eastwood and Williams , 2018 ) . Uncovering the underlying factors of variation has been a long-standing goal in independent component analysis ( ICA ) ( Comon , 1994 ; Bell and Sejnowski , 1995 ) , which provides an identifiable solution for disentangling data mixed via an invertible linear generator receiving at most one Gaussian factor as input . Recent unsupervised approaches for nonlinear generators have largely been based on Variational Autoencoders ( VAEs ) ( Kingma and Welling , 2013 ) and have assumed that the data is independent and identically distributed ( i.i.d . ) ( Locatello et al. , 2018 ) , even though nonlinear methods that make this i.i.d . assumption have been proven to be non-identifiable ( Hyvärinen and Pajunen , 1999 ; Locatello et al. , 2018 ) . Nonetheless , the bottom-up approach of starting with a nonlinear generator that produces well-controlled data has led to considerable achievements in understanding nonlinear disentanglement in VAEs ( Higgins et al. , 2017 ; Burgess et al. , 2018 ; Rolinek et al. , 2019 ; Chen et al. , 2018 ) , consolidating ideas from neural computation and machine learning ( Khemakhem et al. , 2020a ) , and seeking a principled definition of disentanglement ( Ridgeway , 2016 ; Higgins et al. , 2018 ; Eastwood and Williams , 2018 ) . Recently , Hyvärinen and colleagues ( Hyvärinen and Morioka , 2016 ; 2017 ; Hyvärinen et al. , 2018 ) showed that a solution to identifiable nonlinear ICA can be found by assuming that generative factors are conditioned on an additional observed variable , such as past states or the time index itself . This contribution was generalized by Khemakhem et al . ( 2020a ) past the nonlinear ICA domain to any consistent parameter estimation method for deep latent-variable models , including the VAE framework . However , the theoretical assumptions underlying this branch of work do not account for the sparse transitions we observe in the statistics of natural scenes , which we discuss in further detail in appendix F.1.1 . Another branch of work requires some form of supervision to demonstrate disentanglement ( Szabó et al. , 2017 ; Shu et al. , 2019 ; Locatello et al. , 2020 ) . We select two of the above approaches , that are both different in their formulation and state-of-the-art in their respective empirical settings , Hyvärinen and Morioka ( 2017 ) and Locatello et al . ( 2020 ) , for our experiments below . The motivation of our method and dataset contributions is to address the limitations of previous approaches and to enable unsupervised disentanglement learning in more naturalistic scenarios.1 The fact that physical processes bind generative factors in temporally adjacent natural video segments has been thoroughly explored for learning in neural networks ( Hinton , 1990 ; Földiák , 1991 ; Mitchison , 1991 ; Wiskott and Sejnowski , 2002 ; Denton and Birodkar , 2017 ) . We propose a method that uses time information in the form of an L1-sparse temporal prior , which is motivated by the natural scene measurements presented above as well as by previous work ( Simoncelli and Olshausen , 2001 ; Olshausen , 2003 ; Hyvärinen et al. , 2003 ; Cadieu and Olshausen , 2012 ) . Such a prior would intuitively allow for sharp changes in some latent factors , while most other factors remain unchanged between adjacent time-points . Almost all similar methods are variants of slow feature analysis ( SFA , Wiskott and Sejnowski , 2002 ) , which measure slowness in terms of the Euclidean ( i.e . L2 , or log Gaussian ) distance between temporally adjacent encodings . Related to our approach , a probabilistic interpretation of SFA has been previously proposed ( Turner and Sahani , 2007 ) , as well as extensions to variational inference ( Grathwohl and Wilson , 2016 ) . Additionally , Hashimoto ( 2003 ) suggested that a sparse ( Cauchy ) slowness prior improves correspondence to biological complex cells over the L2 slowness prior in a two-layer model . However , to the best of our knowledge , an L1 temporal prior has previously only been used in deep auto-encoder frameworks when applied to semi-supervised tasks ( Mobahi et al. , 2009 ; Zou et al. , 2012 ) , and was mentioned in Cadieu and Olshausen ( 2012 ) , who used an L2 prior , but claimed that an L1 prior performed similarly on their task . Similar to Hyvärinen et al . ( Hyvärinen and Morioka , 2016 ; Hyvärinen et al. , 2018 ) , we only assume that the latent factors are temporally dependent , thus avoiding assuming knowledge of the number of factors where the two observations differ ( Shu et al. , 2019 ; Locatello et al. , 2020 ) . Most of the standard datasets for disentanglement ( dSprites ( Matthey et al. , 2017 ) , Cars3D ( Reed et al. , 2015 ) , SmallNORB ( LeCun et al. , 2004 ) , Shapes3D ( Kim and Mnih , 2018 ) , MPI3D ( Gondal et al. , 2019 ) ) have been compiled into a disentanglement library ( DisLib ) by Locatello et al . ( 2018 ) . However , all of the DisLib datasets are limited in that the data generating process is independent and identically distributed ( i.i.d . ) and all generative factors are assumed to be discrete . In a follow-up study , Locatello et al . ( 2020 ) proposed combining pairs of images such that only k factors change , as this matches their modeling assumptions required to prove identifiability . Here , k ∈ U { 1 , D− 1 } and D denotes the number of ground-truth factors , which are then sampled uniformly . We additionally use the measurements from Fig . 1 to construct datasets for evaluating disentanglement that have time transitions which directly correspond to natural dynamics . 1As in slow feature analysis , we consider learning from videos without labels as unsupervised . 3 THEORY . 3.1 GENERATIVE MODEL . We have provided evidence to support the hypothesis that generative factors of natural videos have sparse temporal transitions ( see Fig . 1 ) . To model this process , we assume temporally adjacent input pairs ( xt−1 , xt ) coming from a nonlinear generator that maps factors to images x = g ( z ) , where generative factors are dependent over time : p ( zt , zt−1 ) = p ( zt\|zt−1 ) p ( zt−1 ) . ( 1 ) Assume the observed data ( xt , xt−1 ) comes from the following generative process , where different latent factors are assumed to be independent ( cf . Appendix F.2 ) : x = g ( z ) , p ( zt−1 ) = d∏ i=1 p ( zt−1 , i ) , p ( zt\|zt−1 ) = d∏ i=1 αλ 2Γ ( 1/α ) exp− ( λ\|zt , i − zt−1 , i\|α ) , ( 2 ) where λ is the distribution rate , p ( zt−1 ) is a factorized Gaussian prior N ( 0 , I ) ( as in Kingma and Welling , 2013 ) and p ( zt\|zt−1 ) is a factorized generalized Laplace distribution ( Subbotin , 1923 ) with shape parameter α , which determines the shape and especially the kurtosis of the function.2 Intuitively , smaller α implies larger kurtosis and sparser temporal transitions of the generative factors ( special cases are Gaussian , α = 2 , and Laplacian , α = 1 ) . Critically , for our proof we assume α < 2 to ensure that temporal transitions are sparse . The novelty of our approach lies in our explicit modeling of sparse transitions that cover the statistics of natural data , which results in a stronger identifiability proof than previously achieved ( see Appendix F.1.1 for a more detailed comparison with Hyvärinen and Morioka , 2017 ; Khemakhem et al. , 2020a ) .	The paper addresses the problem of disentangling the underlying generative factors from data with a particular focus on dynamic natural data. It provides evidence that transitions of objects in natural movies can be characterised by temporally sparse distributions. A novel proof based on a sparse prior on temporally adjacent observations is provided, allowing to recover true latent variables up to permutations and sign flips, and improving the disentangling performance over existing methods. Two new datasets with measured natural dynamics are also proposed to enable evaluation on more realistic scenarios.	SP:2b10bfa0723c412667263a1db3e1d950a8e361c5
Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding	1 INTRODUCTION . Natural scene understanding can be achieved by decomposing the signal into its underlying factors of variation . An intuitive approach for this problem assumes that a visual representation of the world can be constructed via a generative process that receives factors as input and produces natural signals as output ( Bengio et al. , 2013 ) . This analogy is justified by the fact that our world is composed of distinct entities that can vary independently , but with regularity imposed by physics . What makes the approach appealing is that it formalizes representation learning by directly comparing representations to underlying ground-truth states , as opposed to the indirect evaluation of benchmarking against heuristic downstream tasks ( e.g . object recognition ) . However , the core issue with this approach is non-identifiability , which means a set of possible solutions may all appear equally valid to the model , while only one identifies the true generative factors . Our work is motivated by the question of whether the statistics of natural data will allow for the formulation of an identifiable model . Our core observation that enables us to make progress in ∗‡Equal contribution . Code : https : //github.com/bethgelab/slow_disentanglement addressing this question is that generative factors of natural data have sparse transitions . To estimate these generative factors , we compute statistics on measured transitions of area and position for object masks from large-scale , natural , unstructured videos . Specifically , we extracted over 300,000 object segmentation mask transitions from YouTube-VOS ( Xu et al. , 2018 ; Yang et al. , 2019 ) and KITTI-MOTS ( Voigtlaender et al. , 2019 ; Geiger et al. , 2012 ; Milan et al. , 2016 ) ( discussed in detail in Appendix D ) . We fit generalized Laplace distributions to the collected data ( Eq . 2 ) , which we indicate with orange lines in Fig . 1 . We see empirically that all marginal distributions of temporal transitions are highly sparse and that there exist complex dependencies between natural factors ( e.g . motion typically affects both position and apparent size ) . In this study , we focus on the sparse marginals , which we believe constitutes an important advance that sets the stage for solving further issues and eventually applying the technology to real-world problems . With this information at hand , we are able to provide a stronger proof for capturing the underlying generative factors of the data up to permutations and sign flips that is not covered by previous work ( Hyvärinen and Morioka , 2016 ; 2017 ; Khemakhem et al. , 2020a ) . Thus , we present the first work , to the best of our knowledge , which proposes a theoretically grounded solution that covers the statistics observed in real videos . Our contributions are : With measurements from unstructured natural video annotations we provide evidence that natural generative factors undergo sparse changes across time . We provide a proof of identifiability that relies on the observed sparse innovations to identify nonlinearly mixed sources up to a permutation and sign-flips , which we then validate with practical estimation methods for empirical comparisons . We leverage the natural scene information to create novel datasets where the latent transitions between frames follow natural statistics . These datasets provide a benchmark to evaluate how well models can uncover the true latent generative factors in the presence of realistic dynamics . We demonstrate improved disentanglement over previous models on existing datasets and our contributed ones with quantitative metrics from both the disentanglement ( Locatello et al. , 2018 ) and the nonlinear ICA community ( Hyvärinen and Morioka , 2016 ) . We show via numerous visualization techniques that the learned representations for competing models have important differences , even when quantitative metrics suggest that they are performing equally well . 2 RELATED WORK – DISENTANGLEMENT AND NONLINEAR ICA . Disentangled representation learning has its roots in blind source separation ( Cardoso , 1989 ; Jutten and Herault , 1991 ) and shares goals with fields such as inverse graphics ( Kulkarni et al. , 2015 ; Yildirim et al. , 2020 ; Barron and Malik , 2012 ) and developing models of invariant neural computation ( Hyvärinen and Hoyer , 2000 ; Wiskott and Sejnowski , 2002 ; Sohl-Dickstein et al. , 2010 ) ( see Bengio et al. , 2013 , for a review ) . A disentangled representation would be valuable for a wide variety of machine learning applications , including sample efficiency for downstream tasks ( Locatello et al. , 2018 ; Gao et al. , 2019 ) , fairness ( Locatello et al. , 2019 ; Creager et al. , 2019 ) and interpretability ( Bengio et al. , 2013 ; Higgins et al. , 2017 ; Adel et al. , 2018 ) . Since there is no agreed upon definition of disentanglement in the literature , we adopt two common measurable criteria : i ) each encoding element represents a single generative factor and ii ) the values of generative factors are trivially decodable from the encoding ( Ridgeway and Mozer , 2018 ; Eastwood and Williams , 2018 ) . Uncovering the underlying factors of variation has been a long-standing goal in independent component analysis ( ICA ) ( Comon , 1994 ; Bell and Sejnowski , 1995 ) , which provides an identifiable solution for disentangling data mixed via an invertible linear generator receiving at most one Gaussian factor as input . Recent unsupervised approaches for nonlinear generators have largely been based on Variational Autoencoders ( VAEs ) ( Kingma and Welling , 2013 ) and have assumed that the data is independent and identically distributed ( i.i.d . ) ( Locatello et al. , 2018 ) , even though nonlinear methods that make this i.i.d . assumption have been proven to be non-identifiable ( Hyvärinen and Pajunen , 1999 ; Locatello et al. , 2018 ) . Nonetheless , the bottom-up approach of starting with a nonlinear generator that produces well-controlled data has led to considerable achievements in understanding nonlinear disentanglement in VAEs ( Higgins et al. , 2017 ; Burgess et al. , 2018 ; Rolinek et al. , 2019 ; Chen et al. , 2018 ) , consolidating ideas from neural computation and machine learning ( Khemakhem et al. , 2020a ) , and seeking a principled definition of disentanglement ( Ridgeway , 2016 ; Higgins et al. , 2018 ; Eastwood and Williams , 2018 ) . Recently , Hyvärinen and colleagues ( Hyvärinen and Morioka , 2016 ; 2017 ; Hyvärinen et al. , 2018 ) showed that a solution to identifiable nonlinear ICA can be found by assuming that generative factors are conditioned on an additional observed variable , such as past states or the time index itself . This contribution was generalized by Khemakhem et al . ( 2020a ) past the nonlinear ICA domain to any consistent parameter estimation method for deep latent-variable models , including the VAE framework . However , the theoretical assumptions underlying this branch of work do not account for the sparse transitions we observe in the statistics of natural scenes , which we discuss in further detail in appendix F.1.1 . Another branch of work requires some form of supervision to demonstrate disentanglement ( Szabó et al. , 2017 ; Shu et al. , 2019 ; Locatello et al. , 2020 ) . We select two of the above approaches , that are both different in their formulation and state-of-the-art in their respective empirical settings , Hyvärinen and Morioka ( 2017 ) and Locatello et al . ( 2020 ) , for our experiments below . The motivation of our method and dataset contributions is to address the limitations of previous approaches and to enable unsupervised disentanglement learning in more naturalistic scenarios.1 The fact that physical processes bind generative factors in temporally adjacent natural video segments has been thoroughly explored for learning in neural networks ( Hinton , 1990 ; Földiák , 1991 ; Mitchison , 1991 ; Wiskott and Sejnowski , 2002 ; Denton and Birodkar , 2017 ) . We propose a method that uses time information in the form of an L1-sparse temporal prior , which is motivated by the natural scene measurements presented above as well as by previous work ( Simoncelli and Olshausen , 2001 ; Olshausen , 2003 ; Hyvärinen et al. , 2003 ; Cadieu and Olshausen , 2012 ) . Such a prior would intuitively allow for sharp changes in some latent factors , while most other factors remain unchanged between adjacent time-points . Almost all similar methods are variants of slow feature analysis ( SFA , Wiskott and Sejnowski , 2002 ) , which measure slowness in terms of the Euclidean ( i.e . L2 , or log Gaussian ) distance between temporally adjacent encodings . Related to our approach , a probabilistic interpretation of SFA has been previously proposed ( Turner and Sahani , 2007 ) , as well as extensions to variational inference ( Grathwohl and Wilson , 2016 ) . Additionally , Hashimoto ( 2003 ) suggested that a sparse ( Cauchy ) slowness prior improves correspondence to biological complex cells over the L2 slowness prior in a two-layer model . However , to the best of our knowledge , an L1 temporal prior has previously only been used in deep auto-encoder frameworks when applied to semi-supervised tasks ( Mobahi et al. , 2009 ; Zou et al. , 2012 ) , and was mentioned in Cadieu and Olshausen ( 2012 ) , who used an L2 prior , but claimed that an L1 prior performed similarly on their task . Similar to Hyvärinen et al . ( Hyvärinen and Morioka , 2016 ; Hyvärinen et al. , 2018 ) , we only assume that the latent factors are temporally dependent , thus avoiding assuming knowledge of the number of factors where the two observations differ ( Shu et al. , 2019 ; Locatello et al. , 2020 ) . Most of the standard datasets for disentanglement ( dSprites ( Matthey et al. , 2017 ) , Cars3D ( Reed et al. , 2015 ) , SmallNORB ( LeCun et al. , 2004 ) , Shapes3D ( Kim and Mnih , 2018 ) , MPI3D ( Gondal et al. , 2019 ) ) have been compiled into a disentanglement library ( DisLib ) by Locatello et al . ( 2018 ) . However , all of the DisLib datasets are limited in that the data generating process is independent and identically distributed ( i.i.d . ) and all generative factors are assumed to be discrete . In a follow-up study , Locatello et al . ( 2020 ) proposed combining pairs of images such that only k factors change , as this matches their modeling assumptions required to prove identifiability . Here , k ∈ U { 1 , D− 1 } and D denotes the number of ground-truth factors , which are then sampled uniformly . We additionally use the measurements from Fig . 1 to construct datasets for evaluating disentanglement that have time transitions which directly correspond to natural dynamics . 1As in slow feature analysis , we consider learning from videos without labels as unsupervised . 3 THEORY . 3.1 GENERATIVE MODEL . We have provided evidence to support the hypothesis that generative factors of natural videos have sparse temporal transitions ( see Fig . 1 ) . To model this process , we assume temporally adjacent input pairs ( xt−1 , xt ) coming from a nonlinear generator that maps factors to images x = g ( z ) , where generative factors are dependent over time : p ( zt , zt−1 ) = p ( zt\|zt−1 ) p ( zt−1 ) . ( 1 ) Assume the observed data ( xt , xt−1 ) comes from the following generative process , where different latent factors are assumed to be independent ( cf . Appendix F.2 ) : x = g ( z ) , p ( zt−1 ) = d∏ i=1 p ( zt−1 , i ) , p ( zt\|zt−1 ) = d∏ i=1 αλ 2Γ ( 1/α ) exp− ( λ\|zt , i − zt−1 , i\|α ) , ( 2 ) where λ is the distribution rate , p ( zt−1 ) is a factorized Gaussian prior N ( 0 , I ) ( as in Kingma and Welling , 2013 ) and p ( zt\|zt−1 ) is a factorized generalized Laplace distribution ( Subbotin , 1923 ) with shape parameter α , which determines the shape and especially the kurtosis of the function.2 Intuitively , smaller α implies larger kurtosis and sparser temporal transitions of the generative factors ( special cases are Gaussian , α = 2 , and Laplacian , α = 1 ) . Critically , for our proof we assume α < 2 to ensure that temporal transitions are sparse . The novelty of our approach lies in our explicit modeling of sparse transitions that cover the statistics of natural data , which results in a stronger identifiability proof than previously achieved ( see Appendix F.1.1 for a more detailed comparison with Hyvärinen and Morioka , 2017 ; Khemakhem et al. , 2020a ) .	The paper starts with an observation that temporal transitions in sequences of natural images are sparse, which is supported by data collected from two big datasets (youtube-vots and kitti-mots). This suggests using a sparse prior for temporal transitions of latent variables when modelling naturalistic scenes. The authors then introduce SlowVAE, a model for temporal independent component analysis (ICA), that depends on such a sparsity prior, and prove that latent variables are identifiable under this model up to permutation and sign flips, which, they say, is stronger than any previous result. Additionally, this work introduces a number of datasets of increasing complexity (and similarity to natural datasets) for testing disentanglement as well as it performs a large scale and detailed evaluation of the introduced model.	SP:2b10bfa0723c412667263a1db3e1d950a8e361c5
Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding	1 INTRODUCTION . Natural scene understanding can be achieved by decomposing the signal into its underlying factors of variation . An intuitive approach for this problem assumes that a visual representation of the world can be constructed via a generative process that receives factors as input and produces natural signals as output ( Bengio et al. , 2013 ) . This analogy is justified by the fact that our world is composed of distinct entities that can vary independently , but with regularity imposed by physics . What makes the approach appealing is that it formalizes representation learning by directly comparing representations to underlying ground-truth states , as opposed to the indirect evaluation of benchmarking against heuristic downstream tasks ( e.g . object recognition ) . However , the core issue with this approach is non-identifiability , which means a set of possible solutions may all appear equally valid to the model , while only one identifies the true generative factors . Our work is motivated by the question of whether the statistics of natural data will allow for the formulation of an identifiable model . Our core observation that enables us to make progress in ∗‡Equal contribution . Code : https : //github.com/bethgelab/slow_disentanglement addressing this question is that generative factors of natural data have sparse transitions . To estimate these generative factors , we compute statistics on measured transitions of area and position for object masks from large-scale , natural , unstructured videos . Specifically , we extracted over 300,000 object segmentation mask transitions from YouTube-VOS ( Xu et al. , 2018 ; Yang et al. , 2019 ) and KITTI-MOTS ( Voigtlaender et al. , 2019 ; Geiger et al. , 2012 ; Milan et al. , 2016 ) ( discussed in detail in Appendix D ) . We fit generalized Laplace distributions to the collected data ( Eq . 2 ) , which we indicate with orange lines in Fig . 1 . We see empirically that all marginal distributions of temporal transitions are highly sparse and that there exist complex dependencies between natural factors ( e.g . motion typically affects both position and apparent size ) . In this study , we focus on the sparse marginals , which we believe constitutes an important advance that sets the stage for solving further issues and eventually applying the technology to real-world problems . With this information at hand , we are able to provide a stronger proof for capturing the underlying generative factors of the data up to permutations and sign flips that is not covered by previous work ( Hyvärinen and Morioka , 2016 ; 2017 ; Khemakhem et al. , 2020a ) . Thus , we present the first work , to the best of our knowledge , which proposes a theoretically grounded solution that covers the statistics observed in real videos . Our contributions are : With measurements from unstructured natural video annotations we provide evidence that natural generative factors undergo sparse changes across time . We provide a proof of identifiability that relies on the observed sparse innovations to identify nonlinearly mixed sources up to a permutation and sign-flips , which we then validate with practical estimation methods for empirical comparisons . We leverage the natural scene information to create novel datasets where the latent transitions between frames follow natural statistics . These datasets provide a benchmark to evaluate how well models can uncover the true latent generative factors in the presence of realistic dynamics . We demonstrate improved disentanglement over previous models on existing datasets and our contributed ones with quantitative metrics from both the disentanglement ( Locatello et al. , 2018 ) and the nonlinear ICA community ( Hyvärinen and Morioka , 2016 ) . We show via numerous visualization techniques that the learned representations for competing models have important differences , even when quantitative metrics suggest that they are performing equally well . 2 RELATED WORK – DISENTANGLEMENT AND NONLINEAR ICA . Disentangled representation learning has its roots in blind source separation ( Cardoso , 1989 ; Jutten and Herault , 1991 ) and shares goals with fields such as inverse graphics ( Kulkarni et al. , 2015 ; Yildirim et al. , 2020 ; Barron and Malik , 2012 ) and developing models of invariant neural computation ( Hyvärinen and Hoyer , 2000 ; Wiskott and Sejnowski , 2002 ; Sohl-Dickstein et al. , 2010 ) ( see Bengio et al. , 2013 , for a review ) . A disentangled representation would be valuable for a wide variety of machine learning applications , including sample efficiency for downstream tasks ( Locatello et al. , 2018 ; Gao et al. , 2019 ) , fairness ( Locatello et al. , 2019 ; Creager et al. , 2019 ) and interpretability ( Bengio et al. , 2013 ; Higgins et al. , 2017 ; Adel et al. , 2018 ) . Since there is no agreed upon definition of disentanglement in the literature , we adopt two common measurable criteria : i ) each encoding element represents a single generative factor and ii ) the values of generative factors are trivially decodable from the encoding ( Ridgeway and Mozer , 2018 ; Eastwood and Williams , 2018 ) . Uncovering the underlying factors of variation has been a long-standing goal in independent component analysis ( ICA ) ( Comon , 1994 ; Bell and Sejnowski , 1995 ) , which provides an identifiable solution for disentangling data mixed via an invertible linear generator receiving at most one Gaussian factor as input . Recent unsupervised approaches for nonlinear generators have largely been based on Variational Autoencoders ( VAEs ) ( Kingma and Welling , 2013 ) and have assumed that the data is independent and identically distributed ( i.i.d . ) ( Locatello et al. , 2018 ) , even though nonlinear methods that make this i.i.d . assumption have been proven to be non-identifiable ( Hyvärinen and Pajunen , 1999 ; Locatello et al. , 2018 ) . Nonetheless , the bottom-up approach of starting with a nonlinear generator that produces well-controlled data has led to considerable achievements in understanding nonlinear disentanglement in VAEs ( Higgins et al. , 2017 ; Burgess et al. , 2018 ; Rolinek et al. , 2019 ; Chen et al. , 2018 ) , consolidating ideas from neural computation and machine learning ( Khemakhem et al. , 2020a ) , and seeking a principled definition of disentanglement ( Ridgeway , 2016 ; Higgins et al. , 2018 ; Eastwood and Williams , 2018 ) . Recently , Hyvärinen and colleagues ( Hyvärinen and Morioka , 2016 ; 2017 ; Hyvärinen et al. , 2018 ) showed that a solution to identifiable nonlinear ICA can be found by assuming that generative factors are conditioned on an additional observed variable , such as past states or the time index itself . This contribution was generalized by Khemakhem et al . ( 2020a ) past the nonlinear ICA domain to any consistent parameter estimation method for deep latent-variable models , including the VAE framework . However , the theoretical assumptions underlying this branch of work do not account for the sparse transitions we observe in the statistics of natural scenes , which we discuss in further detail in appendix F.1.1 . Another branch of work requires some form of supervision to demonstrate disentanglement ( Szabó et al. , 2017 ; Shu et al. , 2019 ; Locatello et al. , 2020 ) . We select two of the above approaches , that are both different in their formulation and state-of-the-art in their respective empirical settings , Hyvärinen and Morioka ( 2017 ) and Locatello et al . ( 2020 ) , for our experiments below . The motivation of our method and dataset contributions is to address the limitations of previous approaches and to enable unsupervised disentanglement learning in more naturalistic scenarios.1 The fact that physical processes bind generative factors in temporally adjacent natural video segments has been thoroughly explored for learning in neural networks ( Hinton , 1990 ; Földiák , 1991 ; Mitchison , 1991 ; Wiskott and Sejnowski , 2002 ; Denton and Birodkar , 2017 ) . We propose a method that uses time information in the form of an L1-sparse temporal prior , which is motivated by the natural scene measurements presented above as well as by previous work ( Simoncelli and Olshausen , 2001 ; Olshausen , 2003 ; Hyvärinen et al. , 2003 ; Cadieu and Olshausen , 2012 ) . Such a prior would intuitively allow for sharp changes in some latent factors , while most other factors remain unchanged between adjacent time-points . Almost all similar methods are variants of slow feature analysis ( SFA , Wiskott and Sejnowski , 2002 ) , which measure slowness in terms of the Euclidean ( i.e . L2 , or log Gaussian ) distance between temporally adjacent encodings . Related to our approach , a probabilistic interpretation of SFA has been previously proposed ( Turner and Sahani , 2007 ) , as well as extensions to variational inference ( Grathwohl and Wilson , 2016 ) . Additionally , Hashimoto ( 2003 ) suggested that a sparse ( Cauchy ) slowness prior improves correspondence to biological complex cells over the L2 slowness prior in a two-layer model . However , to the best of our knowledge , an L1 temporal prior has previously only been used in deep auto-encoder frameworks when applied to semi-supervised tasks ( Mobahi et al. , 2009 ; Zou et al. , 2012 ) , and was mentioned in Cadieu and Olshausen ( 2012 ) , who used an L2 prior , but claimed that an L1 prior performed similarly on their task . Similar to Hyvärinen et al . ( Hyvärinen and Morioka , 2016 ; Hyvärinen et al. , 2018 ) , we only assume that the latent factors are temporally dependent , thus avoiding assuming knowledge of the number of factors where the two observations differ ( Shu et al. , 2019 ; Locatello et al. , 2020 ) . Most of the standard datasets for disentanglement ( dSprites ( Matthey et al. , 2017 ) , Cars3D ( Reed et al. , 2015 ) , SmallNORB ( LeCun et al. , 2004 ) , Shapes3D ( Kim and Mnih , 2018 ) , MPI3D ( Gondal et al. , 2019 ) ) have been compiled into a disentanglement library ( DisLib ) by Locatello et al . ( 2018 ) . However , all of the DisLib datasets are limited in that the data generating process is independent and identically distributed ( i.i.d . ) and all generative factors are assumed to be discrete . In a follow-up study , Locatello et al . ( 2020 ) proposed combining pairs of images such that only k factors change , as this matches their modeling assumptions required to prove identifiability . Here , k ∈ U { 1 , D− 1 } and D denotes the number of ground-truth factors , which are then sampled uniformly . We additionally use the measurements from Fig . 1 to construct datasets for evaluating disentanglement that have time transitions which directly correspond to natural dynamics . 1As in slow feature analysis , we consider learning from videos without labels as unsupervised . 3 THEORY . 3.1 GENERATIVE MODEL . We have provided evidence to support the hypothesis that generative factors of natural videos have sparse temporal transitions ( see Fig . 1 ) . To model this process , we assume temporally adjacent input pairs ( xt−1 , xt ) coming from a nonlinear generator that maps factors to images x = g ( z ) , where generative factors are dependent over time : p ( zt , zt−1 ) = p ( zt\|zt−1 ) p ( zt−1 ) . ( 1 ) Assume the observed data ( xt , xt−1 ) comes from the following generative process , where different latent factors are assumed to be independent ( cf . Appendix F.2 ) : x = g ( z ) , p ( zt−1 ) = d∏ i=1 p ( zt−1 , i ) , p ( zt\|zt−1 ) = d∏ i=1 αλ 2Γ ( 1/α ) exp− ( λ\|zt , i − zt−1 , i\|α ) , ( 2 ) where λ is the distribution rate , p ( zt−1 ) is a factorized Gaussian prior N ( 0 , I ) ( as in Kingma and Welling , 2013 ) and p ( zt\|zt−1 ) is a factorized generalized Laplace distribution ( Subbotin , 1923 ) with shape parameter α , which determines the shape and especially the kurtosis of the function.2 Intuitively , smaller α implies larger kurtosis and sparser temporal transitions of the generative factors ( special cases are Gaussian , α = 2 , and Laplacian , α = 1 ) . Critically , for our proof we assume α < 2 to ensure that temporal transitions are sparse . The novelty of our approach lies in our explicit modeling of sparse transitions that cover the statistics of natural data , which results in a stronger identifiability proof than previously achieved ( see Appendix F.1.1 for a more detailed comparison with Hyvärinen and Morioka , 2017 ; Khemakhem et al. , 2020a ) .	This paper introduces a novel VAE-based model with the aim to improve unsupervised disentangling of latent factors in visual data. This model differs from previous disentangling models in that it takes short (2-frame) videos as input instead of static images. The model is equipped with a Laplace prior over the dynamics of the video to help it align its representation to axes of sparse temporal dynamics. The intuition is that the temporal dynamics of natural visual stimuli vary sparsely according to some choice of factors, and that choice of factors is exactly what “disentangled” should refer to. The authors show that their model achieves better disentangling than previous static-image methods according to a number of metrics.	SP:2b10bfa0723c412667263a1db3e1d950a8e361c5
