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+# ViSER: Video-Specific Surface Embeddings for Articulated 3D Shape Reconstruction
+
+Gengshan Yang1 Deqing Sun2 Varun Jampani2 Daniel Vlasic2 Forrester Cole2 Ce Liu4∗ Deva Ramanan1,3
+
+1Carnegie Mellon University
+
+2Google Research 3Argo AI 4Microsoft Azure AI
+
+ViSER-webpage
+
+# Abstract
+
+We introduce ViSER, a method for recovering articulated 3D shapes and dense 3D trajectories from monocular videos. Previous work on high-quality reconstruction of dynamic 3D shapes typically relies on multiple synchronized cameras, strong category-specific priors, or 2D keypoint supervision. We show that none of these are required if one can reliably estimate long-range correspondences in a video, making use of only 2D object masks and two-frame optical flow as inputs. ViSER infers correspondences by matching 2D pixels to a canonical, deformable 3D mesh via video-specific surface embeddings that capture the view-independent appearance features of each surface point. These embeddings behave as a continuous set of keypoint descriptors defined over the mesh surface, which can be used to establish dense long-range correspondences across pixels. The surface embeddings are implemented as coordinate-based MLPs that are fit to each video via self-supervised losses. Experimental results show that ViSER compares favorably against prior work on challenging videos of humans with loose clothing and unusual poses as well as animal videos from DAVIS and YTVOS.
+
+# 1 Introduction
+
+Reconstructing the world from a sequence of monocular frames is a long-standing task in computer vision. While there has been tremendous progress in reconstructing rigid scenes (via SfM and SLAM [7, 39, 43], or recent techniques based on neural rendering [28]), reconstructing dynamic scenes with articulated objects remains elusive. For example, given a monocular video, it is still challenging to reconstruct an everyday scene of a moving person with loose clothing. In this work, we tackle the problem of estimating the deforming mesh of articulated objects given a segmented monocular video of that object. Our method avoids the use of any mesh templates or category-specific priors and generalizes to unknown deformable articulated objects in the wild.
+
+Nonrigid shape recovery is highly under-constrained due to fundamental ambiguities between shape, appearance, and time-varying deformation. Current approaches for addressing these challenges fall into two camps: better data “likelihoods” or better “priors”. The first camp extracts richer sensor data, via multi-camera studio setups [15] or depth sensors [30], but requires substantial efforts to work in the wild. The second camp makes use of category-level priors over object shapes [18, 20] and is particularly effective for human reconstruction. However, building such models requires considerable offline efforts in the form of registered 3D scans [26] or manual keypoint annotations [12], both of which are difficult to scale to arbitrary object categories.
+
+![](images/c5024c4b9aaa8a762cdc7d035eea0072af4cfb73583565fe4600659f978d57ee.jpg)  
+Figure 1: Given a long video (or multiple short videos), ViSER jointly learns articulated 3D shapes (represented as a mesh with vertices $\bar { \bf V }$ and faces $\mathbf { F }$ ) and joint pixel-surface embeddings (including a surface embedding $\mathbf { F _ { S } }$ and a pixel embedding $\mathbf { F _ { I } }$ ) that establishes dense long-range pixel correspondences over time. As a result, ViSER produces accurate shapes, long term trajectories and meaningful part segmentation.
+
+In this work, we use a practical but less explored variant of the data-likelihood camp: we use multiple frames of a video rather than multiple cameras or depth sensors. This considerably complicates analysis for dynamic, non-rigid scenes. Nonrigid structure-from-motion (NRSfM) [4, 38] attempts to constrain the problem by relying on motion correspondences such as 2D point tracks. While 2D correspondences over short time scales (i.e., optical flow) are relatively robust to extract, correspondences over long time scales are notoriously difficult to estimate because of appearance variations arising from viewpoint changes, occlusion and fast motion. In practice, this limits the applicability of NRSfM methods to controlled lab sequences.
+
+We propose ViSER (Video-Specific Surface Embeddings for Reconstruction), which establishes long-range correspondence and reconstructs articulated 3D shapes from a monocular video. Fig. 1 shows a sample outdoor video and the corresponding ViSER results. The key insight behind ViSER is to force long-range video pixel correspondences to be consistent with an underlying canonical 3D mesh through the use of video-specific embeddings that capture the pixel appearance of each surface point. These embeddings behave as a continuous set of keypoint descriptors defined over the surface mesh, learned with coordinate-based MLPs that are fit to each video via self-supervised losses. ViSER simultaneously optimizes the image CNN, surface MLP, and 3D shape so as to fit the observed video frames. It reconstructs state-of-the-art articulated 3D shape and 3D trajectories without using category-specific priors, making it easily scalable to diverse videos including humans with challenging clothing and poses as well as animals. Lastly, we demonstrate that ViSER recovers meaningful part segmentation and blend skinning weights from videos, which typically require considerable manual effort from 3D artists.
+
+# 2 Related Work
+
+Low-level correspondence. Optical flow is a well-studied representation for short-term correspondence between adjacent frames of a video. After decades of research, recent CNN models [40, 42, 48] for optical flow have achieved an impressive level of accuracy as evidenced by the Sintel and KITTI benchmarks [5, 9]. However, it is challenging to concatenate optical flow for reliable long-range correspondence due to occlusions and strong appearance changes [33, 37, 41]. ViSER does not concatenate optical flow but use it as a constraint to establishes long-range correspondence.
+
+The layered approach [6, 14, 45] segments a video into different moving objects with coherent motion, thereby establishing long-range correspondence for every frames through the shared layers. Early layered methods assume parameter motion for each layer and can only handle limited scenes. Unwrap Mosiacs [32] uses a dense 2D-to-2D mapping from a texture map to every input frame, and editing operations on the texture map naturally transfers to each individual frame. However, the 2D representation cannot flexibly model complex 3D phenomena, such as occlusions.
+
+![](images/85a0b0d499cfbf36b2e52416eddbbd27a71c5a591aa10f920a05d7b53bc2d6ab.jpg)  
+Figure 2: We learn a joint pixel-surface embedding space for dense correspondence between pixels in video frames $I _ { t }$ and points on a canonical 3D surface $( \bar { \mathbf { V } } , \mathbf { \check { F } } )$ . Such embedding space is optimized through “top-down” differentiable rendering $\mathcal { R } ( \cdot )$ and “bottom-up” correspondence matching $\hat { \bf S } [ x , y ]$ (Sec 3.2). We introduce a 3D matching loss to optimize the embeddings, where the matched surface locations are encouraged to be close to the rendered surface locations. The embedding further enables articulated shape optimization through a 2D-3D-2D cycle reprojection: pixel $[ x , y ] $ matched surface ${ \hat { \mathbf { S } } } [ x , y ] \to$ re-projected pixel $\pi ( \hat { \mathbf { S } } [ x , y ] )$ (Sec. 3.3).
+
+Dense pose and surface mappings. DensePose [12] directly maps pixels to the 3D surface of a human body model. It requires large amounts of training data with annotated image-to-surface correspondence and is hard to generalize to other categories. Articulation-aware Canonical Surface Mapping (A-CSM) [20] uses geometric cycle consistency for learning to map pixels to corresponding points on a template shape without using keypoint annotations. However, it requires a pre-defined template shape for each category. Continuous Surface Embeddings (CSE) [29] establishes dense correspondences between image pixels and 3D object geometry by predicting an embedding vector of the corresponding vertex in the object mesh for each pixel in a 2D image. While applicable to multiple categories, CSE requires annotations and only applies to categories in the training set. ViSER requires neither a template shape nor annotations to work on categories in the wild.
+
+Nonrigid shape reconstruction. One way to accurately reconstruct articulated shapes is to rely on rich sensor data, e.g., multi-view [15] or depth sensors [30], which requires substantial efforts to setup and reconstruct objects in the wild. For monocular videos/images, one popular approach is to adopt strong 3D shape and pose priors [18, 26, 35, 36, 53, 54] but it works well only on limited categories, whose 3D data are easy to collect. To deal with more nonrigid object categories, a recent trend is to learn a category-level 3D shape model from a collection of images or videos with 2D annotations, such as keypoints and object silhouettes [10, 16, 20, 22, 23, 44, 46, 50]. Although they are able to reconstruct more object categories, such as birds and quadruped animals, the reconstruction usually lacks details, and the level of deformation recovered tends to be low.
+
+Category-agnostic methods, such as nonrigid structure from motion (NRSfM) methods [4, 11, 19, 38] reconstruct nonrigid 3D shapes from a set of 2D point trajectories. However, due to the difficulty in obtaining accurate long-range correspondences [37, 41] they do not work well for videos in the wild. A recent work, LASR [49], uses two-frame optical flow to reconstruct articulate shapes from a monocular video with differentiable rendering. Despite the promising results, LASR does not reason about long-range correspondences and can only reliably reconstruct what is visible in a short video. ViSER establishes reliable long-range correspondence that are robust to moderate shape variations and appearance changes. Thus, ViSER can obtain much higher-quality reconstruction by using either a long video or several videos of a category.
+
+# 3 Approach
+
+Fig. 2 provides an overview of our approach, which follows a typical framework of differentiable rendering [16, 25]. Borrowing the notation from LASR [49], we formalize our task as follows. Given a set of video observations including RGB pixel color, segmentation masks, and optical flow estimates $\{ I _ { t } , S _ { t } , u _ { t } \} _ { t = \{ 0 , \ldots , T \} }$ , our goal is to recover a set of shape and motion parameters $\{ \mathbf { S } , \mathbf { D } _ { t } \}$ that produce reconstructions $\{ \hat { I } _ { t } , \hat { S } _ { t } , \hat { u } _ { t } \} _ { t = \{ 0 , \dots , T \} }$ that match the video observations. We refer to supplementary material for a complete list of notations defined in the paper.
+
+# 3.1 Preliminaries
+
+We represent an object’s shape as a triangular mesh $\mathbf { S } = \{ \bar { \mathbf { V } } , \mathbf { F } \}$ with canonical vertices $\bar { \mathbf { V } } \in \mathbb { R } ^ { 3 \times N }$ and a fixed topology (edge connectivity) $\mathbf { \bar { F } } \in \mathbb { R } ^ { 3 \times M }$ . To render an object, we displace mesh vertices with motion parameters $\mathbf { D } _ { t }$ , apply a perspective projection with camera intrinsics $\mathbf { K } _ { t }$ , and rasterize.
+
+We model vertex motion with root body transformations $\mathbf { G } _ { 0 }$ and object articulations $\{ \mathbf { G _ { 1 } } , \cdots , \mathbf { G _ { B } } \}$ using linear blend skinning (LBS) [20, 21]. LBS constrains vertex motion by linearly blending $B$ rigid “bone” transformations with a skinning weight matrix $\mathbf { W } \in \mathbb { R } ^ { B \times N }$ , transforming the canonical shape into frame $t$ as
+
+$$
+\mathbf { V } _ { i , t } = \mathbf { G _ { 0 , t } } \left( \sum _ { b } \mathbf { W } _ { b , i } \mathbf { G } _ { b , t } \right) \bar { \mathbf { V } } _ { i }
+$$
+
+where $i$ is the vertex index, $b$ is the bone index. Similar to LASR, the root body and bone transformations are represented as the outputs of a pose CNN given an input image, $( \mathbf { G _ { 0 } } , \cdot \cdot \cdot , \mathbf { G _ { B } } ) = \psi _ { p } ( I _ { t } )$ .
+
+We define a set of surface properties for rendering, including vertex 3D coordinates, textures and features, and rasterize them in a differentiable manner [25]. We denote the differentiable rendering function that renders the property $\mathbf { C }$ defined on a canonical surface to an image as $\mathcal { R } ( \mathbf { C } ; \mathbf { V } , \mathbf { W } , \mathbf { G } )$ , which executes the blending skinning function in Eq. (1) and softly blends the surface property based on their depth and barycentric coordinates [25]. For simplicity, we omit the shape, skinning, and motion parameters parameters and write the differentiable rendering function as $\mathcal { R } ( \mathbf { C } )$ . To render optical flow, we rasterize and project vertex coordinates in two consecutive frames and compute their 2D displacements [49]. Such renderings are compared against video observations to compute gradients for updating model parameters.
+
+# 3.2 Video-specific Surface Embedding
+
+Pixel-surface embeddings. We learn pixel and surface embeddings that map corresponding pixels in different frames to the same point on a canonical 3D surface. Intuitively, consider a particular region on the canonical surface mesh that is the “nose” of an articulated human. The surface embedding captures a descriptor for the nose, which can then be matched to pixel-level descriptors at each frame.
+
+Given an input image $I _ { t }$ , the pixel-wise descriptor embedding is computed by a U-Net [34] encoder:
+
+$$
+\mathbf { F _ { I } } [ x , y , t ] = \psi _ { e } ( I _ { t } ) [ x , y ] \in \mathbb { R } ^ { 1 6 } ,
+$$
+
+where $[ x , y , t ]$ are pixel locations at frame $t$ . The surface embedding is computed by a positionencoded MLP:
+
+$$
+\mathbf { F _ { S } } ( X , Y , Z ) = \phi _ { e } ( X , Y , Z ) \in \mathbb { R } ^ { 1 6 } ,
+$$
+
+where $\phi _ { e } ( \cdot )$ is an MLP defined over 3D points $( X , Y , Z )$ in the canonical space, augmented with Fourier positional encoding [28]. The two embeddings are optimized on test videos such that pixels representing the same surface location in different frames are mapped to the same canonical surface point [20].
+
+Correspondence via soft-argmax regression. Given the pixel and surface embeddings, we construct a per-frame cost volume $D ( \mathbf { F _ { I } } , \mathbf { F _ { S } } )$ of size $H \times W \times N _ { s }$ over pixels and surface points (we randomly sample $N _ { s } = 2 0 0$ surface points at each step) by considering their cosine feature distances,
+
+$$
+D ( \mathbf { F } _ { \mathbf { I } } , \mathbf { F } _ { \mathbf { S } } ) [ x , y , i ] = 1 - \cos \big ( \mathbf { F } _ { \mathbf { I } } [ x , y ] , \mathbf { F } _ { \mathbf { S } } ( X _ { i } , Y _ { i } , Z _ { i } ) \big ) .
+$$
+
+Normalizing the cost volume over the surface point dimension yields a softmax “heatmap” over surface points that potentially match to pixel $( x , y )$ , as shown in Fig. 3 (Left):
+
+$$
+\sigma _ { \left( x , y \right) } [ i ] = \frac { e ^ { - D [ x , y , i ] / \tau } } { \sum _ { j } e ^ { - D [ x , y , j ] / \tau } } ,
+$$
+
+where $\tau$ is a temperature scaling parameter that is jointly optimized with the feature embeddings. To output a single surface point for pixel $( x , y )$ , we can compute a “soft” argmax [17, 48] by taking the expectation of the softmax distribution over the 3D locations of the points samples,
+
+$$
+\hat { \mathbf { S } } [ x , y ] = \sum _ { i } \sigma _ { ( x , y ) } [ i ] ( X _ { i } , Y _ { i } , Z _ { i } ) ,
+$$
+
+where $( X _ { i } , Y _ { i } , Z _ { i } )$ is the i-th sampled surface point and $\sigma _ { ( } x , y ) [ i ]$ is the matching probability of pixel $( x , y )$ over the sampled points $i \in \{ 1 , 2 , \ldots , N _ { s } \}$ .
+
+![](images/2ebf45b5b43939195594f2e62ce87906cefbbdcdadb94eac45fe28f932ffe5f0.jpg)  
+Figure 3: Pixel-surface embeddings establish a continuous mapping between pixels and points on a canonical surface. Left: Given a query pixel at $( \mathbf { x } , \mathbf { y } )$ , we match it to a set of canonical surface points, where the matching distribution is used to regress a continuous mapping to the canonical surface. Right: Given a query surface point (X,Y,Z), a matching distribution over pixels can be computed. Warm color indicates high matching probability.
+
+We can also normalize the $H \times W \times N _ { s }$ cost volume over spatial positions to capture a distribution of pixel locations that match to each surface point $( X _ { i } , Y _ { i } , Z _ { i } )$ :
+
+$$
+\sigma _ { ( X _ { i } , Y _ { i } , Z _ { i } ) } [ x , y ] = \frac { e ^ { - D [ x , y , i ] / \tau } } { \sum _ { [ x , y ] } e ^ { - D [ x , y , j ] / \tau } } .
+$$
+
+and compute a similar soft argmax mapping of surface points to pixels, as shown in Fig. 3 (Right).
+
+Relation to keypoints. The output of classic keypoint detectors are often represented as $K$ -channel heatmaps over the pixel grid, where $K$ is the number of keypoints. To define dense keypoints, one may increase the number of channels, which is computationally heavy. Similar to CSE [29], we represent dense keypoints as low-dimensional pixel-surface embeddings, which establishes a mapping between pixels and a canonical 3D surface, but far more efficiently. DensePose [12] and CSM [20] use an alternative pixel-to-surface mapping that regresses a surface coordinate at every pixel. In contrast, our pixel-surface embedding captures multimodal uncertainties over keypoints; for example, $\sigma _ { ( x , y ) } [ i ]$ can capture the fact that a particular pixel matches well to both the left and right ankle, as visualized in Fig. 3, while a regressor may “regress” to the mean of the two surface coordinates.