Automated Concatenation of Embeddings for Structured Prediction	1 INTRODUCTION . Recent developments on pretrained contextualized embeddings have significantly improved the performance of structured prediction tasks in natural language processing . Approaches based on contextualized embeddings , such as ELMo ( Peters et al. , 2018 ) , Flair ( Akbik et al. , 2018 ) , BERT ( Devlin et al. , 2019 ) , and XLM-R ( Conneau et al. , 2020 ) , have been consistently raising the state-of-the-art for various structured prediction tasks . Concurrently , research has also showed that word representations based on the concatenation of multiple pretrained contextualized embeddings and traditional non-contextualized embeddings ( such as word2vec ( Mikolov et al. , 2013 ) and character embeddings ( Santos & Zadrozny , 2014 ) ) can further improve performance ( Peters et al. , 2018 ; Akbik et al. , 2018 ; Straková et al. , 2019 ; He & Choi , 2020 ) . Given the ever-increasing number of embedding learning methods that operate on different granularities ( e.g. , word , subword , or character level ) and with different model architectures , choosing the best embeddings to concatenate for a specific task becomes non-trivial , and exploring all possible concatenations can be prohibitively demanding in computing resources . Neural architecture search ( NAS ) is an active area of research in deep learning to automatically search for better model architectures , and has achieved state-of-the-art performance on various tasks in computer vision , such as image classification ( Real et al. , 2019 ) , semantic segmentation ( Liu et al. , 2019a ) , and object detection ( Ghiasi et al. , 2019 ) . In natural language processing , NAS has been successfully applied to find better RNN structures ( Zoph & Le , 2017 ; Pham et al. , 2018b ) and recently better transformer structures ( So et al. , 2019 ; Zhu et al. , 2020 ) . In this paper , we propose the Automated Concatenation of Embeddings ( ACE ) approach to automate the process of finding better concatenations of embeddings for structured prediction tasks , formulated as an NAS problem . In this approach , an iterative search process is guided by a controller based on its belief that models the effectiveness of individual embedding candidates in consideration for a specific task . At each step , the controller samples a concatenation of embeddings according to the belief model and feeds the concatenated word representations as inputs to a task model , which in turn is trained on the task dataset and returns the model accuracy as a reward signal to update the belief model . We use the policy gradient algorithm ( Williams , 1992 ) in reinforcement learning ( Sutton & Barto , 1992 ) to solve the optimization problem . In order to improve the efficiency of the search process , we also design a special reward function by accumulating all the rewards based on the transformation between the current concatenation and all previously sampled concatenations . Our approach is different from previous work on NAS in the following aspects : 1 . Unlike most previous work , we focus on searching for better word representations rather than better model architectures . 2 . We design a unique search space for the embedding concatenation search . Instead of using RNN as in previous work of Zoph & Le ( 2017 ) , we design a more straightforward controller to generate the embedding concatenation . We design a novel reward function in the objective of optimization to better evaluate the effectiveness of each concatenated embeddings . 3 . Our approach is efficient and practical . ACE can find a strong word representation on a single GPU with only a few GPU-hours for structured prediction tasks , while a lot of the NAS approaches require dozens of or even thousands of GPU-hours to search for good neural architecture . 4 . The task model from ACE achieves high accuracy without the need for retraining , while in previous work of NAS the resulting neural network usually requires retraining from scratch . Empirical results show that ACE outperforms strong baselines . Furthermore , we show that when ACE is applied to concatenate pretrained contextualized embeddings which are already fine-tuned on specific tasks , we can achieve state-of-the-art or competitive accuracy on 6 structured prediction tasks including Named Entity Recognition ( Sundheim , 1995 ) , Part-Of-Speech tagging ( DeRose , 1988 ) , chunking ( Tjong Kim Sang & Buchholz , 2000 ) , aspect extraction ( Hu & Liu , 2004 ) , syntactic dependency parsing ( Tesnière , 1959 ) and semantic dependency parsing ( Oepen et al. , 2014 ) over 21 datasets . Besides , we also analyze the advantage of ACE and reward function design over the baselines and show the advantage of ACE over ensemble models . 2 RELATED WORK . 2.1 EMBEDDINGS . Non-contextualized embeddings , such as word2vec ( Mikolov et al. , 2013 ) , GloVe ( Pennington et al. , 2014 ) , and fastText ( Bojanowski et al. , 2017 ) , help lots of NLP tasks . Character embeddings ( Santos & Zadrozny , 2014 ) are trained together with the task and applied in many structured prediction tasks ( Ma & Hovy , 2016 ; Lample et al. , 2016 ; Dozat & Manning , 2018 ) . For pretrained contextualized embeddings , ELMo ( Peters et al. , 2018 ) , a pretrained contextualized word embedding generated with multiple Bidirectional LSTM layers , significantly outperforms previous state-of-the-art approaches on several NLP tasks . Following this idea , Akbik et al . ( 2018 ) proposed Flair embeddings , which is a kind of contextualized character embeddings and achieved strong performance in sequence labeling tasks . Recently , Devlin et al . ( 2019 ) proposed BERT , which encodes contextualized sub-word information by Transformers and significantly improves the performance on a lot of NLP tasks . Much research such as RoBERTa ( Liu et al. , 2019c ) has focused on improving BERT model ’ s performance through stronger masking strategies . Moreover , multilingual contextualized embeddings become popular . Pires et al . ( 2019 ) and Wu & Dredze ( 2019 ) showed that Multilingual BERT ( M-BERT ) could learn a good multilingual representation effectively with strong crosslingual zero-shot transfer performance in various tasks . Conneau et al . ( 2020 ) proposed XLM-R , which is trained on a larger multilingual corpus and significantly outperforms M-BERT on various multilingual tasks . 2.2 NEURAL ARCHITECTURE SEARCH . Recent progress on deep learning has shown that network architecture design is crucial to the model performance . However , designing a strong neural architecture for each task requires enormous efforts , high level of knowledge , and experiences over the task domain . Therefore , automatic design of neural architecture is desired . A crucial part of NAS is search space design , which defines the discoverable NAS space . Previous work ( Baker et al. , 2017 ; Zoph & Le , 2017 ; Xie & Yuille , 2017 ) designs a global search space ( Elsken et al. , 2019 ) which incorporates structures from handcrafted architectures . For example , Zoph & Le ( 2017 ) designed a chained-structured search space with skip connections . The global search space usually has a considerable degree of freedom . As an example , the approach of Zoph & Le ( 2017 ) takes 22,400 GPU-hours to search on CIFAR-10 dataset . Based on the observation that existing hand-crafted architectures contain repeated structures ( Szegedy et al. , 2016 ; He et al. , 2016 ; Huang et al. , 2017 ) , Zoph et al . ( 2018 ) explored cell-based search space which can reduce the search time to 2,000 GPU-hours . In recent NAS research , reinforcement learning and evolutionary algorithms are the most usual approaches . In reinforcement learning , the agent ’ s actions are the generation of neural architectures and the action space is identical to the search space . Previous work usually applies an RNN layer ( Zoph & Le , 2017 ; Zhong et al. , 2018 ; Zoph et al. , 2018 ) or use Markov Decision Process ( Baker et al. , 2017 ) to decide the hyper-parameter of each structure and decide the input order of each structure . Evolutionary algorithms have been applied to architecture search for many decades ( Miller et al. , 1989 ; Angeline et al. , 1994 ; Stanley & Miikkulainen , 2002 ; Floreano et al. , 2008 ; Jozefowicz et al. , 2015 ) . The algorithm repeatedly generates new populations through recombination and mutation operations and selects survivors through competing among the population . Recent work with evolutionary algorithms differ in the method on parent/survivor selection and population generation . For example , Real et al . ( 2017 ) , Liu et al . ( 2018a ) , Wistuba ( 2018 ) and Real et al . ( 2019 ) applied tournament selection ( Goldberg & Deb , 1991 ) for the parent selection while Xie & Yuille ( 2017 ) keeps all parents . Suganuma et al . ( 2017 ) and Elsken et al . ( 2018 ) chose the best model while Real et al . ( 2019 ) chose several latest models as survivors . 3 AUTOMATED CONCATENATION OF EMBEDDINGS . In ACE , a task model and a controller interact with each other repeatedly . The task model predicts the task output , while the controller searches for better embedding concatenation as the word representation for the task model to achieve higher accuracy . Given an embedding concatenation generated from the controller , the task model is trained over the task data and returns a reward to the controller . The controller receives the reward to update its parameter and samples a new embedding concatenation for the task model . Figure 1 shows the general architecture of our approach . 3.1 TASK MODEL . For the tasks model , we emphasis on sequence-structured and graph-structured outputs . Given a structured prediction task with input sentence x and structured output y , we can calculate the probability distribution P ( y\|x ) by : P ( y\|x ) = exp ( Score ( x , y ) ) ∑ y′∈Y ( x ) exp ( Score ( x , y′ ) ) where Y ( x ) represents all possible output structures given the input sentence x . Depending on different structured prediction tasks , the output structure y can be label sequences , trees , graphs or other structures . In this paper , we use sequence-structured and graph-structured outputs as two exemplar structured prediction tasks . We use BiLSTM-CRF model ( Ma & Hovy , 2016 ; Lample et al. , 2016 ) for sequence-structured outputs and use BiLSTM-Biaffine model ( Dozat & Manning , 2017 ) for graph-structured outputs : P seq ( y\|x ) = BiLSTM-CRF ( V , y ) ; P graph ( y\|x ) = BiLSTM-Biaffine ( V , y ) where V = [ v1 ; · · · ; vn ] , V ∈ Rd×n is a matrix of the word representations for the input sentence xwith nwords , d is the hidden size of the concatenation of all embeddings . The word representation vi of i-th word is a concatenation of L types of word embeddings : vli = embed l i ( x ) ; vi = [ v 1 i ; v 2 i ; . . . ; v L i ] where embedl is the model of l-th embeddings , vi ∈ Rd , vli ∈ Rd l . dl is the hidden size of embedl . 3.2 SEARCH SPACE DESIGN . The neural architecture search space can be represented as a set of neural networks ( Elsken et al. , 2019 ) . A neural network can be represented as a directed acyclic graph with a set of nodes and directed edges . Each node represents an operation , while each edge represents the inputs and outputs between these nodes . In ACE , we represent each embedding candidate as a node . The input to the nodes is the input sentence x , and the outputs are the embeddings vl . Since we concatenate the embeddings as the word representation of the task model , there is no connection between nodes in our search space . Without considering the connections between nodes , the search space can be significantly reduced . For each node , there are a lot of options to extract word features . Taking BERT embeddings as an example , Devlin et al . ( 2019 ) concatenated the last four layers as word features while Kondratyuk & Straka ( 2019 ) applied a weighted sum of all twelve layers . However , the empirical results ( Devlin et al. , 2019 ) do not show a significant difference in accuracy . We follow the typical usage for each embedding to further reduce the search space . As a result , each embedding only has a fixed operation and the resulting search space contains 2L−1 possible combinations of nodes . In NAS , weight sharing ( Pham et al. , 2018a ) shares the weight of structures in training different neural architectures to reduce the training cost . In comparison , we fixed the weight of pretrained embedding candidates in ACE except for the character embeddings . Instead of sharing the parameters of the embeddings , we share the parameters of the task models at each step of search . However , the hidden size of word representation varies over the concatenations , making the weight sharing of structured prediction models difficult . Instead of deciding whether each node exists in the graph , we keep all nodes in the search space and add an additional operation for each node to indicate whether the embedding is masked out . To represent the selected concatenation , we use a binary vector a = [ a1 , · · · , al , · · · , aL ] as an mask to mask out the embeddings which are not selected : vi = [ v 1 i a1 ; . . . ; v l ial ; . . . ; v L i aL ] ( 1 ) where al is a binary variable . Since the input V is applied to a linear layer in the BiLSTM layer , multiplying the mask with the embeddings is equivalent to directly concatenating the selected embeddings : W = [ W1 ; W2 ; . . . ; WL ] ; W > vi = L∑ l=1 W > l v l ial ; W ∈ Rd×h andWl ∈ Rd l×h ( 2 ) Therefore , the model weights can be shared after applying the embedding mask to all embedding candidates ’ concatenation . Another benefit of our search space design is that we can remove the unused embedding candidates and the corresponding weights inW for a lighter task model after the best concatenation is found by ACE .	This paper explores a way of learning how to automatically construct a concatenated set of embeddings for structured prediction tasks in NLP. The paper's model takes up to L embeddings concatenated together and feeds them into standard models (BiLSTM-CRFs or the BiLSTM-Biaffine technique of Dozat and Manning) to tackle problems like POS tagging, NER, dependency parsing, and more. Search over embedding concaneations is expressed as a search over binary masks of length L. The controller for this search is parameterized by an independent Bernoulli for each mask position. The paper's approach learns the controller parameters with policy gradient, where the reward function is (a modified version of) the accuracy on the development set for the given task. This modified reward uses all samples throughout training to effectively get a more fine-grained baseline for the current timestep based on prior samples. Notably, the paper uses embeddings that are already fine-tuned for each task, as fine-tuning the concatenated embeddings is hard due to divergent step sizes and steep computational requirements.	SP:481614a6302380b81ac1248b92c4c12ae366a827
Automated Concatenation of Embeddings for Structured Prediction	1 INTRODUCTION . Recent developments on pretrained contextualized embeddings have significantly improved the performance of structured prediction tasks in natural language processing . Approaches based on contextualized embeddings , such as ELMo ( Peters et al. , 2018 ) , Flair ( Akbik et al. , 2018 ) , BERT ( Devlin et al. , 2019 ) , and XLM-R ( Conneau et al. , 2020 ) , have been consistently raising the state-of-the-art for various structured prediction tasks . Concurrently , research has also showed that word representations based on the concatenation of multiple pretrained contextualized embeddings and traditional non-contextualized embeddings ( such as word2vec ( Mikolov et al. , 2013 ) and character embeddings ( Santos & Zadrozny , 2014 ) ) can further improve performance ( Peters et al. , 2018 ; Akbik et al. , 2018 ; Straková et al. , 2019 ; He & Choi , 2020 ) . Given the ever-increasing number of embedding learning methods that operate on different granularities ( e.g. , word , subword , or character level ) and with different model architectures , choosing the best embeddings to concatenate for a specific task becomes non-trivial , and exploring all possible concatenations can be prohibitively demanding in computing resources . Neural architecture search ( NAS ) is an active area of research in deep learning to automatically search for better model architectures , and has achieved state-of-the-art performance on various tasks in computer vision , such as image classification ( Real et al. , 2019 ) , semantic segmentation ( Liu et al. , 2019a ) , and object detection ( Ghiasi et al. , 2019 ) . In natural language processing , NAS has been successfully applied to find better RNN structures ( Zoph & Le , 2017 ; Pham et al. , 2018b ) and recently better transformer structures ( So et al. , 2019 ; Zhu et al. , 2020 ) . In this paper , we propose the Automated Concatenation of Embeddings ( ACE ) approach to automate the process of finding better concatenations of embeddings for structured prediction tasks , formulated as an NAS problem . In this approach , an iterative search process is guided by a controller based on its belief that models the effectiveness of individual embedding candidates in consideration for a specific task . At each step , the controller samples a concatenation of embeddings according to the belief model and feeds the concatenated word representations as inputs to a task model , which in turn is trained on the task dataset and returns the model accuracy as a reward signal to update the belief model . We use the policy gradient algorithm ( Williams , 1992 ) in reinforcement learning ( Sutton & Barto , 1992 ) to solve the optimization problem . In order to improve the efficiency of the search process , we also design a special reward function by accumulating all the rewards based on the transformation between the current concatenation and all previously sampled concatenations . Our approach is different from previous work on NAS in the following aspects : 1 . Unlike most previous work , we focus on searching for better word representations rather than better model architectures . 2 . We design a unique search space for the embedding concatenation search . Instead of using RNN as in previous work of Zoph & Le ( 2017 ) , we design a more straightforward controller to generate the embedding concatenation . We design a novel reward function in the objective of optimization to better evaluate the effectiveness of each concatenated embeddings . 3 . Our approach is efficient and practical . ACE can find a strong word representation on a single GPU with only a few GPU-hours for structured prediction tasks , while a lot of the NAS approaches require dozens of or even thousands of GPU-hours to search for good neural architecture . 4 . The task model from ACE achieves high accuracy without the need for retraining , while in previous work of NAS the resulting neural network usually requires retraining from scratch . Empirical results show that ACE outperforms strong baselines . Furthermore , we show that when ACE is applied to concatenate pretrained contextualized embeddings which are already fine-tuned on specific tasks , we can achieve state-of-the-art or competitive accuracy on 6 structured prediction tasks including Named Entity Recognition ( Sundheim , 1995 ) , Part-Of-Speech tagging ( DeRose , 1988 ) , chunking ( Tjong Kim Sang & Buchholz , 2000 ) , aspect extraction ( Hu & Liu , 2004 ) , syntactic dependency parsing ( Tesnière , 1959 ) and semantic dependency parsing ( Oepen et al. , 2014 ) over 21 datasets . Besides , we also analyze the advantage of ACE and reward function design over the baselines and show the advantage of ACE over ensemble models . 2 RELATED WORK . 2.1 EMBEDDINGS . Non-contextualized embeddings , such as word2vec ( Mikolov et al. , 2013 ) , GloVe ( Pennington et al. , 2014 ) , and fastText ( Bojanowski et al. , 2017 ) , help lots of NLP tasks . Character embeddings ( Santos & Zadrozny , 2014 ) are trained together with the task and applied in many structured prediction tasks ( Ma & Hovy , 2016 ; Lample et al. , 2016 ; Dozat & Manning , 2018 ) . For pretrained contextualized embeddings , ELMo ( Peters et al. , 2018 ) , a pretrained contextualized word embedding generated with multiple Bidirectional LSTM layers , significantly outperforms previous state-of-the-art approaches on several NLP tasks . Following this idea , Akbik et al . ( 2018 ) proposed Flair embeddings , which is a kind of contextualized character embeddings and achieved strong performance in sequence labeling tasks . Recently , Devlin et al . ( 2019 ) proposed BERT , which encodes contextualized sub-word information by Transformers and significantly improves the performance on a lot of NLP tasks . Much research such as RoBERTa ( Liu et al. , 2019c ) has focused on improving BERT model ’ s performance through stronger masking strategies . Moreover , multilingual contextualized embeddings become popular . Pires et al . ( 2019 ) and Wu & Dredze ( 2019 ) showed that Multilingual BERT ( M-BERT ) could learn a good multilingual representation effectively with strong crosslingual zero-shot transfer performance in various tasks . Conneau et al . ( 2020 ) proposed XLM-R , which is trained on a larger multilingual corpus and significantly outperforms M-BERT on various multilingual tasks . 2.2 NEURAL ARCHITECTURE SEARCH . Recent progress on deep learning has shown that network architecture design is crucial to the model performance . However , designing a strong neural architecture for each task requires enormous efforts , high level of knowledge , and experiences over the task domain . Therefore , automatic design of neural architecture is desired . A crucial part of NAS is search space design , which defines the discoverable NAS space . Previous work ( Baker et al. , 2017 ; Zoph & Le , 2017 ; Xie & Yuille , 2017 ) designs a global search space ( Elsken et al. , 2019 ) which incorporates structures from handcrafted architectures . For example , Zoph & Le ( 2017 ) designed a chained-structured search space with skip connections . The global search space usually has a considerable degree of freedom . As an example , the approach of Zoph & Le ( 2017 ) takes 22,400 GPU-hours to search on CIFAR-10 dataset . Based on the observation that existing hand-crafted architectures contain repeated structures ( Szegedy et al. , 2016 ; He et al. , 2016 ; Huang et al. , 2017 ) , Zoph et al . ( 2018 ) explored cell-based search space which can reduce the search time to 2,000 GPU-hours . In recent NAS research , reinforcement learning and evolutionary algorithms are the most usual approaches . In reinforcement learning , the agent ’ s actions are the generation of neural architectures and the action space is identical to the search space . Previous work usually applies an RNN layer ( Zoph & Le , 2017 ; Zhong et al. , 2018 ; Zoph et al. , 2018 ) or use Markov Decision Process ( Baker et al. , 2017 ) to decide the hyper-parameter of each structure and decide the input order of each structure . Evolutionary algorithms have been applied to architecture search for many decades ( Miller et al. , 1989 ; Angeline et al. , 1994 ; Stanley & Miikkulainen , 2002 ; Floreano et al. , 2008 ; Jozefowicz et al. , 2015 ) . The algorithm repeatedly generates new populations through recombination and mutation operations and selects survivors through competing among the population . Recent work with evolutionary algorithms differ in the method on parent/survivor selection and population generation . For example , Real et al . ( 2017 ) , Liu et al . ( 2018a ) , Wistuba ( 2018 ) and Real et al . ( 2019 ) applied tournament selection ( Goldberg & Deb , 1991 ) for the parent selection while Xie & Yuille ( 2017 ) keeps all parents . Suganuma et al . ( 2017 ) and Elsken et al . ( 2018 ) chose the best model while Real et al . ( 2019 ) chose several latest models as survivors . 3 AUTOMATED CONCATENATION OF EMBEDDINGS . In ACE , a task model and a controller interact with each other repeatedly . The task model predicts the task output , while the controller searches for better embedding concatenation as the word representation for the task model to achieve higher accuracy . Given an embedding concatenation generated from the controller , the task model is trained over the task data and returns a reward to the controller . The controller receives the reward to update its parameter and samples a new embedding concatenation for the task model . Figure 1 shows the general architecture of our approach . 3.1 TASK MODEL . For the tasks model , we emphasis on sequence-structured and graph-structured outputs . Given a structured prediction task with input sentence x and structured output y , we can calculate the probability distribution P ( y\|x ) by : P ( y\|x ) = exp ( Score ( x , y ) ) ∑ y′∈Y ( x ) exp ( Score ( x , y′ ) ) where Y ( x ) represents all possible output structures given the input sentence x . Depending on different structured prediction tasks , the output structure y can be label sequences , trees , graphs or other structures . In this paper , we use sequence-structured and graph-structured outputs as two exemplar structured prediction tasks . We use BiLSTM-CRF model ( Ma & Hovy , 2016 ; Lample et al. , 2016 ) for sequence-structured outputs and use BiLSTM-Biaffine model ( Dozat & Manning , 2017 ) for graph-structured outputs : P seq ( y\|x ) = BiLSTM-CRF ( V , y ) ; P graph ( y\|x ) = BiLSTM-Biaffine ( V , y ) where V = [ v1 ; · · · ; vn ] , V ∈ Rd×n is a matrix of the word representations for the input sentence xwith nwords , d is the hidden size of the concatenation of all embeddings . The word representation vi of i-th word is a concatenation of L types of word embeddings : vli = embed l i ( x ) ; vi = [ v 1 i ; v 2 i ; . . . ; v L i ] where embedl is the model of l-th embeddings , vi ∈ Rd , vli ∈ Rd l . dl is the hidden size of embedl . 3.2 SEARCH SPACE DESIGN . The neural architecture search space can be represented as a set of neural networks ( Elsken et al. , 2019 ) . A neural network can be represented as a directed acyclic graph with a set of nodes and directed edges . Each node represents an operation , while each edge represents the inputs and outputs between these nodes . In ACE , we represent each embedding candidate as a node . The input to the nodes is the input sentence x , and the outputs are the embeddings vl . Since we concatenate the embeddings as the word representation of the task model , there is no connection between nodes in our search space . Without considering the connections between nodes , the search space can be significantly reduced . For each node , there are a lot of options to extract word features . Taking BERT embeddings as an example , Devlin et al . ( 2019 ) concatenated the last four layers as word features while Kondratyuk & Straka ( 2019 ) applied a weighted sum of all twelve layers . However , the empirical results ( Devlin et al. , 2019 ) do not show a significant difference in accuracy . We follow the typical usage for each embedding to further reduce the search space . As a result , each embedding only has a fixed operation and the resulting search space contains 2L−1 possible combinations of nodes . In NAS , weight sharing ( Pham et al. , 2018a ) shares the weight of structures in training different neural architectures to reduce the training cost . In comparison , we fixed the weight of pretrained embedding candidates in ACE except for the character embeddings . Instead of sharing the parameters of the embeddings , we share the parameters of the task models at each step of search . However , the hidden size of word representation varies over the concatenations , making the weight sharing of structured prediction models difficult . Instead of deciding whether each node exists in the graph , we keep all nodes in the search space and add an additional operation for each node to indicate whether the embedding is masked out . To represent the selected concatenation , we use a binary vector a = [ a1 , · · · , al , · · · , aL ] as an mask to mask out the embeddings which are not selected : vi = [ v 1 i a1 ; . . . ; v l ial ; . . . ; v L i aL ] ( 1 ) where al is a binary variable . Since the input V is applied to a linear layer in the BiLSTM layer , multiplying the mask with the embeddings is equivalent to directly concatenating the selected embeddings : W = [ W1 ; W2 ; . . . ; WL ] ; W > vi = L∑ l=1 W > l v l ial ; W ∈ Rd×h andWl ∈ Rd l×h ( 2 ) Therefore , the model weights can be shared after applying the embedding mask to all embedding candidates ’ concatenation . Another benefit of our search space design is that we can remove the unused embedding candidates and the corresponding weights inW for a lighter task model after the best concatenation is found by ACE .	This paper introduced an interesting application of reinforcement learning in the selection of concatenation of contextual/non-contextual word embeddings. It is clever to limit the search space on the selection of embedding sources rather than search the whole network structure, as the current strategy is much easier in the training step. The author(s) conducted many experiments that compared many other models (including SOTA models, and ablation study). Those results are pretty good and impressive.	SP:481614a6302380b81ac1248b92c4c12ae366a827
Automated Concatenation of Embeddings for Structured Prediction	1 INTRODUCTION . Recent developments on pretrained contextualized embeddings have significantly improved the performance of structured prediction tasks in natural language processing . Approaches based on contextualized embeddings , such as ELMo ( Peters et al. , 2018 ) , Flair ( Akbik et al. , 2018 ) , BERT ( Devlin et al. , 2019 ) , and XLM-R ( Conneau et al. , 2020 ) , have been consistently raising the state-of-the-art for various structured prediction tasks . Concurrently , research has also showed that word representations based on the concatenation of multiple pretrained contextualized embeddings and traditional non-contextualized embeddings ( such as word2vec ( Mikolov et al. , 2013 ) and character embeddings ( Santos & Zadrozny , 2014 ) ) can further improve performance ( Peters et al. , 2018 ; Akbik et al. , 2018 ; Straková et al. , 2019 ; He & Choi , 2020 ) . Given the ever-increasing number of embedding learning methods that operate on different granularities ( e.g. , word , subword , or character level ) and with different model architectures , choosing the best embeddings to concatenate for a specific task becomes non-trivial , and exploring all possible concatenations can be prohibitively demanding in computing resources . Neural architecture search ( NAS ) is an active area of research in deep learning to automatically search for better model architectures , and has achieved state-of-the-art performance on various tasks in computer vision , such as image classification ( Real et al. , 2019 ) , semantic segmentation ( Liu et al. , 2019a ) , and object detection ( Ghiasi et al. , 2019 ) . In natural language processing , NAS has been successfully applied to find better RNN structures ( Zoph & Le , 2017 ; Pham et al. , 2018b ) and recently better transformer structures ( So et al. , 2019 ; Zhu et al. , 2020 ) . In this paper , we propose the Automated Concatenation of Embeddings ( ACE ) approach to automate the process of finding better concatenations of embeddings for structured prediction tasks , formulated as an NAS problem . In this approach , an iterative search process is guided by a controller based on its belief that models the effectiveness of individual embedding candidates in consideration for a specific task . At each step , the controller samples a concatenation of embeddings according to the belief model and feeds the concatenated word representations as inputs to a task model , which in turn is trained on the task dataset and returns the model accuracy as a reward signal to update the belief model . We use the policy gradient algorithm ( Williams , 1992 ) in reinforcement learning ( Sutton & Barto , 1992 ) to solve the optimization problem . In order to improve the efficiency of the search process , we also design a special reward function by accumulating all the rewards based on the transformation between the current concatenation and all previously sampled concatenations . Our approach is different from previous work on NAS in the following aspects : 1 . Unlike most previous work , we focus on searching for better word representations rather than better model architectures . 2 . We design a unique search space for the embedding concatenation search . Instead of using RNN as in previous work of Zoph & Le ( 2017 ) , we design a more straightforward controller to generate the embedding concatenation . We design a novel reward function in the objective of optimization to better evaluate the effectiveness of each concatenated embeddings . 3 . Our approach is efficient and practical . ACE can find a strong word representation on a single GPU with only a few GPU-hours for structured prediction tasks , while a lot of the NAS approaches require dozens of or even thousands of GPU-hours to search for good neural architecture . 4 . The task model from ACE achieves high accuracy without the need for retraining , while in previous work of NAS the resulting neural network usually requires retraining from scratch . Empirical results show that ACE outperforms strong baselines . Furthermore , we show that when ACE is applied to concatenate pretrained contextualized embeddings which are already fine-tuned on specific tasks , we can achieve state-of-the-art or competitive accuracy on 6 structured prediction tasks including Named Entity Recognition ( Sundheim , 1995 ) , Part-Of-Speech tagging ( DeRose , 1988 ) , chunking ( Tjong Kim Sang & Buchholz , 2000 ) , aspect extraction ( Hu & Liu , 2004 ) , syntactic dependency parsing ( Tesnière , 1959 ) and semantic dependency parsing ( Oepen et al. , 2014 ) over 21 datasets . Besides , we also analyze the advantage of ACE and reward function design over the baselines and show the advantage of ACE over ensemble models . 2 RELATED WORK . 2.1 EMBEDDINGS . Non-contextualized embeddings , such as word2vec ( Mikolov et al. , 2013 ) , GloVe ( Pennington et al. , 2014 ) , and fastText ( Bojanowski et al. , 2017 ) , help lots of NLP tasks . Character embeddings ( Santos & Zadrozny , 2014 ) are trained together with the task and applied in many structured prediction tasks ( Ma & Hovy , 2016 ; Lample et al. , 2016 ; Dozat & Manning , 2018 ) . For pretrained contextualized embeddings , ELMo ( Peters et al. , 2018 ) , a pretrained contextualized word embedding generated with multiple Bidirectional LSTM layers , significantly outperforms previous state-of-the-art approaches on several NLP tasks . Following this idea , Akbik et al . ( 2018 ) proposed Flair embeddings , which is a kind of contextualized character embeddings and achieved strong performance in sequence labeling tasks . Recently , Devlin et al . ( 2019 ) proposed BERT , which encodes contextualized sub-word information by Transformers and significantly improves the performance on a lot of NLP tasks . Much research such as RoBERTa ( Liu et al. , 2019c ) has focused on improving BERT model ’ s performance through stronger masking strategies . Moreover , multilingual contextualized embeddings become popular . Pires et al . ( 2019 ) and Wu & Dredze ( 2019 ) showed that Multilingual BERT ( M-BERT ) could learn a good multilingual representation effectively with strong crosslingual zero-shot transfer performance in various tasks . Conneau et al . ( 2020 ) proposed XLM-R , which is trained on a larger multilingual corpus and significantly outperforms M-BERT on various multilingual tasks . 2.2 NEURAL ARCHITECTURE SEARCH . Recent progress on deep learning has shown that network architecture design is crucial to the model performance . However , designing a strong neural architecture for each task requires enormous efforts , high level of knowledge , and experiences over the task domain . Therefore , automatic design of neural architecture is desired . A crucial part of NAS is search space design , which defines the discoverable NAS space . Previous work ( Baker et al. , 2017 ; Zoph & Le , 2017 ; Xie & Yuille , 2017 ) designs a global search space ( Elsken et al. , 2019 ) which incorporates structures from handcrafted architectures . For example , Zoph & Le ( 2017 ) designed a chained-structured search space with skip connections . The global search space usually has a considerable degree of freedom . As an example , the approach of Zoph & Le ( 2017 ) takes 22,400 GPU-hours to search on CIFAR-10 dataset . Based on the observation that existing hand-crafted architectures contain repeated structures ( Szegedy et al. , 2016 ; He et al. , 2016 ; Huang et al. , 2017 ) , Zoph et al . ( 2018 ) explored cell-based search space which can reduce the search time to 2,000 GPU-hours . In recent NAS research , reinforcement learning and evolutionary algorithms are the most usual approaches . In reinforcement learning , the agent ’ s actions are the generation of neural architectures and the action space is identical to the search space . Previous work usually applies an RNN layer ( Zoph & Le , 2017 ; Zhong et al. , 2018 ; Zoph et al. , 2018 ) or use Markov Decision Process ( Baker et al. , 2017 ) to decide the hyper-parameter of each structure and decide the input order of each structure . Evolutionary algorithms have been applied to architecture search for many decades ( Miller et al. , 1989 ; Angeline et al. , 1994 ; Stanley & Miikkulainen , 2002 ; Floreano et al. , 2008 ; Jozefowicz et al. , 2015 ) . The algorithm repeatedly generates new populations through recombination and mutation operations and selects survivors through competing among the population . Recent work with evolutionary algorithms differ in the method on parent/survivor selection and population generation . For example , Real et al . ( 2017 ) , Liu et al . ( 2018a ) , Wistuba ( 2018 ) and Real et al . ( 2019 ) applied tournament selection ( Goldberg & Deb , 1991 ) for the parent selection while Xie & Yuille ( 2017 ) keeps all parents . Suganuma et al . ( 2017 ) and Elsken et al . ( 2018 ) chose the best model while Real et al . ( 2019 ) chose several latest models as survivors . 3 AUTOMATED CONCATENATION OF EMBEDDINGS . In ACE , a task model and a controller interact with each other repeatedly . The task model predicts the task output , while the controller searches for better embedding concatenation as the word representation for the task model to achieve higher accuracy . Given an embedding concatenation generated from the controller , the task model is trained over the task data and returns a reward to the controller . The controller receives the reward to update its parameter and samples a new embedding concatenation for the task model . Figure 1 shows the general architecture of our approach . 3.1 TASK MODEL . For the tasks model , we emphasis on sequence-structured and graph-structured outputs . Given a structured prediction task with input sentence x and structured output y , we can calculate the probability distribution P ( y\|x ) by : P ( y\|x ) = exp ( Score ( x , y ) ) ∑ y′∈Y ( x ) exp ( Score ( x , y′ ) ) where Y ( x ) represents all possible output structures given the input sentence x . Depending on different structured prediction tasks , the output structure y can be label sequences , trees , graphs or other structures . In this paper , we use sequence-structured and graph-structured outputs as two exemplar structured prediction tasks . We use BiLSTM-CRF model ( Ma & Hovy , 2016 ; Lample et al. , 2016 ) for sequence-structured outputs and use BiLSTM-Biaffine model ( Dozat & Manning , 2017 ) for graph-structured outputs : P seq ( y\|x ) = BiLSTM-CRF ( V , y ) ; P graph ( y\|x ) = BiLSTM-Biaffine ( V , y ) where V = [ v1 ; · · · ; vn ] , V ∈ Rd×n is a matrix of the word representations for the input sentence xwith nwords , d is the hidden size of the concatenation of all embeddings . The word representation vi of i-th word is a concatenation of L types of word embeddings : vli = embed l i ( x ) ; vi = [ v 1 i ; v 2 i ; . . . ; v L i ] where embedl is the model of l-th embeddings , vi ∈ Rd , vli ∈ Rd l . dl is the hidden size of embedl . 3.2 SEARCH SPACE DESIGN . The neural architecture search space can be represented as a set of neural networks ( Elsken et al. , 2019 ) . A neural network can be represented as a directed acyclic graph with a set of nodes and directed edges . Each node represents an operation , while each edge represents the inputs and outputs between these nodes . In ACE , we represent each embedding candidate as a node . The input to the nodes is the input sentence x , and the outputs are the embeddings vl . Since we concatenate the embeddings as the word representation of the task model , there is no connection between nodes in our search space . Without considering the connections between nodes , the search space can be significantly reduced . For each node , there are a lot of options to extract word features . Taking BERT embeddings as an example , Devlin et al . ( 2019 ) concatenated the last four layers as word features while Kondratyuk & Straka ( 2019 ) applied a weighted sum of all twelve layers . However , the empirical results ( Devlin et al. , 2019 ) do not show a significant difference in accuracy . We follow the typical usage for each embedding to further reduce the search space . As a result , each embedding only has a fixed operation and the resulting search space contains 2L−1 possible combinations of nodes . In NAS , weight sharing ( Pham et al. , 2018a ) shares the weight of structures in training different neural architectures to reduce the training cost . In comparison , we fixed the weight of pretrained embedding candidates in ACE except for the character embeddings . Instead of sharing the parameters of the embeddings , we share the parameters of the task models at each step of search . However , the hidden size of word representation varies over the concatenations , making the weight sharing of structured prediction models difficult . Instead of deciding whether each node exists in the graph , we keep all nodes in the search space and add an additional operation for each node to indicate whether the embedding is masked out . To represent the selected concatenation , we use a binary vector a = [ a1 , · · · , al , · · · , aL ] as an mask to mask out the embeddings which are not selected : vi = [ v 1 i a1 ; . . . ; v l ial ; . . . ; v L i aL ] ( 1 ) where al is a binary variable . Since the input V is applied to a linear layer in the BiLSTM layer , multiplying the mask with the embeddings is equivalent to directly concatenating the selected embeddings : W = [ W1 ; W2 ; . . . ; WL ] ; W > vi = L∑ l=1 W > l v l ial ; W ∈ Rd×h andWl ∈ Rd l×h ( 2 ) Therefore , the model weights can be shared after applying the embedding mask to all embedding candidates ’ concatenation . Another benefit of our search space design is that we can remove the unused embedding candidates and the corresponding weights inW for a lighter task model after the best concatenation is found by ACE .	This paper proposes to automate the concatenation of word embeddings (obtained using different strategies) to produce powerful word representations for a given downstream task. To this end, the paper develops an approach based on Neural Architecture Search, wherein the search space is comprised of embedding candidates obtained using different concatenations. Using an accuracy-based reward function, it is showed that ACE can determine more effective concatenations. ACE is evaluated using extensive experiments with different tasks and datasets, and it outperforms the two baselines (to different degrees) -- random search and concatenating all embeddings with no subselection.	SP:481614a6302380b81ac1248b92c4c12ae366a827
GAN2GAN: Generative Noise Learning for Blind Denoising with Single Noisy Images	1 INTRODUCTION . Image denoising is one of the oldest problems in image processing and low-level computer vision , yet it still attracts lots of attention due to the fundamental nature of the problem . A vast number of algorithms have been proposed over the past several decades , and recently , the CNN-based methods , e.g. , Cha & Moon ( 2019 ) ; Zhang et al . ( 2017 ) ; Tai et al . ( 2017 ) ; Liu et al . ( 2018 ) , became the throne-holders in terms of the PSNR performance . The main approach of the most CNN-based denoisers is to apply the discriminative learning framework with ( clean , noisy ) image pairs and known noise distribution assumption . While being effective , such framework also possesses a couple of limitations that become critical in practice ; the assumed noise distribution may be mismatched to the actual noise in the data or obtaining the noise-free clean target images is not always possible or very expensive , e.g. , medical imaging ( CT or MRI ) or astrophotographs . Several attempts have been made to resolve above issues . For the noise uncertainty , the so-called blind training have been proposed . Namely , a denoiser can be trained with a composite training set that contains images corrupted with multiple , pre-defined noise levels or distributions , and such blindly trained denoisers , e.g. , DnCNN-B in Zhang et al . ( 2017 ) , were shown to alleviate the mismatch scenarios to some extent . However , the second limitation , i.e. , the requirement of clean images for building the training set , still remains . As an attempt to address this second limitation , Lehtinen et al . ( 2018 ) recently proposed the Noise2Noise ( N2N ) method . It has been shown that a denoiser , which has a negligible performance loss , can be trained without the clean target images , as long as two independent noisy image realizations for the same underlying clean image are available . Despite its ∗Corresponding author ( E-mail : tsmoon @ snu.ac.kr ) effectiveness , the requirement of the two independently realized noisy image pair for a single clean image , which may hardly be available in practice , is a critical limiting factor for N2N . In this paper , we consider a setting in which neither of above approach is applicable , namely , the pure unsupervised blind denoising setting where only single distinct noisy images are available for training . Namely , nothing is known about the noise other than it being zero-mean , additive , and independent of the clean image , and neither the clean target images for blind training nor the noisy image pairs for N2N training is available . While some recent work , e.g. , Krull et al . ( 2019 ) ; Batson & Royer ( 2019 ) ; Laine et al . ( 2019 ) , took the self-supervised learning ( SSL ) approach for the same setting , we take a generative learning approach . The crux of our method is to first learn a Wasserstein GAN ( Arjovsky et al. , 2017 ) -based generative model that can 1 ) learn and simulate the noise in the given noisy images and 2 ) generate rough , initially denoised images . Using such generative model , we then synthesize noisy image pairs by corrupting each of the initially denoised images with the simulated noise twice and use them to train a CNN denoiser as in the N2N training ( i.e. , Noisy N2N ) . We further show that iterative N2N training with refined denoised images can significantly improve the final denoising performance . We dubbed our method as GAN2GAN ( Generated-Artifical-Noise to Generated-Artificial-Noise ) and show that the denoiser trained with our method can achieve ( sometimes , even outperform ) the performance of the standard supervised-trained or N2N-trained blind denoisers for the white Gaussian noise case . Furthermore , for mixture/correlated noise or real-world noise in microscopy/CT images , for which the exact distributions are hard to know a priori , we show our denoiser significantly outperforms those standard blind denoisers , which are mismatch-trained with white Gaussian noise , as well as other baselines that operate in the same condition as ours : the SSL baseline , N2V ( Krull et al. , 2019 ) , and a more conventional BM3D ( Dabov et al. , 2007 ) . 2 RELATED WORK . Several works have been proposed to overcome the limitation of the vanilla supervised learning based denoising . As mentioned above , Noise2Self ( N2S ) ( Batson & Royer , 2019 ) and Noise2Void ( N2V ) ( Krull et al. , 2019 ) recently applied self-supervised learning ( SSL ) approach to train a denoiser only with single noisy images . Their settings exactly coincide with ours , but we show later that our GAN2GAN significantly outperforms them . More recently , Laine et al . ( 2019 ) improved N2V by incorporating specific noise likelihood models with Bayesian framework , however , their method required to know the exact noise model and could not be applied to more general , unknown noise settings . Similarly , Soltanayev & Chun ( 2018 ) proposed SURE ( Stein ’ s Unbiased Risk Estimator ) based denoiser that can also be trained with single noisy images , but it worked only with the Gaussian noise . Their work was extended in Zhussip et al . ( 2019 ) , but it required noisy image pairs as in N2N as well as the Gaussian noise constraint . Chen et al . ( 2018 ) devised GCBD method to learn and generate noise in the given noisy images using W-GAN Arjovsky et al . ( 2017 ) and utilized the unpaired clean images to build a supervised training set . Our GAN2GAN is related to Chen et al . ( 2018 ) , but we significantly improve their noise learning step and do not use the clean data at all . Table 1 summarizes and compares the settings among the above mentioned recent baselines . We clearly see that only our GAN2GAN and N2V do not utilize any “ sidekicks ” that other methods use . Additionally , there are recently published papers on blind image denoising but these also have a difference with ours . Anwar & Barnes ( 2019 ) ; Zhang et al . ( 2018 ) suggest effective CNN architectures for denoising , however , they only consider the setting in which clean images are necessary for training . Zamir et al . ( 2020 ) considers the denoising of specific camera settings , and it also requires clean sRGB images as well as the knowledge of the noise level . Thus , it can not be applied to the complete blind setting as ours , in which no information on the specific noise distribution or clean images is available . More classical denoising methods are capable of denoising solely based on the single noisy images by applying various principles , e.g. , filtering-based Buades et al . ( 2005 ) ; Dabov et al . ( 2007 ) , optimization-based Elad & Aharon ( 2006 ) ; Mairal et al . ( 2009 ) , Wavelet-based Donoho & Johnstone ( 1995 ) , and effective prior-based Zoran & Weiss ( 2011 ) . Those methods typically are , however , computationally intensive during the inference time and can not be trained from a separate set of noisy images , which limits their denoising performance . Another line of recent work worth mentioning is the deep learning-based priors or regularizers , e.g. , Ulyanov et al . ( 2018 ) ; Yeh et al . ( 2018 ) ; Lunz et al . ( 2018 ) , but their PSNRs still fell short of the supervised trained CNN-based denoisers . 3 MOTIVATION . In order to develop the core intuition for motivating our method , we first consider a simple , singleletter Gaussian noise setting . Let Z = X +N be the noisy observation of X ∼ N ( 0 , σ2X ) , corrupted by the N ∼ N ( 0 , σ2N ) . It is well known that the minimum MSE ( MMSE ) estimator of X given Z is f∗MMSE ( Z ) = E ( X\|Z ) = σ2X σ2X+σ 2 N Z . We now identify the optimality of N2N in this setting . N2N Assume that we have two i.i.d . copies of the noise N : N1 and N2 . Then , let Z1 = X +N1 and Z2 = X +N2 be the two independent noisy observation pairs of X . The N2N in this setting corresponds to obtaining the MMSE estimator of Z2 given Z1 , fN2N ( Z1 ) , argmin f E ( Z2 − f ( Z1 ) ) 2 = E ( Z2\|Z1 ) = E ( X +N2\|Z1 ) ( a ) = E ( X\|Z1 ) = σ2X σ2X + σ 2 N Z1 , ( 1 ) in which ( a ) follows fromN2 being independent ofZ1 . Note ( 1 ) has the exact same form as f∗MMSE ( Z ) , hence , estimating X with fN2N ( Z ) also achieves the MMSE , in line with ( Lehtinen et al. , 2018 ) . “ Noisy ” N2N Now , consider the case in which we again have the two i.i.d . N1 and N2 , but the noisy observations are of a noisy version of X . Namely , let X ′ = X +N0 , in which N0 ∼ N ( 0 , σ20 ) , and denote Z ′1 = X ′ +N1 and Z ′2 = X ′ +N2 as the noisy observation pairs . Then , we can define a “ Noisy ” N2N estimator as the MMSE estimator of Z ′2 given Z ′ 1 , fNoisy N2N ( Z ′ 1 , y ) , argmin f E ( Z ′2 − f ( Z ′1 ) ) 2 = E ( X ′\|Z ′1 ) = σ2X ( 1 + y ) σ2X ( 1 + y ) + σ 2 N Z ′1 , ( 2 ) in which we denote y , σ20/σ 2 X and assume that 0 ≤ y < 1 . Note clearly ( 2 ) coincides with ( 1 ) when y = σ20 = 0 . Following N2N , ( 2 ) is essentially estimating X ′ based on Z ′ = X ′ + N . An interesting subtle question is what happens when we use the mapping fNoisy N2N ( Z , y ) for estimating X given Z = X +N , not X ′ given Z ′ . Our theorem below , of which proof is in the Supplementary Material ( S.M . ) , shows that for a sufficiently large σ20 , fNoisy N2N ( Z , y ) gives a better estimate of X than X ′ . Theorem 1 Consider the single-letter Gaussian setting and fNoisy N2N ( Z , y ) obtained in ( 2 ) . Also , assume 0 < y < 1 . Then , there exists some y0 s.t . ∀y ∈ ( y0 , 1 ) , E ( X − fNoisy N2N ( Z , y ) ) 2 < σ20 . Theorem 1 provides a simple , but useful , intuition that motivates our method ; if simulating the noise in the images is possible , we may carry out the N2N training iteratively , provided that a rough noisy estimate of the clean image is initially available . Namely , we can first simulate the noise to generate noisy observation pairs of the initial noisy estimate , then do the Noisy N2N training with them to obtain a denoiser that may result in a better estimate of the clean image when applied to the actual noisy image subject to denoising ( as in Theorem 1 ) . Then , we can refine the estimates by iterating the Noisy N2N training with the generated noisy observation pairs of the previous step ’ s estimate of the clean image , until convergence . To check whether above intuition is valid , we carry out a feasibility experiment . Figure 1 shows the denoising results on BSD68 ( Roth & Black , 2009 ) for Gaussian noise with σ = 25 . The blue line is the PSNR of the N2N model trained with noisy observation pairs of the clean images in the BSD training set , serving as an upper bound . The orange line , in contrast , is the PSNR of the Noisy N2N1 model that is trained with the noisy observation pairs of the noisy estimates for the clean images , which were set to be another Gaussian noise-corrupted training images . The standard deviations ( σ0 ) of the Gaussian for generating the noisy estimates are given in the horizontal axis , and the corresponding PSNRs of the estimates are given in the parentheses . Although Noisy N2N1 clearly lies much lower than the N2N upper bound , we note its PSNR is still higher than that of the initial noisy estimates , which is in line with Theorem 1 . Now , if we iterate the Noisy N2N with the previous step ’ s denoised images ( i.e. , Noisy-N2N2/Noisy-N2N3 for second/third iterations , respectively ) , we observe that the PSNR significantly improves and approaches the ordinary N2N for most of the initial σ0 values . Thus , we observe the intuition from Theorem 1 generalizes well to the image denoising case in an ideal setting , where the noise can be perfectly simulated , and the initial noisy estimates are Gaussian corrupted versions . The remaining question is whether we can also obtain similar results for the blind image denoising setting . We show our generative model-based approach in details in the next section .	This paper proposes a framework to train a network to remove noises, which are zero-mean, additive and independent of the clean image, with only noisy images and without knowing the noise statistics. They mathematically prove that a network, which is trained from pairs of images generated by adding simulated noises into the noisy image, can remove noises from the input noisy image. The proposed framework can remove the noises from the input noisy image. Then, it adds simulated noises into the denoised image to generate a pair of images and train the next network to further remove noises and it does this process iteratively. The proposed framework follows GCBD and utilizes flat textureless regions to train a network to simulate noises and proposes a wavelet based method to effectively distinguish flat regions from the ones that contain high-frequency repeating patterns. The experimental results show that the proposed method has good performance under simulated Gaussian noises as well as WT and CT datasets.	SP:dc8873528c8b4cccafa3d45fe0b244a2d0d99e9b
GAN2GAN: Generative Noise Learning for Blind Denoising with Single Noisy Images	1 INTRODUCTION . Image denoising is one of the oldest problems in image processing and low-level computer vision , yet it still attracts lots of attention due to the fundamental nature of the problem . A vast number of algorithms have been proposed over the past several decades , and recently , the CNN-based methods , e.g. , Cha & Moon ( 2019 ) ; Zhang et al . ( 2017 ) ; Tai et al . ( 2017 ) ; Liu et al . ( 2018 ) , became the throne-holders in terms of the PSNR performance . The main approach of the most CNN-based denoisers is to apply the discriminative learning framework with ( clean , noisy ) image pairs and known noise distribution assumption . While being effective , such framework also possesses a couple of limitations that become critical in practice ; the assumed noise distribution may be mismatched to the actual noise in the data or obtaining the noise-free clean target images is not always possible or very expensive , e.g. , medical imaging ( CT or MRI ) or astrophotographs . Several attempts have been made to resolve above issues . For the noise uncertainty , the so-called blind training have been proposed . Namely , a denoiser can be trained with a composite training set that contains images corrupted with multiple , pre-defined noise levels or distributions , and such blindly trained denoisers , e.g. , DnCNN-B in Zhang et al . ( 2017 ) , were shown to alleviate the mismatch scenarios to some extent . However , the second limitation , i.e. , the requirement of clean images for building the training set , still remains . As an attempt to address this second limitation , Lehtinen et al . ( 2018 ) recently proposed the Noise2Noise ( N2N ) method . It has been shown that a denoiser , which has a negligible performance loss , can be trained without the clean target images , as long as two independent noisy image realizations for the same underlying clean image are available . Despite its ∗Corresponding author ( E-mail : tsmoon @ snu.ac.kr ) effectiveness , the requirement of the two independently realized noisy image pair for a single clean image , which may hardly be available in practice , is a critical limiting factor for N2N . In this paper , we consider a setting in which neither of above approach is applicable , namely , the pure unsupervised blind denoising setting where only single distinct noisy images are available for training . Namely , nothing is known about the noise other than it being zero-mean , additive , and independent of the clean image , and neither the clean target images for blind training nor the noisy image pairs for N2N training is available . While some recent work , e.g. , Krull et al . ( 2019 ) ; Batson & Royer ( 2019 ) ; Laine et al . ( 2019 ) , took the self-supervised learning ( SSL ) approach for the same setting , we take a generative learning approach . The crux of our method is to first learn a Wasserstein GAN ( Arjovsky et al. , 2017 ) -based generative model that can 1 ) learn and simulate the noise in the given noisy images and 2 ) generate rough , initially denoised images . Using such generative model , we then synthesize noisy image pairs by corrupting each of the initially denoised images with the simulated noise twice and use them to train a CNN denoiser as in the N2N training ( i.e. , Noisy N2N ) . We further show that iterative N2N training with refined denoised images can significantly improve the final denoising performance . We dubbed our method as GAN2GAN ( Generated-Artifical-Noise to Generated-Artificial-Noise ) and show that the denoiser trained with our method can achieve ( sometimes , even outperform ) the performance of the standard supervised-trained or N2N-trained blind denoisers for the white Gaussian noise case . Furthermore , for mixture/correlated noise or real-world noise in microscopy/CT images , for which the exact distributions are hard to know a priori , we show our denoiser significantly outperforms those standard blind denoisers , which are mismatch-trained with white Gaussian noise , as well as other baselines that operate in the same condition as ours : the SSL baseline , N2V ( Krull et al. , 2019 ) , and a more conventional BM3D ( Dabov et al. , 2007 ) . 2 RELATED WORK . Several works have been proposed to overcome the limitation of the vanilla supervised learning based denoising . As mentioned above , Noise2Self ( N2S ) ( Batson & Royer , 2019 ) and Noise2Void ( N2V ) ( Krull et al. , 2019 ) recently applied self-supervised learning ( SSL ) approach to train a denoiser only with single noisy images . Their settings exactly coincide with ours , but we show later that our GAN2GAN significantly outperforms them . More recently , Laine et al . ( 2019 ) improved N2V by incorporating specific noise likelihood models with Bayesian framework , however , their method required to know the exact noise model and could not be applied to more general , unknown noise settings . Similarly , Soltanayev & Chun ( 2018 ) proposed SURE ( Stein ’ s Unbiased Risk Estimator ) based denoiser that can also be trained with single noisy images , but it worked only with the Gaussian noise . Their work was extended in Zhussip et al . ( 2019 ) , but it required noisy image pairs as in N2N as well as the Gaussian noise constraint . Chen et al . ( 2018 ) devised GCBD method to learn and generate noise in the given noisy images using W-GAN Arjovsky et al . ( 2017 ) and utilized the unpaired clean images to build a supervised training set . Our GAN2GAN is related to Chen et al . ( 2018 ) , but we significantly improve their noise learning step and do not use the clean data at all . Table 1 summarizes and compares the settings among the above mentioned recent baselines . We clearly see that only our GAN2GAN and N2V do not utilize any “ sidekicks ” that other methods use . Additionally , there are recently published papers on blind image denoising but these also have a difference with ours . Anwar & Barnes ( 2019 ) ; Zhang et al . ( 2018 ) suggest effective CNN architectures for denoising , however , they only consider the setting in which clean images are necessary for training . Zamir et al . ( 2020 ) considers the denoising of specific camera settings , and it also requires clean sRGB images as well as the knowledge of the noise level . Thus , it can not be applied to the complete blind setting as ours , in which no information on the specific noise distribution or clean images is available . More classical denoising methods are capable of denoising solely based on the single noisy images by applying various principles , e.g. , filtering-based Buades et al . ( 2005 ) ; Dabov et al . ( 2007 ) , optimization-based Elad & Aharon ( 2006 ) ; Mairal et al . ( 2009 ) , Wavelet-based Donoho & Johnstone ( 1995 ) , and effective prior-based Zoran & Weiss ( 2011 ) . Those methods typically are , however , computationally intensive during the inference time and can not be trained from a separate set of noisy images , which limits their denoising performance . Another line of recent work worth mentioning is the deep learning-based priors or regularizers , e.g. , Ulyanov et al . ( 2018 ) ; Yeh et al . ( 2018 ) ; Lunz et al . ( 2018 ) , but their PSNRs still fell short of the supervised trained CNN-based denoisers . 3 MOTIVATION . In order to develop the core intuition for motivating our method , we first consider a simple , singleletter Gaussian noise setting . Let Z = X +N be the noisy observation of X ∼ N ( 0 , σ2X ) , corrupted by the N ∼ N ( 0 , σ2N ) . It is well known that the minimum MSE ( MMSE ) estimator of X given Z is f∗MMSE ( Z ) = E ( X\|Z ) = σ2X σ2X+σ 2 N Z . We now identify the optimality of N2N in this setting . N2N Assume that we have two i.i.d . copies of the noise N : N1 and N2 . Then , let Z1 = X +N1 and Z2 = X +N2 be the two independent noisy observation pairs of X . The N2N in this setting corresponds to obtaining the MMSE estimator of Z2 given Z1 , fN2N ( Z1 ) , argmin f E ( Z2 − f ( Z1 ) ) 2 = E ( Z2\|Z1 ) = E ( X +N2\|Z1 ) ( a ) = E ( X\|Z1 ) = σ2X σ2X + σ 2 N Z1 , ( 1 ) in which ( a ) follows fromN2 being independent ofZ1 . Note ( 1 ) has the exact same form as f∗MMSE ( Z ) , hence , estimating X with fN2N ( Z ) also achieves the MMSE , in line with ( Lehtinen et al. , 2018 ) . “ Noisy ” N2N Now , consider the case in which we again have the two i.i.d . N1 and N2 , but the noisy observations are of a noisy version of X . Namely , let X ′ = X +N0 , in which N0 ∼ N ( 0 , σ20 ) , and denote Z ′1 = X ′ +N1 and Z ′2 = X ′ +N2 as the noisy observation pairs . Then , we can define a “ Noisy ” N2N estimator as the MMSE estimator of Z ′2 given Z ′ 1 , fNoisy N2N ( Z ′ 1 , y ) , argmin f E ( Z ′2 − f ( Z ′1 ) ) 2 = E ( X ′\|Z ′1 ) = σ2X ( 1 + y ) σ2X ( 1 + y ) + σ 2 N Z ′1 , ( 2 ) in which we denote y , σ20/σ 2 X and assume that 0 ≤ y < 1 . Note clearly ( 2 ) coincides with ( 1 ) when y = σ20 = 0 . Following N2N , ( 2 ) is essentially estimating X ′ based on Z ′ = X ′ + N . An interesting subtle question is what happens when we use the mapping fNoisy N2N ( Z , y ) for estimating X given Z = X +N , not X ′ given Z ′ . Our theorem below , of which proof is in the Supplementary Material ( S.M . ) , shows that for a sufficiently large σ20 , fNoisy N2N ( Z , y ) gives a better estimate of X than X ′ . Theorem 1 Consider the single-letter Gaussian setting and fNoisy N2N ( Z , y ) obtained in ( 2 ) . Also , assume 0 < y < 1 . Then , there exists some y0 s.t . ∀y ∈ ( y0 , 1 ) , E ( X − fNoisy N2N ( Z , y ) ) 2 < σ20 . Theorem 1 provides a simple , but useful , intuition that motivates our method ; if simulating the noise in the images is possible , we may carry out the N2N training iteratively , provided that a rough noisy estimate of the clean image is initially available . Namely , we can first simulate the noise to generate noisy observation pairs of the initial noisy estimate , then do the Noisy N2N training with them to obtain a denoiser that may result in a better estimate of the clean image when applied to the actual noisy image subject to denoising ( as in Theorem 1 ) . Then , we can refine the estimates by iterating the Noisy N2N training with the generated noisy observation pairs of the previous step ’ s estimate of the clean image , until convergence . To check whether above intuition is valid , we carry out a feasibility experiment . Figure 1 shows the denoising results on BSD68 ( Roth & Black , 2009 ) for Gaussian noise with σ = 25 . The blue line is the PSNR of the N2N model trained with noisy observation pairs of the clean images in the BSD training set , serving as an upper bound . The orange line , in contrast , is the PSNR of the Noisy N2N1 model that is trained with the noisy observation pairs of the noisy estimates for the clean images , which were set to be another Gaussian noise-corrupted training images . The standard deviations ( σ0 ) of the Gaussian for generating the noisy estimates are given in the horizontal axis , and the corresponding PSNRs of the estimates are given in the parentheses . Although Noisy N2N1 clearly lies much lower than the N2N upper bound , we note its PSNR is still higher than that of the initial noisy estimates , which is in line with Theorem 1 . Now , if we iterate the Noisy N2N with the previous step ’ s denoised images ( i.e. , Noisy-N2N2/Noisy-N2N3 for second/third iterations , respectively ) , we observe that the PSNR significantly improves and approaches the ordinary N2N for most of the initial σ0 values . Thus , we observe the intuition from Theorem 1 generalizes well to the image denoising case in an ideal setting , where the noise can be perfectly simulated , and the initial noisy estimates are Gaussian corrupted versions . The remaining question is whether we can also obtain similar results for the blind image denoising setting . We show our generative model-based approach in details in the next section .	This paper addresses a challenging task of blind image denoising where a single noisy image is provided with assumption that it is zero mean, additive and independent from the original image content. This is mostly the real-world scenario. Different from the recent N2N training, the authors propose a GAN2GAN based method since this blind setting cannot be trained by N2N. N2N or deterministic training needs explicit or implicit knowledge of clean image in order to be trained whereas the GAN2GAN method does not, leading to more realistic and efficient training. This method first attempts to simulate noise given the noisy image, generate rough and noisy estimates of the clean image, and iteratively train a denoiser with synthetic noisy pairs from the generator. For blind denoising, this work produces impressive results for synthetic and real world blind denoising.	SP:dc8873528c8b4cccafa3d45fe0b244a2d0d99e9b