+
+# 3.3 Learning Embeddings and Articulated Shapes
+
+Next we will introduce the loss functions that enable learning both embeddings and articulated shapes from monocular videos without a pre-defined shape template or annotated correspondence. To learn non-degenerate embeddings and overcome the local optima issue in differertiable renderers, we carefully construct a 3D matching loss and a 2D cycle loss.
+
+3D match loss. Arguably, the simplest loss to learn embeddings is to minimize the difference between the rendered surface features and the observed pixel features:
+
+$$
+L _ { \mathrm { f e a t u r e - c o n s i s t e n c y } } = \sum _ { x , y } { \Big ( } 1 - \cos ( \mathcal { R } ( \mathbf { F } _ { \mathbf { S } } ) [ x , y ] , \mathbf { F } _ { \mathbf { I } } [ x , y ] ) { \Big ) } ,
+$$
+
+where $\cos ( \cdot )$ denotes the inner product between two normalized vectors, and $\mathcal { R } ( \mathbf { F _ { S } } )$ is the differentiably rendered surface descriptors. However, the feature consistency loss admits a trivial solution, where all pixel and surface features are the same constant (yielding zero error). To address this, we introduce a 3D matching loss that ensures pixel embeddings only match to surface embeddings rendered at the pixel location:
+
+$$
+L _ { \mathrm { m a t c h i n g } } = \sum _ { x , y } \left. \mathcal { R } ( \bar { \mathbf { V } } ) [ x , y ] - \hat { \mathbf { S } } [ x , y ] \right. _ { 2 }
+$$
+
+where $\mathcal { R } ( \bar { \bf V } )$ is the rendered 3D surface location and $\hat { \mathbf { S } } [ x , y ]$ is the estimated pixel-to-surface mapping from Eq. (6), computed through sampling and computing the softmax distributions $\sigma [ i ]$ over surface points [17]. To minimize the loss, the embeddings of surface points that do not project to $( x , y )$ will be pulled away from the pixel embedding of $( x , y )$ in a contrastive way [13].
+
+2D cycle loss. The match loss aims to learn pixel-surface embeddings that are consistent over video frames and discriminative over difference surface locations. However for articulation optimization, the match loss suffers from bad local optima issue similar to other losses based on differentiable rendering [25]. For instance, when the rendering of a body part is outside the ground-truth object silhouette, a gradient update of articulation parameters would likely not incur a lower loss.
+
+To guide articulated 3D shape learning using the learned pixel-surface embeddings, we further define a cycle-based re-projection loss, inspired by prior approaches in 3D model fitting with keypoints [3] and canonical surface mappings [20]. Given an input image, we establish a 2D-3D mapping by extracting a pixel embedding and matching it to surface embedding. Then, we compute the expected surface coordinate $\hat { \mathbf { S } } [ x , y ]$ at every pixel using Eq. (6), and ensure the differentiably rendered canonical surface coordinate lands back on the original pixel coordinate $( x , y )$ ,
+
+$$
+L _ { \mathrm { r e p r o j } } = \sum _ { x , y } \left\| \mathcal { R } ( \hat { \mathbf { S } } [ x , y ] ) - ( x , y ) \right\| _ { 2 } .
+$$
+
+Reconstruction loss. Finally, we make use of reconstruction losses to ensure that generated images, masks, and flows match their estimated counterparts:
+
+$$
+L _ { \mathrm { r e c o n } } = \beta _ { 1 } \lvert | \hat { S } _ { t } ^ { i } - S _ { t } \rvert | _ { 2 } ^ { 2 } + \beta _ { 2 } \lvert | \hat { I } _ { t } ^ { i } - I _ { t } \rvert | _ { 2 } ^ { 2 } + \beta _ { 3 } \sigma _ { t } \lvert | \hat { u } _ { t } ^ { i } - u _ { t } \rvert | _ { 2 } + \beta _ { 4 } \mathrm { p d i s t } ( \hat { I } _ { t } , I _ { t } )
+$$
+
+where $\{ \beta _ { 1 } , \cdot \cdot \cdot , \beta _ { 4 } \}$ are weights empirically chosen, $\sigma _ { t }$ is the normalized confidence map for flow measurement, and pdist $( \cdot , \cdot )$ is the perceptual distance [51] measured by an ImageNet-pretrained AlexNet. The reconstruction losses ensure the match between rendered and observed optical flow, texture and silhouette images.
+
+Regularization. To avoid degenerate shapes, we use mesh Laplacian regularization [16, 49] to enforce the recovered shape to be smooth, and as-rigid-as-possible (ARAP) regularization to enforce the deformation to be locally rigid [44]. Different from prior work that only preserves the length of edges after articulation, we encourage both the area and length of faces to be the same after articulation. The area preserving term is defined as
+
+$$
+L _ { \mathrm { A R A P - a r e a } } = \sum _ { i = 1 } ^ { | E | } \sum _ { j \in N _ { i } } \mid \left| \mathbf { E _ { i } ^ { t } } \times \mathbf { E _ { j } ^ { t } } \right| - \left| \mathbf { E _ { i } ^ { t + 1 } } \times \mathbf { E _ { j } ^ { t + 1 } } \right| \mid ,
+$$
+
+where $| E |$ is the number of edges and $N _ { i }$ the indices of neighbouring edges.
+
+# 3.4 Representing Surface Properties with MLPs
+
+By extending surface embedding MLPs with additional dimensions, we can model other surface properties including textures and even surface-based geometric deformations. Compared to explicitly defined textures, such continuous implicit representations have the capacity to encode arbitrary amount of details and are empirically easier to optimize.
+
+Surface appearance. The appearance of the object is represented by a coordinate-MLP queried at points on the canonical mesh surface. To handle view-dependent appearance (such as shadow and lighting), we further concatenate the Fourier features of the $( X , Y , Z )$ coordinates with a frame appearance code, as the input to the texture MLP,
+
+$$
+\mathbf { C _ { i , t } } = \phi _ { t e x } ( \mathcal { F } ( \bar { \mathbf { V } } _ { i } ) , \omega _ { t } ) \in \mathbb { R } ^ { 3 } ,
+$$
+
+where $\bar { \bf V } _ { i }$ is the i-th canonical mesh vertex, which is passed through a Fourier encoder $\mathcal F ( \cdot )$ as used in NeRF [28], and concatenated with $\omega _ { t }$ , a 64-dimensional frame appearance code associated each image frame $t$ , predicted from a ResNet-18, as $\omega _ { t } = \psi _ { t e x } ( I _ { t } )$ .
+
+Instance shape deformation fields. To deal with videos of multiple instances of the same category, as experiments in Sec. 4.3, we model shape variations across instances by a continuous surface deformation field defined on the canonical surface. Similar to the surface texture, we represent the surface deformation field by a shape MLP,
+
+$$
+\mathbf { V _ { i , k } } = \bar { \mathbf { V _ { i } } } + \phi _ { s h a p e } ( \mathcal { F } ( \bar { \mathbf { V } } _ { i } ) , \alpha _ { k } ) \in \mathbb { R } ^ { 3 } ,
+$$
+
+where $\mathbf { V _ { k } }$ is the rest shape of instance $k$ and $\alpha _ { k }$ is a video-specific 64-dimensional shape code that is randomly initialized and optimized together with the shape MLP.
+
+# 4 Experiments
+
+We evaluate ViSER in three different scenarios where objects are highly articulating, making it challenging to reconstruct and estimate long-range correspondences. First, we consider long human videos with loose clothing and unusual poses. Next, we evaluate on videos of articulated animals for which accurate shape templates are missing. Finally, we analyze a multi-video variant of ViSER that learns a single model from multiple videos of the same category. All scenarios require jointly establishing long-range correspondences and reconstructing articulated 3D shapes at the same time.
+
+Table 1: 2D Keypoint transfer accuracy on athletic videos. Methods with ∗ use keypoint annotations to train. Best results are in bold.   
+
+<table><tr><td>Method</td><td>break-1</td><td>break-2</td><td>dance</td><td>parkour</td><td>ballet-1</td><td>ballet-2</td><td>ballet-3</td><td>Ave.</td></tr><tr><td>*DensePose CSE [29]</td><td>56.0</td><td>13.2</td><td>77.2</td><td>85.9</td><td>45.6</td><td>49.0</td><td>64.5</td><td>55.9</td></tr><tr><td>*VIBE+SMPLify [18]</td><td>37.1</td><td>8.2</td><td>70.4</td><td>83.8</td><td>55.4</td><td>53.0</td><td>78.8</td><td>55.2</td></tr><tr><td>LASR [49]</td><td>29.1</td><td>18.1</td><td>56.6</td><td>49.8</td><td>44.5</td><td>47.4</td><td>48.6</td><td>42.0</td></tr><tr><td>ViSER (Ours)</td><td>70.5</td><td>22.5</td><td>80.7</td><td>62.9</td><td>52.7</td><td>56.1</td><td>59.9</td><td>57.9</td></tr></table>
+
+Table 2: 2D Keypoint transfer accuracy on multiple elephant videos. Best results are in bold.   
+
+<table><tr><td>Method</td><td>inner</td><td>across</td></tr><tr><td>CSE [29]</td><td>55.7</td><td>52.2</td></tr><tr><td>Flow-VCN[48]</td><td>51.1</td><td>41.2</td></tr><tr><td>LASR [49]</td><td>57.8</td><td>-</td></tr><tr><td>ViSER (Ours)</td><td>80.4</td><td>68.9</td></tr></table>
+
+Table 3: 2D Keypoint transfer accuracy on BADJA dataset. Best results are in bold.   
+
+<table><tr><td>Method</td><td>camel</td><td>dog</td><td>cows</td><td>horse</td><td>bear</td><td>Ave.</td></tr><tr><td>CSE [29]</td><td>48.8</td><td>38.6</td><td>63.8</td><td>60.2</td><td>76.6</td><td>57.6</td></tr><tr><td>Flow-VCN [48]</td><td>47.9</td><td>25.7</td><td>60.7</td><td>14.4</td><td>63.8</td><td>42.5</td></tr><tr><td>N-NRSfM[38]</td><td>67.8</td><td>17.9</td><td>70.0</td><td>8.7</td><td>60.2</td><td>44.9</td></tr><tr><td>LASR [49]</td><td>81.9</td><td>65.8</td><td>83.7</td><td>49.3</td><td>85.1</td><td>73.2</td></tr><tr><td>ViSER (Ours)</td><td>80.1</td><td>73.8</td><td>82.9</td><td>76.3</td><td>87.3</td><td>80.1</td></tr></table>
+
+Optimization details We use the AdamW [27] optimizer with a batch of 4 consecutive image pairs. We reconstruct a long video sequence in an incremental manner similar to classic SfM. First, we use an initial set of around 20 consecutive frames to initialize the shape and pixel surface embeddings. The initial set is selected such that the viewpoint coverage is large enough. Then we gradually add in new frames. When a new frame is added, we first apply the 2D cycle loss $L _ { r e p r o j }$ to optimize its articulations, and then jointly optimize all frames with all losses. Empirically, simultaneously optimizing all the video frames produces unstable results of root body poses (or equivalently camera poses).
+
+# 4.1 Athletic Video Reconstruction
+
+Dataset. To evaluate ViSER on long-videos, we construct an athletic video dataset that is challenging due to loose clothing and unusual body poses. It consists of four videos from DAVIS [31] and three ballet videos. All videos are segmented and manually annotated with keypoints following the MSCOCO format [24]. We only use keypoint annotations for evaluation purposes.
+
+Metrics. Due to the lack of ground-truth 3D data for challenging athletic human videos, we use 2D keypoint transfer as a proxy metric [2, 52]. Given any two frames from a video, the goal is to transfer an annotated 2D keypoint from one frame to another. The accuracy is measured by percentage of correctly transferred keypoints over all T(T-1) pairs of frames in a T-frame video. A transferred keypoint is marked as correct when its distance to the ground-truth annotation is lower than $d _ { t h } = 0 . 2 \sqrt { | \boldsymbol { S } | }$ , where $| S |$ is the area of the ground-truth silhouette [2]. In general, a more accurate reconstruction leads to a higher transfer accuracy.
+
+Baselines. To compare with template-based approaches for video human reconstruction, we use VIBE with SMPLify temporal smoothing [18]. To compare with template-free methods, we use LASR [49], which also reconstructs articulated shapes using the same input setting as ours. To transfer keypoints from a reference frame to a target frame, we back-project the annotated keypoint in the reference frame to the canonical surface, and then project the intersected 3D point to the target frame. We also compare against Densepose CSE [29], which produces dense pixel-to-surface correspondences for a given category, but does not produce 3D reconstructions. To transfer keypoints for Densepose CSE, we compute pixelwise surface mappings for both frames and find the best matching w.r.t.geodesic distance on the surface. We further qualitatively compare against a state-ofthe-art human reconstruction method, PiFUHD [36] in Fig. 4, which only produces reconstruction, but not correspondence.
+
+![](images/fbe091753c2f0495ced8cb953c6fbf2f59e8ee2d1da65f5d598e70f3690e086d.jpg)  
+Figure 4: Qualitative comparisons for athletic video articulated shape reconstruction. Compared to methods that uses shape and pose priors (VIBE $^ +$ SMPLify and PiFUHD), our method achieves comparable performance for common appearance and poses, and does much better on unusual poses such as break-dancers.
+
+![](images/166855c4415d8702b655a7dc42e4809baf2c3b46ebfba6bba2cf6a0b7a6990f9.jpg)  
+Figure 5: Qualitative comparisons for elephant shape reconstruction from multiple videos. Notice that ViSER is able to take advantage of multiple videos to improve the category-level shape reconstruction but also reconstruct instance-specific details (as shown in red circles).
+
+![](images/16aec4a0675c4cd1f54df8a58f8265aa7b4db992bddc92c4e4686871ea10a196.jpg)  
+Figure 6: Part segmentation results. Colors are determined by hard-assigning vertices to the closest rigid bones.
+
+![](images/8d21ac3a175e69d94bbae955b587e18f58587f3cb36c3a814a1ad4d8dd719e12.jpg)  
+Figure 7: Comparison between single video ViSER and multi-video ViSER in terms of reconstructing YTVOS elephants. We find using multiple videos helps reconstructing the body parts that may be occluded in a single video. While single-video ViSER reconstructs a flattened shape and misses the hidden rear leg of the elephant, multiple video ViSER reconstructs a more plausible shape and recovers both the two rear limbs.
+
+Results. Fig. 4 shows visual reconstruction results on sample videos and for different techniques. ViSER estimates reconstructions that are more faithful to the input than the baselines, especially when the humans have unusual poses like in the first two rows. The accurate long-range correspondence enables ViSER to reconstruct finer details than LASR that does not explicitly try to estimate longrange correspondences. We summarize quantitative comparisons in Tab. 1. There is a moderate performance gap between ViSER and template-based methods when the input fits the latter, such as parkour with tight clothing and usual pose. Note that the supervised Densepose CSE and OpenPose methods fail on breakdance videos due to the novel pose, and also do not work well on ballet dancers due to loose clothing. As a result, template-based approaches that rely on accurate pose recognition, such as VIBE [18] fails. In contrast. our method does not suffer from such poor out-of-distribution generalization. By establishing long-range correspondences, ViSER achieves higher keypoint transfer accuracy and better 3D reconstruction than LASR.
+
+# 4.2 Reconstructing Animals from a Video
+
+We use BADJA [2] to evaluate ViSER on animal videos including camel, cow, dog, bear and horse. Similar to the athletic human video dataset, we compare against template-free methods such as LASR and neural-dense-NRSfM (N-NRSfM) [38]. Similar to LASR and our setup, N-NRSfM learns a video-specific model for object shape, deformation and camera parameters from multi-frame optical flow estimations [8]. We further report performance comparison with dense correspondence methods such as CSE and an optical flow method, VCN. We use the CSE model trained on corresponding animal categories (except that we use the horse model for camel), and the “robust” model of VCN [1], which is the input to our method. As shown in Tab. 3, ViSER achieves better or similar accuracy on all five animal videos compared to LASR and N-NRSfM. While the input optical flow is not robust at estimating long-range correspondences, our method integrates local optical flow to a dense long-range correspondences via a canonical shape, and achieves much better keypoint transfer accuracy. Note that CSE performs well for categories it has been trained on, such as cow, horse and bear, but performs poorly on novel animal categories, such as camel and a novel breed of dog.
+
+# 4.3 Multi-video Shape and Correspondence
+
+We curate a set of seven videos of different elephants from YTVOS [47] for multi-video shape and correspondence recovery. The annotations will be released for further research. We treat multiple videos as a single long video with strong appearance changes and shape variations. In the multi-video setup, We evaluate keypoint transfer accuracy on both the same instance (with video frames) and over different instances (across video frames), as denoted by “inner” and “across”. Quantitative results in Tab. 2 shows that ViSER is more accurate than the baseline methods in both cross-video keypoint transfer and inner-video keypoint transfer by a large margin, without using any keypoint annotations or pre-defined shape templates. Fig. 5 show visual result comparisons. While LASR recovers the visible surfaces in a video, it cannot infer the invisible parts. In contrast, our method is able to take advantage of multiple videos from the same category and produce a much better shape reconstruction. Note that LASR cannot handle multiple videos as it requires optical flow computed between every adjacent frame pairs. ViSER, on the other hand, also uses correspondences via estimated 3D shape, thereby allowing the use of multiple videos even when the optical flow is missing across videos.