GAN2GAN: Generative Noise Learning for Blind Denoising with Single Noisy Images	1 INTRODUCTION . Image denoising is one of the oldest problems in image processing and low-level computer vision , yet it still attracts lots of attention due to the fundamental nature of the problem . A vast number of algorithms have been proposed over the past several decades , and recently , the CNN-based methods , e.g. , Cha & Moon ( 2019 ) ; Zhang et al . ( 2017 ) ; Tai et al . ( 2017 ) ; Liu et al . ( 2018 ) , became the throne-holders in terms of the PSNR performance . The main approach of the most CNN-based denoisers is to apply the discriminative learning framework with ( clean , noisy ) image pairs and known noise distribution assumption . While being effective , such framework also possesses a couple of limitations that become critical in practice ; the assumed noise distribution may be mismatched to the actual noise in the data or obtaining the noise-free clean target images is not always possible or very expensive , e.g. , medical imaging ( CT or MRI ) or astrophotographs . Several attempts have been made to resolve above issues . For the noise uncertainty , the so-called blind training have been proposed . Namely , a denoiser can be trained with a composite training set that contains images corrupted with multiple , pre-defined noise levels or distributions , and such blindly trained denoisers , e.g. , DnCNN-B in Zhang et al . ( 2017 ) , were shown to alleviate the mismatch scenarios to some extent . However , the second limitation , i.e. , the requirement of clean images for building the training set , still remains . As an attempt to address this second limitation , Lehtinen et al . ( 2018 ) recently proposed the Noise2Noise ( N2N ) method . It has been shown that a denoiser , which has a negligible performance loss , can be trained without the clean target images , as long as two independent noisy image realizations for the same underlying clean image are available . Despite its ∗Corresponding author ( E-mail : tsmoon @ snu.ac.kr ) effectiveness , the requirement of the two independently realized noisy image pair for a single clean image , which may hardly be available in practice , is a critical limiting factor for N2N . In this paper , we consider a setting in which neither of above approach is applicable , namely , the pure unsupervised blind denoising setting where only single distinct noisy images are available for training . Namely , nothing is known about the noise other than it being zero-mean , additive , and independent of the clean image , and neither the clean target images for blind training nor the noisy image pairs for N2N training is available . While some recent work , e.g. , Krull et al . ( 2019 ) ; Batson & Royer ( 2019 ) ; Laine et al . ( 2019 ) , took the self-supervised learning ( SSL ) approach for the same setting , we take a generative learning approach . The crux of our method is to first learn a Wasserstein GAN ( Arjovsky et al. , 2017 ) -based generative model that can 1 ) learn and simulate the noise in the given noisy images and 2 ) generate rough , initially denoised images . Using such generative model , we then synthesize noisy image pairs by corrupting each of the initially denoised images with the simulated noise twice and use them to train a CNN denoiser as in the N2N training ( i.e. , Noisy N2N ) . We further show that iterative N2N training with refined denoised images can significantly improve the final denoising performance . We dubbed our method as GAN2GAN ( Generated-Artifical-Noise to Generated-Artificial-Noise ) and show that the denoiser trained with our method can achieve ( sometimes , even outperform ) the performance of the standard supervised-trained or N2N-trained blind denoisers for the white Gaussian noise case . Furthermore , for mixture/correlated noise or real-world noise in microscopy/CT images , for which the exact distributions are hard to know a priori , we show our denoiser significantly outperforms those standard blind denoisers , which are mismatch-trained with white Gaussian noise , as well as other baselines that operate in the same condition as ours : the SSL baseline , N2V ( Krull et al. , 2019 ) , and a more conventional BM3D ( Dabov et al. , 2007 ) . 2 RELATED WORK . Several works have been proposed to overcome the limitation of the vanilla supervised learning based denoising . As mentioned above , Noise2Self ( N2S ) ( Batson & Royer , 2019 ) and Noise2Void ( N2V ) ( Krull et al. , 2019 ) recently applied self-supervised learning ( SSL ) approach to train a denoiser only with single noisy images . Their settings exactly coincide with ours , but we show later that our GAN2GAN significantly outperforms them . More recently , Laine et al . ( 2019 ) improved N2V by incorporating specific noise likelihood models with Bayesian framework , however , their method required to know the exact noise model and could not be applied to more general , unknown noise settings . Similarly , Soltanayev & Chun ( 2018 ) proposed SURE ( Stein ’ s Unbiased Risk Estimator ) based denoiser that can also be trained with single noisy images , but it worked only with the Gaussian noise . Their work was extended in Zhussip et al . ( 2019 ) , but it required noisy image pairs as in N2N as well as the Gaussian noise constraint . Chen et al . ( 2018 ) devised GCBD method to learn and generate noise in the given noisy images using W-GAN Arjovsky et al . ( 2017 ) and utilized the unpaired clean images to build a supervised training set . Our GAN2GAN is related to Chen et al . ( 2018 ) , but we significantly improve their noise learning step and do not use the clean data at all . Table 1 summarizes and compares the settings among the above mentioned recent baselines . We clearly see that only our GAN2GAN and N2V do not utilize any “ sidekicks ” that other methods use . Additionally , there are recently published papers on blind image denoising but these also have a difference with ours . Anwar & Barnes ( 2019 ) ; Zhang et al . ( 2018 ) suggest effective CNN architectures for denoising , however , they only consider the setting in which clean images are necessary for training . Zamir et al . ( 2020 ) considers the denoising of specific camera settings , and it also requires clean sRGB images as well as the knowledge of the noise level . Thus , it can not be applied to the complete blind setting as ours , in which no information on the specific noise distribution or clean images is available . More classical denoising methods are capable of denoising solely based on the single noisy images by applying various principles , e.g. , filtering-based Buades et al . ( 2005 ) ; Dabov et al . ( 2007 ) , optimization-based Elad & Aharon ( 2006 ) ; Mairal et al . ( 2009 ) , Wavelet-based Donoho & Johnstone ( 1995 ) , and effective prior-based Zoran & Weiss ( 2011 ) . Those methods typically are , however , computationally intensive during the inference time and can not be trained from a separate set of noisy images , which limits their denoising performance . Another line of recent work worth mentioning is the deep learning-based priors or regularizers , e.g. , Ulyanov et al . ( 2018 ) ; Yeh et al . ( 2018 ) ; Lunz et al . ( 2018 ) , but their PSNRs still fell short of the supervised trained CNN-based denoisers . 3 MOTIVATION . In order to develop the core intuition for motivating our method , we first consider a simple , singleletter Gaussian noise setting . Let Z = X +N be the noisy observation of X ∼ N ( 0 , σ2X ) , corrupted by the N ∼ N ( 0 , σ2N ) . It is well known that the minimum MSE ( MMSE ) estimator of X given Z is f∗MMSE ( Z ) = E ( X\|Z ) = σ2X σ2X+σ 2 N Z . We now identify the optimality of N2N in this setting . N2N Assume that we have two i.i.d . copies of the noise N : N1 and N2 . Then , let Z1 = X +N1 and Z2 = X +N2 be the two independent noisy observation pairs of X . The N2N in this setting corresponds to obtaining the MMSE estimator of Z2 given Z1 , fN2N ( Z1 ) , argmin f E ( Z2 − f ( Z1 ) ) 2 = E ( Z2\|Z1 ) = E ( X +N2\|Z1 ) ( a ) = E ( X\|Z1 ) = σ2X σ2X + σ 2 N Z1 , ( 1 ) in which ( a ) follows fromN2 being independent ofZ1 . Note ( 1 ) has the exact same form as f∗MMSE ( Z ) , hence , estimating X with fN2N ( Z ) also achieves the MMSE , in line with ( Lehtinen et al. , 2018 ) . “ Noisy ” N2N Now , consider the case in which we again have the two i.i.d . N1 and N2 , but the noisy observations are of a noisy version of X . Namely , let X ′ = X +N0 , in which N0 ∼ N ( 0 , σ20 ) , and denote Z ′1 = X ′ +N1 and Z ′2 = X ′ +N2 as the noisy observation pairs . Then , we can define a “ Noisy ” N2N estimator as the MMSE estimator of Z ′2 given Z ′ 1 , fNoisy N2N ( Z ′ 1 , y ) , argmin f E ( Z ′2 − f ( Z ′1 ) ) 2 = E ( X ′\|Z ′1 ) = σ2X ( 1 + y ) σ2X ( 1 + y ) + σ 2 N Z ′1 , ( 2 ) in which we denote y , σ20/σ 2 X and assume that 0 ≤ y < 1 . Note clearly ( 2 ) coincides with ( 1 ) when y = σ20 = 0 . Following N2N , ( 2 ) is essentially estimating X ′ based on Z ′ = X ′ + N . An interesting subtle question is what happens when we use the mapping fNoisy N2N ( Z , y ) for estimating X given Z = X +N , not X ′ given Z ′ . Our theorem below , of which proof is in the Supplementary Material ( S.M . ) , shows that for a sufficiently large σ20 , fNoisy N2N ( Z , y ) gives a better estimate of X than X ′ . Theorem 1 Consider the single-letter Gaussian setting and fNoisy N2N ( Z , y ) obtained in ( 2 ) . Also , assume 0 < y < 1 . Then , there exists some y0 s.t . ∀y ∈ ( y0 , 1 ) , E ( X − fNoisy N2N ( Z , y ) ) 2 < σ20 . Theorem 1 provides a simple , but useful , intuition that motivates our method ; if simulating the noise in the images is possible , we may carry out the N2N training iteratively , provided that a rough noisy estimate of the clean image is initially available . Namely , we can first simulate the noise to generate noisy observation pairs of the initial noisy estimate , then do the Noisy N2N training with them to obtain a denoiser that may result in a better estimate of the clean image when applied to the actual noisy image subject to denoising ( as in Theorem 1 ) . Then , we can refine the estimates by iterating the Noisy N2N training with the generated noisy observation pairs of the previous step ’ s estimate of the clean image , until convergence . To check whether above intuition is valid , we carry out a feasibility experiment . Figure 1 shows the denoising results on BSD68 ( Roth & Black , 2009 ) for Gaussian noise with σ = 25 . The blue line is the PSNR of the N2N model trained with noisy observation pairs of the clean images in the BSD training set , serving as an upper bound . The orange line , in contrast , is the PSNR of the Noisy N2N1 model that is trained with the noisy observation pairs of the noisy estimates for the clean images , which were set to be another Gaussian noise-corrupted training images . The standard deviations ( σ0 ) of the Gaussian for generating the noisy estimates are given in the horizontal axis , and the corresponding PSNRs of the estimates are given in the parentheses . Although Noisy N2N1 clearly lies much lower than the N2N upper bound , we note its PSNR is still higher than that of the initial noisy estimates , which is in line with Theorem 1 . Now , if we iterate the Noisy N2N with the previous step ’ s denoised images ( i.e. , Noisy-N2N2/Noisy-N2N3 for second/third iterations , respectively ) , we observe that the PSNR significantly improves and approaches the ordinary N2N for most of the initial σ0 values . Thus , we observe the intuition from Theorem 1 generalizes well to the image denoising case in an ideal setting , where the noise can be perfectly simulated , and the initial noisy estimates are Gaussian corrupted versions . The remaining question is whether we can also obtain similar results for the blind image denoising setting . We show our generative model-based approach in details in the next section .	This paper proposed a new method for blind image denoising using a "GAN2GAN" network. Different from previous work noise2noise, the proposed method only needs single noisy images to train the network, without noisy pairs. Given a noisy dataset, a GAN generator is trained using the real noisy but smooth patches. With the noise generator, a pair of generated noise samples are added on the noisy image content to train a denoiser. This is trained iteratively, so that the image content is better and better after cleaned by the trained denoiser. The final results on synthetic and real data showed improvements over the baselines.	SP:dc8873528c8b4cccafa3d45fe0b244a2d0d99e9b
Learning not to learn: Nature versus nurture in silico	1 INTRODUCTION . The ’ nature versus nurture ’ debate ( e.g. , Mutti et al. , 1996 ; Tabery , 2014 ) – the question of which aspects of behavior are ’ hard-coded ’ by evolution , and which are learned from experience – is one of the oldest and most controversial debates in biology . Evolutionary principles prescribe that hard-coded behavioral routines should be those for which there is no benefit in adaptation . This is believed to be the case for behaviors whose evolutionary advantage varies little among individuals of a species . Mating instincts or flight reflexes are general solutions that rarely present an evolutionary disadvantage . On the other hand , features of the environment that vary substantially for individuals of a species potentially ask for adaptive behavior ( Buss , 2015 ) . Naturally , the same principles should not only apply to biological but also to artificial agents . But how can a reinforcement learning agent differentiate between these two behavioral regimes ? A promising approach to automatically learn rules of adaptation that facilitate environment-specific specialization is meta-learning ( Schmidhuber , 1987 ; Thrun & Pratt , 1998 ) . At its core lies the idea of using generic optimization methods to learn inductive biases for a given ensemble of tasks . In this approach , the inductive bias usually has its own set of parameters ( e.g. , weights in a recurrent network ; Hochreiter et al. , 2001 ) that are optimized on the whole task ensemble , that is , on a long , ’ evolutionary ’ time scale . These parameters in turn control how a different set of parameters ( e.g. , activities in the network ) are updated on a much faster time scale . These rapidly adapting parameters then allow the system to adapt to a specific task at hand . Notably , the parameters of the system that are subject to ’ nature ’ – i.e. , those that shape the inductive bias and are common across tasks – and those that are subject to ’ nurture ’ are usually predefined from the start . In this work , we use the memory-based meta-learning approach for a different goal , namely to acquire a qualitative understanding of which aspects of behavior should be hard-coded and which should be adaptive . Our hypothesis is that meta-learning can not only learn efficient learning algorithms , but can also decide not to be adaptive at all , and to instead apply a generic heuristic to the whole ensemble of tasks . Phrased in the language of biology , meta-learning can decide whether to hard-code a behavior or to render it adaptive , based on the range of environments the individuals of a species could encounter . We study the dependence of the meta-learned algorithm on three central features of the metareinforcement learning problem : • Ecological uncertainty : How diverse is the range of tasks the agent could encounter ? • Task complexity : How long does it take to learn the optimal strategy for the task at hand ? Note that this could be different from the time it takes to execute the optimal strategy . • Expected lifetime : How much time can the agent spend on exploration and exploitation ? Using analytical and numerical analyses , we show that non-adaptive behaviors are optimal in two cases – when the optimal policy varies little across the tasks within the task ensemble and when the time it takes to learn the optimal policy is too long to allow a sufficient exploitation of the learned policy . Our results suggest that not only the design of the meta-task distribution , but also the lifetime of the agent can have strong effects on the meta-learned algorithm of RNN-based agents . In particular , we find highly nonlinear and potentially discontinuous effects of ecological uncertainty , task complexity and lifetime on the optimal algorithm . As a consequence , a meta-learned adaptation strategy that was optimized , e.g. , for a given lifetime may not generalize well to other lifetimes . This is essential for research questions that are interested in the conducted adaptation behavior , including curriculum design , safe exploration as well as human-in-the-loop applications . Our work may provide a principled way of examining the constraint-dependence of meta-learned inductive biases . The remainder of this paper is structured as follows : First , we review the background in memorybased meta-reinforcement learning and contrast the related literature . Afterwards , we analyze a Gaussian multi-arm bandit setting , which allows us to analytically disentangle the behavioral impact of ecological uncertainty , task complexity and lifetime . Our derivation of the lifetime-dependent Bayes optimal exploration reveals a highly non-linear interplay of these three factors . We show numerically that memory-based meta-learning reproduces our theoretical results and can learn not to learn . Furthermore , we extend our analysis to more complicated exploration problems . Throughout , we analyze the resulting recurrent dynamics of the network and the representations associated with learning and non-adaptive strategies . 2 RELATED WORK & BACKGROUND . Meta-learning or ’ learning to learn ’ ( e.g. , Schmidhuber , 1987 ; Thrun & Pratt , 1998 ; Hochreiter et al. , 2001 ; Duan et al. , 2016 ; Wang et al. , 2016 ; Finn et al. , 2017 ) has been proposed as a computational framework for acquiring task distribution-specific learning rules . During a costly outer loop optimization , an agent crafts a niche-specific adaptation strategy , which is applicable to an engineered task distribution . At inference time , the acquired inner loop learning algorithm is executed for a fixed amount of timesteps ( lifetime ) on a test task . This framework has successfully been applied to a range of applications such as the meta-learning of optimization updates ( Andrychowicz et al. , 2016 ; Flennerhag et al. , 2018 ; 2019 ) , agent ( Rabinowitz et al. , 2018 ) and world models ( Nagabandi et al. , 2018 ) and explicit models of memory ( Santoro et al. , 2016 ; Bartunov et al. , 2019 ) . Already , early work by Schmidhuber ( 1987 ) suggested an evolutionary perspective on recursively learning the rules of learning . This perspective holds the promise of explaining the emergence of mechanisms underlying both natural and artificial behaviors . Furthermore , a similarity between the hidden activations of LSTM-based meta-learners and the recurrent activity of neurons in the prefrontal cortex ( Wang et al. , 2018 ) has recently been suggested . Previous work has shown that LSTM-based meta-learning is capable of distilling a sequential integration algorithm akin to amortized Bayesian inference ( Ortega et al. , 2019 ; Rabinowitz , 2019 ; Mikulik et al. , 2020 ) . Here we investigate when the integration of information might not be the optimal strategy to meta-learn . We analytically characterize a task regime in which not adapting to sensory information is optimal . Furthermore , we study whether LSTM-based meta-learning is capable of inferring when to learn and when to execute a non-adaptive program . Rabinowitz ( 2019 ) previously studied the outer loop learning dynamics and found differences across several tasks , the origin of which is however not fully understood . Our work may provide an explanation for these different meta-learning dynamics and the dependence on the task distribution as well as the time horizon of adaptation . Our work is most closely related to Pardo et al . ( 2017 ) and Zintgraf et al . ( 2019 ) . Pardo et al . ( 2017 ) study the impact of fixed time limits and time-awareness on deep reinforcement learning agents . They propose using a timestamp as part of the state representation in order to avoid state-aliasing and the non-Markovianity resulting from a finite horizon treatment of an infinite horizon problem . Our setting differs in several aspects . First , we study the case of meta-reinforcement learning where the agent has to learn within a single lifetime . Second , we focus on a finite horizon perspective with limited adaptation . Zintgraf et al . ( 2019 ) , on the other hand , investigate meta reinforcement-learning for Bayes-adaptive Markov Decision Processes and introduce a novel architecture that disentangles task-specific belief representations from policy representations . Similarly to our work , Zintgraf et al . ( 2019 ) are interested in using the meta-learning framework to distill Bayes optimal exploration behavior . While their adaptation setup extends over multiple episodes , we focus on single lifetime adaption and analytically analyze when it is beneficial to learn in the first place . Finally , our work extends upon the efforts of computational ethology ( Stephens , 1991 ) and experimental evolution ( Dunlap & Stephens , 2009 ; 2016 ; Marcus et al. , 2018 ) , which aims to characterize the conditions under which behavioral plasticity may evolve . Their work shows that both environmental change and the predictability of the environment shape the selection pressure , which evolves adaptive traits . Our work is based on memory-based meta-learning with function approximation and aims to extend these original findings to task distributions for which no analytical solution may be available . 3 LEARNING NOT TO LEARN . To disentangle the influence of ecological uncertainty , task complexity , and lifetime on the nature of the meta-learned strategy , we first focus on a minimal two-arm Gaussian bandit task , which allows for an analytical solution . The agent experiences episodes consisting of T arm pulls , representing the lifetime of the agent . The statistics of the bandit are constant during each episode , but vary between episodes . To keep it simple , one of the two arms is deterministic and always returns a reward of 0 . The task distribution is represented by the variable expected reward of the other arm , which is sampled at the beginning of an episode , from a Gaussian distribution with mean -1 and standard deviation σp , i.e . µ ∼ N ( −1 , σ2p ) . The standard deviation σp controls the uncertainty of the ecological niche . For σp 1 , the deterministic arm is almost always the better option . For σp 1 , the chances of either arm being the best in the given episode are largely even . While the mean µ remains constant for the lifetime T of the agent , the reward obtained in a given trial is stochastic and is sampled from a second Gaussian , r ∼ N ( µ , σl ) . This trial-to-trial variability controls how many pulls the agent needs to estimate the mean reward of the stochastic arm . The standard deviation σl hence controls how quickly the agent can learn the optimal policy . We therefore use it as a proxy for task complexity . In this simple setting , the optimal meta-learned strategy can be calculated analytically . The optimal exploration strategy is to initially explore the stochastic arm for a given trial number n. Afterwards , it chooses the best arm based on its maximum a posteriori-estimate of the remaining episode return . The optimal amount of exploration trials n ? can then be derived analytically : 1 n ? = arg max n E [ T∑ t=1 rt\|n , T , σl , σp ] = arg max n [ −n+ Eµ , r [ ( T − n ) × µ× p ( µ̂ > 0 ) ] ] , where µ̂ is the estimate of the mean reward of the stochatic arm after the n exploration trials . We find two distinct types of behavior ( left-hand side of figure 1 ) : A regime in which learning via exploration is effective and a second regime in which not learning is the optimal behavior . It may be optimal not to learn for two reasons : First , the ecological uncertainty may be so small that it is very unlikely that the stochastic first arm is better . Second , if the trial-to-trial variability is too large relative to the range 1Please refer to the supplementary material for a detailed derivation of this analytical result as well as the hyperparameters of the numerical experiments . of potential ecological niches , so that it may simply not be possible to integrate sufficient information given a limited lifespan . We make two observations : 1 . There exists a hard nonlinear threshold between learning and not learning behaviors described by the ratio of σl and σp . If σl is too large , the value of exploration ( or the reduction in uncertainty ) is too small to be profitable within the remaining lifetime of the agent . Instead , it is advantageous to hard-code a heuristic choice . 2 . The two regimes consistently exist across different lifetimes . As the lifetime grows , the learning regime becomes more and more prevalent . Given a sufficient amount of time , learning by exploring the uncertain arm is the best strategy . Is the common meta-learning framework capable of reproducing these different qualitative behaviors and performing Bayes optimal amortized inference across the entire spectrum of meta-task distributions ? Or differently put : Can memory-based meta-learning yield agents that do not only learn to learn but that also learn not to learn ? To answer this question , we train LSTM-based RL2 ( Wang et al. , 2016 ) agents with the standard synchronous actor-critic ( Mnih et al. , 2016 ) setup on the same grid of ecological uncertainties σp and ” task complexities ” σl . The input xt to the network at time t consists of the action of the previous timestep , a monotonically increasing timestamp within the current episode and crucially the reward of the previous timestep , xt = { at−1 , φ ( t ) , rt−1 } . The recurrent weight dynamics of the inner loop can then implement an internal learning algorithm that integrates previous experiences . After collecting a set of trajectories , we optimize the weights and initial condition of the hidden state with an outer loop gradient descent update to minimize the common actor-critic objective . We obtain the amount of meta-learned exploration by testing the RL2 agents on hold-out bandits for which we set σp = 0 and only vary σl . Thereby , it is ensured that the deterministic arm is the better arm . We can then define the number of exploration trials as the pulls from the suboptimal stochastic arm . We observe that meta-learning is capable of yielding agents that behave according to our derived theory of a Bayes optimal agent , which explicitly knows the given lifetime as well as uncertainties σl , σp ( figure 1 ) . Importantly , the meta-learned behavior also falls into two regimes : A regime in which the meta-learned strategy resembles a learning algorithm and a regime in which the recurrent dynamics encode a hard-coded choice of the deterministic arm . Furthermore , the edge between the two meta-learned regimes shifts with the agent ’ s lifetime as predicted by the Bayesian theory . As the lifetime increases , wider ecological niches at higher levels of task complexity become solvable and the strategy of learning profitable . In the Bayesian model , the edge between the two regimes is located at parameter values where the learning strategy and the non-learning strategy perform equally well . Because these two strategies are very distinct , we wondered whether the reward landscape for the memory-based meta-learner has two local maxima corresponding to the two strategies ( figure 2 ) . To test this , we trained N = 1000 networks with different initial conditions , for task parameters close to the edge , but in the regime where the theoretically optimal strategy would be to learn . We then evaluated for each network the number of explorative pulls of the stochastic arm , averaged across 100 episodes . The distribution of the number of explorative pulls across the 1000 networks shows i ) a peak at zero exploration and ii ) a broad tail of mean explorative pulls ( figure 2 ) , suggesting that there are indeed two classes of networks . One class never pulls the stochastic arm , i.e. , those networks adopt a non-learning strategy . The other class learns . For task parameters further away from the edge , this bimodality disappears . The two behavioral regimes are characterized by distinct recurrent dynamics of the trained LSTM agents . The two left-most columns of figure 3 display the policy entropy and hidden state statistics for a network trained on a σl , σp-combination associated with the regime in which learning is the optimal behavior . We differentiate between the case in which the deterministic arm is the better one ( µ < 0 ) and the case in which the second arm should be preferred ( µ > 0 ) . In both cases the agent first explores in order to identify the better arm . Moreover , the hidden dynamics appear to display two different attractors , which correspond to either of the arms being the better choice . The better arm can clearly be identified from the PCA-dimensionality reduced hidden state dynamics ( bottom row of figure 3 ) . The two right-most columns of figure 3 , on the other hand , depict the same statistics for a network that was meta-trained on the regime in which the optimal strategy is not to learn . Indeed , the agent always chooses the deterministic arm , regardless of whether it is the better choice . Accordingly , the network dynamics seem to fall into a single attractor . We examined how these strategies evolve over the course of meta-training and find that there are two phases : After an initial period of universal random behavior across all conditions , the distinct behavioral regimes emerge ( supplementary figure 9 ) . We note that this observation may be partially caused by the linear annealing of the entropy regularization coefficient in the actor-critic objective which we found to be crucial in training the networks . In summary , we observe that the meta-learned strategy shows a highly nonlinear , partially discontinuous dependence on task parameters . In transition regions between strategies , we find local maxima in the reward landscape that correspond to different learning strategies . In the simple bandit setting , these local maxima correspond to a learning and a non-learning strategy , respectively , hence providing a minimal model for a sharp nature-nurture trade-off . Next , we investigate whether these insights generalize to more complex domains by studying spatial reasoning .	This paper provides an analysis of RNN-based meta learning approaches. In particular, it investigates the strategies learned via meta-learning, contrasting strategies involving task-dependent learning vs heuristic or hard-coded solutions. Empirical evidence in two sets of experiments, on a 2-armed bandit toy task and a grid-world navigation task, show that hard-coded strategies can be a function of training task distribution and task complexity as well as task horizon.	SP:b70f1ee4f3fb0fd89e76cf8f09b038cfe13e7e89