+
+Table 4: Ablation study on keypoint transfer. Best results are in bold.   
+
+<table><tr><td>Method</td><td>break-1</td><td>elephants-inner</td><td>elephants-cross</td></tr><tr><td>Full</td><td>70.5</td><td>80.4</td><td>68.9</td></tr><tr><td>w/o matching loss Lmatching, Eq. (9)</td><td>36.2</td><td>51.3</td><td>42.6</td></tr><tr><td>w/o reprojection loss Lreproj, Eq. (10)</td><td>38.3</td><td>80.1</td><td>62.5</td></tr><tr><td>CSM regression [20]</td><td>47.1</td><td>77.4</td><td>63.3</td></tr></table>
+
+Benefit of Using Multiple Videos. To examine the benefits of using multiple videos, we further compare multi-video ViSER with single-video ViSER, as shown in Fig. 7. We find using multiple videos helps reconstructing the body parts that may be occluded in a single video.
+
+# 4.4 Part Discovery and Ablations
+
+Part discovery. ViSER can discover detailed 3D part segmentation without any manual annotation, as shown in Fig. 6. After training either on a collection of videos or a long video, ViSER can segment the 3D shape into meaningful parts, such as the trunk of the elephants and the feet of the dancer.
+
+Ablation study. We perform an ablation study on break-1 and elephants, as shown in Tab. 4. Without the contrastive matching loss, the pixel-surface embedding converges to a trivial solution with a significant decrease of accuracy. Removing the re-projection loss leads to much lower keypoint transfer (KPT) accuracy on the breakdance-1 sequence and cross-video KPT accuracy on the elephant videos. Likely the surface reprojection loss plays an important role in learning correct articulation that follows the bottom-up dense keypoint predictions. This may effectively avoid the local minimum issue for the differentiable rendering optimization. Finally, replacing the pixel-surface embedding with direct CSM regression [20] does not reason about distribution of possible matches and results in worse performance.
+
+Limitations. We find ViSER to be sensitive to the random initialization of network parameters. We run optimization with different random seeds for initializing the network parameters and find some perform considerably worse than the others, due to the convergence to bad local optima. Although in practice, one could spot the convergence to a bad local optimum by visualizing the articulated shapes and re-run the optimization with a different random seed, an automatic method for selecting the best model parameters over different trials is desired. We leave how to make the optimization of ViSER robust for future research.
+
+ViSER also relies on optical flow to kick-start with a reasonable initial shape and pose for learning pixel-surface embeddings. Although recent optical flow models generalize well in many scenarios, they may fail when a video is of low resolution or contains significant motion blur. In such challenging cases, using category shape and pose priors to initialize ViSER would be a promising direction.
+
+# 5 Conclusions
+
+We have introduced ViSER, a method to reconstruct articulate shapes, dense trajectories, and object parts from monocular videos. ViSER establishes long-range correspondence by matching 2D pixels to a canonical 3D mesh via learned video-specific surface embeddings. Experimental results show that ViSER, without a template shape or keypoint annotations, compares favorably against prior work on challenging human and animal videos. ViSER shows that it could be fruitful to reconstruct articulate shapes for categories in the wild, and we hope to see more work in this direction.
+
+Broader impact. ViSER has many potential applications, e.g., in robotics, AR/VR, and film industry, but may be used for malicious purposes, e.g., producing fake videos or extracting bio-metric information without prior consent. ViSER is only suitable to offline applications as it takes about several hours to process a 80-frame video on one NVIDIA P100 GPU.
+
+# Acknowledgments
+
+This work was supported by Google Cloud Platform (GCP) awards received from Google and the CMU Argo AI Center for Autonomous Vehicle Research. We thank William T. Freeman and many others from CMU and Google for valuable feedback.
+
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parse/train/B1lxH20qtX/B1lxH20qtX.md

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+# LEARNING TO CONTROL SELF-ASSEMBLING MORPHOLOGIES: A STUDY OF GENERALIZATION VIA MODULARITY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Much of contemporary sensorimotor learning assumes that one is already given a complex agent (e.g., a robotic arm) and the goal is to learn to control it. In contrast, this paper investigates a modular co-evolution strategy: a collection of primitive agents learns to self-assemble into increasingly complex collectives in order to solve control tasks. Each primitive agent consists of a limb and a neural controller. Limbs may choose to link up to form collectives, with linking being treated as a dynamic action. When two limbs link, a joint is added between them, actuated by the ‘parent’ limb’s controller. This forms a new ‘single’ agent, which may further link with other agents. In this way, complex morphologies can emerge, controlled by a policy whose architecture is in explicit correspondence with the morphology. In experiments, we demonstrate that agents with these modular and dynamic topologies generalize better to test-time environments compared to static and monolithic baselines. Project videos are available at https:// doubleblindICLR19.github.io/self-assembly/.
+
+# 1 INTRODUCTION
+
+Only a tiny fraction of the Earth’s biomass is composed of higher-level organisms capable of complex sensorimotor actions of the kind popular in contemporary robotics research (navigation, pick and place, etc). A large portion is primitive single-celled organisms, such as bacteria (Bar-On et al., 2018). Possibly the single most pivotal event in the history of evolution was the point when single-celled organisms switched from always competing with each other for resources to sometimes cooperating, first by forming colonies, and later by merging into multicellular organisms (Alberts et al., 1994). These modular self-assemblies were successful because they combined the high adaptability of single-celled organisms while making it possible for vastly more complex behaviours to emerge. Like many researchers before us (Murata & Kurokawa, 2007; Sims, 1994; Tu & Terzopoulos, 1994; Yim et al., 2000; 2007), we are inspired by the biology of multicellular evolution as a model for emergent complexity in artificial agents. Unlike most previous work however, we are primarily focused on modularity as a way of improving generalization to novel environmental conditions.
+
+In this paper, we present a study of modular self-assemblies of primitive agents — “limbs” which can link up to solve a shared task. The limbs have the option to bind together by adding a joint that connects their morphologies (Figure 1a), and when they do so, they pass messages and share rewards. Each limb comes with a simple neural net that controls the torque applied to its joints. Linking and unlinking is treated as a dynamic action, so that the limb assembly can change shape during a single episode of the simulation. This setup has previously been explored in robotics as “self-reconfiguring modular robots” (Stoy et al., 2010). However, unlike prior work on such robots, where the control policies are hand-defined, we show how to learn the policies and study the generalization properties that emerge.
+
+To make this problem computationally tractable, we do not allow the limb assemblies to form cycles in morphology. Limbs pass messages to their neighbors in this graph in order to coordinate behavior. All limbs share a common policy function, parametrized by a neural network, which takes the messages from adjacent limbs as input and outputs a torque to rotate the limb in addition to the linking/un-linking action. We call the aggregate neural network a Dynamic Graph Network (DGN)
+
+![](images/7e38b385fb3bd76766da4a06a8e23300b7c416b8411920bae8c26e48e432e04c.jpg)  
+Figure 1: We study the modular co-evolution of control and morphology where a collection of primitive agents self-assemble to form complex collectives to perform given tasks. (a) Each primitive agent is a limb containing a cylindrical body and a configurable motor. These limbs can connect with each other using the attached motor as a joint. (b) We illustrate our dynamic agents in four environments / tasks: standing up, locomotion, manipulation (pushing), and sumo wrestling. See project videos at https://doubleblindICLR19.github.io/self-assembly/.
+
+since it is a graph neural network (Scarselli et al., 2009) that can dynamically change topology as a function of its own outputs.
+
+We test our limb assemblies on four tasks: standing, locomotion, pushing and wrestling, shown in Figure 1b. We find that DGNs enable a single modular policy to control multiple possible morphologies, even those unseen during training. For example, a 6-limb policy, trained to build a 6-limb tower, can be applied at test time on 12 limbs, and results in a 12-limb tower. Not only are the policies robust to changes in number of limbs, they also generalize well to novel test-time environmental conditions, such as added wind, or new landscapes. These results together demonstrate that our modular and dynamic self-assembling agents have advantages toward generalization to new environments and tasks. Our main contributions are:
+
+• Training primitive agents that self-assemble into complex morphologies to jointly solve control tasks.   
+Formulating morphological search as a reinforcement learning problem, where linking and unlinking are treated as actions.   
+• Representing policy via a graph whose topology matches the agent’s physical structure.   
+• Demonstrating that these self-assembling agents both train and generalize better than fixedmorphology baselines.
+
+# 2 ENVIRONMENT AND AGENTS
+
+Investigating the co-evolution of control (i.e., software) and morphology (i.e., hardware) is not supported within standard benchmark environments typically used for sensorimotor control, requiring us to create our own. We opted for a minimalist design for our agents, the environment, and the reward structure, which is crucial to ensuring that the emergence of limb assemblies with complex morphologies is not forced, but happens naturally.
+
+Environment Structure Our environment contains an arena where a collection of primitive agent limbs can self-assemble to perform control tasks. This arena is a ground surface equipped with gravity and friction. The arena can be procedurally changed to generate a variety of novel terrains by changing the height of each tile on the ground (see Figure 1b). To evaluate the generalization properties of our agents, we generate a series of novels terrains. This include generating bumpy terrain by randomizing the height of nearby tiles, stairs terrain by incrementally increasing height of each row of tiles, hurdles terrain by changing height of each row of tiles, gaps terrain by removing alternate row of tiles, etc. Some variations also include putting the arena ‘under water’ which basically amounts to increased drag (i.e. buoyancy). We start our environment with a set of six primitive limb agents on the ground which can assemble to form collectives to perform complex tasks.
+
+Agent Structure All our primitive limb agents share the same simple structure: a cylindrical body with a configurable motor on one end. One end of the cylinder is free and the other end contains a configurable motor. The free-end of the limb can link up with the motor-end of the other limb, and then the motor acts as a joint between two limbs with three degrees of rotation. Hence, one can refer to the motor-end of the cylindrical limb as a parent-end and the free end as a child-end. Multiple limbs can attach their child-end to the parent-end of another limb, as shown in Figure 1(a), to allow for complex graph morphologies to emerge. The limb of the parent-end controls the torques of joint. The un-linking action can be easily implemented by detaching two limbs, but the linking action has to deal with the ambiguity of which limb to connect to (if at all). To resolve these modeling issues, we implement the linking action by attaching the closest limb within a small radius around the parent-node. If no other limb is present within the threshold range, the linking action has no effect.
+
+The primitive limb agents are dropped in an environment to jointly solve a given control task. One key component of the self-assembling agent setup that makes it different from typical multi-agent scenarios (Wooldridge, 2009) is that if some agents assemble to form a collective, the resulting morphology becomes a new single agent and all limbs within the morphology maximize a joint reward function. The output action space of each primitive agent contains the continuous torque values that are to be applied to the motor connected to the agent, and are denoted by $\{ \tau _ { \alpha } , \tau _ { \beta } , \tau _ { \gamma } \}$ for three degrees of rotation. In addition to the torque controls, each limb can decide to attach another link at its parent-end, or decide to unlink its child-end if already connected to other limb. The linking and unlinking decisions are binary. This complementary role assignment of child and parent ends, i.e., parent can only link and child can only unlink, makes it possible to decentralize the control across limbs in a self-assembly.
+
+In our self-assembling setup, each agent limb only has access to its local sensory information and does not know about other limbs. The sensory input of each agent includes its own dynamics, i.e., the location of the limb in 3-D euclidean coordinates, its velocity, angular rotation and angular velocity. Each end of the limb also has a trinary touch sensor to detect whether the end of the cylinder is touching 1) the floor, 2) another limb, or 3) nothing. Additionally, we also provide our limbs with a very simple point depth sensor that captures the surface height on a $9 \times 9$ grid around the projection of center of limb on the surface. One essential requirement to operationalize this setup is an efficient simulator to allow simultaneous simulation of several of these primitive limb agents. We implement our environments in the Unity ML (Juliani et al., 2018) framework, which is one of the dominant platforms for designing realistic games. For computational reasons, we do not allow the emergence of cycles in the self-assembling agents by not allowing the limbs to link up with already attached limbs within the same morphology. However, our setup is trivially extensible to general graphs.
+
+# 3 LEARNING TO CONTROL SELF-ASSEMBLING MORPHOLOGIES
+
+Consider a set of primitive limbs indexed by $i$ in $\{ 1 , 2 , \ldots , n \}$ , which are dropped in the environment arena $\mathcal { E }$ to perform a given continuous control task. If needed, these limbs can assemble to form complex collectives in order to improve their performance on the task. The task is represented by a reward function $r _ { t }$ and the goal of the limbs is to maximize the discounted sum of rewards over time $t$ . If some limbs assemble to form a collective, the resulting morphology effectively becomes a single agent with a joint network to maximize the joint reward of the connected limbs. Further, the reward of an assembled morphology is a function of the whole morphology and not the individual agent limbs. For instance, in the task of learning to stand up, the reward is the height of the individual limbs if they are separate, but is the height of the whole morphology if those limbs have assembled into a collective. We now discuss our proposed formulation for learning to control these self-assembling agents.
+
+![](images/d4957739eb7c79e6a980308079d88025f7454b0ce228924b33db7531a74c9a58.jpg)  
+Figure 2: High-level visualization of our method. A set of primitive ’limbs’ learn to self-assemble into morphologies where each limb is represented by a neural network linked via graph of physical edges. The inset on right shows the message-passing diagram for each node. Project videos at https://doubleblindICLR19.github.io/self-assembly/.
+
+# 3.1 CO-EVOLUTION: LINKING/UNLINKING AS AN ACTION
+
+To learn a modular controller policy that could generalize to novel setups, our agents must learn the controller jointly as the morphology evolves over time. The limbs should simultaneously decide which torques to apply to their respective motors, while taking into account the connected morphology. Our hypothesis is that if a controller policy could learn in a modular fashion over iterations of increasingly sophisticated morphologies (see Figure 3b), it could learn to be robust and generalizable to diverse situations. So, how can we optimize control and morphology under a common end-to-end framework?
+
+We propose to treat the decision of linking and unlinking as additional actions of our primitive limb agents. The total action space $a _ { t }$ at each iteration $t$ can be denoted as $\left\{ \tau _ { \alpha } , \tau _ { \beta } , \tau _ { \gamma } , \sigma _ { l i n k } , \sigma _ { u n l i n k } \right\}$ where $\tau _ { * }$ denote the raw continuous torque values to be applied at the motor and $\sigma _ { * }$ denote the binary actions whether to connect another limb at the parent-end or disconnect the child-end from the other already attached limb. This simple view of morphological evolution allows us to use ideas from learning-driven control, in particular, reinforcement learning (Sutton & Barto, 1998).
+
+# 3.2 MODULARITY: SELF-ASSEMBLING AGENT AS A GRAPH OF LIMBS
+
+Integration of control and morphology in a common framework is only the first step. The key question is how to model this controller policy such that it is modular and reuses information across generations of morphologies. Let $a _ { t } ^ { i }$ be the action space and $s _ { t } ^ { i }$ be the local sensory input-space of the agent $i$ . One naive approach to maximizing the reward is to simply combine the states of the limbs into the input-space output all the actions jointly using a single network. Formally, the policy is simply $\vec { a } _ { t } = \bar { [ } a _ { t } ^ { 0 } , \bar { a _ { t } ^ { 1 } } \cdot \cdot \cdot a _ { t } ^ { n } \bar { ] } = \Pi ( s _ { t } ^ { 0 } , s _ { t } ^ { 0 } \ldots , s _ { t } ^ { \bar { n } } )$ . This interprets the self-assemblies as a single monolithic agent, ignoring the graphical structure. This is the current approach to solve many control problems, e.g., Mujoco environments like humanoid (Brockman et al., 2016) where the policy $\Pi$ is trained to maximize the sum of discounted rewards using reinforcement learning.
+
+In this work, we represent the policy of the agent via a graph neural network (Scarselli et al., 2009) in such a way that it explicitly corresponds to the morphology of the agent. Let’s consider the collection of primitive agent limbs as graph $G$ where each node is denoted by to the primitive limb agent $i$ . Two limbs being physically connected by a joint is analogous to having an edge in the graph. At a joint, the limb which connects itself via its parent-end acts as a parent-node in the corresponding edge, and the other limbs which connect to that joint via child-ends are child-nodes. The parent-node (i.e., the agent with the parent-end) controls the torque of the edge (i.e., the joint motor), as described in Section-2.
+
+# 3.3 DYNAMIC GRAPH NETWORKS (DGN)
+
+Each primitive limb node $i$ has a policy controller of its own, which is represented by a neural network $\pi _ { \theta } ^ { i }$ and receives a corresponding reward $r _ { t } ^ { i }$ for each time step $t$ . We represent the policy of the self-assembled agent by the aggregated neural network that is connected in the same graphical manner as the physical morphology. The edge connectivity of the graph is represented in the overall graph policy by passing messages that flow from each limb network to the other limbs physically connected to it via a joint. The parameters $\theta$ are shared across each primitive limb agent allowing the overall policy of the graph to be modular with respect to each node. However, recall that the agent morphologies are dynamic, i.e., the connectivity of the limbs changes based on policy outputs. This changes the edge connectivity of the corresponding graph network at every timestep, depending on the actions predicted by each limb controller network in the previous timestep. Hence, we call this aggregate neural net a Dynamic Graph Network (DGN) since it is a graph neural network that can dynamically change topology as a function of its own outputs in the previous iteration.