Learning not to learn: Nature versus nurture in silico	1 INTRODUCTION . The ’ nature versus nurture ’ debate ( e.g. , Mutti et al. , 1996 ; Tabery , 2014 ) – the question of which aspects of behavior are ’ hard-coded ’ by evolution , and which are learned from experience – is one of the oldest and most controversial debates in biology . Evolutionary principles prescribe that hard-coded behavioral routines should be those for which there is no benefit in adaptation . This is believed to be the case for behaviors whose evolutionary advantage varies little among individuals of a species . Mating instincts or flight reflexes are general solutions that rarely present an evolutionary disadvantage . On the other hand , features of the environment that vary substantially for individuals of a species potentially ask for adaptive behavior ( Buss , 2015 ) . Naturally , the same principles should not only apply to biological but also to artificial agents . But how can a reinforcement learning agent differentiate between these two behavioral regimes ? A promising approach to automatically learn rules of adaptation that facilitate environment-specific specialization is meta-learning ( Schmidhuber , 1987 ; Thrun & Pratt , 1998 ) . At its core lies the idea of using generic optimization methods to learn inductive biases for a given ensemble of tasks . In this approach , the inductive bias usually has its own set of parameters ( e.g. , weights in a recurrent network ; Hochreiter et al. , 2001 ) that are optimized on the whole task ensemble , that is , on a long , ’ evolutionary ’ time scale . These parameters in turn control how a different set of parameters ( e.g. , activities in the network ) are updated on a much faster time scale . These rapidly adapting parameters then allow the system to adapt to a specific task at hand . Notably , the parameters of the system that are subject to ’ nature ’ – i.e. , those that shape the inductive bias and are common across tasks – and those that are subject to ’ nurture ’ are usually predefined from the start . In this work , we use the memory-based meta-learning approach for a different goal , namely to acquire a qualitative understanding of which aspects of behavior should be hard-coded and which should be adaptive . Our hypothesis is that meta-learning can not only learn efficient learning algorithms , but can also decide not to be adaptive at all , and to instead apply a generic heuristic to the whole ensemble of tasks . Phrased in the language of biology , meta-learning can decide whether to hard-code a behavior or to render it adaptive , based on the range of environments the individuals of a species could encounter . We study the dependence of the meta-learned algorithm on three central features of the metareinforcement learning problem : • Ecological uncertainty : How diverse is the range of tasks the agent could encounter ? • Task complexity : How long does it take to learn the optimal strategy for the task at hand ? Note that this could be different from the time it takes to execute the optimal strategy . • Expected lifetime : How much time can the agent spend on exploration and exploitation ? Using analytical and numerical analyses , we show that non-adaptive behaviors are optimal in two cases – when the optimal policy varies little across the tasks within the task ensemble and when the time it takes to learn the optimal policy is too long to allow a sufficient exploitation of the learned policy . Our results suggest that not only the design of the meta-task distribution , but also the lifetime of the agent can have strong effects on the meta-learned algorithm of RNN-based agents . In particular , we find highly nonlinear and potentially discontinuous effects of ecological uncertainty , task complexity and lifetime on the optimal algorithm . As a consequence , a meta-learned adaptation strategy that was optimized , e.g. , for a given lifetime may not generalize well to other lifetimes . This is essential for research questions that are interested in the conducted adaptation behavior , including curriculum design , safe exploration as well as human-in-the-loop applications . Our work may provide a principled way of examining the constraint-dependence of meta-learned inductive biases . The remainder of this paper is structured as follows : First , we review the background in memorybased meta-reinforcement learning and contrast the related literature . Afterwards , we analyze a Gaussian multi-arm bandit setting , which allows us to analytically disentangle the behavioral impact of ecological uncertainty , task complexity and lifetime . Our derivation of the lifetime-dependent Bayes optimal exploration reveals a highly non-linear interplay of these three factors . We show numerically that memory-based meta-learning reproduces our theoretical results and can learn not to learn . Furthermore , we extend our analysis to more complicated exploration problems . Throughout , we analyze the resulting recurrent dynamics of the network and the representations associated with learning and non-adaptive strategies . 2 RELATED WORK & BACKGROUND . Meta-learning or ’ learning to learn ’ ( e.g. , Schmidhuber , 1987 ; Thrun & Pratt , 1998 ; Hochreiter et al. , 2001 ; Duan et al. , 2016 ; Wang et al. , 2016 ; Finn et al. , 2017 ) has been proposed as a computational framework for acquiring task distribution-specific learning rules . During a costly outer loop optimization , an agent crafts a niche-specific adaptation strategy , which is applicable to an engineered task distribution . At inference time , the acquired inner loop learning algorithm is executed for a fixed amount of timesteps ( lifetime ) on a test task . This framework has successfully been applied to a range of applications such as the meta-learning of optimization updates ( Andrychowicz et al. , 2016 ; Flennerhag et al. , 2018 ; 2019 ) , agent ( Rabinowitz et al. , 2018 ) and world models ( Nagabandi et al. , 2018 ) and explicit models of memory ( Santoro et al. , 2016 ; Bartunov et al. , 2019 ) . Already , early work by Schmidhuber ( 1987 ) suggested an evolutionary perspective on recursively learning the rules of learning . This perspective holds the promise of explaining the emergence of mechanisms underlying both natural and artificial behaviors . Furthermore , a similarity between the hidden activations of LSTM-based meta-learners and the recurrent activity of neurons in the prefrontal cortex ( Wang et al. , 2018 ) has recently been suggested . Previous work has shown that LSTM-based meta-learning is capable of distilling a sequential integration algorithm akin to amortized Bayesian inference ( Ortega et al. , 2019 ; Rabinowitz , 2019 ; Mikulik et al. , 2020 ) . Here we investigate when the integration of information might not be the optimal strategy to meta-learn . We analytically characterize a task regime in which not adapting to sensory information is optimal . Furthermore , we study whether LSTM-based meta-learning is capable of inferring when to learn and when to execute a non-adaptive program . Rabinowitz ( 2019 ) previously studied the outer loop learning dynamics and found differences across several tasks , the origin of which is however not fully understood . Our work may provide an explanation for these different meta-learning dynamics and the dependence on the task distribution as well as the time horizon of adaptation . Our work is most closely related to Pardo et al . ( 2017 ) and Zintgraf et al . ( 2019 ) . Pardo et al . ( 2017 ) study the impact of fixed time limits and time-awareness on deep reinforcement learning agents . They propose using a timestamp as part of the state representation in order to avoid state-aliasing and the non-Markovianity resulting from a finite horizon treatment of an infinite horizon problem . Our setting differs in several aspects . First , we study the case of meta-reinforcement learning where the agent has to learn within a single lifetime . Second , we focus on a finite horizon perspective with limited adaptation . Zintgraf et al . ( 2019 ) , on the other hand , investigate meta reinforcement-learning for Bayes-adaptive Markov Decision Processes and introduce a novel architecture that disentangles task-specific belief representations from policy representations . Similarly to our work , Zintgraf et al . ( 2019 ) are interested in using the meta-learning framework to distill Bayes optimal exploration behavior . While their adaptation setup extends over multiple episodes , we focus on single lifetime adaption and analytically analyze when it is beneficial to learn in the first place . Finally , our work extends upon the efforts of computational ethology ( Stephens , 1991 ) and experimental evolution ( Dunlap & Stephens , 2009 ; 2016 ; Marcus et al. , 2018 ) , which aims to characterize the conditions under which behavioral plasticity may evolve . Their work shows that both environmental change and the predictability of the environment shape the selection pressure , which evolves adaptive traits . Our work is based on memory-based meta-learning with function approximation and aims to extend these original findings to task distributions for which no analytical solution may be available . 3 LEARNING NOT TO LEARN . To disentangle the influence of ecological uncertainty , task complexity , and lifetime on the nature of the meta-learned strategy , we first focus on a minimal two-arm Gaussian bandit task , which allows for an analytical solution . The agent experiences episodes consisting of T arm pulls , representing the lifetime of the agent . The statistics of the bandit are constant during each episode , but vary between episodes . To keep it simple , one of the two arms is deterministic and always returns a reward of 0 . The task distribution is represented by the variable expected reward of the other arm , which is sampled at the beginning of an episode , from a Gaussian distribution with mean -1 and standard deviation σp , i.e . µ ∼ N ( −1 , σ2p ) . The standard deviation σp controls the uncertainty of the ecological niche . For σp 1 , the deterministic arm is almost always the better option . For σp 1 , the chances of either arm being the best in the given episode are largely even . While the mean µ remains constant for the lifetime T of the agent , the reward obtained in a given trial is stochastic and is sampled from a second Gaussian , r ∼ N ( µ , σl ) . This trial-to-trial variability controls how many pulls the agent needs to estimate the mean reward of the stochastic arm . The standard deviation σl hence controls how quickly the agent can learn the optimal policy . We therefore use it as a proxy for task complexity . In this simple setting , the optimal meta-learned strategy can be calculated analytically . The optimal exploration strategy is to initially explore the stochastic arm for a given trial number n. Afterwards , it chooses the best arm based on its maximum a posteriori-estimate of the remaining episode return . The optimal amount of exploration trials n ? can then be derived analytically : 1 n ? = arg max n E [ T∑ t=1 rt\|n , T , σl , σp ] = arg max n [ −n+ Eµ , r [ ( T − n ) × µ× p ( µ̂ > 0 ) ] ] , where µ̂ is the estimate of the mean reward of the stochatic arm after the n exploration trials . We find two distinct types of behavior ( left-hand side of figure 1 ) : A regime in which learning via exploration is effective and a second regime in which not learning is the optimal behavior . It may be optimal not to learn for two reasons : First , the ecological uncertainty may be so small that it is very unlikely that the stochastic first arm is better . Second , if the trial-to-trial variability is too large relative to the range 1Please refer to the supplementary material for a detailed derivation of this analytical result as well as the hyperparameters of the numerical experiments . of potential ecological niches , so that it may simply not be possible to integrate sufficient information given a limited lifespan . We make two observations : 1 . There exists a hard nonlinear threshold between learning and not learning behaviors described by the ratio of σl and σp . If σl is too large , the value of exploration ( or the reduction in uncertainty ) is too small to be profitable within the remaining lifetime of the agent . Instead , it is advantageous to hard-code a heuristic choice . 2 . The two regimes consistently exist across different lifetimes . As the lifetime grows , the learning regime becomes more and more prevalent . Given a sufficient amount of time , learning by exploring the uncertain arm is the best strategy . Is the common meta-learning framework capable of reproducing these different qualitative behaviors and performing Bayes optimal amortized inference across the entire spectrum of meta-task distributions ? Or differently put : Can memory-based meta-learning yield agents that do not only learn to learn but that also learn not to learn ? To answer this question , we train LSTM-based RL2 ( Wang et al. , 2016 ) agents with the standard synchronous actor-critic ( Mnih et al. , 2016 ) setup on the same grid of ecological uncertainties σp and ” task complexities ” σl . The input xt to the network at time t consists of the action of the previous timestep , a monotonically increasing timestamp within the current episode and crucially the reward of the previous timestep , xt = { at−1 , φ ( t ) , rt−1 } . The recurrent weight dynamics of the inner loop can then implement an internal learning algorithm that integrates previous experiences . After collecting a set of trajectories , we optimize the weights and initial condition of the hidden state with an outer loop gradient descent update to minimize the common actor-critic objective . We obtain the amount of meta-learned exploration by testing the RL2 agents on hold-out bandits for which we set σp = 0 and only vary σl . Thereby , it is ensured that the deterministic arm is the better arm . We can then define the number of exploration trials as the pulls from the suboptimal stochastic arm . We observe that meta-learning is capable of yielding agents that behave according to our derived theory of a Bayes optimal agent , which explicitly knows the given lifetime as well as uncertainties σl , σp ( figure 1 ) . Importantly , the meta-learned behavior also falls into two regimes : A regime in which the meta-learned strategy resembles a learning algorithm and a regime in which the recurrent dynamics encode a hard-coded choice of the deterministic arm . Furthermore , the edge between the two meta-learned regimes shifts with the agent ’ s lifetime as predicted by the Bayesian theory . As the lifetime increases , wider ecological niches at higher levels of task complexity become solvable and the strategy of learning profitable . In the Bayesian model , the edge between the two regimes is located at parameter values where the learning strategy and the non-learning strategy perform equally well . Because these two strategies are very distinct , we wondered whether the reward landscape for the memory-based meta-learner has two local maxima corresponding to the two strategies ( figure 2 ) . To test this , we trained N = 1000 networks with different initial conditions , for task parameters close to the edge , but in the regime where the theoretically optimal strategy would be to learn . We then evaluated for each network the number of explorative pulls of the stochastic arm , averaged across 100 episodes . The distribution of the number of explorative pulls across the 1000 networks shows i ) a peak at zero exploration and ii ) a broad tail of mean explorative pulls ( figure 2 ) , suggesting that there are indeed two classes of networks . One class never pulls the stochastic arm , i.e. , those networks adopt a non-learning strategy . The other class learns . For task parameters further away from the edge , this bimodality disappears . The two behavioral regimes are characterized by distinct recurrent dynamics of the trained LSTM agents . The two left-most columns of figure 3 display the policy entropy and hidden state statistics for a network trained on a σl , σp-combination associated with the regime in which learning is the optimal behavior . We differentiate between the case in which the deterministic arm is the better one ( µ < 0 ) and the case in which the second arm should be preferred ( µ > 0 ) . In both cases the agent first explores in order to identify the better arm . Moreover , the hidden dynamics appear to display two different attractors , which correspond to either of the arms being the better choice . The better arm can clearly be identified from the PCA-dimensionality reduced hidden state dynamics ( bottom row of figure 3 ) . The two right-most columns of figure 3 , on the other hand , depict the same statistics for a network that was meta-trained on the regime in which the optimal strategy is not to learn . Indeed , the agent always chooses the deterministic arm , regardless of whether it is the better choice . Accordingly , the network dynamics seem to fall into a single attractor . We examined how these strategies evolve over the course of meta-training and find that there are two phases : After an initial period of universal random behavior across all conditions , the distinct behavioral regimes emerge ( supplementary figure 9 ) . We note that this observation may be partially caused by the linear annealing of the entropy regularization coefficient in the actor-critic objective which we found to be crucial in training the networks . In summary , we observe that the meta-learned strategy shows a highly nonlinear , partially discontinuous dependence on task parameters . In transition regions between strategies , we find local maxima in the reward landscape that correspond to different learning strategies . In the simple bandit setting , these local maxima correspond to a learning and a non-learning strategy , respectively , hence providing a minimal model for a sharp nature-nurture trade-off . Next , we investigate whether these insights generalize to more complex domains by studying spatial reasoning .	This paper observes that in meta-RL (and evolutionary biology), sometimes it is advantageous to learn behaviors that adapt to the particular task, while other times not adapting to the task, and instead relying on a task-agnostic “hard-coded” behavior is sufficient. While much meta-RL research typically focuses on the former setting, this paper studies when it is not necessary to learn adaptive behaviors. Specifically, this paper presents three main findings: (i) whether or not it is optimal to learn adaptive behavior strongly depends on the horizon of the task and complexity of learning such adaptive behaviors — if the horizon is too short, or if the adaptive behavior requires complex exploration, then exploring the new task to learn adaptive behaviors may not be worth it; (ii) existing meta-RL agents are capable of choosing not to learn adaptive behaviors, when it is optimal to do so; (iii) existing meta-RL agents generalize poorly to tasks with varying horizon-lengths.	SP:b70f1ee4f3fb0fd89e76cf8f09b038cfe13e7e89
Learning not to learn: Nature versus nurture in silico	1 INTRODUCTION . The ’ nature versus nurture ’ debate ( e.g. , Mutti et al. , 1996 ; Tabery , 2014 ) – the question of which aspects of behavior are ’ hard-coded ’ by evolution , and which are learned from experience – is one of the oldest and most controversial debates in biology . Evolutionary principles prescribe that hard-coded behavioral routines should be those for which there is no benefit in adaptation . This is believed to be the case for behaviors whose evolutionary advantage varies little among individuals of a species . Mating instincts or flight reflexes are general solutions that rarely present an evolutionary disadvantage . On the other hand , features of the environment that vary substantially for individuals of a species potentially ask for adaptive behavior ( Buss , 2015 ) . Naturally , the same principles should not only apply to biological but also to artificial agents . But how can a reinforcement learning agent differentiate between these two behavioral regimes ? A promising approach to automatically learn rules of adaptation that facilitate environment-specific specialization is meta-learning ( Schmidhuber , 1987 ; Thrun & Pratt , 1998 ) . At its core lies the idea of using generic optimization methods to learn inductive biases for a given ensemble of tasks . In this approach , the inductive bias usually has its own set of parameters ( e.g. , weights in a recurrent network ; Hochreiter et al. , 2001 ) that are optimized on the whole task ensemble , that is , on a long , ’ evolutionary ’ time scale . These parameters in turn control how a different set of parameters ( e.g. , activities in the network ) are updated on a much faster time scale . These rapidly adapting parameters then allow the system to adapt to a specific task at hand . Notably , the parameters of the system that are subject to ’ nature ’ – i.e. , those that shape the inductive bias and are common across tasks – and those that are subject to ’ nurture ’ are usually predefined from the start . In this work , we use the memory-based meta-learning approach for a different goal , namely to acquire a qualitative understanding of which aspects of behavior should be hard-coded and which should be adaptive . Our hypothesis is that meta-learning can not only learn efficient learning algorithms , but can also decide not to be adaptive at all , and to instead apply a generic heuristic to the whole ensemble of tasks . Phrased in the language of biology , meta-learning can decide whether to hard-code a behavior or to render it adaptive , based on the range of environments the individuals of a species could encounter . We study the dependence of the meta-learned algorithm on three central features of the metareinforcement learning problem : • Ecological uncertainty : How diverse is the range of tasks the agent could encounter ? • Task complexity : How long does it take to learn the optimal strategy for the task at hand ? Note that this could be different from the time it takes to execute the optimal strategy . • Expected lifetime : How much time can the agent spend on exploration and exploitation ? Using analytical and numerical analyses , we show that non-adaptive behaviors are optimal in two cases – when the optimal policy varies little across the tasks within the task ensemble and when the time it takes to learn the optimal policy is too long to allow a sufficient exploitation of the learned policy . Our results suggest that not only the design of the meta-task distribution , but also the lifetime of the agent can have strong effects on the meta-learned algorithm of RNN-based agents . In particular , we find highly nonlinear and potentially discontinuous effects of ecological uncertainty , task complexity and lifetime on the optimal algorithm . As a consequence , a meta-learned adaptation strategy that was optimized , e.g. , for a given lifetime may not generalize well to other lifetimes . This is essential for research questions that are interested in the conducted adaptation behavior , including curriculum design , safe exploration as well as human-in-the-loop applications . Our work may provide a principled way of examining the constraint-dependence of meta-learned inductive biases . The remainder of this paper is structured as follows : First , we review the background in memorybased meta-reinforcement learning and contrast the related literature . Afterwards , we analyze a Gaussian multi-arm bandit setting , which allows us to analytically disentangle the behavioral impact of ecological uncertainty , task complexity and lifetime . Our derivation of the lifetime-dependent Bayes optimal exploration reveals a highly non-linear interplay of these three factors . We show numerically that memory-based meta-learning reproduces our theoretical results and can learn not to learn . Furthermore , we extend our analysis to more complicated exploration problems . Throughout , we analyze the resulting recurrent dynamics of the network and the representations associated with learning and non-adaptive strategies . 2 RELATED WORK & BACKGROUND . Meta-learning or ’ learning to learn ’ ( e.g. , Schmidhuber , 1987 ; Thrun & Pratt , 1998 ; Hochreiter et al. , 2001 ; Duan et al. , 2016 ; Wang et al. , 2016 ; Finn et al. , 2017 ) has been proposed as a computational framework for acquiring task distribution-specific learning rules . During a costly outer loop optimization , an agent crafts a niche-specific adaptation strategy , which is applicable to an engineered task distribution . At inference time , the acquired inner loop learning algorithm is executed for a fixed amount of timesteps ( lifetime ) on a test task . This framework has successfully been applied to a range of applications such as the meta-learning of optimization updates ( Andrychowicz et al. , 2016 ; Flennerhag et al. , 2018 ; 2019 ) , agent ( Rabinowitz et al. , 2018 ) and world models ( Nagabandi et al. , 2018 ) and explicit models of memory ( Santoro et al. , 2016 ; Bartunov et al. , 2019 ) . Already , early work by Schmidhuber ( 1987 ) suggested an evolutionary perspective on recursively learning the rules of learning . This perspective holds the promise of explaining the emergence of mechanisms underlying both natural and artificial behaviors . Furthermore , a similarity between the hidden activations of LSTM-based meta-learners and the recurrent activity of neurons in the prefrontal cortex ( Wang et al. , 2018 ) has recently been suggested . Previous work has shown that LSTM-based meta-learning is capable of distilling a sequential integration algorithm akin to amortized Bayesian inference ( Ortega et al. , 2019 ; Rabinowitz , 2019 ; Mikulik et al. , 2020 ) . Here we investigate when the integration of information might not be the optimal strategy to meta-learn . We analytically characterize a task regime in which not adapting to sensory information is optimal . Furthermore , we study whether LSTM-based meta-learning is capable of inferring when to learn and when to execute a non-adaptive program . Rabinowitz ( 2019 ) previously studied the outer loop learning dynamics and found differences across several tasks , the origin of which is however not fully understood . Our work may provide an explanation for these different meta-learning dynamics and the dependence on the task distribution as well as the time horizon of adaptation . Our work is most closely related to Pardo et al . ( 2017 ) and Zintgraf et al . ( 2019 ) . Pardo et al . ( 2017 ) study the impact of fixed time limits and time-awareness on deep reinforcement learning agents . They propose using a timestamp as part of the state representation in order to avoid state-aliasing and the non-Markovianity resulting from a finite horizon treatment of an infinite horizon problem . Our setting differs in several aspects . First , we study the case of meta-reinforcement learning where the agent has to learn within a single lifetime . Second , we focus on a finite horizon perspective with limited adaptation . Zintgraf et al . ( 2019 ) , on the other hand , investigate meta reinforcement-learning for Bayes-adaptive Markov Decision Processes and introduce a novel architecture that disentangles task-specific belief representations from policy representations . Similarly to our work , Zintgraf et al . ( 2019 ) are interested in using the meta-learning framework to distill Bayes optimal exploration behavior . While their adaptation setup extends over multiple episodes , we focus on single lifetime adaption and analytically analyze when it is beneficial to learn in the first place . Finally , our work extends upon the efforts of computational ethology ( Stephens , 1991 ) and experimental evolution ( Dunlap & Stephens , 2009 ; 2016 ; Marcus et al. , 2018 ) , which aims to characterize the conditions under which behavioral plasticity may evolve . Their work shows that both environmental change and the predictability of the environment shape the selection pressure , which evolves adaptive traits . Our work is based on memory-based meta-learning with function approximation and aims to extend these original findings to task distributions for which no analytical solution may be available . 3 LEARNING NOT TO LEARN . To disentangle the influence of ecological uncertainty , task complexity , and lifetime on the nature of the meta-learned strategy , we first focus on a minimal two-arm Gaussian bandit task , which allows for an analytical solution . The agent experiences episodes consisting of T arm pulls , representing the lifetime of the agent . The statistics of the bandit are constant during each episode , but vary between episodes . To keep it simple , one of the two arms is deterministic and always returns a reward of 0 . The task distribution is represented by the variable expected reward of the other arm , which is sampled at the beginning of an episode , from a Gaussian distribution with mean -1 and standard deviation σp , i.e . µ ∼ N ( −1 , σ2p ) . The standard deviation σp controls the uncertainty of the ecological niche . For σp 1 , the deterministic arm is almost always the better option . For σp 1 , the chances of either arm being the best in the given episode are largely even . While the mean µ remains constant for the lifetime T of the agent , the reward obtained in a given trial is stochastic and is sampled from a second Gaussian , r ∼ N ( µ , σl ) . This trial-to-trial variability controls how many pulls the agent needs to estimate the mean reward of the stochastic arm . The standard deviation σl hence controls how quickly the agent can learn the optimal policy . We therefore use it as a proxy for task complexity . In this simple setting , the optimal meta-learned strategy can be calculated analytically . The optimal exploration strategy is to initially explore the stochastic arm for a given trial number n. Afterwards , it chooses the best arm based on its maximum a posteriori-estimate of the remaining episode return . The optimal amount of exploration trials n ? can then be derived analytically : 1 n ? = arg max n E [ T∑ t=1 rt\|n , T , σl , σp ] = arg max n [ −n+ Eµ , r [ ( T − n ) × µ× p ( µ̂ > 0 ) ] ] , where µ̂ is the estimate of the mean reward of the stochatic arm after the n exploration trials . We find two distinct types of behavior ( left-hand side of figure 1 ) : A regime in which learning via exploration is effective and a second regime in which not learning is the optimal behavior . It may be optimal not to learn for two reasons : First , the ecological uncertainty may be so small that it is very unlikely that the stochastic first arm is better . Second , if the trial-to-trial variability is too large relative to the range 1Please refer to the supplementary material for a detailed derivation of this analytical result as well as the hyperparameters of the numerical experiments . of potential ecological niches , so that it may simply not be possible to integrate sufficient information given a limited lifespan . We make two observations : 1 . There exists a hard nonlinear threshold between learning and not learning behaviors described by the ratio of σl and σp . If σl is too large , the value of exploration ( or the reduction in uncertainty ) is too small to be profitable within the remaining lifetime of the agent . Instead , it is advantageous to hard-code a heuristic choice . 2 . The two regimes consistently exist across different lifetimes . As the lifetime grows , the learning regime becomes more and more prevalent . Given a sufficient amount of time , learning by exploring the uncertain arm is the best strategy . Is the common meta-learning framework capable of reproducing these different qualitative behaviors and performing Bayes optimal amortized inference across the entire spectrum of meta-task distributions ? Or differently put : Can memory-based meta-learning yield agents that do not only learn to learn but that also learn not to learn ? To answer this question , we train LSTM-based RL2 ( Wang et al. , 2016 ) agents with the standard synchronous actor-critic ( Mnih et al. , 2016 ) setup on the same grid of ecological uncertainties σp and ” task complexities ” σl . The input xt to the network at time t consists of the action of the previous timestep , a monotonically increasing timestamp within the current episode and crucially the reward of the previous timestep , xt = { at−1 , φ ( t ) , rt−1 } . The recurrent weight dynamics of the inner loop can then implement an internal learning algorithm that integrates previous experiences . After collecting a set of trajectories , we optimize the weights and initial condition of the hidden state with an outer loop gradient descent update to minimize the common actor-critic objective . We obtain the amount of meta-learned exploration by testing the RL2 agents on hold-out bandits for which we set σp = 0 and only vary σl . Thereby , it is ensured that the deterministic arm is the better arm . We can then define the number of exploration trials as the pulls from the suboptimal stochastic arm . We observe that meta-learning is capable of yielding agents that behave according to our derived theory of a Bayes optimal agent , which explicitly knows the given lifetime as well as uncertainties σl , σp ( figure 1 ) . Importantly , the meta-learned behavior also falls into two regimes : A regime in which the meta-learned strategy resembles a learning algorithm and a regime in which the recurrent dynamics encode a hard-coded choice of the deterministic arm . Furthermore , the edge between the two meta-learned regimes shifts with the agent ’ s lifetime as predicted by the Bayesian theory . As the lifetime increases , wider ecological niches at higher levels of task complexity become solvable and the strategy of learning profitable . In the Bayesian model , the edge between the two regimes is located at parameter values where the learning strategy and the non-learning strategy perform equally well . Because these two strategies are very distinct , we wondered whether the reward landscape for the memory-based meta-learner has two local maxima corresponding to the two strategies ( figure 2 ) . To test this , we trained N = 1000 networks with different initial conditions , for task parameters close to the edge , but in the regime where the theoretically optimal strategy would be to learn . We then evaluated for each network the number of explorative pulls of the stochastic arm , averaged across 100 episodes . The distribution of the number of explorative pulls across the 1000 networks shows i ) a peak at zero exploration and ii ) a broad tail of mean explorative pulls ( figure 2 ) , suggesting that there are indeed two classes of networks . One class never pulls the stochastic arm , i.e. , those networks adopt a non-learning strategy . The other class learns . For task parameters further away from the edge , this bimodality disappears . The two behavioral regimes are characterized by distinct recurrent dynamics of the trained LSTM agents . The two left-most columns of figure 3 display the policy entropy and hidden state statistics for a network trained on a σl , σp-combination associated with the regime in which learning is the optimal behavior . We differentiate between the case in which the deterministic arm is the better one ( µ < 0 ) and the case in which the second arm should be preferred ( µ > 0 ) . In both cases the agent first explores in order to identify the better arm . Moreover , the hidden dynamics appear to display two different attractors , which correspond to either of the arms being the better choice . The better arm can clearly be identified from the PCA-dimensionality reduced hidden state dynamics ( bottom row of figure 3 ) . The two right-most columns of figure 3 , on the other hand , depict the same statistics for a network that was meta-trained on the regime in which the optimal strategy is not to learn . Indeed , the agent always chooses the deterministic arm , regardless of whether it is the better choice . Accordingly , the network dynamics seem to fall into a single attractor . We examined how these strategies evolve over the course of meta-training and find that there are two phases : After an initial period of universal random behavior across all conditions , the distinct behavioral regimes emerge ( supplementary figure 9 ) . We note that this observation may be partially caused by the linear annealing of the entropy regularization coefficient in the actor-critic objective which we found to be crucial in training the networks . In summary , we observe that the meta-learned strategy shows a highly nonlinear , partially discontinuous dependence on task parameters . In transition regions between strategies , we find local maxima in the reward landscape that correspond to different learning strategies . In the simple bandit setting , these local maxima correspond to a learning and a non-learning strategy , respectively , hence providing a minimal model for a sharp nature-nurture trade-off . Next , we investigate whether these insights generalize to more complex domains by studying spatial reasoning .	.** The authors investigate the question of when the optimal behavior for an agent is to learn from experience versus when the optimal behavior is to apply the same (memorized) policy in every scenario. They begin by introducing a simple bandits environment wherein they derive the optimal policy and identify regimes in which it involves memorization vs. learning. Then they train an RL^2 agent and verify that it behaves as expected in these regimes. Next, they expand their approach to a slightly more complicated gridworld environment which does not have an analytic solution to the question. The agent behaves as expected in the gridworld environment.	SP:b70f1ee4f3fb0fd89e76cf8f09b038cfe13e7e89