+
+DGN Optimization A typical rollout of our self-assembling agents during an episode of training contains a sequence of torques $\tau _ { t } ^ { i }$ and the linking actions $\boldsymbol { \sigma } _ { t } ^ { i }$ for each limb at each timestep $t$ . The policy parameter $\theta$ is optimized to jointly maximize the reward for each network limb:
+
+$$
+\operatorname* { m a x } _ { \theta } \sum _ { i = \{ 1 , 2 . . . , n \} } \mathbb { E } _ { \vec { a } ^ { i } \sim \pi _ { \theta } ^ { i } } [ \Sigma _ { t } r _ { t } ^ { i } ]
+$$
+
+We optimize this objective via reinforcement learning, in particular the policy gradient method PPO (Schulman et al., 2017).
+
+DGN Connectivity The topology is captured in the DGN by passing messages through the edges between individual network nodes. These messages allow each node to take into account its context relative to other nodes, and are supposed to convey information about the neighbouring policy network nodes in the graph. Since the parameters of these limb networks are shared across each node, these messages can be seen as context information that may inform the policy of its role in the corresponding connected component of graph. The aggregated flow through the whole graph can be encapsulated by passing these contextual messages in topological order (no cycles). One can either do a top-down pass, beginning from the root node (i.e., the node with no parents) to the leaf nodes, or do bottom-up pass, from leaves to root node. This idea is inspired from classical work on Bayesian graph networks where message passing is used for belief-propagation (Jordan, 2003). However, when the graph contains cycles, this idea can be easily extended by performing message-passing iteratively through the cycle until convergence, similar to loopy-belief-propagation in Bayesian graphs (Murphy et al., 1999). We now discuss these message-passing strategies:
+
+(a) Top-down message passing: Instead of defining $\pi _ { \theta } ^ { i }$ to be just as a function of state, $\pi _ { \theta } ^ { i } : s _ { t } ^ { i } \to a _ { t } ^ { i }$ we pass each limb’s policy network the information about its parent node as well. Formally, one can redefine $\pi _ { \theta } ^ { i }$ as $\pi _ { \theta } ^ { i } : [ \bar { s } _ { t } ^ { i } , m _ { t } ^ { \bar { p } _ { i } } ]  a _ { t } ^ { i }$ where $p _ { i }$ is the parent of node $i$ . However, this also implies that each network node should pass context information as messages to its children networks for them to take it as input. So, we need to define $m _ { t } ^ { i }$ which is the output of each node $i$ , and which is passed as the input context message to all its children. We simply append this to the output of $\pi _ { \theta } ^ { i }$ . Thus, we finally define $\pi _ { \theta } ^ { i } : [ s _ { t } ^ { i } , m _ { t } ^ { p _ { i } } ]  [ a _ { t } ^ { i } , m _ { t } ^ { i } ]$ . If $i$ has no parents (i.e, root), a vector of zeros is passed in $m _ { t } ^ { p _ { i } }$ . This is computed recursively until the messages reach the leaf nodes.
+
+$( b )$ Bottom-up message passing: In this strategy, messages are passed from leaf nodes to root, i.e., each agent gets information from its children, but not from its parent. Similar to top-down, we redefine $\pi _ { \theta } ^ { i }$ as $\pi _ { \theta } ^ { i } : [ s _ { t } ^ { i } , m _ { t } ^ { C _ { i } } ]  [ a _ { t } ^ { i } , m _ { t } ^ { i } ]$ where $m _ { t } ^ { i }$ is the output message of policy that goes into the parent limb and $m _ { t } ^ { C _ { i } }$ t t  t t tis the aggregated input messages from all the children nodes, i.e, $\begin{array} { r } { m _ { t } ^ { C _ { i } } = \sum _ { c \in C _ { i } } m _ { t } ^ { c } } \end{array}$ . If $i$ has no children (i.e, root), a vector of zeros is passed in $m _ { t } ^ { C _ { i } }$ . Messages are passed recursively until the root node.
+
+(c) Bottom-up then top-down message passing: In this strategy, we pass messages both ways: bottomup, then top-down. In the absence of cycles in graph, a one-way pass (either top-down or bottom-up) is sufficient to capture the aggregated information, similar to Bayesian trees (Jordan, 2003). Even though both-way message-passing is redundant, we still explore it as an alternative since it might help in learning when the agent grows too complex. This is implemented by dividing the policy into two parts, each responsible for one direction of message passing, i.e., the parameters $\theta = [ \theta _ { 1 } , \theta _ { 2 } ]$ . First the bottom-up message passing is formulated as $\pi _ { \theta _ { 1 } } ^ { i } : [ s _ { t } ^ { i } , m _ { t } ^ { C _ { i } } ]  m _ { t } ^ { i }$ where the sensory input $s _ { t } ^ { i }$ and input messages $m _ { t } ^ { C _ { i } }$ 1    are used to generate outgoing messages to the parent node. In the top-down pass, messages from the parent are used, in addition with the agent’s own message, to output its action: $\pi _ { \theta _ { 2 } } ^ { i } : [ m _ { t } ^ { i } , m _ { t } ^ { p _ { i } } ]  \bar { [ a _ { t } ^ { i } , \hat { m } _ { t } ^ { i } ] }$ where $\hat { m } _ { t } ^ { i }$ are the messages passed to the children nodes.
+
+(d) No message passing: Note that for some environments or tasks, the context from the other nodes might not be a necessary requirement for effective control.In such scenarios, passing messages might creates an extra-overhead for training a DGN. Importantly, even with no messages being passed, the DGN framework still allows for coordination between limbs. This is because the control and morphology are still learned jointly in a modular mannner through the course of an episode i.e. the morphology and control in each timestep t depends explicitly on the physical morphology and the torques at previous timestep t 1. To implement the no message passing variant of DGN, we simply zero-out the messages $\bar { m } _ { t } ^ { p _ { i } } , m _ { t } ^ { i }$ at each timestep $t$ . This is similar to a typical cooperative multi-agent setup (Wooldridge, 2009) where each limb makes its own decisions in response to the previous actions of the other agents. However, our setup differs in that our agents may physically join up, rather than just coordinate behavior.
+
+# 4 IMPLEMENTATION DETAILS AND BASELINES
+
+Implementation Details: We use PPO (Schulman et al., 2017) as the underlying reinforcement learning method to optimize Equation 1. Limb policies are represented by fully-connected neural network and trained with a learning rate of $3 e - 4$ , discount factor of 0.995 and entropy coefficient of 0.01. Each episode is 5000 steps long at training and 1200 steps long at testing. Across all the tasks, the number of limbs at training is kept fixed to 6. Limbs start each episode disconnected and located just above the ground plane at random locations, as shown in Figure 3b. During generalization to novel scenarios, we experiment with changing the number of limbs to 12 or 3 to test the same policy without any further finetuning. All of our tasks require the agent to output continuous raw torque control values.
+
+Baselines We compare the role of the above four message passing strategies in DGN across a variety of tasks. Different strategies may work well in different scenarios. We further compare how well these dynamic morphologies perform in comparison to a learned monolithic policy for both dynamic and fixed morphologies. In particular, we compare to a (a) Monolithic Policy, Dynamic Graph: in this baseline, our agents are still dynamic and self-assemble to perform the task, however, their controller is represented by a single monolithic policy that takes as input the combined state of all agents and outputs actions for each of them. (b) Monolithic Policy, Fixed Graph: For each task, a hand-designed morphology is constructed from the limbs and trained using a single monolithic policy that takes as input the combined state of all agents and outputs the actions for all agents. The agents are not able to combine or separate This can be compared to a standard robotics setup in which a morphology is predefined and then a policy is learned to control it. Note that one cannot generalize Monolithic Policy baselines to scenarios where the number of limbs vary as it would change the action and state space of the policy.
+
+For the Fixed Graph baseline, we chose the fixed morphology to be a straight line chain of 6-limbs (i.e., a linear morphology) in all the experiments including the task of standing up and locomotion. This linear-chain may be optimal for standing as tall as possible, but it is not necessarily optimal for learning to stand; the same would hold for locomotion. Further, note that, the best performing DGN variants also converges to linear-chain morphology (shown in Figure 3b and video results on the project website) to achieve the best reward in case of standing up task. Moreover, one can confirm that the locomotion task is also solvable with linear-morphology because one of the DGN ablation methods converged to a linear-morphology while doing well at locomotion (see video).
+
+# 5 EXPERIMENTS: EMERGENT MORPHOLOGIES AND GENERALIZATION
+
+We test the co-evolution of morphology and control across four tasks where self-assembling agents learn to: (a) stand up, (b) perform locomotion, (c) perform manipulation, and (d) fight in a sumo wrestling environment. There are two primary objectives of our investigation. The first is to determine
+
+![](images/bfb386f1ec4e547d15833dea54e0d14ba3d0bfcf03b8740d85a6cdcd1f3669bc.jpg)  
+Figure 3: Training of self-assembling agents: (a) The training performance of different methods for joint training of control and morphology for the task of learning to stand up. The generalization performance of these policies across new scenarios is shown in Table 1. (b) The gradual co-evolution of controller as well as the morphology of self-assembling agents over the course of training.
+
+<table><tr><td>Environment</td><td colspan="4">DGN</td><td colspan="2">Monolithic Policy</td></tr><tr><td></td><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td colspan="7">Training Environment</td></tr><tr><td>Standing Up</td><td>15253</td><td>13486</td><td>17518</td><td>12470</td><td>4104</td><td>5351</td></tr><tr><td colspan="7">Zero-Shot Generalization</td></tr><tr><td>More (2x) Limbs</td><td>15006 (98%)</td><td>14429 (107%)</td><td>19796 (113%)</td><td>14084 (113%)</td><td></td><td></td></tr><tr><td>Fewer(.5x) Limbs</td><td>11730 (77%)</td><td>9842 (73%)</td><td>10839 (62%)</td><td>9070 (73%)</td><td></td><td></td></tr><tr><td>Water +2x Limbs</td><td>16642 (109%)</td><td>14192 (105%)</td><td>16871 (96%)</td><td>13360 (107%)</td><td></td><td></td></tr><tr><td>Winds</td><td>14654 (96%)</td><td>12116 (90%)</td><td>16803 (96%)</td><td>12560 (101%)</td><td>3923 (96%)</td><td>4531 (85%)</td></tr><tr><td>Strong Winds</td><td>14727 (97%)</td><td>13416 (99%)</td><td>15853 (90%)</td><td>12257 (98%)</td><td>3937 (96%)</td><td>4961 (93%)</td></tr></table>
+
+Table 1: Testing generalization for the standing up task. We show quantitative evaluation of the generalization ability of the learned policies. For each of the methods, we first pick the best performing model from the training run and then evaluate it on each of the novel scenarios without any further finetuning, i.e., in a zero-shot manner. We report first the score attained by the self-assembling agent and then report, in parenthesis, the percentage of training performance retained upon transfer. The higher the numbers, the better it is.
+
+if such a modular co-evolution results in the emergence of complex self-assembling agents. The second is to evaluate if the emerged modular controller generalizes to novel scenarios.
+
+# 5.1 TASK: STANDING UP
+
+In this task, each agent’s reward is proportional to the highest vertical point in its combined morphology, i.e., the limb assemblies should try to maximize their $Y$ -axis height. Limbs have an incentive to self-assemble since the potential reward scales with the number of agents in the body, given that the agent can learn the controller for it. The learning process begins by six-limbs falling on the ground randomly, as shown in Figure 3b. In the beginning, each agent learns independently of others but these limbs learn to self-assemble to form a complex agent after training. Figure 3a compares different methods in terms of their performance on the task of standing as high as possible. We found that our DGN policy variants perform significantly better than the monolithic policies for the standing up task. In particular, the bottom-up and up-then-down message passing strategies attain the highest reward. To verify the implementation of our monolithic policy with fixed morphology, we show its ablation with varying number of limbs in Section A.1 in the supplementary.
+
+However, the key question is whether the learned policy generalizes to novel scenarios. We investigate it by testing the learned policies without any further finetuning, i.e. zero-shot generalization, in novel scenarios: adding two times the number of limbs, reducing the number of limbs by half, increasing drag (i.e., ‘under water’) and number of limbs at the same time, and adding varying strength of random pushes-n-pulls (i.e., ‘wind’). As the results in Table 1 show, DGN achieves similar performance as it did on the training environment, despite never having seen these scenarios before. Interestingly, the DGN variants seem to generalize better than the fixed-graph policies (last column). Monolithic
+
+![](images/1fe61865162a0548a6d693284f20ac66f155fbbfaf92f3057aa955697cfdc8ee.jpg)  
+Figure 4: Training self-assembling agents: We show the performance of different methods for joint training of control and morphology for three tasks: standing up in the presence of wind and random push-n-pulls (left), locomotion in bumpy terrain (center) and manipulation (pushing) of two objects (right). These policies generalize to novel scenarios as shown in respective tables.
+
+<table><tr><td rowspan="2">Environment</td><td colspan="4">DGN</td><td colspan="2">Monolithic Policy</td></tr><tr><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td>Training Environment</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Standing Up in Wind</td><td>16339</td><td>18423</td><td></td><td>17237</td><td>4176</td><td>4500</td></tr><tr><td>Zero-Shot Generalization</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(S)trong Winds</td><td>15649 (96%)</td><td>17384 (94%)</td><td></td><td></td><td>4010 (96%)</td><td>4507 (100%)</td></tr><tr><td>2x Limbs +(S)Winds</td><td>16250 (99%)</td><td>15351 (83%)</td><td></td><td>15728 (91%)</td><td></td><td></td></tr><tr><td>Water+2x(L)+(S)Winds</td><td>17254 (106%)</td><td>17068 (93%)</td><td></td><td>16592 (96%)</td><td>一</td><td></td></tr></table>
+
+Table 2: Testing generalization for the standing up task in the presence of random push-n-pulls (i.e. ‘wind’). The best performing model from the training is evaluated on each of the novel scenarios without any further finetuning. The score attained by the self-assembling agent is reported first and then, in parenthesis, the percentage of training performance retained upon transfer. The bottom-up DGN failed due to some experimental error and will be reported in the final version of paper.
+
+policy baselines cannot be generalized to more or fewer limbs due to the fixed action and state space. A better understanding of these results may be obtained by looking at the dynamically combining morphologies in the project video.
+
+# 5.2 TASK: STANDING UP IN THE PRESENCE OF RANDOM PUSH-N-PULLS (WIND)
+
+The task in this case is same as the previous one of learning to stand up. However, unlike in the previous subsection, here we also trained in the presence of random push-n-pulls (i.e., ‘wind’) with hope of making the learned morphologies even more robust. The training performance in Figure 4a show the superior performance of DGN with respect to the baselines. The generalization results, in Table 2, show that the DGN both-ways messaging passing variant is the most robust. This may be because in the presence of distractors, communication both ways can be helpful since a random force on a single limb affects all other attached limbs.
+
+# 5.3 LOCOMOTION TASK
+
+The reward function in this environment is defined as the distance covered by the agent along an axis, in particular, the limbs are rewarded is proportional to their velocity along the $X$ -axis. The training environment is a bumpy terrain (shown in Figure 1(b)) and the training performance is shown in Figure 4b. Our DGN variants significantly outperform the monolithic baselines (see supplementary, Section A.1, for ablation). Interestingly, DGN variant with no message passing performs the best. Upon in-depth investigation, we found that it is possible to do well on this locomotion task with a large variety of morphologies, unlike the task of standing up where a tower is strongly preferrable. Here, any morphology with sufficient height and forward velocity is able to make competitive progress in locomotion (see videos), and thus reducing message-passing to an unnecessary overhead. As discussed in Section 3.3, no message passing merely implies the absence of context to the limbs, but the DGN aggregated policy is still modular and jointly learned with the morphology over the episode.
+
+Table 3: Testing generalization for the locomotion task. The best performing model from the training is evaluated on each of the novel scenarios without any further finetuning. The score attained by the self-assembling agent is reported first and then, in parenthesis, the percentage of training performance retained upon transfer.   