For self-supervised learning, Rationality implies generalization, provably	1 INTRODUCTION . The current standard approach for classification is “ end-to-end supervised learning ” where one fits a complex ( e.g. , a deep neural network ) classifier to the given training set ( Tan & Le , 2019 ; He et al. , 2016 ) . However , modern classifiers are heavily over parameterized , and as demonstrated by Zhang et al . ( 2017 ) , can fit 100 % of their training set even when given random labels as inputs ( in which case test performance is no better than chance ) . Hence , the training performance of such methods is by itself no indication of their performance on new unseen test points . In this work , we study a different class of supervised learning procedures that have recently attracted significant interest . These classifiers are obtained by : ( i ) performing pre-training with a selfsupervised task ( i.e. , without labels ) to obtain a complex representation of the data points , and then ( ii ) fitting a simple ( e.g. , linear ) classifier on the representation and the labels . Such “ Self-Supervised + Simple ” ( SSS for short ) algorithms are commonly used in natural language processing tasks ( Devlin et al. , 2018 ; Brown et al. , 2020 ) , and have recently found uses in other domains as well ( Ravanelli et al. , 2020 ; Liu et al. , 2019 ) . Compared to standard “ end-to-end supervised learning ” , SSS algorithms have several practical advantages . In particular , SSS algorithms can incorporate additional unlabeled data , the representation obtained can be useful for multiple downstream tasks , and they can have improved out-of-distribution performance ( Hendrycks et al. , 2019 ) . Moreover , recent works show that even without additional unlabeled data , SSS algorithms can get close to state-of-art accuracy in several classification tasks ( Chen et al. , 2020b ; He et al. , 2020 ; Misra & Maaten , 2020 ; ∗Equal contribution . Email : { ybansal , galkaplun } @ g.harvard.edu †Email : b @ boazbarak.org . Tian et al. , 2019 ) . For instance , SimCLRv2 ( Chen et al. , 2020b ) achieves 79.8 % top-1 performance on ImageNet with a variant of ResNet-152 , on par with the end-to-end supervised accuracy of this architecture at 80.5 % . We show that SSS algorithms have another advantage over standard supervised learning—they often have a small generalization gap between their train and test accuracy , and we prove non-vacuous bounds on this gap . We stress that SSS algorithms use over-parameterized models to extract the representation , and reuse the same training data to learn a simple classifier on this representation . Thus , the final classifier they produce has high complexity by most standard measures , and it is by no means apriori evident that their generalization gap will be small . Our bound is obtained by first noting that the generalization gap of every training algorithm is bounded by the sum of three quantities , which we name the Robustness gap , Rationality gap , and Memorization gap ( we call this the RRM bound , see Fact I ) . We now describe these gaps at a high level , deferring the formal definitions to Section 2 . All three gaps involve comparison with a setting where we inject label noise by replacing a small fraction η of the labels with random values . The robustness gap corresponds to the amount by which training performance degrades by noise injection . That is , it equals the difference between the standard expected training accuracy ( with no label noise ) and the expected training accuracy in the noisy setting ; in both cases , we measure accuracy with respect to the original ( uncorrupted ) labels . The robustness gap is nearly always small , and sometimes provably so ( see Section 3 ) . The rationality gap corresponds to the difference between performance on the noisy training samples ( on which the training algorithm gets the wrong label ) and test samples ( on which it doesn ’ t get any label at all ) , again with respect to uncorrupted labels . An optimal Bayesian procedure would have zero rationality gap , and indeed this gap is typically zero or small in practice . Since it is a nonstandard quantity , We discuss the rationality gap in Section 3.1 , and explain assuming it is small is both well-founded and does not trivialize the question of generalization . The memorization gap , which often accounts for the lion ’ s share of the generalization gap , corresponds to the difference in the noisy experiment between the training accuracy on the entire train set and the training accuracy on the samples that received the wrong label ( both measured with respect to uncorrupted labels ) . The memorization gap can be thought of as quantifying the extent to which the classifier can “ memorize ” noisy labels , or act differently on the noisy points compared to the overall train set . The memorization gap is large in standard “ end-to-end supervised training ” . In contrast , our main theoretical result is that for SSS algorithms , the memorization gap is small if the simple classifier has small complexity , independently of the complexity of the representation . As long as the simple classifier is under-parameterized ( i.e. , its complexity is asymptotically smaller than the sample size ) , our bound on the memorization gap tends to zero . When combined with small rationality and robustness , we get concrete non-vacuous generalization bounds for various SSS algorithms on the CIFAR-10 and ImageNet datasets ( see Figures 1 and 4 ) . In a nutshell , our results are the following : Theoretical contributions .. 1 . Our main theoretical result ( Theorem II ) is that the memorization gap of an SSS algorithm is bounded byO ( √ C/n ) whereC is the complexity of the simple classifier produced in the “ simple fit ” stage . This bound is oblivious to the complexity of the representation produced in the pre-training and does not make any assumptions on the relationship between the representation learning method and the supervised learning task . One way to interpret this result is that we give a rigorous bound on the generalization gap of SSS algorithms , under the assumptions that the robustness and rationality gaps are bounded by some small constant ( e.g. , 5 % ) . As mentioned below , these assumptions hold widely in practice across many different classifiers . Moreover , these assumptions are nontrivial and do not “ assume away the difficulty ” . Indeed , there are many natural examples of training algorithms for which these assumptions hold but the generalization gap is large . Last , making some assumptions is necessary for a generalization bound to hold for SSS algorithms ; see Remark 3.1 and Appendix E. 2 . We also give a theoretical justification for the assumption of a small rationality gap , by proving that a positive rationality gap corresponds to “ leaving performance on the table ” , in the sense that we can transform a learning procedure with a large rationality gap into a procedure with better test performance ( Theorem 3.2 ) . Empirical contributions . We complement the theoretical results above with an extensive empirical study of several SSS and end-to-end algorithms on both the CIFAR-10 and ImageNet datasets . 1 . We study several top-performing SSS architectures , and show that they all exhibit relatively small generalization gaps on both CIFAR-10 and ImageNet . We stress that we consider the case where the same data is used for both representation learning and classification , and hence it is by no means a-priori obvious that these algorithms should have small generalization gaps . See Figures 1 and 4 for sample results and Section 4 for more details . 2 . We also show that the results of Zhang et al . ( 2017 ) do not replicate to SSS algorithms , in the sense that such algorithms , despite using an over-parameterized representation , are not able to fit random label noise . 3 . We understake an empirical study of the robustness , rationality , and memorization gaps for both SSS and end-to-end supervised learning algorithms . We show that the robustness and rationality gaps are small for all these algorithms , while the memorization gap is small for SSS algorithms but can be large for end-to-end supervised learning . We show that the RRM bound is typically non-vacuous , and in fact , often close to tight , for a variety of SSS algorithms on the CIFAR-10 and ImageNet datasets , including SimCLR ( which achieves test errors close to its supervised counterparts ) . 4 . We demonstrate that replacing the memorization gap with the upper bound of Theorem II yields a non-vacuous generalization bound for a variety of SSS algorithms on CIFAR-10 and ImageNet . Moreover , this bound gets tighter with more data augmentation . Related Work . There are many works on generalization bounds for supervised learning ( e.g. , Golowich et al . ( 2018 ) ; Neyshabur et al . ( 2017 ) ; Bartlett et al . ( 2017 ) ; Dziugaite & Roy ( 2017 ) ; Neyshabur et al . ( 2018 ) ; Cao & Gu ( 2019 ) , and references therein ) . The related work section of Arora et al . ( 2019 ) contains an extensive discussion of such bounds , and why more often than not the assumptions used do not hold in practice . Indeed , many such bounds give vacuous guarantees for modern architectures ( such as the ones considered in this paper ) that have the capacity to memorize their entire training set ( Zhang et al. , 2017 ) . Some non-vacuous bounds are known ; e.g. , Zhou et al . ( 2019 ) gave a 96.5 % bound on the error of MobileNet on ImageNet . Belkin et al . ( 2019 ) ; Nagarajan & Kolter ( 2019 ) showed some barriers for generalization gaps for standard end-to-end supervised learning . Similarly , standard approaches such as Rademacher complexity can not directly bound SSS algorithms ’ generalization gap ( see Remark 3.1 ) . Recently , Saunshi et al . ( 2019 ) and Lee et al . ( 2020 ) gave generalization bounds for self-supervised based classifiers . The two works considered special cases of SSS algorithms , such as contrastive learning and pre-text tasks . Both works make strong statistical assumptions of ( exact or approximate ) conditional independence relating the pre-training and classification tasks . For example , if the pre-training task is obtained by splitting a given image x into two pieces ( x1 , x2 ) and predicting x2 from x1 , then Lee et al . ( 2020 ) ’ s results require x1 and x2 to be approximately independent conditioned on their class y . However , in many realistic cases , the two parts of the same image will share a significant amount of information not explained by the label . Our work applies to general SSS algorithms without such statistical assumptions , at the expense of assuming bounds on the robustness and rationality gaps . There have been works providing rigorous bounds on the robustness gap or related quantities ( See Section 3. ) . However , as far as we know , the rationality gap has not been explicitly defined or studied before . We provide a brief exposition of the various types of SSS methods in Section 4 , and a more detailed discussion in Appendix D.1 . Paper Organization . Section 2 contains formal definitions and statements of our results . Section 3 provides an overview of prior work and our new results on the three gaps of the RRM bound . In Section 4 , we describe our experimental setup and detail our empirical results . Section 5 concludes the paper and discusses important open questions . We defer proofs and additional experimental results to the appendix . Appendix B contains the proof of Theorem II , while Appendix C contains the proof of Theorem 3.2 . Appendix D fully details our experimental setup.1 Notation . We use capital letters ( e.g. , X ) for random variables , lower case letters ( e.g. , x ) for a single value , and bold font ( e.g. , x ) for tuples ( which will typically have dimension corresponding to the number of samples , denoted by n ) . We use xi for the i-th element of the tuple x . We use calligraphic letters ( e.g. , X , D ) for both sets and distributions .	The present paper aims to understand the generalization capability of self-supervised learning algorithms that fine-tune a simple linear classifier to the labels. Analyzing generalization in this case is challenging due to a data re-use problem: the same training data that is used for self-supervised learning is also used to fit the labels. The paper addresses this issue by implicitly conditioning on the training covariates x and then deriving generalization bounds that depend only on (hypothetical) noise to the labels y. The paper show that, empirically, the dominant factor in generalization error is a certain quantity called the "memorization gap", which can also be upper-bounded via theoretical analysis (the theoretical bound seems to be loose by about a factor of 4 compared to the empirical measurement, but is still non-vacuous in many cases). Interestingly, this is not the case for standard supervised learning, likely to the higher-complexity models used to fit the labels; in that case the memorization gap is high, but a different gap (called the "rationality gap") is large in magnitude but negative.	SP:280743806ee639df87f4d7de86f287fb455e3ff3
For self-supervised learning, Rationality implies generalization, provably	1 INTRODUCTION . The current standard approach for classification is “ end-to-end supervised learning ” where one fits a complex ( e.g. , a deep neural network ) classifier to the given training set ( Tan & Le , 2019 ; He et al. , 2016 ) . However , modern classifiers are heavily over parameterized , and as demonstrated by Zhang et al . ( 2017 ) , can fit 100 % of their training set even when given random labels as inputs ( in which case test performance is no better than chance ) . Hence , the training performance of such methods is by itself no indication of their performance on new unseen test points . In this work , we study a different class of supervised learning procedures that have recently attracted significant interest . These classifiers are obtained by : ( i ) performing pre-training with a selfsupervised task ( i.e. , without labels ) to obtain a complex representation of the data points , and then ( ii ) fitting a simple ( e.g. , linear ) classifier on the representation and the labels . Such “ Self-Supervised + Simple ” ( SSS for short ) algorithms are commonly used in natural language processing tasks ( Devlin et al. , 2018 ; Brown et al. , 2020 ) , and have recently found uses in other domains as well ( Ravanelli et al. , 2020 ; Liu et al. , 2019 ) . Compared to standard “ end-to-end supervised learning ” , SSS algorithms have several practical advantages . In particular , SSS algorithms can incorporate additional unlabeled data , the representation obtained can be useful for multiple downstream tasks , and they can have improved out-of-distribution performance ( Hendrycks et al. , 2019 ) . Moreover , recent works show that even without additional unlabeled data , SSS algorithms can get close to state-of-art accuracy in several classification tasks ( Chen et al. , 2020b ; He et al. , 2020 ; Misra & Maaten , 2020 ; ∗Equal contribution . Email : { ybansal , galkaplun } @ g.harvard.edu †Email : b @ boazbarak.org . Tian et al. , 2019 ) . For instance , SimCLRv2 ( Chen et al. , 2020b ) achieves 79.8 % top-1 performance on ImageNet with a variant of ResNet-152 , on par with the end-to-end supervised accuracy of this architecture at 80.5 % . We show that SSS algorithms have another advantage over standard supervised learning—they often have a small generalization gap between their train and test accuracy , and we prove non-vacuous bounds on this gap . We stress that SSS algorithms use over-parameterized models to extract the representation , and reuse the same training data to learn a simple classifier on this representation . Thus , the final classifier they produce has high complexity by most standard measures , and it is by no means apriori evident that their generalization gap will be small . Our bound is obtained by first noting that the generalization gap of every training algorithm is bounded by the sum of three quantities , which we name the Robustness gap , Rationality gap , and Memorization gap ( we call this the RRM bound , see Fact I ) . We now describe these gaps at a high level , deferring the formal definitions to Section 2 . All three gaps involve comparison with a setting where we inject label noise by replacing a small fraction η of the labels with random values . The robustness gap corresponds to the amount by which training performance degrades by noise injection . That is , it equals the difference between the standard expected training accuracy ( with no label noise ) and the expected training accuracy in the noisy setting ; in both cases , we measure accuracy with respect to the original ( uncorrupted ) labels . The robustness gap is nearly always small , and sometimes provably so ( see Section 3 ) . The rationality gap corresponds to the difference between performance on the noisy training samples ( on which the training algorithm gets the wrong label ) and test samples ( on which it doesn ’ t get any label at all ) , again with respect to uncorrupted labels . An optimal Bayesian procedure would have zero rationality gap , and indeed this gap is typically zero or small in practice . Since it is a nonstandard quantity , We discuss the rationality gap in Section 3.1 , and explain assuming it is small is both well-founded and does not trivialize the question of generalization . The memorization gap , which often accounts for the lion ’ s share of the generalization gap , corresponds to the difference in the noisy experiment between the training accuracy on the entire train set and the training accuracy on the samples that received the wrong label ( both measured with respect to uncorrupted labels ) . The memorization gap can be thought of as quantifying the extent to which the classifier can “ memorize ” noisy labels , or act differently on the noisy points compared to the overall train set . The memorization gap is large in standard “ end-to-end supervised training ” . In contrast , our main theoretical result is that for SSS algorithms , the memorization gap is small if the simple classifier has small complexity , independently of the complexity of the representation . As long as the simple classifier is under-parameterized ( i.e. , its complexity is asymptotically smaller than the sample size ) , our bound on the memorization gap tends to zero . When combined with small rationality and robustness , we get concrete non-vacuous generalization bounds for various SSS algorithms on the CIFAR-10 and ImageNet datasets ( see Figures 1 and 4 ) . In a nutshell , our results are the following : Theoretical contributions .. 1 . Our main theoretical result ( Theorem II ) is that the memorization gap of an SSS algorithm is bounded byO ( √ C/n ) whereC is the complexity of the simple classifier produced in the “ simple fit ” stage . This bound is oblivious to the complexity of the representation produced in the pre-training and does not make any assumptions on the relationship between the representation learning method and the supervised learning task . One way to interpret this result is that we give a rigorous bound on the generalization gap of SSS algorithms , under the assumptions that the robustness and rationality gaps are bounded by some small constant ( e.g. , 5 % ) . As mentioned below , these assumptions hold widely in practice across many different classifiers . Moreover , these assumptions are nontrivial and do not “ assume away the difficulty ” . Indeed , there are many natural examples of training algorithms for which these assumptions hold but the generalization gap is large . Last , making some assumptions is necessary for a generalization bound to hold for SSS algorithms ; see Remark 3.1 and Appendix E. 2 . We also give a theoretical justification for the assumption of a small rationality gap , by proving that a positive rationality gap corresponds to “ leaving performance on the table ” , in the sense that we can transform a learning procedure with a large rationality gap into a procedure with better test performance ( Theorem 3.2 ) . Empirical contributions . We complement the theoretical results above with an extensive empirical study of several SSS and end-to-end algorithms on both the CIFAR-10 and ImageNet datasets . 1 . We study several top-performing SSS architectures , and show that they all exhibit relatively small generalization gaps on both CIFAR-10 and ImageNet . We stress that we consider the case where the same data is used for both representation learning and classification , and hence it is by no means a-priori obvious that these algorithms should have small generalization gaps . See Figures 1 and 4 for sample results and Section 4 for more details . 2 . We also show that the results of Zhang et al . ( 2017 ) do not replicate to SSS algorithms , in the sense that such algorithms , despite using an over-parameterized representation , are not able to fit random label noise . 3 . We understake an empirical study of the robustness , rationality , and memorization gaps for both SSS and end-to-end supervised learning algorithms . We show that the robustness and rationality gaps are small for all these algorithms , while the memorization gap is small for SSS algorithms but can be large for end-to-end supervised learning . We show that the RRM bound is typically non-vacuous , and in fact , often close to tight , for a variety of SSS algorithms on the CIFAR-10 and ImageNet datasets , including SimCLR ( which achieves test errors close to its supervised counterparts ) . 4 . We demonstrate that replacing the memorization gap with the upper bound of Theorem II yields a non-vacuous generalization bound for a variety of SSS algorithms on CIFAR-10 and ImageNet . Moreover , this bound gets tighter with more data augmentation . Related Work . There are many works on generalization bounds for supervised learning ( e.g. , Golowich et al . ( 2018 ) ; Neyshabur et al . ( 2017 ) ; Bartlett et al . ( 2017 ) ; Dziugaite & Roy ( 2017 ) ; Neyshabur et al . ( 2018 ) ; Cao & Gu ( 2019 ) , and references therein ) . The related work section of Arora et al . ( 2019 ) contains an extensive discussion of such bounds , and why more often than not the assumptions used do not hold in practice . Indeed , many such bounds give vacuous guarantees for modern architectures ( such as the ones considered in this paper ) that have the capacity to memorize their entire training set ( Zhang et al. , 2017 ) . Some non-vacuous bounds are known ; e.g. , Zhou et al . ( 2019 ) gave a 96.5 % bound on the error of MobileNet on ImageNet . Belkin et al . ( 2019 ) ; Nagarajan & Kolter ( 2019 ) showed some barriers for generalization gaps for standard end-to-end supervised learning . Similarly , standard approaches such as Rademacher complexity can not directly bound SSS algorithms ’ generalization gap ( see Remark 3.1 ) . Recently , Saunshi et al . ( 2019 ) and Lee et al . ( 2020 ) gave generalization bounds for self-supervised based classifiers . The two works considered special cases of SSS algorithms , such as contrastive learning and pre-text tasks . Both works make strong statistical assumptions of ( exact or approximate ) conditional independence relating the pre-training and classification tasks . For example , if the pre-training task is obtained by splitting a given image x into two pieces ( x1 , x2 ) and predicting x2 from x1 , then Lee et al . ( 2020 ) ’ s results require x1 and x2 to be approximately independent conditioned on their class y . However , in many realistic cases , the two parts of the same image will share a significant amount of information not explained by the label . Our work applies to general SSS algorithms without such statistical assumptions , at the expense of assuming bounds on the robustness and rationality gaps . There have been works providing rigorous bounds on the robustness gap or related quantities ( See Section 3. ) . However , as far as we know , the rationality gap has not been explicitly defined or studied before . We provide a brief exposition of the various types of SSS methods in Section 4 , and a more detailed discussion in Appendix D.1 . Paper Organization . Section 2 contains formal definitions and statements of our results . Section 3 provides an overview of prior work and our new results on the three gaps of the RRM bound . In Section 4 , we describe our experimental setup and detail our empirical results . Section 5 concludes the paper and discusses important open questions . We defer proofs and additional experimental results to the appendix . Appendix B contains the proof of Theorem II , while Appendix C contains the proof of Theorem 3.2 . Appendix D fully details our experimental setup.1 Notation . We use capital letters ( e.g. , X ) for random variables , lower case letters ( e.g. , x ) for a single value , and bold font ( e.g. , x ) for tuples ( which will typically have dimension corresponding to the number of samples , denoted by n ) . We use xi for the i-th element of the tuple x . We use calligraphic letters ( e.g. , X , D ) for both sets and distributions .	The paper analyzes the generalization gap for self-supervised learning. This paper's contribution includes the proposal of decomposing the generalization bound into three terms: robustness, rationality, and memorization (RRM). The three terms explain the generalization gap with some different perspectives. With the RRM decomposition's help, it proves that since SSS doesn't memorize data, small robustness and small rationality gap will naturally guarantee a small generalization gap.	SP:280743806ee639df87f4d7de86f287fb455e3ff3