+
+<table><tr><td>Environment</td><td colspan="4">DGN</td><td colspan="2">Monolithic Policy</td></tr><tr><td></td><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td colspan="7">Training Environment 一</td></tr><tr><td>Locomotion</td><td>3.91</td><td>6.87</td><td>8.71</td><td>9.0</td><td>0.96</td><td>2.96</td></tr><tr><td colspan="7">Zero-Shot Generalization</td></tr><tr><td>More (2x) Limbs</td><td>4.01 (103%)</td><td>4.29 (63%)</td><td>5.47 (63%)</td><td>9.19 (102%)</td><td></td><td></td></tr><tr><td>Fewer (.5x)Limbs</td><td>3.52 (90%)</td><td>4.49 (65%)</td><td>6.64 (76%)</td><td>8.2 (91%)</td><td></td><td></td></tr><tr><td>Water+2xLimbs</td><td>2.64 (68%)</td><td>3.54 (52%)</td><td>6.57 (75%)</td><td>7.2 (80%)</td><td></td><td></td></tr><tr><td>Hurdles</td><td>1.84 (47%)</td><td>3.66 (53%)</td><td>6.39 (73%)</td><td>5.56 (62%)</td><td>-0.77 (-79%)</td><td>-3.12 (-104%)</td></tr><tr><td>Gaps in Terrain</td><td>1.84 (47%)</td><td>2.8 (41%)</td><td>3.25 (37%)</td><td>4.17 (46%)</td><td>-0.32 (-33%)</td><td>2.09 (71%)</td></tr><tr><td>Bi-modal Bumps</td><td>2.97 (76%)</td><td>4.55 (66%)</td><td>6.62 (76%)</td><td>6.15 (68%)</td><td>-0.56 (-57%)</td><td>-0.44 (-14%)</td></tr><tr><td>Stairs</td><td>1.0 (26%)</td><td>4.25 (62%)</td><td>6.6 (76%)</td><td>8.59 (95%)</td><td>-8.8 (-912%)</td><td>-3.65 (-122%)</td></tr><tr><td>Inside Valley</td><td>4.37 (112%)</td><td>6.55 (95%)</td><td>5.29 (61%)</td><td>6.21 (69%)</td><td>0.47 (48%)</td><td>-1.35 (-45%)</td></tr></table>
+
+<table><tr><td>Environment</td><td colspan="4">DGN</td><td colspan="2">Monolithic Policy</td></tr><tr><td></td><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td colspan="7">Training Environment</td></tr><tr><td>Manipulation</td><td>-7985</td><td>-7861</td><td>-8482</td><td>-9603</td><td>-8773</td><td>-7725</td></tr><tr><td colspan="7">Zero-Shot Generalization</td></tr><tr><td>More (2x) Limbs</td><td>-14319 (-179%)</td><td>-14894 (-189%)</td><td>-9969 (-118%)</td><td>-10879 (-112%)</td><td></td><td></td></tr><tr><td>Water +2x Limbs</td><td>-10724 (-134%)</td><td>-13278 (-169%)</td><td>-12368 (-146%)</td><td>-10362 (-108%)</td><td></td><td></td></tr></table>
+
+Table 4: Testing generalization for the manipulation task. The score attained by the self-assembling agent is reported first and then, in parenthesis, the percentage of training performance retained.
+
+We evaluate the learned policy without any further finetuning on several scenarios: more limbs, fewer limbs, more limbs under water, a terrain with hurdles of a certain height, a terrain with gaps between platforms, a bumpy terrain with a bi-modal distribution of bump heights, stairs, and an environment with a valley surrounded by walls on both sides. These environments are procedurally generated as discussed in Section 2. Across these novel environments, the modular policies learned by DGN tend to generalize better than the monolithic agent policies, as indicated in Table 3.
+
+# 5.4 TASK: MANIPULATION OF TWO OBJECTS
+
+The agents are dropped inside a room containing two objects and the goal is to decrease the distance between the objects, as shown in Figure 1(b). The reward for the agents is the negative distance between the objects, so as to encourage the behavior of pushing the blocks together. The training plots are shown in Figure 4c and the generalization results are shown in Table 4. This is a very hard task due to the sparse reward problem as agents only get reward if they move the block. Interestingly, the learned policies do not work well enough in this environment, and only learn to slightly move the blocks (see video). We believe this task requires more reward engineering than just the distance, and we will update the improved results in the final version.
+
+# 5.5 TASK: SUMO WRESTLING BETWEEN TWO TEAMS
+
+In this task, we divide the limbs into two teams of 6 limbs each and drop them into an arena to fight. Each team gets rewarded if any opponent limb falls out of the arena. The agents are trained via competitive self-play (Bansal et al., 2017; Tesauro, 1995). This is in contrast to the previous “single-team” tasks for self-assembling agents, i.e., standing, locomotion and manipulation. We present it as an additional result demonstrating the wider applicability of the method. However, it is non-trivial to measure the performance in self-play as the game is zero-sum, and rewards therefore do not increase over time. Instead, we refer the readers to the qualitative results in the video. The policies learned by the self-assembling agents demonstrate some interesting behaviors, but there is a lot of room for improvement in future research. We will release these environments upon acceptance.
+
+# 6 RELATED WORK
+
+Morphologenesis and self-reconfiguring modular robots The idea of modular and selfassembling agents goes back at least to Von Neumman’s Theory of Self-Reproducing Automata (Von Neumann et al., 1966). In robotics, such systems have been termed “self-reconfiguring modular robots” (Murata & Kurokawa, 2007; Stoy et al., 2010). There has been a lot of work in the modular robotics community in designing real hardware robotic modules that can be docked with each other to form complex robotic morphologies (Daudelin et al., 2018; Gilpin et al., 2008; Romanishin et al., 2013; Wright et al., 2007; Yim et al., 2000). Our main contribution is to approach this problem from a learning perspective, in particular deep RL, and study the resulting generalization properties.
+
+A variety of alternative approaches have also been proposed to optimize agent morphologies, including genetic algorithms that search over a generative grammar (Sims, 1994), as well as directly optimizing over morphology parameters with RL (Schaff et al., 2018). One key difference between these approaches and our own is that we achieve morphogenesis via dynamic actions (linking), which agents take during their lifetimes, whereas the past approaches treat morphology as an optimization target to be updated between generations or episodes. Since the physical morphology also defines the connectivity of the policy net, our proposed algorithm can also be viewed as performing a kind of neural architecture search (Zoph & Le, 2016) in physical agents.
+
+Graph neural networks Encoding graphical structures into neural networks has been used for a large number of applications, including quantum chemistry (Gilmer et al., 2017), semi-supervised classification (Kipf & Welling, 2016), and representation learning (Yang et al., 2018). The works most similar to ours involve learning control policies. For example, Nervenet (Wang et al., 2018) represents individual limbs and joints as nodes in a graph and demonstrates multi-limb generalization, just like our system does. However, the morphologies on which Nervenet operates are not learned jointly with the policy. hand-defined to be compositional in nature. Others (Battaglia et al., 2018; Huang et al., 2018) have shown that graph neural networks can also be applied to inference models as well as to planning. Many of these past works implement some variant of Graph Neural Networks (Scarselli et al., 2009) which operate on general graphs. Our method leverages the constraint that the morphologies can always be represented as a rooted tree in order to simplify the message passing.
+
+# 7 DISCUSSION
+
+Modeling intelligent agents as modular, self-assembling morphologies has long been a very appealing idea. The efforts to create practical systems to evolve artificial agents goes back at least two decades to the beautiful work of Karl Sims (Sims, 1994). In this paper, we are revisiting these ideas using the contemporary machinery of deep networks and reinforcement learning. Examining the problem in the context of machine learning, rather than optimization, we are particularly interested in modularity as a key to generalization, in terms of improving adaptability and robustness to novel environmental conditions. Poor generalization is the Achilles heel of modern robotics research, and the hope is that this could be a promising direction in addressing this key issue. We demonstrated a number of promising experimental results, suggesting that modularity does indeed improve generalization in simulated agents. While these are just the initial steps, we believe that the proposed research direction is promising and its exploration will be fruitful to the research community. To encourage follow-up work, we will release all code, models, and environments online once the paper is published.
+
+# REFERENCES
+
+Bruce Alberts, Dennis Bray, Julian Lewis, Martin Raff, Keith Roberts, and James D Watson. Molecular Biology of the Cell. Garland Publishing, New York, 1994. 1
+
+Trapit Bansal, Jakub Pachocki, Szymon Sidor, Ilya Sutskever, and Igor Mordatch. Emergent complexity via multi-agent competition. CoRR, 2017. 9
+
+Yinon M Bar-On, Rob Phillips, and Ron Milo. The biomass distribution on earth. Proceedings of the National Academy of Sciences, 2018. 1
+
+Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al. Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261, 2018. 10
+
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv:1606.01540, 2016. 4
+
+Jonathan Daudelin, Gangyuan Jing, Tarik Tosun, Mark Yim, Hadas Kress-Gazit, and Mark Campbell. An integrated system for perception-driven autonomy with modular robots. Science Robotics, 2018. 10
+
+Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. arXiv preprint arXiv:1704.01212, 2017. 10
+
+Kyle Gilpin, Keith Kotay, Daniela Rus, and Iuliu Vasilescu. Miche: Modular shape formation by self-disassembly. IJRR, 2008. 10
+
+De-An Huang, Suraj Nair, Danfei Xu, Yuke Zhu, Animesh Garg, Li Fei-Fei, Silvio Savarese, and Juan Carlos Niebles. Neural task graphs: Generalizing to unseen tasks from a single video demonstration. arXiv preprint arXiv:1807.03480, 2018. 10
+
+Michael I Jordan. An introduction to probabilistic graphical models, 2003. 5
+
+Arthur Juliani, Vincent-Pierre Berges, Esh Vckay, Yuan Gao, Hunter Henry, Marwan Mattar, and Danny Lange. Unity: A general platform for intelligent agents. arXiv preprint arXiv:1809.02627, 2018. 3
+
+Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016. 10
+
+Satoshi Murata and Haruhsa Kurokawa. Self-reconfigurable robots. IEEE Robotics & Automation Magazine, 2007. 1, 10
+
+Kevin P Murphy, Yair Weiss, and Michael I Jordan. Loopy belief propagation for approximate inference: An empirical study. In Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence, 1999. 5
+
+John W Romanishin, Kyle Gilpin, and Daniela Rus. M-blocks: Momentum-driven, magnetic modular robots. In IROS, 2013. 10
+
+Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. IEEE Transactions on Neural Network, 2009. 2, 4, 10
+
+Charles B. Schaff, David Yunis, Ayan Chakrabarti, and Matthew R. Walter. Jointly learning to construct and control agents using deep reinforcement learning. CoRR, abs/1801.01432, 2018. URL http://arxiv.org/abs/1801.01432. 10
+
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. abs/1707.06347, 2017. 5, 6
+
+Karl Sims. Evolving virtual creatures. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques, 1994. 1, 10
+
+Kasper Stoy, David Brandt, David J Christensen, and David Brandt. Self-reconfigurable robots: an introduction. Mit Press Cambridge, 2010. 1, 10
+
+Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press Cambridge, 1998. 4
+
+Gerald Tesauro. Temporal difference learning and td-gammon. Communications of the ACM, 1995. 9
+
+Xiaoyuan Tu and Demetri Terzopoulos. Artificial fishes: Physics, locomotion, perception, behavior. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques, 1994. 1
+
+John Von Neumann, Arthur W Burks, et al. Theory of self-reproducing automata. IEEE Transactions on Neural Networks, 1966. 10
+
+Tingwu Wang, Renjie Liao, Jimmy Ba, and Sanja Fidler. Nervenet: Learning structured policy with graph neural networks. 2018. 10
+
+Michael Wooldridge. An introduction to multiagent systems. John Wiley & Sons, 2009. 3, 6
+
+Cornell Wright, Aaron Johnson, Aaron Peck, Zachary McCord, Allison Naaktgeboren, Philip Gianfortoni, Manuel Gonzalez-Rivero, Ross Hatton, and Howie Choset. Design of a modular snake robot. In IROS, 2007. 10
+
+Zhilin Yang, Bhuwan Dhingra, Kaiming He, William W Cohen, Ruslan Salakhutdinov, Yann LeCun, et al. Glomo: Unsupervisedly learned relational graphs as transferable representations. arXiv preprint arXiv:1806.05662, 2018. 10
+
+Mark Yim, David G Duff, and Kimon D Roufas. Polybot: a modular reconfigurable robot. In ICRA, 2000. 1, 10
+
+Mark Yim, Wei-Min Shen, Behnam Salemi, Daniela Rus, Mark Moll, Hod Lipson, Eric Klavins, and Gregory S Chirikjian. Modular self-reconfigurable robot systems [grand challenges of robotics]. IEEE Robotics & Automation Magazine, 2007. 1
+
+Barret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. arXiv preprint arXiv:1611.01578, 2016. 10
+
+# A SUPPLEMENTARY MATERIAL
+
+# A.1 PERFORMANCE OF FIXED-GRAPH BASELINE VS. NUMBER OF LIMBS
+
+To verify whether the training of Monolithic Policy w/ Fixed Graph is working, we ran it on standing up and locomotion tasks across varying number of limbs. We show in Figure 5 that the baseline performs well with less number of limbs which suggests that the reason for failure in 6-limbs case is indeed the morphology graph being fixed, and not the implementation of this baseline.
+
+![](images/557a4a6ce0d15f4b84d5a8b914890ee9239f60306ec2c59bdc87490c9cd4f9df.jpg)  
+Figure 5: The performance of Monolithic Policy w/ Fixed Graph baseline as the number of limbs varies in the two tasks: standing up (left) and locomotion (right). This shows that the monolithic baseline works well with less (1-3 limbs), but fails with 6 limbs during training.
+
+# A.2 GENERALIZATION OF LEARNED POLICIES AT DIFFERENT TRAINING INTERVALS
+
+In this section, we show the generalization plots corresponding to the Tables 1, 2, 3, 4. To plot generalization, we pick the trained model from different training intervals and plot them across new environments without finetuning at all, in a zero-shot manner.
+
+![](images/322766492b77d96530e5903ed73afb31edaa04cae3792ec8c6af045946583aba.jpg)  
+Figure 6: Generalization for the task of Standing Up: Performance of different methods across novel scenarios without any finetuning.
+
+![](images/eb7fee8482b89b8ee0b9039eb096786efc56ea0b0ec3381bac2344f7712f8612.jpg)  
+Figure 7: Generalization for the task of Standing Up w/ Wind: Performance of different methods across novel scenarios without any finetuning.
+
+![](images/a186ebeb262fe5356a9dd99cc84b81f64dfb84f9b3a6038762e6469904d93d95.jpg)  
+Figure 8: Generalization for the task of Locomotion: Performance of different methods across novel scenarios without any finetuning.
+
+![](images/49da75b9bfd538f716f7a485b431ad4be669f90d45e75a9aeb1957942217161f.jpg)  
+Figure 9: Generalization for the task of Manipulation: Performance of different methods across novel scenarios without any finetuning.

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+    {
+        "type": "text",
+        "text": "LEARNING TO CONTROL SELF-ASSEMBLING MORPHOLOGIES: A STUDY OF GENERALIZATION VIA MODULARITY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
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+            171
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            195,
+            398,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            260,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Much of contemporary sensorimotor learning assumes that one is already given a complex agent (e.g., a robotic arm) and the goal is to learn to control it. In contrast, this paper investigates a modular co-evolution strategy: a collection of primitive agents learns to self-assemble into increasingly complex collectives in order to solve control tasks. Each primitive agent consists of a limb and a neural controller. Limbs may choose to link up to form collectives, with linking being treated as a dynamic action. When two limbs link, a joint is added between them, actuated by the ‘parent’ limb’s controller. This forms a new ‘single’ agent, which may further link with other agents. In this way, complex morphologies can emerge, controlled by a policy whose architecture is in explicit correspondence with the morphology. In experiments, we demonstrate that agents with these modular and dynamic topologies generalize better to test-time environments compared to static and monolithic baselines. Project videos are available at https:// doubleblindICLR19.github.io/self-assembly/. ",
+        "bbox": [
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+            292,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            336,
+            531
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Only a tiny fraction of the Earth’s biomass is composed of higher-level organisms capable of complex sensorimotor actions of the kind popular in contemporary robotics research (navigation, pick and place, etc). A large portion is primitive single-celled organisms, such as bacteria (Bar-On et al., 2018). Possibly the single most pivotal event in the history of evolution was the point when single-celled organisms switched from always competing with each other for resources to sometimes cooperating, first by forming colonies, and later by merging into multicellular organisms (Alberts et al., 1994). These modular self-assemblies were successful because they combined the high adaptability of single-celled organisms while making it possible for vastly more complex behaviours to emerge. Like many researchers before us (Murata & Kurokawa, 2007; Sims, 1994; Tu & Terzopoulos, 1994; Yim et al., 2000; 2007), we are inspired by the biology of multicellular evolution as a model for emergent complexity in artificial agents. Unlike most previous work however, we are primarily focused on modularity as a way of improving generalization to novel environmental conditions. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a study of modular self-assemblies of primitive agents — “limbs” which can link up to solve a shared task. The limbs have the option to bind together by adding a joint that connects their morphologies (Figure 1a), and when they do so, they pass messages and share rewards. Each limb comes with a simple neural net that controls the torque applied to its joints. Linking and unlinking is treated as a dynamic action, so that the limb assembly can change shape during a single episode of the simulation. This setup has previously been explored in robotics as “self-reconfiguring modular robots” (Stoy et al., 2010). However, unlike prior work on such robots, where the control policies are hand-defined, we show how to learn the policies and study the generalization properties that emerge. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To make this problem computationally tractable, we do not allow the limb assemblies to form cycles in morphology. Limbs pass messages to their neighbors in this graph in order to coordinate behavior. All limbs share a common policy function, parametrized by a neural network, which takes the messages from adjacent limbs as input and outputs a torque to rotate the limb in addition to the linking/un-linking action. We call the aggregate neural network a Dynamic Graph Network (DGN) ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/7e38b385fb3bd76766da4a06a8e23300b7c416b8411920bae8c26e48e432e04c.jpg",
+        "image_caption": [
+            "Figure 1: We study the modular co-evolution of control and morphology where a collection of primitive agents self-assemble to form complex collectives to perform given tasks. (a) Each primitive agent is a limb containing a cylindrical body and a configurable motor. These limbs can connect with each other using the attached motor as a joint. (b) We illustrate our dynamic agents in four environments / tasks: standing up, locomotion, manipulation (pushing), and sumo wrestling. See project videos at https://doubleblindICLR19.github.io/self-assembly/. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            104,
+            820,
+            363
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "since it is a graph neural network (Scarselli et al., 2009) that can dynamically change topology as a function of its own outputs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We test our limb assemblies on four tasks: standing, locomotion, pushing and wrestling, shown in Figure 1b. We find that DGNs enable a single modular policy to control multiple possible morphologies, even those unseen during training. For example, a 6-limb policy, trained to build a 6-limb tower, can be applied at test time on 12 limbs, and results in a 12-limb tower. Not only are the policies robust to changes in number of limbs, they also generalize well to novel test-time environmental conditions, such as added wind, or new landscapes. These results together demonstrate that our modular and dynamic self-assembling agents have advantages toward generalization to new environments and tasks. Our main contributions are: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Training primitive agents that self-assemble into complex morphologies to jointly solve control tasks.   \nFormulating morphological search as a reinforcement learning problem, where linking and unlinking are treated as actions.   \n• Representing policy via a graph whose topology matches the agent’s physical structure.   \n• Demonstrating that these self-assembling agents both train and generalize better than fixedmorphology baselines. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 ENVIRONMENT AND AGENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            777,
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+            792
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Investigating the co-evolution of control (i.e., software) and morphology (i.e., hardware) is not supported within standard benchmark environments typically used for sensorimotor control, requiring us to create our own. We opted for a minimalist design for our agents, the environment, and the reward structure, which is crucial to ensuring that the emergence of limb assemblies with complex morphologies is not forced, but happens naturally. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Environment Structure Our environment contains an arena where a collection of primitive agent limbs can self-assemble to perform control tasks. This arena is a ground surface equipped with gravity and friction. The arena can be procedurally changed to generate a variety of novel terrains by changing the height of each tile on the ground (see Figure 1b). To evaluate the generalization properties of our agents, we generate a series of novels terrains. This include generating bumpy terrain by randomizing the height of nearby tiles, stairs terrain by incrementally increasing height of each row of tiles, hurdles terrain by changing height of each row of tiles, gaps terrain by removing alternate row of tiles, etc. Some variations also include putting the arena ‘under water’ which basically amounts to increased drag (i.e. buoyancy). We start our environment with a set of six primitive limb agents on the ground which can assemble to form collectives to perform complex tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Agent Structure All our primitive limb agents share the same simple structure: a cylindrical body with a configurable motor on one end. One end of the cylinder is free and the other end contains a configurable motor. The free-end of the limb can link up with the motor-end of the other limb, and then the motor acts as a joint between two limbs with three degrees of rotation. Hence, one can refer to the motor-end of the cylindrical limb as a parent-end and the free end as a child-end. Multiple limbs can attach their child-end to the parent-end of another limb, as shown in Figure 1(a), to allow for complex graph morphologies to emerge. The limb of the parent-end controls the torques of joint. The un-linking action can be easily implemented by detaching two limbs, but the linking action has to deal with the ambiguity of which limb to connect to (if at all). To resolve these modeling issues, we implement the linking action by attaching the closest limb within a small radius around the parent-node. If no other limb is present within the threshold range, the linking action has no effect. ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The primitive limb agents are dropped in an environment to jointly solve a given control task. One key component of the self-assembling agent setup that makes it different from typical multi-agent scenarios (Wooldridge, 2009) is that if some agents assemble to form a collective, the resulting morphology becomes a new single agent and all limbs within the morphology maximize a joint reward function. The output action space of each primitive agent contains the continuous torque values that are to be applied to the motor connected to the agent, and are denoted by $\\{ \\tau _ { \\alpha } , \\tau _ { \\beta } , \\tau _ { \\gamma } \\}$ for three degrees of rotation. In addition to the torque controls, each limb can decide to attach another link at its parent-end, or decide to unlink its child-end if already connected to other limb. The linking and unlinking decisions are binary. This complementary role assignment of child and parent ends, i.e., parent can only link and child can only unlink, makes it possible to decentralize the control across limbs in a self-assembly. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In our self-assembling setup, each agent limb only has access to its local sensory information and does not know about other limbs. The sensory input of each agent includes its own dynamics, i.e., the location of the limb in 3-D euclidean coordinates, its velocity, angular rotation and angular velocity. Each end of the limb also has a trinary touch sensor to detect whether the end of the cylinder is touching 1) the floor, 2) another limb, or 3) nothing. Additionally, we also provide our limbs with a very simple point depth sensor that captures the surface height on a $9 \\times 9$ grid around the projection of center of limb on the surface. One essential requirement to operationalize this setup is an efficient simulator to allow simultaneous simulation of several of these primitive limb agents. We implement our environments in the Unity ML (Juliani et al., 2018) framework, which is one of the dominant platforms for designing realistic games. For computational reasons, we do not allow the emergence of cycles in the self-assembling agents by not allowing the limbs to link up with already attached limbs within the same morphology. However, our setup is trivially extensible to general graphs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 LEARNING TO CONTROL SELF-ASSEMBLING MORPHOLOGIES",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a set of primitive limbs indexed by $i$ in $\\{ 1 , 2 , \\ldots , n \\}$ , which are dropped in the environment arena $\\mathcal { E }$ to perform a given continuous control task. If needed, these limbs can assemble to form complex collectives in order to improve their performance on the task. The task is represented by a reward function $r _ { t }$ and the goal of the limbs is to maximize the discounted sum of rewards over time $t$ . If some limbs assemble to form a collective, the resulting morphology effectively becomes a single agent with a joint network to maximize the joint reward of the connected limbs. Further, the reward of an assembled morphology is a function of the whole morphology and not the individual agent limbs. For instance, in the task of learning to stand up, the reward is the height of the individual limbs if they are separate, but is the height of the whole morphology if those limbs have assembled into a collective. We now discuss our proposed formulation for learning to control these self-assembling agents. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/d4957739eb7c79e6a980308079d88025f7454b0ce228924b33db7531a74c9a58.jpg",
+        "image_caption": [
+            "Figure 2: High-level visualization of our method. A set of primitive ’limbs’ learn to self-assemble into morphologies where each limb is represented by a neural network linked via graph of physical edges. The inset on right shows the message-passing diagram for each node. Project videos at https://doubleblindICLR19.github.io/self-assembly/. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 CO-EVOLUTION: LINKING/UNLINKING AS AN ACTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            428,
+            593,
+            443
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To learn a modular controller policy that could generalize to novel setups, our agents must learn the controller jointly as the morphology evolves over time. The limbs should simultaneously decide which torques to apply to their respective motors, while taking into account the connected morphology. Our hypothesis is that if a controller policy could learn in a modular fashion over iterations of increasingly sophisticated morphologies (see Figure 3b), it could learn to be robust and generalizable to diverse situations. So, how can we optimize control and morphology under a common end-to-end framework? ",
+        "bbox": [
+            174,
+            455,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose to treat the decision of linking and unlinking as additional actions of our primitive limb agents. The total action space $a _ { t }$ at each iteration $t$ can be denoted as $\\left\\{ \\tau _ { \\alpha } , \\tau _ { \\beta } , \\tau _ { \\gamma } , \\sigma _ { l i n k } , \\sigma _ { u n l i n k } \\right\\}$ where $\\tau _ { * }$ denote the raw continuous torque values to be applied at the motor and $\\sigma _ { * }$ denote the binary actions whether to connect another limb at the parent-end or disconnect the child-end from the other already attached limb. This simple view of morphological evolution allows us to use ideas from learning-driven control, in particular, reinforcement learning (Sutton & Barto, 1998). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 MODULARITY: SELF-ASSEMBLING AGENT AS A GRAPH OF LIMBS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Integration of control and morphology in a common framework is only the first step. The key question is how to model this controller policy such that it is modular and reuses information across generations of morphologies. Let $a _ { t } ^ { i }$ be the action space and $s _ { t } ^ { i }$ be the local sensory input-space of the agent $i$ . One naive approach to maximizing the reward is to simply combine the states of the limbs into the input-space output all the actions jointly using a single network. Formally, the policy is simply $\\vec { a } _ { t } = \\bar { [ } a _ { t } ^ { 0 } , \\bar { a _ { t } ^ { 1 } } \\cdot \\cdot \\cdot a _ { t } ^ { n } \\bar { ] } = \\Pi ( s _ { t } ^ { 0 } , s _ { t } ^ { 0 } \\ldots , s _ { t } ^ { \\bar { n } } )$ . This interprets the self-assemblies as a single monolithic agent, ignoring the graphical structure. This is the current approach to solve many control problems, e.g., Mujoco environments like humanoid (Brockman et al., 2016) where the policy $\\Pi$ is trained to maximize the sum of discounted rewards using reinforcement learning. ",
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+        "text": "In this work, we represent the policy of the agent via a graph neural network (Scarselli et al., 2009) in such a way that it explicitly corresponds to the morphology of the agent. Let’s consider the collection of primitive agent limbs as graph $G$ where each node is denoted by to the primitive limb agent $i$ . Two limbs being physically connected by a joint is analogous to having an edge in the graph. At a joint, the limb which connects itself via its parent-end acts as a parent-node in the corresponding edge, and the other limbs which connect to that joint via child-ends are child-nodes. The parent-node (i.e., the agent with the parent-end) controls the torque of the edge (i.e., the joint motor), as described in Section-2. ",
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+    {
+        "type": "text",
+        "text": "3.3 DYNAMIC GRAPH NETWORKS (DGN) ",
+        "text_level": 1,
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+        "text": "Each primitive limb node $i$ has a policy controller of its own, which is represented by a neural network $\\pi _ { \\theta } ^ { i }$ and receives a corresponding reward $r _ { t } ^ { i }$ for each time step $t$ . We represent the policy of the self-assembled agent by the aggregated neural network that is connected in the same graphical manner as the physical morphology. The edge connectivity of the graph is represented in the overall graph policy by passing messages that flow from each limb network to the other limbs physically connected to it via a joint. The parameters $\\theta$ are shared across each primitive limb agent allowing the overall policy of the graph to be modular with respect to each node. However, recall that the agent morphologies are dynamic, i.e., the connectivity of the limbs changes based on policy outputs. This changes the edge connectivity of the corresponding graph network at every timestep, depending on the actions predicted by each limb controller network in the previous timestep. Hence, we call this aggregate neural net a Dynamic Graph Network (DGN) since it is a graph neural network that can dynamically change topology as a function of its own outputs in the previous iteration. ",
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+        "text": "DGN Optimization A typical rollout of our self-assembling agents during an episode of training contains a sequence of torques $\\tau _ { t } ^ { i }$ and the linking actions $\\boldsymbol { \\sigma } _ { t } ^ { i }$ for each limb at each timestep $t$ . The policy parameter $\\theta$ is optimized to jointly maximize the reward for each network limb: ",
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+    {
+        "type": "equation",
+        "img_path": "images/7dcb3506c58ba54b97c3a3edc27d5a6570afa8fc3a81f2f6db870db8c85d8444.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\sum _ { i = \\{ 1 , 2 . . . , n \\} } \\mathbb { E } _ { \\vec { a } ^ { i } \\sim \\pi _ { \\theta } ^ { i } } [ \\Sigma _ { t } r _ { t } ^ { i } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We optimize this objective via reinforcement learning, in particular the policy gradient method PPO (Schulman et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "DGN Connectivity The topology is captured in the DGN by passing messages through the edges between individual network nodes. These messages allow each node to take into account its context relative to other nodes, and are supposed to convey information about the neighbouring policy network nodes in the graph. Since the parameters of these limb networks are shared across each node, these messages can be seen as context information that may inform the policy of its role in the corresponding connected component of graph. The aggregated flow through the whole graph can be encapsulated by passing these contextual messages in topological order (no cycles). One can either do a top-down pass, beginning from the root node (i.e., the node with no parents) to the leaf nodes, or do bottom-up pass, from leaves to root node. This idea is inspired from classical work on Bayesian graph networks where message passing is used for belief-propagation (Jordan, 2003). However, when the graph contains cycles, this idea can be easily extended by performing message-passing iteratively through the cycle until convergence, similar to loopy-belief-propagation in Bayesian graphs (Murphy et al., 1999). We now discuss these message-passing strategies: ",
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+    {
+        "type": "text",
+        "text": "(a) Top-down message passing: Instead of defining $\\pi _ { \\theta } ^ { i }$ to be just as a function of state, $\\pi _ { \\theta } ^ { i } : s _ { t } ^ { i } \\to a _ { t } ^ { i }$ we pass each limb’s policy network the information about its parent node as well. Formally, one can redefine $\\pi _ { \\theta } ^ { i }$ as $\\pi _ { \\theta } ^ { i } : [ \\bar { s } _ { t } ^ { i } , m _ { t } ^ { \\bar { p } _ { i } } ]  a _ { t } ^ { i }$ where $p _ { i }$ is the parent of node $i$ . However, this also implies that each network node should pass context information as messages to its children networks for them to take it as input. So, we need to define $m _ { t } ^ { i }$ which is the output of each node $i$ , and which is passed as the input context message to all its children. We simply append this to the output of $\\pi _ { \\theta } ^ { i }$ . Thus, we finally define $\\pi _ { \\theta } ^ { i } : [ s _ { t } ^ { i } , m _ { t } ^ { p _ { i } } ]  [ a _ { t } ^ { i } , m _ { t } ^ { i } ]$ . If $i$ has no parents (i.e, root), a vector of zeros is passed in $m _ { t } ^ { p _ { i } }$ . This is computed recursively until the messages reach the leaf nodes. ",