For self-supervised learning, Rationality implies generalization, provably	1 INTRODUCTION . The current standard approach for classification is “ end-to-end supervised learning ” where one fits a complex ( e.g. , a deep neural network ) classifier to the given training set ( Tan & Le , 2019 ; He et al. , 2016 ) . However , modern classifiers are heavily over parameterized , and as demonstrated by Zhang et al . ( 2017 ) , can fit 100 % of their training set even when given random labels as inputs ( in which case test performance is no better than chance ) . Hence , the training performance of such methods is by itself no indication of their performance on new unseen test points . In this work , we study a different class of supervised learning procedures that have recently attracted significant interest . These classifiers are obtained by : ( i ) performing pre-training with a selfsupervised task ( i.e. , without labels ) to obtain a complex representation of the data points , and then ( ii ) fitting a simple ( e.g. , linear ) classifier on the representation and the labels . Such “ Self-Supervised + Simple ” ( SSS for short ) algorithms are commonly used in natural language processing tasks ( Devlin et al. , 2018 ; Brown et al. , 2020 ) , and have recently found uses in other domains as well ( Ravanelli et al. , 2020 ; Liu et al. , 2019 ) . Compared to standard “ end-to-end supervised learning ” , SSS algorithms have several practical advantages . In particular , SSS algorithms can incorporate additional unlabeled data , the representation obtained can be useful for multiple downstream tasks , and they can have improved out-of-distribution performance ( Hendrycks et al. , 2019 ) . Moreover , recent works show that even without additional unlabeled data , SSS algorithms can get close to state-of-art accuracy in several classification tasks ( Chen et al. , 2020b ; He et al. , 2020 ; Misra & Maaten , 2020 ; ∗Equal contribution . Email : { ybansal , galkaplun } @ g.harvard.edu †Email : b @ boazbarak.org . Tian et al. , 2019 ) . For instance , SimCLRv2 ( Chen et al. , 2020b ) achieves 79.8 % top-1 performance on ImageNet with a variant of ResNet-152 , on par with the end-to-end supervised accuracy of this architecture at 80.5 % . We show that SSS algorithms have another advantage over standard supervised learning—they often have a small generalization gap between their train and test accuracy , and we prove non-vacuous bounds on this gap . We stress that SSS algorithms use over-parameterized models to extract the representation , and reuse the same training data to learn a simple classifier on this representation . Thus , the final classifier they produce has high complexity by most standard measures , and it is by no means apriori evident that their generalization gap will be small . Our bound is obtained by first noting that the generalization gap of every training algorithm is bounded by the sum of three quantities , which we name the Robustness gap , Rationality gap , and Memorization gap ( we call this the RRM bound , see Fact I ) . We now describe these gaps at a high level , deferring the formal definitions to Section 2 . All three gaps involve comparison with a setting where we inject label noise by replacing a small fraction η of the labels with random values . The robustness gap corresponds to the amount by which training performance degrades by noise injection . That is , it equals the difference between the standard expected training accuracy ( with no label noise ) and the expected training accuracy in the noisy setting ; in both cases , we measure accuracy with respect to the original ( uncorrupted ) labels . The robustness gap is nearly always small , and sometimes provably so ( see Section 3 ) . The rationality gap corresponds to the difference between performance on the noisy training samples ( on which the training algorithm gets the wrong label ) and test samples ( on which it doesn ’ t get any label at all ) , again with respect to uncorrupted labels . An optimal Bayesian procedure would have zero rationality gap , and indeed this gap is typically zero or small in practice . Since it is a nonstandard quantity , We discuss the rationality gap in Section 3.1 , and explain assuming it is small is both well-founded and does not trivialize the question of generalization . The memorization gap , which often accounts for the lion ’ s share of the generalization gap , corresponds to the difference in the noisy experiment between the training accuracy on the entire train set and the training accuracy on the samples that received the wrong label ( both measured with respect to uncorrupted labels ) . The memorization gap can be thought of as quantifying the extent to which the classifier can “ memorize ” noisy labels , or act differently on the noisy points compared to the overall train set . The memorization gap is large in standard “ end-to-end supervised training ” . In contrast , our main theoretical result is that for SSS algorithms , the memorization gap is small if the simple classifier has small complexity , independently of the complexity of the representation . As long as the simple classifier is under-parameterized ( i.e. , its complexity is asymptotically smaller than the sample size ) , our bound on the memorization gap tends to zero . When combined with small rationality and robustness , we get concrete non-vacuous generalization bounds for various SSS algorithms on the CIFAR-10 and ImageNet datasets ( see Figures 1 and 4 ) . In a nutshell , our results are the following : Theoretical contributions .. 1 . Our main theoretical result ( Theorem II ) is that the memorization gap of an SSS algorithm is bounded byO ( √ C/n ) whereC is the complexity of the simple classifier produced in the “ simple fit ” stage . This bound is oblivious to the complexity of the representation produced in the pre-training and does not make any assumptions on the relationship between the representation learning method and the supervised learning task . One way to interpret this result is that we give a rigorous bound on the generalization gap of SSS algorithms , under the assumptions that the robustness and rationality gaps are bounded by some small constant ( e.g. , 5 % ) . As mentioned below , these assumptions hold widely in practice across many different classifiers . Moreover , these assumptions are nontrivial and do not “ assume away the difficulty ” . Indeed , there are many natural examples of training algorithms for which these assumptions hold but the generalization gap is large . Last , making some assumptions is necessary for a generalization bound to hold for SSS algorithms ; see Remark 3.1 and Appendix E. 2 . We also give a theoretical justification for the assumption of a small rationality gap , by proving that a positive rationality gap corresponds to “ leaving performance on the table ” , in the sense that we can transform a learning procedure with a large rationality gap into a procedure with better test performance ( Theorem 3.2 ) . Empirical contributions . We complement the theoretical results above with an extensive empirical study of several SSS and end-to-end algorithms on both the CIFAR-10 and ImageNet datasets . 1 . We study several top-performing SSS architectures , and show that they all exhibit relatively small generalization gaps on both CIFAR-10 and ImageNet . We stress that we consider the case where the same data is used for both representation learning and classification , and hence it is by no means a-priori obvious that these algorithms should have small generalization gaps . See Figures 1 and 4 for sample results and Section 4 for more details . 2 . We also show that the results of Zhang et al . ( 2017 ) do not replicate to SSS algorithms , in the sense that such algorithms , despite using an over-parameterized representation , are not able to fit random label noise . 3 . We understake an empirical study of the robustness , rationality , and memorization gaps for both SSS and end-to-end supervised learning algorithms . We show that the robustness and rationality gaps are small for all these algorithms , while the memorization gap is small for SSS algorithms but can be large for end-to-end supervised learning . We show that the RRM bound is typically non-vacuous , and in fact , often close to tight , for a variety of SSS algorithms on the CIFAR-10 and ImageNet datasets , including SimCLR ( which achieves test errors close to its supervised counterparts ) . 4 . We demonstrate that replacing the memorization gap with the upper bound of Theorem II yields a non-vacuous generalization bound for a variety of SSS algorithms on CIFAR-10 and ImageNet . Moreover , this bound gets tighter with more data augmentation . Related Work . There are many works on generalization bounds for supervised learning ( e.g. , Golowich et al . ( 2018 ) ; Neyshabur et al . ( 2017 ) ; Bartlett et al . ( 2017 ) ; Dziugaite & Roy ( 2017 ) ; Neyshabur et al . ( 2018 ) ; Cao & Gu ( 2019 ) , and references therein ) . The related work section of Arora et al . ( 2019 ) contains an extensive discussion of such bounds , and why more often than not the assumptions used do not hold in practice . Indeed , many such bounds give vacuous guarantees for modern architectures ( such as the ones considered in this paper ) that have the capacity to memorize their entire training set ( Zhang et al. , 2017 ) . Some non-vacuous bounds are known ; e.g. , Zhou et al . ( 2019 ) gave a 96.5 % bound on the error of MobileNet on ImageNet . Belkin et al . ( 2019 ) ; Nagarajan & Kolter ( 2019 ) showed some barriers for generalization gaps for standard end-to-end supervised learning . Similarly , standard approaches such as Rademacher complexity can not directly bound SSS algorithms ’ generalization gap ( see Remark 3.1 ) . Recently , Saunshi et al . ( 2019 ) and Lee et al . ( 2020 ) gave generalization bounds for self-supervised based classifiers . The two works considered special cases of SSS algorithms , such as contrastive learning and pre-text tasks . Both works make strong statistical assumptions of ( exact or approximate ) conditional independence relating the pre-training and classification tasks . For example , if the pre-training task is obtained by splitting a given image x into two pieces ( x1 , x2 ) and predicting x2 from x1 , then Lee et al . ( 2020 ) ’ s results require x1 and x2 to be approximately independent conditioned on their class y . However , in many realistic cases , the two parts of the same image will share a significant amount of information not explained by the label . Our work applies to general SSS algorithms without such statistical assumptions , at the expense of assuming bounds on the robustness and rationality gaps . There have been works providing rigorous bounds on the robustness gap or related quantities ( See Section 3. ) . However , as far as we know , the rationality gap has not been explicitly defined or studied before . We provide a brief exposition of the various types of SSS methods in Section 4 , and a more detailed discussion in Appendix D.1 . Paper Organization . Section 2 contains formal definitions and statements of our results . Section 3 provides an overview of prior work and our new results on the three gaps of the RRM bound . In Section 4 , we describe our experimental setup and detail our empirical results . Section 5 concludes the paper and discusses important open questions . We defer proofs and additional experimental results to the appendix . Appendix B contains the proof of Theorem II , while Appendix C contains the proof of Theorem 3.2 . Appendix D fully details our experimental setup.1 Notation . We use capital letters ( e.g. , X ) for random variables , lower case letters ( e.g. , x ) for a single value , and bold font ( e.g. , x ) for tuples ( which will typically have dimension corresponding to the number of samples , denoted by n ) . We use xi for the i-th element of the tuple x . We use calligraphic letters ( e.g. , X , D ) for both sets and distributions .	This paper gives a new perspective on generalization, motivated by the success of self-supervised learning, especially on noisy data. They view the generalization error as consisting of 3 independent components: robustness, rationality and memorization. Informally, robustness measures the degradation in training accuracy due to the addition of noise. rationality measures the gap between noisy training and test accuracy and memorization is the gap between the training error on the uncorrupted and corrupted examples. The main point of the paper is that the memorization gap is smaller for self-supervised, simple algorithms compared to full supervised algorithms. They prove that when the noise is defined by a small fraction of labels being randomly flipped, then the memorization error of such simple algorithms can be bounded in terms of their information-theoretic complexity, independent of the representation they produce. (The proof is simple once the definitions are set up properly).	SP:280743806ee639df87f4d7de86f287fb455e3ff3
Direct Evolutionary Optimization of Variational Autoencoders with Binary Latents	Discrete latent variables are considered important to model the generation process of real world data , which has motivated research on Variational Autoencoders ( VAEs ) with discrete latents . However , standard VAE training is not possible in this case , which has motivated different strategies to manipulate discrete distributions in order to train discrete VAEs similarly to conventional ones . Here we ask if it is also possible to keep the discrete nature of the latents fully intact by applying a direct discrete optimization for the encoding model . The studied approach is consequently strongly diverting from standard VAE training by altogether sidestepping absolute standard VAE mechanisms such as sampling approximation , reparameterization trick and amortization . Discrete optimization is realized in a variational setting using truncated posteriors in conjunction with evolutionary algorithms ( using a recently suggested approach ) . For VAEs with binary latents , we first show how such a discrete variational method ( A ) ties into gradient ascent for network weights and ( B ) uses the decoder network to select latent states for training . More conventional amortized training is , as may be expected , more efficient than direct discrete optimization , and applicable to large neural networks . However , we here find direct optimization to be efficiently scalable to hundreds of latent variables using smaller networks . More importantly , we find the effectiveness of direct optimization to be highly competitive in ‘ zero-shot ’ learning ( where high effectiveness for small networks is required ) . In contrast to large supervised neural networks , the here investigated VAEs can , e.g. , denoise a single image without previous training on clean data and/or training on large image datasets . More generally , the studied approach shows that training of VAEs is indeed possible without sampling-based approximation and reparameterization , which may be interesting for the analysis of VAE-training in general . In the regime of few data , direct optimization , furthermore , makes VAEs competitive for denoising where they have previously been outperformed by non-generative approaches . 1 INTRODUCTION AND RELATED WORK . Variational autoencoders ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) are prominent and very actively researched models for unsupervised learning . VAEs , in their many different variations , have successfully been applied to a large number of tasks including semi-supervised learning ( e.g . Maaløe et al. , 2016 ) , anomaly detection ( e.g . An & Cho , 2015 ; Kiran et al. , 2018 ) , sentence interpolation ( Bowman et al. , 2016 ) , music interpolation ( Roberts et al. , 2018 ) and drug response prediction ( Rampasek et al. , 2017 ) . The success of VAEs rests on a series of methods that enable the derivation of scalable training algorithms to optimize their model parameters ( discussed further below ) . A desired feature when applying VAEs to a given problem is that their latent variables ( i.e. , the encoder output variables ) correspond to meaningful properties of the data , ideally to those latent causes that have originally generated the data . However , many real-world datasets suggest the use of discrete latents as they often describe the data generation process more naturally . For instance , the presence or absence of objects in images is best described by binary latents ( e.g . Jojic & Frey , 2001 ) . Discrete latents are also a popular choice in modeling sounds ; for instance , describing piano sounds may naturally involve binary latents : keys are pressed or not ( e.g . Titsias & Lázaro-Gredilla , 2011 ; Goodfellow et al. , 2013 ; Sheikh et al. , 2014 ) . The success of standard forms of VAEs has consequently spurred research on novel formulations that feature discrete latents ( e.g . Rolfe , 2016 ; Khoshaman & Amin , 2018 ; Roy et al. , 2018 ; Sadeghi et al. , 2019 ; Vahdat et al. , 2019 ) . The objective of VAE training is the optimization of a generative data model which parameterizes a given data distribution . Typically we seek model parameters Θ of a VAE that maximize the data log-likelihood , L ( Θ ) = ∑ n log ( pΘ ( ~x ( n ) ) ) , where we denote by ~x ( 1 : N ) a set of N observed data points , and where pΘ ( ~x ) denotes the modeled data distribution . Like conventional autoencoders ( e.g. , Bengio et al. , 2007 ) , VAEs use a deep neural network ( DNN ) to generate ( or decode ) observables ~x from a latent code ~z . Unlike conventional autoencoders , however , the generation of data ~x is not deterministic but it takes the form of a probabilistic generative model . For VAEs with binary latent variables , as they will be of interest here , we consider the following VAE generative model : pΘ ( ~z ) = Bern ( ~z ; ~π ) = ∏ h ( πzhh ( 1− πh ) ( 1−zh ) ) , pΘ ( ~x \|~z ) = N ( ~x ; ~µ ( ~z ; W ) , σ2I ) , ( 1 ) where ~z ∈ { 0 , 1 } H is a binary code and the non-linear function ~µ ( ~z ; W ) is a DNN that outputs the mean of the Gaussian distribution . pΘ ( ~x \|~z ) is commonly referred to as decoder . The set of model parameters is Θ = { ~π , W , σ2 } , where W incorporates DNN weights and biases . We assume homoscedasticity of the Gaussian distribution , but note that there is no obstacle to generalizing the model by inserting a DNN non-linearity that outputs a correlation matrix . Similarly , the algorithm could easily be generalized to different noise distributions should the task at hand call for it . For the purpose of this work , however , we will focus on as elementary as possible VAEs , with the form shown in Eqn . ( 1 ) . Given standard or binary-latent VAEs , essentially all learning algorithms seek to approximately maximize the log-likelihood using the following series of methods ( we elaborate in the appendix ) : ( A ) Instead of the log-likelihood , a variational lower-bound ( a.k.a . ELBO ) is optimized . ( B ) VAE posteriors are approximated by an encoding model , that is a specific distribution ( often Gaussian ) parameterized by one or more DNNs . ( C ) The variational parameters of the encoder are optimized using gradient ascent on the lower bound , where the gradient is evaluated based on sampling and reparameterization trick to obtain sufficiently low-variance and yet efficiently computable estimates . ( D ) Using samples from the encoder , the parameters of the decoder are optimized using gradient ascent on the variational lower bound . Optimization procedures for VAEs with discrete latents follow the same steps ( Points A to D ) . However , discrete or binary latents pose substantial further obstacles in learning , mainly due to the fact that backpropagation through discrete variables is generally not possible ( Rolfe , 2016 ; Bengio et al. , 2013 ) . In order to maintain the general VAE framework for encoder optimization , different groups have therefore suggested different possible solutions : work by Rolfe ( 2016 ) , for instance , extends VAEs with discrete latents by auxiliary continuous latents such that gradients can still be computed . Work on the concrete distribution ( Maddison et al. , 2016 ) or Gumbel-softmax distribution ( Jang et al. , 2016 ) proposes newly defined continuous distributions that contain discrete distributions as limit cases . Work by Lorberbom et al . ( 2019 ) merges the Gumbel-Max reparameterization with the use of direct loss minimization for gradient estimation , enabling efficient training on structured latent spaces . Finally , work by van den Oord et al . ( 2017 ) , and Roy et al . ( 2018 ) combines VAEs with a vector quantization ( VQ ) stage in the latent layer . Latents become discrete through quantization but gradients for learning are adapted from latent values before they are processed by the VQ stage . All methods have the goal of treating discrete distributions such that standard VAE training as developed for continuous latents can still be applied . These techniques interact during training with the standard methods ( Points A-D ) already in place for VAE optimization . Furthermore , they add further types of design decisions and hyper-parameters , for example parameters for annealing from softened discrete distributions to the ( hard ) original distributions for discrete latents . For discrete VAEs , it may consequently be a desirable goal to investigate alternative , more direct optimization procedures that do not require a softening of discrete distributions or the use of other indirect solutions . Such a direct approach is challenging , however , because once DNNs are used to define the encoding model ( Point B ) standard tricks to estimate gradients ( Point C ) seem unavoidable . A direct optimization procedure , as is investigated here , consequently has to substantially change VAE training . For the data model ( 1 ) we will maintain the variational setting and a decoding model with DNNs as non-linearity ( Points A and D ) . However , we will not use an encoder model parameterized by DNNs ( Point B ) . Instead , the variational bound will be increased w.r.t . the encoder model by using a discrete optimization approach . The procedure does not require gradients to be computed for the encoder such that discrete latents are addressed without the use of reparameterization trick and sampling approximations . 2 TRUNCATED VARIATIONAL OPTIMIZATION . Let us consider the variational lower bound of the likelihood . If we denote by q ( n ) Φ ( ~z ) the variational distributions with parameters Φ ( n ) , and by 〈 h ( ~z ) 〉 q ( n ) Φ = ∑ ~z q ( n ) Φ ( ~z ) h ( ~z ) expectation values w.r.t . to q ( n ) Φ ( ~z ) , then the lower bound can be written as : F ( Φ , Θ ) = ∑ n 〈 log ( pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ) 〉 q ( n ) Φ − ∑ n 〈 log ( q ( n ) Φ ( ~z ) ) 〉 q ( n ) Φ , ( 2 ) The general challenge for the maximization of F ( Φ , Θ ) is the optimization of the encoding model q ( n ) Φ . VAEs with discrete latents add to this challenge the problem of taking gradients w.r.t . discrete latents . If we seek to avoid derivatives w.r.t . discrete variables , we have to define an alternative encoding model q ( n ) Φ but such an encoding has to remain sufficiently efficient . Considering prior work on generative models with discrete latents , variational distributions based on truncated posteriors offer themselves as such an alternative ( Lücke & Sahani , 2008 ) . Truncated posterior approximations have been shown to be functionally competitive ( e.g . Sheikh et al. , 2014 ; Hughes & Sudderth , 2016 ; Shelton et al. , 2017 ) , and they are able to efficiently train also very large scale models with hundreds or thousands of latents ( e.g . Shelton et al. , 2011 ; Sheikh & Lücke , 2016 ; Forster & Lücke , 2018 ) . However , the important question for training discrete VAEs is if or how truncated variational distributions can be used in gradient-based optimization of neural network parameters . We here , for the first time , address this question noting that all previous approaches relied on closed-form ( or pseudo-closed form ) parameter update equations in an expectation-maximization learning paradigm . Optimization of the Decoding Model . In order to optimize the parameters W of the decoder DNN ~µ ( ~z , W ) , the gradient of the variational bound ( 2 ) w.r.t . W has to be computed . We consequently need , for any VAE , a sufficiently precise and efficient approximation of the expectation value w.r.t . the encoder q ( n ) Φ ( ~z ) . Gradient estimation is of central importance for deep unsupervised learning , and approaches , e.g. , for variance reduction of estimators have played an important role and are dedicated solely to this purpose ( e.g. , Williams , 1992 ) . Reparameterization finally emerged as a key method because it allowed for sufficiently low-variance estimation of gradients based , e.g. , on Gaussian middle-layer units ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) . For discrete VAEs , however , reparameterization requires the introduction of additional manipulations of discrete distributions that we here seek to fully avoid . Instead of using reparameterization or variance reduction , we will compute gradients based on truncated posterior as variational distributions . A truncated posterior has the following form : q ( n ) Φ ( ~z ) : = pΘ ( ~z \| ~x ( n ) ) ∑ ~z ′∈Φ ( n ) pΘ ( ~z ′ \| ~x ( n ) ) = pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ∑ ~z ′∈Φ ( n ) pΘ ( ~x ( n ) \|~z ′ ) pΘ ( ~z ′ ) if ~z ∈ Φ ( n ) , ( 3 ) where for all ~z 6∈ Φ ( n ) the probability q ( n ) Φ ( ~z ) equals zero . That is , a variational distribution q ( n ) Φ ( ~z ) is proportional to the true posteriors in a subset Φ ( n ) , which acts as its variational parameter . We can now compute the gradient of ( 2 ) w.r.t . the decoder weights W which results in : ~∇WF ( Φ , Θ ) = − 1 2σ2 ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ~∇W ‖~x ( n ) − ~µ ( ~z , W ) ‖2 . ( 4 ) The right-hand-side has salient similarities to the standard gradient ascent for VAE decoders . Especially the familiar gradient of the mean squared error ( MSE ) shows that , e.g. , standard automatic differentiation tools can be applied . However , the decisive difference are the weighting factors q ( n ) Φ ( ~z ) . Considering ( 3 ) , in order to compute the weighting factors we require all ~z ∈ Φ ( n ) to be passed through the decoder DNN . As all states of Φ ( n ) anyway have to be passed through the decoder for the MSE term of ( 4 ) , the overall computational complexity is not higher than an estimation of the gradient with samples instead of states in Φ ( n ) ( we elaborate in Appendix A ) . To complete the decoder optimization , update equations for variance σ2 and prior parameters ~π can be computed in closed-form ( compare , e.g. , Shelton et al. , 2011 ) and are given by : σ2 , new = 1DN ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ‖~x ( n ) − ~µ ( ~z , W ) ‖2 , ~πnew = 1 N ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ~z , ( 5 ) where N is the number of samples in the training dataset and D is the number of observables . Optimization of the Encoding Model . After having established that the decoder can be optimized efficiently and by using standard DNN methods , the important question is if the encoder can be trained efficiently . Encoder optimization is usually based on a reformulation of the variational bound ( 2 ) given by : F ( Φ , Θ ) = ∑ n 〈 log ( pΘ ( ~x ( n ) \|~z ) ) 〉 q ( n ) Φ − ∑ nDKL ( q ( n ) Φ ( ~z ) , pΘ ( ~z ) ) . ( 6 ) Centrally for this work , truncated posteriors allow a specific alternative reformulation of the bound that enables efficient optimization . The reformulation recombines the entropy term of the original form ( 2 ) with the first expectation value into a single term , and is given by ( see Lücke , 2019 , for details ) : F ( Φ , Θ ) = ∑ n log ( ∑ ~z∈Φ ( n ) pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ) . ( 7 ) Thanks to the simplified form of the bound , the variational parameters Φ ( n ) of the encoding model can now be sought using direct discrete optimization procedures . More concretely , because of the specific form ( 7 ) , pairwise comparisons of joint probabilities are sufficient to maximize the lower bound : if we update the set Φ ( n ) for a given ~x ( n ) by replacing a state ~z old ∈ Φ ( n ) with a state ~z new 6∈ Φ ( n ) , then F ( Φ , Θ ) increases if and only if : log ( pΘ ( ~x ( n ) , ~z new ) ) > log ( pΘ ( ~x ( n ) , ~z old ) ) . ( 8 ) To obtain intuition for the pairwise comparison , consider its form when inserting the binary VAE ( 1 ) into the left- and right-hand sides . Eliding terms that do not depend on ~z we obtain : ˜log pΘ ( ~x , ~z ) = −‖~x− ~µ ( ~z , W ) ‖2 − 2σ2∑h π̃h zh ( 9 ) where π̃h = log ( ( 1 − πh ) /πh ) . The expression assumes an even more familiar form if we restrict ourselves for a moment to sparse priors π < 12 , i.e. , π̃ > 0 . Criterion ( 8 ) then becomes : ‖~x ( n ) − ~µ ( ~z new , W ) ‖2 + 2σ2π̃ \|~z new\| < ‖~x ( n ) − ~µ ( ~z old , W ) ‖2 + 2σ2π̃ \|~z old\| ( 10 ) where \|~z\| = ∑H h=1 zh . Such functions are routinely encountered in sparse coding or compressive sensing ( Eldar & Kutyniok , 2012 ) : for each set Φ ( n ) we seek those states ~z that are reconstructing ~x ( n ) well while being sparse ( ~z with few non-zero bits ) . For VAEs , ~µ ( ~z new , W ) is a DNN and as such much more flexible in matching the distribution of observables ~x than can be expected from linear mappings . Furthermore , criteria like ( 10 ) usually emerge for maximum a-posteriori ( MAP ) training in sparse coding ( Olshausen & Field , 1996 ) . In contrast , we here seek a population of states ~z in Φ ( n ) for each data point . It is a consequence of the reformulated lower bound ( 7 ) that it remains optimal to evaluate joint probabilities ( as for MAP ) although the constructed population of states Φ ( n ) can capture ( unlike MAP training ) a rich posterior structure . But how can new states ~z new that optimize Φ ( n ) be found efficiently in high-dimensional latent spaces ? Random search and search by sampling has recently been explored for elementary generative models ( Lücke et al. , 2018 ) . Here we will follow another recent suggestion ( Guiraud et al. , 2018 ) and make use of a search based on evolutionary algorithms ( EAs ) . In this setting we interpret sets Φ ( n ) as populations of binary genomes ~z and base the fitness function on Eqn . ( 9 ) . Concretely , using Φ ( n ) as initial parent pool , we apply the following genetic operators in sequence : firstly , parent selection picks Np states from the parent pool . In our numerical experiments we used fitness-proportional parent selection , for which we add an offset ( constant w.r.t . ~z ) to the fitness values in order to make them strictly non-negative . Each of the children undergoes mutation : one or more bits are flipped to further increase offspring diversity . In our experiments we perform random uniform selection of the bits to flip . Crossover could also be employed to increase offspring diversity . We repeat the procedure using the children generated this way as the parent pool , giving birth to multiple generations of candidate states . Finally , we update Φ ( n ) by substituting individuals with low fitness with candidates with higher fitness . The whole procedure can be seen as an evolutionary algorithm with perfect memory or very strong elitism ( individuals with higher fitness never drop out of the gene pool ) . Note that the improvement of the variational lower bound depends on generating as many as possible different children with high fitness over the course of training . We point out that the EAs optimize each Φ ( n ) independently , so this technique can be applied to large datasets in conjunction with stochastic or batch gradient descent on the model parameters Θ : it does not require to keep the full dataset ( or all sets Φ ( n ) ) in memory at a given time . Fig 1 shows how EAs produce new states that are used to update each set Φ ( n ) . The full training procedure for binary VAEs is summarized in Algorithm 1 .	This paper proposes an evolutionary optimization framework for training vartional autoencoders (VAEs) with discrete latents. In contrast to the standard VAE paradigm, the proposed TVAE approach does not require an encoder for amortized inference given the input. The method instead relies on a pool of latent variable samples for each data point to activate the decoder network. The latent variable pools are maintained and iteratively updated to increase the average lower-bound of the marginal log-likelihood of the input data. Experimental results show that non-linear decoders optimized by the TVAE framework outperform their linear counterparts on a denoising task. Further results demonstrate method's competitiveness on zero-shot denoising, where a TVAE decoder is only trained on the noisy input image to reconstruct a smoothed version of the input image.	SP:f741d980c9c560a21298e947f1605dcbab7ceeac
Direct Evolutionary Optimization of Variational Autoencoders with Binary Latents	Discrete latent variables are considered important to model the generation process of real world data , which has motivated research on Variational Autoencoders ( VAEs ) with discrete latents . However , standard VAE training is not possible in this case , which has motivated different strategies to manipulate discrete distributions in order to train discrete VAEs similarly to conventional ones . Here we ask if it is also possible to keep the discrete nature of the latents fully intact by applying a direct discrete optimization for the encoding model . The studied approach is consequently strongly diverting from standard VAE training by altogether sidestepping absolute standard VAE mechanisms such as sampling approximation , reparameterization trick and amortization . Discrete optimization is realized in a variational setting using truncated posteriors in conjunction with evolutionary algorithms ( using a recently suggested approach ) . For VAEs with binary latents , we first show how such a discrete variational method ( A ) ties into gradient ascent for network weights and ( B ) uses the decoder network to select latent states for training . More conventional amortized training is , as may be expected , more efficient than direct discrete optimization , and applicable to large neural networks . However , we here find direct optimization to be efficiently scalable to hundreds of latent variables using smaller networks . More importantly , we find the effectiveness of direct optimization to be highly competitive in ‘ zero-shot ’ learning ( where high effectiveness for small networks is required ) . In contrast to large supervised neural networks , the here investigated VAEs can , e.g. , denoise a single image without previous training on clean data and/or training on large image datasets . More generally , the studied approach shows that training of VAEs is indeed possible without sampling-based approximation and reparameterization , which may be interesting for the analysis of VAE-training in general . In the regime of few data , direct optimization , furthermore , makes VAEs competitive for denoising where they have previously been outperformed by non-generative approaches . 