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+    {
+        "type": "text",
+        "text": "$( b )$ Bottom-up message passing: In this strategy, messages are passed from leaf nodes to root, i.e., each agent gets information from its children, but not from its parent. Similar to top-down, we redefine $\\pi _ { \\theta } ^ { i }$ as $\\pi _ { \\theta } ^ { i } : [ s _ { t } ^ { i } , m _ { t } ^ { C _ { i } } ]  [ a _ { t } ^ { i } , m _ { t } ^ { i } ]$ where $m _ { t } ^ { i }$ is the output message of policy that goes into the parent limb and $m _ { t } ^ { C _ { i } }$ t t  t t tis the aggregated input messages from all the children nodes, i.e, $\\begin{array} { r } { m _ { t } ^ { C _ { i } } = \\sum _ { c \\in C _ { i } } m _ { t } ^ { c } } \\end{array}$ . If $i$ has no children (i.e, root), a vector of zeros is passed in $m _ { t } ^ { C _ { i } }$ . Messages are passed recursively until the root node. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(c) Bottom-up then top-down message passing: In this strategy, we pass messages both ways: bottomup, then top-down. In the absence of cycles in graph, a one-way pass (either top-down or bottom-up) is sufficient to capture the aggregated information, similar to Bayesian trees (Jordan, 2003). Even though both-way message-passing is redundant, we still explore it as an alternative since it might help in learning when the agent grows too complex. This is implemented by dividing the policy into two parts, each responsible for one direction of message passing, i.e., the parameters $\\theta = [ \\theta _ { 1 } , \\theta _ { 2 } ]$ . First the bottom-up message passing is formulated as $\\pi _ { \\theta _ { 1 } } ^ { i } : [ s _ { t } ^ { i } , m _ { t } ^ { C _ { i } } ]  m _ { t } ^ { i }$ where the sensory input $s _ { t } ^ { i }$ and input messages $m _ { t } ^ { C _ { i } }$ 1    are used to generate outgoing messages to the parent node. In the top-down pass, messages from the parent are used, in addition with the agent’s own message, to output its action: $\\pi _ { \\theta _ { 2 } } ^ { i } : [ m _ { t } ^ { i } , m _ { t } ^ { p _ { i } } ]  \\bar { [ a _ { t } ^ { i } , \\hat { m } _ { t } ^ { i } ] }$ where $\\hat { m } _ { t } ^ { i }$ are the messages passed to the children nodes. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "(d) No message passing: Note that for some environments or tasks, the context from the other nodes might not be a necessary requirement for effective control.In such scenarios, passing messages might creates an extra-overhead for training a DGN. Importantly, even with no messages being passed, the DGN framework still allows for coordination between limbs. This is because the control and morphology are still learned jointly in a modular mannner through the course of an episode i.e. the morphology and control in each timestep t depends explicitly on the physical morphology and the torques at previous timestep t 1. To implement the no message passing variant of DGN, we simply zero-out the messages $\\bar { m } _ { t } ^ { p _ { i } } , m _ { t } ^ { i }$ at each timestep $t$ . This is similar to a typical cooperative multi-agent setup (Wooldridge, 2009) where each limb makes its own decisions in response to the previous actions of the other agents. However, our setup differs in that our agents may physically join up, rather than just coordinate behavior. ",
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+    {
+        "type": "text",
+        "text": "4 IMPLEMENTATION DETAILS AND BASELINES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Implementation Details: We use PPO (Schulman et al., 2017) as the underlying reinforcement learning method to optimize Equation 1. Limb policies are represented by fully-connected neural network and trained with a learning rate of $3 e - 4$ , discount factor of 0.995 and entropy coefficient of 0.01. Each episode is 5000 steps long at training and 1200 steps long at testing. Across all the tasks, the number of limbs at training is kept fixed to 6. Limbs start each episode disconnected and located just above the ground plane at random locations, as shown in Figure 3b. During generalization to novel scenarios, we experiment with changing the number of limbs to 12 or 3 to test the same policy without any further finetuning. All of our tasks require the agent to output continuous raw torque control values. ",
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+        "type": "text",
+        "text": "Baselines We compare the role of the above four message passing strategies in DGN across a variety of tasks. Different strategies may work well in different scenarios. We further compare how well these dynamic morphologies perform in comparison to a learned monolithic policy for both dynamic and fixed morphologies. In particular, we compare to a (a) Monolithic Policy, Dynamic Graph: in this baseline, our agents are still dynamic and self-assemble to perform the task, however, their controller is represented by a single monolithic policy that takes as input the combined state of all agents and outputs actions for each of them. (b) Monolithic Policy, Fixed Graph: For each task, a hand-designed morphology is constructed from the limbs and trained using a single monolithic policy that takes as input the combined state of all agents and outputs the actions for all agents. The agents are not able to combine or separate This can be compared to a standard robotics setup in which a morphology is predefined and then a policy is learned to control it. Note that one cannot generalize Monolithic Policy baselines to scenarios where the number of limbs vary as it would change the action and state space of the policy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the Fixed Graph baseline, we chose the fixed morphology to be a straight line chain of 6-limbs (i.e., a linear morphology) in all the experiments including the task of standing up and locomotion. This linear-chain may be optimal for standing as tall as possible, but it is not necessarily optimal for learning to stand; the same would hold for locomotion. Further, note that, the best performing DGN variants also converges to linear-chain morphology (shown in Figure 3b and video results on the project website) to achieve the best reward in case of standing up task. Moreover, one can confirm that the locomotion task is also solvable with linear-morphology because one of the DGN ablation methods converged to a linear-morphology while doing well at locomotion (see video). ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS: EMERGENT MORPHOLOGIES AND GENERALIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We test the co-evolution of morphology and control across four tasks where self-assembling agents learn to: (a) stand up, (b) perform locomotion, (c) perform manipulation, and (d) fight in a sumo wrestling environment. There are two primary objectives of our investigation. The first is to determine ",
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+    {
+        "type": "image",
+        "img_path": "images/bfb386f1ec4e547d15833dea54e0d14ba3d0bfcf03b8740d85a6cdcd1f3669bc.jpg",
+        "image_caption": [
+            "Figure 3: Training of self-assembling agents: (a) The training performance of different methods for joint training of control and morphology for the task of learning to stand up. The generalization performance of these policies across new scenarios is shown in Table 1. (b) The gradual co-evolution of controller as well as the morphology of self-assembling agents over the course of training. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/83cc63dff1bc6fe5a349ad5627c82ad1764ea90487c1bcbeb3438d93e9648fb7.jpg",
+        "table_caption": [],
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+        "table_body": "<table><tr><td>Environment</td><td colspan=\"4\">DGN</td><td colspan=\"2\">Monolithic Policy</td></tr><tr><td></td><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td colspan=\"7\">Training Environment</td></tr><tr><td>Standing Up</td><td>15253</td><td>13486</td><td>17518</td><td>12470</td><td>4104</td><td>5351</td></tr><tr><td colspan=\"7\">Zero-Shot Generalization</td></tr><tr><td>More (2x) Limbs</td><td>15006 (98%)</td><td>14429 (107%)</td><td>19796 (113%)</td><td>14084 (113%)</td><td></td><td></td></tr><tr><td>Fewer(.5x) Limbs</td><td>11730 (77%)</td><td>9842 (73%)</td><td>10839 (62%)</td><td>9070 (73%)</td><td></td><td></td></tr><tr><td>Water +2x Limbs</td><td>16642 (109%)</td><td>14192 (105%)</td><td>16871 (96%)</td><td>13360 (107%)</td><td></td><td></td></tr><tr><td>Winds</td><td>14654 (96%)</td><td>12116 (90%)</td><td>16803 (96%)</td><td>12560 (101%)</td><td>3923 (96%)</td><td>4531 (85%)</td></tr><tr><td>Strong Winds</td><td>14727 (97%)</td><td>13416 (99%)</td><td>15853 (90%)</td><td>12257 (98%)</td><td>3937 (96%)</td><td>4961 (93%)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Testing generalization for the standing up task. We show quantitative evaluation of the generalization ability of the learned policies. For each of the methods, we first pick the best performing model from the training run and then evaluate it on each of the novel scenarios without any further finetuning, i.e., in a zero-shot manner. We report first the score attained by the self-assembling agent and then report, in parenthesis, the percentage of training performance retained upon transfer. The higher the numbers, the better it is. ",
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+    {
+        "type": "text",
+        "text": "if such a modular co-evolution results in the emergence of complex self-assembling agents. The second is to evaluate if the emerged modular controller generalizes to novel scenarios. ",
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+    {
+        "type": "text",
+        "text": "5.1 TASK: STANDING UP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this task, each agent’s reward is proportional to the highest vertical point in its combined morphology, i.e., the limb assemblies should try to maximize their $Y$ -axis height. Limbs have an incentive to self-assemble since the potential reward scales with the number of agents in the body, given that the agent can learn the controller for it. The learning process begins by six-limbs falling on the ground randomly, as shown in Figure 3b. In the beginning, each agent learns independently of others but these limbs learn to self-assemble to form a complex agent after training. Figure 3a compares different methods in terms of their performance on the task of standing as high as possible. We found that our DGN policy variants perform significantly better than the monolithic policies for the standing up task. In particular, the bottom-up and up-then-down message passing strategies attain the highest reward. To verify the implementation of our monolithic policy with fixed morphology, we show its ablation with varying number of limbs in Section A.1 in the supplementary. ",
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+    {
+        "type": "text",
+        "text": "However, the key question is whether the learned policy generalizes to novel scenarios. We investigate it by testing the learned policies without any further finetuning, i.e. zero-shot generalization, in novel scenarios: adding two times the number of limbs, reducing the number of limbs by half, increasing drag (i.e., ‘under water’) and number of limbs at the same time, and adding varying strength of random pushes-n-pulls (i.e., ‘wind’). As the results in Table 1 show, DGN achieves similar performance as it did on the training environment, despite never having seen these scenarios before. Interestingly, the DGN variants seem to generalize better than the fixed-graph policies (last column). Monolithic ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fe61865162a0548a6d693284f20ac66f155fbbfaf92f3057aa955697cfdc8ee.jpg",
+        "image_caption": [
+            "Figure 4: Training self-assembling agents: We show the performance of different methods for joint training of control and morphology for three tasks: standing up in the presence of wind and random push-n-pulls (left), locomotion in bumpy terrain (center) and manipulation (pushing) of two objects (right). These policies generalize to novel scenarios as shown in respective tables. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/879d2537f6461c6ecafd3e83ee1772ff8c8d51e17abbb703008e2d8d0fd89c16.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Environment</td><td colspan=\"4\">DGN</td><td colspan=\"2\">Monolithic Policy</td></tr><tr><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td>Training Environment</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Standing Up in Wind</td><td>16339</td><td>18423</td><td></td><td>17237</td><td>4176</td><td>4500</td></tr><tr><td>Zero-Shot Generalization</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(S)trong Winds</td><td>15649 (96%)</td><td>17384 (94%)</td><td></td><td></td><td>4010 (96%)</td><td>4507 (100%)</td></tr><tr><td>2x Limbs +(S)Winds</td><td>16250 (99%)</td><td>15351 (83%)</td><td></td><td>15728 (91%)</td><td></td><td></td></tr><tr><td>Water+2x(L)+(S)Winds</td><td>17254 (106%)</td><td>17068 (93%)</td><td></td><td>16592 (96%)</td><td>一</td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Testing generalization for the standing up task in the presence of random push-n-pulls (i.e. ‘wind’). The best performing model from the training is evaluated on each of the novel scenarios without any further finetuning. The score attained by the self-assembling agent is reported first and then, in parenthesis, the percentage of training performance retained upon transfer. The bottom-up DGN failed due to some experimental error and will be reported in the final version of paper. ",
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+    {
+        "type": "text",
+        "text": "policy baselines cannot be generalized to more or fewer limbs due to the fixed action and state space. A better understanding of these results may be obtained by looking at the dynamically combining morphologies in the project video. ",
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+    {
+        "type": "text",
+        "text": "5.2 TASK: STANDING UP IN THE PRESENCE OF RANDOM PUSH-N-PULLS (WIND) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The task in this case is same as the previous one of learning to stand up. However, unlike in the previous subsection, here we also trained in the presence of random push-n-pulls (i.e., ‘wind’) with hope of making the learned morphologies even more robust. The training performance in Figure 4a show the superior performance of DGN with respect to the baselines. The generalization results, in Table 2, show that the DGN both-ways messaging passing variant is the most robust. This may be because in the presence of distractors, communication both ways can be helpful since a random force on a single limb affects all other attached limbs. ",
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+    {
+        "type": "text",
+        "text": "5.3 LOCOMOTION TASK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The reward function in this environment is defined as the distance covered by the agent along an axis, in particular, the limbs are rewarded is proportional to their velocity along the $X$ -axis. The training environment is a bumpy terrain (shown in Figure 1(b)) and the training performance is shown in Figure 4b. Our DGN variants significantly outperform the monolithic baselines (see supplementary, Section A.1, for ablation). Interestingly, DGN variant with no message passing performs the best. Upon in-depth investigation, we found that it is possible to do well on this locomotion task with a large variety of morphologies, unlike the task of standing up where a tower is strongly preferrable. Here, any morphology with sufficient height and forward velocity is able to make competitive progress in locomotion (see videos), and thus reducing message-passing to an unnecessary overhead. As discussed in Section 3.3, no message passing merely implies the absence of context to the limbs, but the DGN aggregated policy is still modular and jointly learned with the morphology over the episode. ",
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+        "type": "table",
+        "img_path": "images/189ed2862e80ef626157217157e79f90cc84c705e9526454f557bf8afae4adad.jpg",
+        "table_caption": [
+            "Table 3: Testing generalization for the locomotion task. The best performing model from the training is evaluated on each of the novel scenarios without any further finetuning. The score attained by the self-assembling agent is reported first and then, in parenthesis, the percentage of training performance retained upon transfer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Environment</td><td colspan=\"4\">DGN</td><td colspan=\"2\">Monolithic Policy</td></tr><tr><td></td><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td colspan=\"7\">Training Environment 一</td></tr><tr><td>Locomotion</td><td>3.91</td><td>6.87</td><td>8.71</td><td>9.0</td><td>0.96</td><td>2.96</td></tr><tr><td colspan=\"7\">Zero-Shot Generalization</td></tr><tr><td>More (2x) Limbs</td><td>4.01 (103%)</td><td>4.29 (63%)</td><td>5.47 (63%)</td><td>9.19 (102%)</td><td></td><td></td></tr><tr><td>Fewer (.5x)Limbs</td><td>3.52 (90%)</td><td>4.49 (65%)</td><td>6.64 (76%)</td><td>8.2 (91%)</td><td></td><td></td></tr><tr><td>Water+2xLimbs</td><td>2.64 (68%)</td><td>3.54 (52%)</td><td>6.57 (75%)</td><td>7.2 (80%)</td><td></td><td></td></tr><tr><td>Hurdles</td><td>1.84 (47%)</td><td>3.66 (53%)</td><td>6.39 (73%)</td><td>5.56 (62%)</td><td>-0.77 (-79%)</td><td>-3.12 (-104%)</td></tr><tr><td>Gaps in Terrain</td><td>1.84 (47%)</td><td>2.8 (41%)</td><td>3.25 (37%)</td><td>4.17 (46%)</td><td>-0.32 (-33%)</td><td>2.09 (71%)</td></tr><tr><td>Bi-modal Bumps</td><td>2.97 (76%)</td><td>4.55 (66%)</td><td>6.62 (76%)</td><td>6.15 (68%)</td><td>-0.56 (-57%)</td><td>-0.44 (-14%)</td></tr><tr><td>Stairs</td><td>1.0 (26%)</td><td>4.25 (62%)</td><td>6.6 (76%)</td><td>8.59 (95%)</td><td>-8.8 (-912%)</td><td>-3.65 (-122%)</td></tr><tr><td>Inside Valley</td><td>4.37 (112%)</td><td>6.55 (95%)</td><td>5.29 (61%)</td><td>6.21 (69%)</td><td>0.47 (48%)</td><td>-1.35 (-45%)</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/05f42042335ddfac0c3ecd7781df3a9257cefc46d8197727176ac85d202fec37.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Environment</td><td colspan=\"4\">DGN</td><td colspan=\"2\">Monolithic Policy</td></tr><tr><td></td><td>(up then down)</td><td>(top-down)</td><td>(bottom-up)</td><td>(no msgs)</td><td>(dynamic graph)</td><td>(fixed graph)</td></tr><tr><td colspan=\"7\">Training Environment</td></tr><tr><td>Manipulation</td><td>-7985</td><td>-7861</td><td>-8482</td><td>-9603</td><td>-8773</td><td>-7725</td></tr><tr><td colspan=\"7\">Zero-Shot Generalization</td></tr><tr><td>More (2x) Limbs</td><td>-14319 (-179%)</td><td>-14894 (-189%)</td><td>-9969 (-118%)</td><td>-10879 (-112%)</td><td></td><td></td></tr><tr><td>Water +2x Limbs</td><td>-10724 (-134%)</td><td>-13278 (-169%)</td><td>-12368 (-146%)</td><td>-10362 (-108%)</td><td></td><td></td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 4: Testing generalization for the manipulation task. The score attained by the self-assembling agent is reported first and then, in parenthesis, the percentage of training performance retained. ",