1 INTRODUCTION AND RELATED WORK . Variational autoencoders ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) are prominent and very actively researched models for unsupervised learning . VAEs , in their many different variations , have successfully been applied to a large number of tasks including semi-supervised learning ( e.g . Maaløe et al. , 2016 ) , anomaly detection ( e.g . An & Cho , 2015 ; Kiran et al. , 2018 ) , sentence interpolation ( Bowman et al. , 2016 ) , music interpolation ( Roberts et al. , 2018 ) and drug response prediction ( Rampasek et al. , 2017 ) . The success of VAEs rests on a series of methods that enable the derivation of scalable training algorithms to optimize their model parameters ( discussed further below ) . A desired feature when applying VAEs to a given problem is that their latent variables ( i.e. , the encoder output variables ) correspond to meaningful properties of the data , ideally to those latent causes that have originally generated the data . However , many real-world datasets suggest the use of discrete latents as they often describe the data generation process more naturally . For instance , the presence or absence of objects in images is best described by binary latents ( e.g . Jojic & Frey , 2001 ) . Discrete latents are also a popular choice in modeling sounds ; for instance , describing piano sounds may naturally involve binary latents : keys are pressed or not ( e.g . Titsias & Lázaro-Gredilla , 2011 ; Goodfellow et al. , 2013 ; Sheikh et al. , 2014 ) . The success of standard forms of VAEs has consequently spurred research on novel formulations that feature discrete latents ( e.g . Rolfe , 2016 ; Khoshaman & Amin , 2018 ; Roy et al. , 2018 ; Sadeghi et al. , 2019 ; Vahdat et al. , 2019 ) . The objective of VAE training is the optimization of a generative data model which parameterizes a given data distribution . Typically we seek model parameters Θ of a VAE that maximize the data log-likelihood , L ( Θ ) = ∑ n log ( pΘ ( ~x ( n ) ) ) , where we denote by ~x ( 1 : N ) a set of N observed data points , and where pΘ ( ~x ) denotes the modeled data distribution . Like conventional autoencoders ( e.g. , Bengio et al. , 2007 ) , VAEs use a deep neural network ( DNN ) to generate ( or decode ) observables ~x from a latent code ~z . Unlike conventional autoencoders , however , the generation of data ~x is not deterministic but it takes the form of a probabilistic generative model . For VAEs with binary latent variables , as they will be of interest here , we consider the following VAE generative model : pΘ ( ~z ) = Bern ( ~z ; ~π ) = ∏ h ( πzhh ( 1− πh ) ( 1−zh ) ) , pΘ ( ~x \|~z ) = N ( ~x ; ~µ ( ~z ; W ) , σ2I ) , ( 1 ) where ~z ∈ { 0 , 1 } H is a binary code and the non-linear function ~µ ( ~z ; W ) is a DNN that outputs the mean of the Gaussian distribution . pΘ ( ~x \|~z ) is commonly referred to as decoder . The set of model parameters is Θ = { ~π , W , σ2 } , where W incorporates DNN weights and biases . We assume homoscedasticity of the Gaussian distribution , but note that there is no obstacle to generalizing the model by inserting a DNN non-linearity that outputs a correlation matrix . Similarly , the algorithm could easily be generalized to different noise distributions should the task at hand call for it . For the purpose of this work , however , we will focus on as elementary as possible VAEs , with the form shown in Eqn . ( 1 ) . Given standard or binary-latent VAEs , essentially all learning algorithms seek to approximately maximize the log-likelihood using the following series of methods ( we elaborate in the appendix ) : ( A ) Instead of the log-likelihood , a variational lower-bound ( a.k.a . ELBO ) is optimized . ( B ) VAE posteriors are approximated by an encoding model , that is a specific distribution ( often Gaussian ) parameterized by one or more DNNs . ( C ) The variational parameters of the encoder are optimized using gradient ascent on the lower bound , where the gradient is evaluated based on sampling and reparameterization trick to obtain sufficiently low-variance and yet efficiently computable estimates . ( D ) Using samples from the encoder , the parameters of the decoder are optimized using gradient ascent on the variational lower bound . Optimization procedures for VAEs with discrete latents follow the same steps ( Points A to D ) . However , discrete or binary latents pose substantial further obstacles in learning , mainly due to the fact that backpropagation through discrete variables is generally not possible ( Rolfe , 2016 ; Bengio et al. , 2013 ) . In order to maintain the general VAE framework for encoder optimization , different groups have therefore suggested different possible solutions : work by Rolfe ( 2016 ) , for instance , extends VAEs with discrete latents by auxiliary continuous latents such that gradients can still be computed . Work on the concrete distribution ( Maddison et al. , 2016 ) or Gumbel-softmax distribution ( Jang et al. , 2016 ) proposes newly defined continuous distributions that contain discrete distributions as limit cases . Work by Lorberbom et al . ( 2019 ) merges the Gumbel-Max reparameterization with the use of direct loss minimization for gradient estimation , enabling efficient training on structured latent spaces . Finally , work by van den Oord et al . ( 2017 ) , and Roy et al . ( 2018 ) combines VAEs with a vector quantization ( VQ ) stage in the latent layer . Latents become discrete through quantization but gradients for learning are adapted from latent values before they are processed by the VQ stage . All methods have the goal of treating discrete distributions such that standard VAE training as developed for continuous latents can still be applied . These techniques interact during training with the standard methods ( Points A-D ) already in place for VAE optimization . Furthermore , they add further types of design decisions and hyper-parameters , for example parameters for annealing from softened discrete distributions to the ( hard ) original distributions for discrete latents . For discrete VAEs , it may consequently be a desirable goal to investigate alternative , more direct optimization procedures that do not require a softening of discrete distributions or the use of other indirect solutions . Such a direct approach is challenging , however , because once DNNs are used to define the encoding model ( Point B ) standard tricks to estimate gradients ( Point C ) seem unavoidable . A direct optimization procedure , as is investigated here , consequently has to substantially change VAE training . For the data model ( 1 ) we will maintain the variational setting and a decoding model with DNNs as non-linearity ( Points A and D ) . However , we will not use an encoder model parameterized by DNNs ( Point B ) . Instead , the variational bound will be increased w.r.t . the encoder model by using a discrete optimization approach . The procedure does not require gradients to be computed for the encoder such that discrete latents are addressed without the use of reparameterization trick and sampling approximations . 2 TRUNCATED VARIATIONAL OPTIMIZATION . Let us consider the variational lower bound of the likelihood . If we denote by q ( n ) Φ ( ~z ) the variational distributions with parameters Φ ( n ) , and by 〈 h ( ~z ) 〉 q ( n ) Φ = ∑ ~z q ( n ) Φ ( ~z ) h ( ~z ) expectation values w.r.t . to q ( n ) Φ ( ~z ) , then the lower bound can be written as : F ( Φ , Θ ) = ∑ n 〈 log ( pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ) 〉 q ( n ) Φ − ∑ n 〈 log ( q ( n ) Φ ( ~z ) ) 〉 q ( n ) Φ , ( 2 ) The general challenge for the maximization of F ( Φ , Θ ) is the optimization of the encoding model q ( n ) Φ . VAEs with discrete latents add to this challenge the problem of taking gradients w.r.t . discrete latents . If we seek to avoid derivatives w.r.t . discrete variables , we have to define an alternative encoding model q ( n ) Φ but such an encoding has to remain sufficiently efficient . Considering prior work on generative models with discrete latents , variational distributions based on truncated posteriors offer themselves as such an alternative ( Lücke & Sahani , 2008 ) . Truncated posterior approximations have been shown to be functionally competitive ( e.g . Sheikh et al. , 2014 ; Hughes & Sudderth , 2016 ; Shelton et al. , 2017 ) , and they are able to efficiently train also very large scale models with hundreds or thousands of latents ( e.g . Shelton et al. , 2011 ; Sheikh & Lücke , 2016 ; Forster & Lücke , 2018 ) . However , the important question for training discrete VAEs is if or how truncated variational distributions can be used in gradient-based optimization of neural network parameters . We here , for the first time , address this question noting that all previous approaches relied on closed-form ( or pseudo-closed form ) parameter update equations in an expectation-maximization learning paradigm . Optimization of the Decoding Model . In order to optimize the parameters W of the decoder DNN ~µ ( ~z , W ) , the gradient of the variational bound ( 2 ) w.r.t . W has to be computed . We consequently need , for any VAE , a sufficiently precise and efficient approximation of the expectation value w.r.t . the encoder q ( n ) Φ ( ~z ) . Gradient estimation is of central importance for deep unsupervised learning , and approaches , e.g. , for variance reduction of estimators have played an important role and are dedicated solely to this purpose ( e.g. , Williams , 1992 ) . Reparameterization finally emerged as a key method because it allowed for sufficiently low-variance estimation of gradients based , e.g. , on Gaussian middle-layer units ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) . For discrete VAEs , however , reparameterization requires the introduction of additional manipulations of discrete distributions that we here seek to fully avoid . Instead of using reparameterization or variance reduction , we will compute gradients based on truncated posterior as variational distributions . A truncated posterior has the following form : q ( n ) Φ ( ~z ) : = pΘ ( ~z \| ~x ( n ) ) ∑ ~z ′∈Φ ( n ) pΘ ( ~z ′ \| ~x ( n ) ) = pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ∑ ~z ′∈Φ ( n ) pΘ ( ~x ( n ) \|~z ′ ) pΘ ( ~z ′ ) if ~z ∈ Φ ( n ) , ( 3 ) where for all ~z 6∈ Φ ( n ) the probability q ( n ) Φ ( ~z ) equals zero . That is , a variational distribution q ( n ) Φ ( ~z ) is proportional to the true posteriors in a subset Φ ( n ) , which acts as its variational parameter . We can now compute the gradient of ( 2 ) w.r.t . the decoder weights W which results in : ~∇WF ( Φ , Θ ) = − 1 2σ2 ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ~∇W ‖~x ( n ) − ~µ ( ~z , W ) ‖2 . ( 4 ) The right-hand-side has salient similarities to the standard gradient ascent for VAE decoders . Especially the familiar gradient of the mean squared error ( MSE ) shows that , e.g. , standard automatic differentiation tools can be applied . However , the decisive difference are the weighting factors q ( n ) Φ ( ~z ) . Considering ( 3 ) , in order to compute the weighting factors we require all ~z ∈ Φ ( n ) to be passed through the decoder DNN . As all states of Φ ( n ) anyway have to be passed through the decoder for the MSE term of ( 4 ) , the overall computational complexity is not higher than an estimation of the gradient with samples instead of states in Φ ( n ) ( we elaborate in Appendix A ) . To complete the decoder optimization , update equations for variance σ2 and prior parameters ~π can be computed in closed-form ( compare , e.g. , Shelton et al. , 2011 ) and are given by : σ2 , new = 1DN ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ‖~x ( n ) − ~µ ( ~z , W ) ‖2 , ~πnew = 1 N ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ~z , ( 5 ) where N is the number of samples in the training dataset and D is the number of observables . Optimization of the Encoding Model . After having established that the decoder can be optimized efficiently and by using standard DNN methods , the important question is if the encoder can be trained efficiently . Encoder optimization is usually based on a reformulation of the variational bound ( 2 ) given by : F ( Φ , Θ ) = ∑ n 〈 log ( pΘ ( ~x ( n ) \|~z ) ) 〉 q ( n ) Φ − ∑ nDKL ( q ( n ) Φ ( ~z ) , pΘ ( ~z ) ) . ( 6 ) Centrally for this work , truncated posteriors allow a specific alternative reformulation of the bound that enables efficient optimization . The reformulation recombines the entropy term of the original form ( 2 ) with the first expectation value into a single term , and is given by ( see Lücke , 2019 , for details ) : F ( Φ , Θ ) = ∑ n log ( ∑ ~z∈Φ ( n ) pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ) . ( 7 ) Thanks to the simplified form of the bound , the variational parameters Φ ( n ) of the encoding model can now be sought using direct discrete optimization procedures . More concretely , because of the specific form ( 7 ) , pairwise comparisons of joint probabilities are sufficient to maximize the lower bound : if we update the set Φ ( n ) for a given ~x ( n ) by replacing a state ~z old ∈ Φ ( n ) with a state ~z new 6∈ Φ ( n ) , then F ( Φ , Θ ) increases if and only if : log ( pΘ ( ~x ( n ) , ~z new ) ) > log ( pΘ ( ~x ( n ) , ~z old ) ) . ( 8 ) To obtain intuition for the pairwise comparison , consider its form when inserting the binary VAE ( 1 ) into the left- and right-hand sides . Eliding terms that do not depend on ~z we obtain : ˜log pΘ ( ~x , ~z ) = −‖~x− ~µ ( ~z , W ) ‖2 − 2σ2∑h π̃h zh ( 9 ) where π̃h = log ( ( 1 − πh ) /πh ) . The expression assumes an even more familiar form if we restrict ourselves for a moment to sparse priors π < 12 , i.e. , π̃ > 0 . Criterion ( 8 ) then becomes : ‖~x ( n ) − ~µ ( ~z new , W ) ‖2 + 2σ2π̃ \|~z new\| < ‖~x ( n ) − ~µ ( ~z old , W ) ‖2 + 2σ2π̃ \|~z old\| ( 10 ) where \|~z\| = ∑H h=1 zh . Such functions are routinely encountered in sparse coding or compressive sensing ( Eldar & Kutyniok , 2012 ) : for each set Φ ( n ) we seek those states ~z that are reconstructing ~x ( n ) well while being sparse ( ~z with few non-zero bits ) . For VAEs , ~µ ( ~z new , W ) is a DNN and as such much more flexible in matching the distribution of observables ~x than can be expected from linear mappings . Furthermore , criteria like ( 10 ) usually emerge for maximum a-posteriori ( MAP ) training in sparse coding ( Olshausen & Field , 1996 ) . In contrast , we here seek a population of states ~z in Φ ( n ) for each data point . It is a consequence of the reformulated lower bound ( 7 ) that it remains optimal to evaluate joint probabilities ( as for MAP ) although the constructed population of states Φ ( n ) can capture ( unlike MAP training ) a rich posterior structure . But how can new states ~z new that optimize Φ ( n ) be found efficiently in high-dimensional latent spaces ? Random search and search by sampling has recently been explored for elementary generative models ( Lücke et al. , 2018 ) . Here we will follow another recent suggestion ( Guiraud et al. , 2018 ) and make use of a search based on evolutionary algorithms ( EAs ) . In this setting we interpret sets Φ ( n ) as populations of binary genomes ~z and base the fitness function on Eqn . ( 9 ) . Concretely , using Φ ( n ) as initial parent pool , we apply the following genetic operators in sequence : firstly , parent selection picks Np states from the parent pool . In our numerical experiments we used fitness-proportional parent selection , for which we add an offset ( constant w.r.t . ~z ) to the fitness values in order to make them strictly non-negative . Each of the children undergoes mutation : one or more bits are flipped to further increase offspring diversity . In our experiments we perform random uniform selection of the bits to flip . Crossover could also be employed to increase offspring diversity . We repeat the procedure using the children generated this way as the parent pool , giving birth to multiple generations of candidate states . Finally , we update Φ ( n ) by substituting individuals with low fitness with candidates with higher fitness . The whole procedure can be seen as an evolutionary algorithm with perfect memory or very strong elitism ( individuals with higher fitness never drop out of the gene pool ) . Note that the improvement of the variational lower bound depends on generating as many as possible different children with high fitness over the course of training . We point out that the EAs optimize each Φ ( n ) independently , so this technique can be applied to large datasets in conjunction with stochastic or batch gradient descent on the model parameters Θ : it does not require to keep the full dataset ( or all sets Φ ( n ) ) in memory at a given time . Fig 1 shows how EAs produce new states that are used to update each set Φ ( n ) . The full training procedure for binary VAEs is summarized in Algorithm 1 .	the paper proposes a novel approach to training variational autoencoder models, based on non-parametric form of truncated approximate posterior. Posterior is truncated to have support on a small subset of latent space allowing for exact marginalization. The support of approximate posterior in latent space for each data point $x$ is learned via evolutionary algorithm, minimizing ELBO. Method is applied to denoising tasks for images.	SP:f741d980c9c560a21298e947f1605dcbab7ceeac
Direct Evolutionary Optimization of Variational Autoencoders with Binary Latents	Discrete latent variables are considered important to model the generation process of real world data , which has motivated research on Variational Autoencoders ( VAEs ) with discrete latents . However , standard VAE training is not possible in this case , which has motivated different strategies to manipulate discrete distributions in order to train discrete VAEs similarly to conventional ones . Here we ask if it is also possible to keep the discrete nature of the latents fully intact by applying a direct discrete optimization for the encoding model . The studied approach is consequently strongly diverting from standard VAE training by altogether sidestepping absolute standard VAE mechanisms such as sampling approximation , reparameterization trick and amortization . Discrete optimization is realized in a variational setting using truncated posteriors in conjunction with evolutionary algorithms ( using a recently suggested approach ) . For VAEs with binary latents , we first show how such a discrete variational method ( A ) ties into gradient ascent for network weights and ( B ) uses the decoder network to select latent states for training . More conventional amortized training is , as may be expected , more efficient than direct discrete optimization , and applicable to large neural networks . However , we here find direct optimization to be efficiently scalable to hundreds of latent variables using smaller networks . More importantly , we find the effectiveness of direct optimization to be highly competitive in ‘ zero-shot ’ learning ( where high effectiveness for small networks is required ) . In contrast to large supervised neural networks , the here investigated VAEs can , e.g. , denoise a single image without previous training on clean data and/or training on large image datasets . More generally , the studied approach shows that training of VAEs is indeed possible without sampling-based approximation and reparameterization , which may be interesting for the analysis of VAE-training in general . In the regime of few data , direct optimization , furthermore , makes VAEs competitive for denoising where they have previously been outperformed by non-generative approaches . 1 INTRODUCTION AND RELATED WORK . Variational autoencoders ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) are prominent and very actively researched models for unsupervised learning . VAEs , in their many different variations , have successfully been applied to a large number of tasks including semi-supervised learning ( e.g . Maaløe et al. , 2016 ) , anomaly detection ( e.g . An & Cho , 2015 ; Kiran et al. , 2018 ) , sentence interpolation ( Bowman et al. , 2016 ) , music interpolation ( Roberts et al. , 2018 ) and drug response prediction ( Rampasek et al. , 2017 ) . The success of VAEs rests on a series of methods that enable the derivation of scalable training algorithms to optimize their model parameters ( discussed further below ) . A desired feature when applying VAEs to a given problem is that their latent variables ( i.e. , the encoder output variables ) correspond to meaningful properties of the data , ideally to those latent causes that have originally generated the data . However , many real-world datasets suggest the use of discrete latents as they often describe the data generation process more naturally . For instance , the presence or absence of objects in images is best described by binary latents ( e.g . Jojic & Frey , 2001 ) . Discrete latents are also a popular choice in modeling sounds ; for instance , describing piano sounds may naturally involve binary latents : keys are pressed or not ( e.g . Titsias & Lázaro-Gredilla , 2011 ; Goodfellow et al. , 2013 ; Sheikh et al. , 2014 ) . The success of standard forms of VAEs has consequently spurred research on novel formulations that feature discrete latents ( e.g . Rolfe , 2016 ; Khoshaman & Amin , 2018 ; Roy et al. , 2018 ; Sadeghi et al. , 2019 ; Vahdat et al. , 2019 ) . The objective of VAE training is the optimization of a generative data model which parameterizes a given data distribution . Typically we seek model parameters Θ of a VAE that maximize the data log-likelihood , L ( Θ ) = ∑ n log ( pΘ ( ~x ( n ) ) ) , where we denote by ~x ( 1 : N ) a set of N observed data points , and where pΘ ( ~x ) denotes the modeled data distribution . Like conventional autoencoders ( e.g. , Bengio et al. , 2007 ) , VAEs use a deep neural network ( DNN ) to generate ( or decode ) observables ~x from a latent code ~z . Unlike conventional autoencoders , however , the generation of data ~x is not deterministic but it takes the form of a probabilistic generative model . For VAEs with binary latent variables , as they will be of interest here , we consider the following VAE generative model : pΘ ( ~z ) = Bern ( ~z ; ~π ) = ∏ h ( πzhh ( 1− πh ) ( 1−zh ) ) , pΘ ( ~x \|~z ) = N ( ~x ; ~µ ( ~z ; W ) , σ2I ) , ( 1 ) where ~z ∈ { 0 , 1 } H is a binary code and the non-linear function ~µ ( ~z ; W ) is a DNN that outputs the mean of the Gaussian distribution . pΘ ( ~x \|~z ) is commonly referred to as decoder . The set of model parameters is Θ = { ~π , W , σ2 } , where W incorporates DNN weights and biases . We assume homoscedasticity of the Gaussian distribution , but note that there is no obstacle to generalizing the model by inserting a DNN non-linearity that outputs a correlation matrix . Similarly , the algorithm could easily be generalized to different noise distributions should the task at hand call for it . For the purpose of this work , however , we will focus on as elementary as possible VAEs , with the form shown in Eqn . ( 1 ) . Given standard or binary-latent VAEs , essentially all learning algorithms seek to approximately maximize the log-likelihood using the following series of methods ( we elaborate in the appendix ) : ( A ) Instead of the log-likelihood , a variational lower-bound ( a.k.a . ELBO ) is optimized . ( B ) VAE posteriors are approximated by an encoding model , that is a specific distribution ( often Gaussian ) parameterized by one or more DNNs . ( C ) The variational parameters of the encoder are optimized using gradient ascent on the lower bound , where the gradient is evaluated based on sampling and reparameterization trick to obtain sufficiently low-variance and yet efficiently computable estimates . ( D ) Using samples from the encoder , the parameters of the decoder are optimized using gradient ascent on the variational lower bound . Optimization procedures for VAEs with discrete latents follow the same steps ( Points A to D ) . However , discrete or binary latents pose substantial further obstacles in learning , mainly due to the fact that backpropagation through discrete variables is generally not possible ( Rolfe , 2016 ; Bengio et al. , 2013 ) . In order to maintain the general VAE framework for encoder optimization , different groups have therefore suggested different possible solutions : work by Rolfe ( 2016 ) , for instance , extends VAEs with discrete latents by auxiliary continuous latents such that gradients can still be computed . Work on the concrete distribution ( Maddison et al. , 2016 ) or Gumbel-softmax distribution ( Jang et al. , 2016 ) proposes newly defined continuous distributions that contain discrete distributions as limit cases . Work by Lorberbom et al . ( 2019 ) merges the Gumbel-Max reparameterization with the use of direct loss minimization for gradient estimation , enabling efficient training on structured latent spaces . Finally , work by van den Oord et al . ( 2017 ) , and Roy et al . ( 2018 ) combines VAEs with a vector quantization ( VQ ) stage in the latent layer . Latents become discrete through quantization but gradients for learning are adapted from latent values before they are processed by the VQ stage . All methods have the goal of treating discrete distributions such that standard VAE training as developed for continuous latents can still be applied . These techniques interact during training with the standard methods ( Points A-D ) already in place for VAE optimization . Furthermore , they add further types of design decisions and hyper-parameters , for example parameters for annealing from softened discrete distributions to the ( hard ) original distributions for discrete latents . For discrete VAEs , it may consequently be a desirable goal to investigate alternative , more direct optimization procedures that do not require a softening of discrete distributions or the use of other indirect solutions . Such a direct approach is challenging , however , because once DNNs are used to define the encoding model ( Point B ) standard tricks to estimate gradients ( Point C ) seem unavoidable . A direct optimization procedure , as is investigated here , consequently has to substantially change VAE training . For the data model ( 1 ) we will maintain the variational setting and a decoding model with DNNs as non-linearity ( Points A and D ) . However , we will not use an encoder model parameterized by DNNs ( Point B ) . Instead , the variational bound will be increased w.r.t . the encoder model by using a discrete optimization approach . The procedure does not require gradients to be computed for the encoder such that discrete latents are addressed without the use of reparameterization trick and sampling approximations . 2 TRUNCATED VARIATIONAL OPTIMIZATION . Let us consider the variational lower bound of the likelihood . If we denote by q ( n ) Φ ( ~z ) the variational distributions with parameters Φ ( n ) , and by 〈 h ( ~z ) 〉 q ( n ) Φ = ∑ ~z q ( n ) Φ ( ~z ) h ( ~z ) expectation values w.r.t . to q ( n ) Φ ( ~z ) , then the lower bound can be written as : F ( Φ , Θ ) = ∑ n 〈 log ( pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ) 〉 q ( n ) Φ − ∑ n 〈 log ( q ( n ) Φ ( ~z ) ) 〉 q ( n ) Φ , ( 2 ) The general challenge for the maximization of F ( Φ , Θ ) is the optimization of the encoding model q ( n ) Φ . VAEs with discrete latents add to this challenge the problem of taking gradients w.r.t . discrete latents . If we seek to avoid derivatives w.r.t . discrete variables , we have to define an alternative encoding model q ( n ) Φ but such an encoding has to remain sufficiently efficient . Considering prior work on generative models with discrete latents , variational distributions based on truncated posteriors offer themselves as such an alternative ( Lücke & Sahani , 2008 ) . Truncated posterior approximations have been shown to be functionally competitive ( e.g . Sheikh et al. , 2014 ; Hughes & Sudderth , 2016 ; Shelton et al. , 2017 ) , and they are able to efficiently train also very large scale models with hundreds or thousands of latents ( e.g . Shelton et al. , 2011 ; Sheikh & Lücke , 2016 ; Forster & Lücke , 2018 ) . However , the important question for training discrete VAEs is if or how truncated variational distributions can be used in gradient-based optimization of neural network parameters . We here , for the first time , address this question noting that all previous approaches relied on closed-form ( or pseudo-closed form ) parameter update equations in an expectation-maximization learning paradigm . Optimization of the Decoding Model . In order to optimize the parameters W of the decoder DNN ~µ ( ~z , W ) , the gradient of the variational bound ( 2 ) w.r.t . W has to be computed . We consequently need , for any VAE , a sufficiently precise and efficient approximation of the expectation value w.r.t . the encoder q ( n ) Φ ( ~z ) . Gradient estimation is of central importance for deep unsupervised learning , and approaches , e.g. , for variance reduction of estimators have played an important role and are dedicated solely to this purpose ( e.g. , Williams , 1992 ) . Reparameterization finally emerged as a key method because it allowed for sufficiently low-variance estimation of gradients based , e.g. , on Gaussian middle-layer units ( Kingma & Welling , 2014 ; Rezende et al. , 2014 ) . For discrete VAEs , however , reparameterization requires the introduction of additional manipulations of discrete distributions that we here seek to fully avoid . Instead of using reparameterization or variance reduction , we will compute gradients based on truncated posterior as variational distributions . A truncated posterior has the following form : q ( n ) Φ ( ~z ) : = pΘ ( ~z \| ~x ( n ) ) ∑ ~z ′∈Φ ( n ) pΘ ( ~z ′ \| ~x ( n ) ) = pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ∑ ~z ′∈Φ ( n ) pΘ ( ~x ( n ) \|~z ′ ) pΘ ( ~z ′ ) if ~z ∈ Φ ( n ) , ( 3 ) where for all ~z 6∈ Φ ( n ) the probability q ( n ) Φ ( ~z ) equals zero . That is , a variational distribution q ( n ) Φ ( ~z ) is proportional to the true posteriors in a subset Φ ( n ) , which acts as its variational parameter . We can now compute the gradient of ( 2 ) w.r.t . the decoder weights W which results in : ~∇WF ( Φ , Θ ) = − 1 2σ2 ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ~∇W ‖~x ( n ) − ~µ ( ~z , W ) ‖2 . ( 4 ) The right-hand-side has salient similarities to the standard gradient ascent for VAE decoders . Especially the familiar gradient of the mean squared error ( MSE ) shows that , e.g. , standard automatic differentiation tools can be applied . However , the decisive difference are the weighting factors q ( n ) Φ ( ~z ) . Considering ( 3 ) , in order to compute the weighting factors we require all ~z ∈ Φ ( n ) to be passed through the decoder DNN . As all states of Φ ( n ) anyway have to be passed through the decoder for the MSE term of ( 4 ) , the overall computational complexity is not higher than an estimation of the gradient with samples instead of states in Φ ( n ) ( we elaborate in Appendix A ) . To complete the decoder optimization , update equations for variance σ2 and prior parameters ~π can be computed in closed-form ( compare , e.g. , Shelton et al. , 2011 ) and are given by : σ2 , new = 1DN ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ‖~x ( n ) − ~µ ( ~z , W ) ‖2 , ~πnew = 1 N ∑ n ∑ ~z∈Φ ( n ) q ( n ) Φ ( ~z ) ~z , ( 5 ) where N is the number of samples in the training dataset and D is the number of observables . Optimization of the Encoding Model . After having established that the decoder can be optimized efficiently and by using standard DNN methods , the important question is if the encoder can be trained efficiently . Encoder optimization is usually based on a reformulation of the variational bound ( 2 ) given by : F ( Φ , Θ ) = ∑ n 〈 log ( pΘ ( ~x ( n ) \|~z ) ) 〉 q ( n ) Φ − ∑ nDKL ( q ( n ) Φ ( ~z ) , pΘ ( ~z ) ) . ( 6 ) Centrally for this work , truncated posteriors allow a specific alternative reformulation of the bound that enables efficient optimization . The reformulation recombines the entropy term of the original form ( 2 ) with the first expectation value into a single term , and is given by ( see Lücke , 2019 , for details ) : F ( Φ , Θ ) = ∑ n log ( ∑ ~z∈Φ ( n ) pΘ ( ~x ( n ) \|~z ) pΘ ( ~z ) ) . ( 7 ) Thanks to the simplified form of the bound , the variational parameters Φ ( n ) of the encoding model can now be sought using direct discrete optimization procedures . More concretely , because of the specific form ( 7 ) , pairwise comparisons of joint probabilities are sufficient to maximize the lower bound : if we update the set Φ ( n ) for a given ~x ( n ) by replacing a state ~z old ∈ Φ ( n ) with a state ~z new 6∈ Φ ( n ) , then F ( Φ , Θ ) increases if and only if : log ( pΘ ( ~x ( n ) , ~z new ) ) > log ( pΘ ( ~x ( n ) , ~z old ) ) . ( 8 ) To obtain intuition for the pairwise comparison , consider its form when inserting the binary VAE ( 1 ) into the left- and right-hand sides . Eliding terms that do not depend on ~z we obtain : ˜log pΘ ( ~x , ~z ) = −‖~x− ~µ ( ~z , W ) ‖2 − 2σ2∑h π̃h zh ( 9 ) where π̃h = log ( ( 1 − πh ) /πh ) . The expression assumes an even more familiar form if we restrict ourselves for a moment to sparse priors π < 12 , i.e. , π̃ > 0 . Criterion ( 8 ) then becomes : ‖~x ( n ) − ~µ ( ~z new , W ) ‖2 + 2σ2π̃ \|~z new\| < ‖~x ( n ) − ~µ ( ~z old , W ) ‖2 + 2σ2π̃ \|~z old\| ( 10 ) where \|~z\| = ∑H h=1 zh . Such functions are routinely encountered in sparse coding or compressive sensing ( Eldar & Kutyniok , 2012 ) : for each set Φ ( n ) we seek those states ~z that are reconstructing ~x ( n ) well while being sparse ( ~z with few non-zero bits ) . For VAEs , ~µ ( ~z new , W ) is a DNN and as such much more flexible in matching the distribution of observables ~x than can be expected from linear mappings . Furthermore , criteria like ( 10 ) usually emerge for maximum a-posteriori ( MAP ) training in sparse coding ( Olshausen & Field , 1996 ) . In contrast , we here seek a population of states ~z in Φ ( n ) for each data point . It is a consequence of the reformulated lower bound ( 7 ) that it remains optimal to evaluate joint probabilities ( as for MAP ) although the constructed population of states Φ ( n ) can capture ( unlike MAP training ) a rich posterior structure . But how can new states ~z new that optimize Φ ( n ) be found efficiently in high-dimensional latent spaces ? Random search and search by sampling has recently been explored for elementary generative models ( Lücke et al. , 2018 ) . Here we will follow another recent suggestion ( Guiraud et al. , 2018 ) and make use of a search based on evolutionary algorithms ( EAs ) . In this setting we interpret sets Φ ( n ) as populations of binary genomes ~z and base the fitness function on Eqn . ( 9 ) . Concretely , using Φ ( n ) as initial parent pool , we apply the following genetic operators in sequence : firstly , parent selection picks Np states from the parent pool . In our numerical experiments we used fitness-proportional parent selection , for which we add an offset ( constant w.r.t . ~z ) to the fitness values in order to make them strictly non-negative . Each of the children undergoes mutation : one or more bits are flipped to further increase offspring diversity . In our experiments we perform random uniform selection of the bits to flip . Crossover could also be employed to increase offspring diversity . We repeat the procedure using the children generated this way as the parent pool , giving birth to multiple generations of candidate states . Finally , we update Φ ( n ) by substituting individuals with low fitness with candidates with higher fitness . The whole procedure can be seen as an evolutionary algorithm with perfect memory or very strong elitism ( individuals with higher fitness never drop out of the gene pool ) . Note that the improvement of the variational lower bound depends on generating as many as possible different children with high fitness over the course of training . We point out that the EAs optimize each Φ ( n ) independently , so this technique can be applied to large datasets in conjunction with stochastic or batch gradient descent on the model parameters Θ : it does not require to keep the full dataset ( or all sets Φ ( n ) ) in memory at a given time . Fig 1 shows how EAs produce new states that are used to update each set Φ ( n ) . The full training procedure for binary VAEs is summarized in Algorithm 1 .	This paper proposes a new approach to train VAEs with binary latents, using an evolutionary algorithm to optimise a discrete set of variational parameters rather than the usual amortised variational model trained with gradient-based methods. The authors consider the setting of training a VAE with discrete, Bernoulli distributed latents and a continuous, Gaussian distributed output. They use a novel approach to form and train a posterior on the discrete latent space, parameterising the posterior somewhat implicitly via the sets defining a truncated posterior - in which the approximate posterior is proportional to the true posterior, but only within a subset of all points. Defining the subset of points amounts to parameterising the approximate posterior. Optimisation of these parameters amounts to a discrete search problem, which the authors tackle using an evolutionary algorithm.	SP:f741d980c9c560a21298e947f1605dcbab7ceeac

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