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate the learned policy without any further finetuning on several scenarios: more limbs, fewer limbs, more limbs under water, a terrain with hurdles of a certain height, a terrain with gaps between platforms, a bumpy terrain with a bi-modal distribution of bump heights, stairs, and an environment with a valley surrounded by walls on both sides. These environments are procedurally generated as discussed in Section 2. Across these novel environments, the modular policies learned by DGN tend to generalize better than the monolithic agent policies, as indicated in Table 3. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.4 TASK: MANIPULATION OF TWO OBJECTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The agents are dropped inside a room containing two objects and the goal is to decrease the distance between the objects, as shown in Figure 1(b). The reward for the agents is the negative distance between the objects, so as to encourage the behavior of pushing the blocks together. The training plots are shown in Figure 4c and the generalization results are shown in Table 4. This is a very hard task due to the sparse reward problem as agents only get reward if they move the block. Interestingly, the learned policies do not work well enough in this environment, and only learn to slightly move the blocks (see video). We believe this task requires more reward engineering than just the distance, and we will update the improved results in the final version. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.5 TASK: SUMO WRESTLING BETWEEN TWO TEAMS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this task, we divide the limbs into two teams of 6 limbs each and drop them into an arena to fight. Each team gets rewarded if any opponent limb falls out of the arena. The agents are trained via competitive self-play (Bansal et al., 2017; Tesauro, 1995). This is in contrast to the previous “single-team” tasks for self-assembling agents, i.e., standing, locomotion and manipulation. We present it as an additional result demonstrating the wider applicability of the method. However, it is non-trivial to measure the performance in self-play as the game is zero-sum, and rewards therefore do not increase over time. Instead, we refer the readers to the qualitative results in the video. The policies learned by the self-assembling agents demonstrate some interesting behaviors, but there is a lot of room for improvement in future research. We will release these environments upon acceptance. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Morphologenesis and self-reconfiguring modular robots The idea of modular and selfassembling agents goes back at least to Von Neumman’s Theory of Self-Reproducing Automata (Von Neumann et al., 1966). In robotics, such systems have been termed “self-reconfiguring modular robots” (Murata & Kurokawa, 2007; Stoy et al., 2010). There has been a lot of work in the modular robotics community in designing real hardware robotic modules that can be docked with each other to form complex robotic morphologies (Daudelin et al., 2018; Gilpin et al., 2008; Romanishin et al., 2013; Wright et al., 2007; Yim et al., 2000). Our main contribution is to approach this problem from a learning perspective, in particular deep RL, and study the resulting generalization properties. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A variety of alternative approaches have also been proposed to optimize agent morphologies, including genetic algorithms that search over a generative grammar (Sims, 1994), as well as directly optimizing over morphology parameters with RL (Schaff et al., 2018). One key difference between these approaches and our own is that we achieve morphogenesis via dynamic actions (linking), which agents take during their lifetimes, whereas the past approaches treat morphology as an optimization target to be updated between generations or episodes. Since the physical morphology also defines the connectivity of the policy net, our proposed algorithm can also be viewed as performing a kind of neural architecture search (Zoph & Le, 2016) in physical agents. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks Encoding graphical structures into neural networks has been used for a large number of applications, including quantum chemistry (Gilmer et al., 2017), semi-supervised classification (Kipf & Welling, 2016), and representation learning (Yang et al., 2018). The works most similar to ours involve learning control policies. For example, Nervenet (Wang et al., 2018) represents individual limbs and joints as nodes in a graph and demonstrates multi-limb generalization, just like our system does. However, the morphologies on which Nervenet operates are not learned jointly with the policy. hand-defined to be compositional in nature. Others (Battaglia et al., 2018; Huang et al., 2018) have shown that graph neural networks can also be applied to inference models as well as to planning. Many of these past works implement some variant of Graph Neural Networks (Scarselli et al., 2009) which operate on general graphs. Our method leverages the constraint that the morphologies can always be represented as a rooted tree in order to simplify the message passing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Modeling intelligent agents as modular, self-assembling morphologies has long been a very appealing idea. The efforts to create practical systems to evolve artificial agents goes back at least two decades to the beautiful work of Karl Sims (Sims, 1994). In this paper, we are revisiting these ideas using the contemporary machinery of deep networks and reinforcement learning. Examining the problem in the context of machine learning, rather than optimization, we are particularly interested in modularity as a key to generalization, in terms of improving adaptability and robustness to novel environmental conditions. Poor generalization is the Achilles heel of modern robotics research, and the hope is that this could be a promising direction in addressing this key issue. We demonstrated a number of promising experimental results, suggesting that modularity does indeed improve generalization in simulated agents. While these are just the initial steps, we believe that the proposed research direction is promising and its exploration will be fruitful to the research community. To encourage follow-up work, we will release all code, models, and environments online once the paper is published. ",
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+    {
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+        "text": "A SUPPLEMENTARY MATERIAL ",
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+    {
+        "type": "text",
+        "text": "To verify whether the training of Monolithic Policy w/ Fixed Graph is working, we ran it on standing up and locomotion tasks across varying number of limbs. We show in Figure 5 that the baseline performs well with less number of limbs which suggests that the reason for failure in 6-limbs case is indeed the morphology graph being fixed, and not the implementation of this baseline. ",
+        "bbox": [
+            174,
+            160,
+            825,
+            215
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/557a4a6ce0d15f4b84d5a8b914890ee9239f60306ec2c59bdc87490c9cd4f9df.jpg",
+        "image_caption": [
+            "Figure 5: The performance of Monolithic Policy w/ Fixed Graph baseline as the number of limbs varies in the two tasks: standing up (left) and locomotion (right). This shows that the monolithic baseline works well with less (1-3 limbs), but fails with 6 limbs during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            228,
+            233,
+            769,
+            407
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 GENERALIZATION OF LEARNED POLICIES AT DIFFERENT TRAINING INTERVALS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            488,
+            761,
+            501
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show the generalization plots corresponding to the Tables 1, 2, 3, 4. To plot generalization, we pick the trained model from different training intervals and plot them across new environments without finetuning at all, in a zero-shot manner. ",
+        "bbox": [
+            174,
+            513,
+            826,
+            555
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/322766492b77d96530e5903ed73afb31edaa04cae3792ec8c6af045946583aba.jpg",
+        "image_caption": [
+            "Figure 6: Generalization for the task of Standing Up: Performance of different methods across novel scenarios without any finetuning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            570,
+            810,
+            851
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb7fee8482b89b8ee0b9039eb096786efc56ea0b0ec3381bac2344f7712f8612.jpg",
+        "image_caption": [
+            "Figure 7: Generalization for the task of Standing Up w/ Wind: Performance of different methods across novel scenarios without any finetuning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            104,
+            813,
+            242
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/a186ebeb262fe5356a9dd99cc84b81f64dfb84f9b3a6038762e6469904d93d95.jpg",
+        "image_caption": [
+            "Figure 8: Generalization for the task of Locomotion: Performance of different methods across novel scenarios without any finetuning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            285,
+            812,
+            708
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/49da75b9bfd538f716f7a485b431ad4be669f90d45e75a9aeb1957942217161f.jpg",
+        "image_caption": [
+            "Figure 9: Generalization for the task of Manipulation: Performance of different methods across novel scenarios without any finetuning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            752,
+            802,
+            888
+        ],
+        "page_idx": 13
+    }
+]

parse/train/BJgy96EYvr/BJgy96EYvr.md

@@ -0,0 +1,561 @@
+# INFLUENCE-BASED MULTI-AGENT EXPLORATION
+
+Tonghan Wang∗, Jianhao Wang∗, Yi Wu & Chongjie Zhang
+
+Institute for Interdisciplinary Information Sciences   
+Tsinghua University   
+Beijing, China   
+wangth18@mails.tsinghua.edu.cn, wjh720.eric@gmail.com   
+jxwuyi@openai.com, chongjie@tsinghua.edu.cn
+
+# ABSTRACT
+
+Intrinsically motivated reinforcement learning aims to address the exploration challenge for sparse-reward tasks. However, the study of exploration methods in transition-dependent multi-agent settings is largely absent from the literature. We aim to take a step towards solving this problem. We present two exploration methods: exploration via information-theoretic influence (EITI) and exploration via decision-theoretic influence (EDTI), by exploiting the role of interaction in coordinated behaviors of agents. EITI uses mutual information to capture the interdependence between the transition dynamics of agents. EDTI uses a novel intrinsic reward, called Value of Interaction (VoI), to characterize and quantify the influence of one agent’s behavior on expected returns of other agents. By optimizing EITI or EDTI objective as a regularizer, agents are encouraged to coordinate their exploration and learn policies to optimize the team performance. We show how to optimize these regularizers so that they can be easily integrated with policy gradient reinforcement learning. The resulting update rule draws a connection between coordinated exploration and intrinsic reward distribution. Finally, we empirically demonstrate the significant strength of our methods in a variety of multi-agent scenarios.
+
+# 1 INTRODUCTION
+
+Reinforcement learning algorithms aim to learn a policy that maximizes the accumulative reward from an environment. Many advances of deep reinforcement learning rely on a dense shaped reward function, such as distance to the goal (Mirowski et al., 2016; Wu et al., 2018), scores in games (Mnih et al., 2015) or expert-designed rewards (Wu & Tian, 2016; OpenAI, 2018), but they tend to struggle in many real-world scenarios with sparse rewards (Burda et al., 2019). Therefore, many recent works propose to introduce additional intrinsic incentives to boost exploration, including pseudocounts (Bellemare et al., 2016; Tang et al., 2017; Ostrovski et al., 2017), model-learning improvements (Burda et al., 2019; Pathak et al., 2017; Burda et al., 2018), and information gain (Florensa et al., 2017; Gupta et al., 2018; Hyoungseok Kim, 2019). These works result in significant progress in many challenging tasks such as Montezuma Revenge (Burda et al., 2018), robotic manipulation (Pathak et al., 2018; Riedmiller et al., 2018), and Super Mario games (Burda et al., 2019; Pathak et al., 2017).
+
+Notably, most of the existing breakthroughs on sparse-reward environments have been focusing on single-agent scenarios and leave the exploration problem largely unstudied for multi-agent settings – it is common in real-world applications that multiple agents are required to solve a task in a coordinated fashion (Cao et al., 2012; Nowe et al., 2012; Zhang & Lesser, 2011). This problem has recently ´ attracted attention and several exploration strategies have been proposed for transition-independent cooperative multi-agent settings (Dimakopoulou & Van Roy, 2018; Dimakopoulou et al., 2018; Bargiacchi et al., 2018; Iqbal & Sha, 2019b). Nevertheless, how to explore effectively in more general scenarios with complex reward and transition dependency among cooperative agents remains an open research problem.
+
+This paper aims to take a step towards this goal. Our basic idea is to coordinate agents’ exploration by taking into account their interactions during their learning processes. Configurations where interaction happens (interaction points) lie at critical junctions in the state-action space, through these critical configurations can transit to potentially important under-explored regions. To exploit this idea, we propose exploration strategies where agents start with decentralized exploration driven by their individual curiosity, and are also encouraged to visit interaction points to influence the exploration processes of other agents and help them get more extrinsic and intrinsic rewards. Based on how to quantify influence among agents, we propose two exploration methods. Exploration via information-theoretic influence (EITI) uses mutual information (MI) to capture the interdependence between the transition dynamics of agents. Exploration via decision-theoretic influence (EDTI) goes further and uses a novel measure called value of interaction (VoI) to disentangle the effect of one agent’s state-action pair on the expected (intrinsic) value of other agents. By optimizing MI or VoI as a regularizer to the value function, agents are encouraged to explore state-action pairs where they can exert influences on other agents for learning sophisticated multi-agent cooperation strategies.
+
+To efficiently optimize MI and VoI, we propose augmented policy gradient formulations so that the gradients can be estimated purely from trajectories. The resulting update rule draws a connection between coordinated exploration and the distribution of individual intrinsic rewards among team members, which further explains why our methods are able to facilitate multi-agent exploration.
+
+We demonstrate the effectiveness of our methods on a variety of sparse-reward cooperative multiagent tasks. Empirical results show that both EITI and EDTI allow for the discovery of influential states and EDTI further filter out interactions that have no effects on the performance. Our results also imply that these influential states are implicitly discovered as subgoals in search space that guide and coordinate exploration. The video of experiments is available at https://sites. google.com/view/influence-based-mae/.
+
+# 2 SETTINGS
+
+In our work, we consider a fully cooperative multi-agent task that can be modelled by a factored multi-agent MDP $G = \langle N , S , \overset { \cdot } { A } , T , \overset { \cdot } { r } , h , n \rangle$ , where $\bar { \boldsymbol { N } } \equiv \{ 1 , 2 , . . . , n \}$ is the finite set of agents, ${ \cal S } \equiv \times _ { i \in N } S _ { i }$ is the finite set of joint states and $S _ { i }$ is the state set of agent $i$ . At each timestep, each agent selects an action $a _ { i } \in A _ { i }$ at state $\pmb { s }$ , forming a joint action $\pmb { a } \in A \equiv \times _ { i \in N } A _ { i }$ , resulting in a shared extrinsic reward $r ( s , a )$ for each agent and the next state $s ^ { \prime }$ according to the transition function $T ( s ^ { \prime } | s , a )$ .
+
+The objective of the task is that each agent learns a policy $\pi _ { i } ( a _ { i } | s _ { i } )$ , jointly maximizing team performance. The joint policy ${ \pmb { \pi } } \mathrm { = } \langle \pi _ { 1 } , \ldots , \pi _ { n } \rangle$ induces an action-value function, ${ Q } ^ { e x t , \pi } ( { \pmb s } , { \pmb a } ) =$ $\scriptstyle \mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { h } r ^ { t } | s ^ { 0 } = s , { \boldsymbol { a } } ^ { 0 } = { \boldsymbol { a } } , \pi ]$ , and a value function $V ^ { e x t , \pi } ( s ) { = } \operatorname* { m a x } _ { \pmb { a } } Q ^ { e x t , \pi } ( s , \pmb { a } )$ , where $\tau$ is the $h$
+
+We adopt a centralized training and decentralized execution paradigm, which has been widely used in multi-agent deep reinforcement learning (Foerster et al., 2016; Lowe et al., 2017; Foerster et al., 2018; Rashid et al., 2018). During training, agents are granted access to the states, actions, (intrinsic) rewards, and value functions of other agents, while decentralized execution only requires individual states.
+
+# 3 INFLUENCE-BASED COORDINATED MULTI-AGENT EXPLORATION
+
+Efficient exploration is critical for reinforcement learning, particularly in sparse-reward tasks. Intrinsic motivation (Oudeyer & Kaplan, 2009) is a crucial mechanism for behaviour learning since it provides the driver of exploration. Therefore, to trade off exploration and exploitation, it is common for an RL agent to maximize an objective of the expected extrinsic reward augmented by the expected intrinsic reward. Curiosity is one of the extensively-studied intrinsic rewards to encourage an agent to explore according to its uncertainty about the environment, which can be measured by model prediction error (Burda et al., 2019; Pathak et al., 2017; Burda et al., 2018) or state visitation count (Bellemare et al., 2016; Tang et al., 2017; Ostrovski et al., 2017).
+
+While such an intrinsic motivation as curiosity drives effective individual exploration, it is often not sufficient enough for learning in collaborative multi-agent settings, because it does not take into account agent interactions. To encourage interactions, we propose an influence value aims to quantify one agent’s influence on the exploration processes of other agents. Maximizing this value will encourage agents to visit interaction points more often through which the agent team can reach configurations that are rarely visited by decentralized exploration. In next sections, we will provide two ways to formulate the influence value with such properties, leading to two exploration strategies.
+
+Thus, for each agent $i$ , our overall optimization objective is:
+
+$$
+J _ { \theta _ { i } } [ \pi _ { i } | \pi _ { - i } , p _ { 0 } ] \equiv V ^ { e x t , \pi } ( s _ { 0 } ) + V _ { i } ^ { i n t , \pi } ( s _ { 0 } ) + \beta \cdot I _ { - i | i } ^ { \pi } ,
+$$
+
+where $p _ { 0 } ( s _ { 0 } )$ is the initial state distribution, $\pi _ { - i }$ is the joint policy excluding that of agent $i$ , and $V _ { i } ^ { i n t , \pi } ( s )$ is the intrinsic value function of agent $i$ , $I _ { - i | i } ^ { \pi }$ is the influence value, $\beta > 0$ is a weighting term. In this paper, we use the following notations:
+
+$$
+\begin{array} { r l } & { \displaystyle \tilde { r } _ { i } ( s , \pmb { a } ) = r ( s , \pmb { a } ) + u _ { i } ( s _ { i } , { a } _ { i } ) , } \\ & { \displaystyle V _ { i } ^ { \pi } ( \pmb { s } ) = V ^ { e x t , \pi } ( \pmb { s } ) + V _ { i } ^ { i n t , \pi } ( \pmb { s } ) , } \\ & { \displaystyle Q _ { i } ^ { \pi } ( \pmb { s } , \pmb { a } ) = \tilde { r } _ { i } ( \pmb { s } , \pmb { a } ) + \sum _ { s ^ { \prime } } T ( \pmb { s } ^ { \prime } | \pmb { s } , \pmb { a } ) V _ { i } ^ { \pi } ( \pmb { s } ^ { \prime } ) , } \end{array}
+$$
+
+where $u _ { i } ( s _ { i } , a _ { i } )$ is a curiosity-derived intrinsic reward, $\tilde { r } _ { i } ( s , { \pmb a } )$ is a sum of intrinsic and extrinsic rewards, $V _ { i } ^ { \pi } ( s )$ and $Q _ { i } ^ { \pi } ( s , { \pmb a } )$ here contain both intrinsic and extrinsic rewards.
+
+# 3.1 EXPLORATION VIA INFORMATION-THEORETIC INFLUENCE
+
+One critical problem in our learning framework presented above is to define the influence value $I$ . For simplicity, we start with a two-agent case. The first method we propose is to use mutual information between agents’ trajectories to measure one agent’s influence on other agents’ learning processes. Such mutual information can be defined as information gain of one agent’s state transition given the other’s state and action. Without loss of generality, we define it from the perspective of agent 1:
+
+$$
+M I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \sum _ { s , a , s _ { 2 } ^ { \prime } \in ( S , A , S _ { 2 } ) } p ^ { \pi } ( s , a , s _ { 2 } ^ { \prime } ) \left[ \log p ^ { \pi } ( s _ { 2 } ^ { \prime } | s , a ) - \log p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) \right] ,
+$$
+
+where $\pmb { s } = ( s _ { 1 } , s _ { 2 } )$ is the joint state, $\pmb { a } = ( a _ { 1 } , a _ { 2 } )$ is the joint action, and $S _ { i }$ and $A _ { i }$ are the random variables of state and action of agent $i$ subject to the distribution induced by the joint policy $\pi$ . So we define $I _ { 2 | 1 } ^ { \pi }$ as $M I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } )$ that captures transition interactions between agents. Optimizing this objective encourages agent 1 to visited critical points where it can influence the transition probability of agent 2. We call such an exploration method exploration via informationtheoretic influence (EITI).
+
+Optimizing $M I _ { 2 | 1 } ^ { \pi }$ with respect to the policy parameters $\theta _ { 1 }$ of agent 1 is a little bit challenging, because it is an expectation with respect to a distribution that depends on $\theta _ { 1 }$ . The gradient consists of two terms:
+
+$$
+\begin{array} { r l r } {  { \nabla _ { \theta _ { 1 } } M I ^ { \boldsymbol { \pi } } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \sum _ { \boldsymbol { s } , \boldsymbol { a } , \boldsymbol { s } _ { 2 } ^ { \prime } \in ( S , A , S _ { 2 } ) } \nabla _ { \theta _ { 1 } } ( p ^ { \boldsymbol { \pi } } ( \boldsymbol { s } , \boldsymbol { a } , \boldsymbol { s } _ { 2 } ^ { \prime } ) ) \log \frac { p ( s _ { 2 } ^ { \prime } | \boldsymbol { s } , \boldsymbol { a } ) } { p ^ { \pi } ( s _ { 2 } ^ { \prime } | \boldsymbol { s } _ { 2 } , \boldsymbol { a } _ { 2 } ) } } } \\ & { } & { + \sum _ { \boldsymbol { s } , \boldsymbol { a } , \boldsymbol { s } _ { 2 } ^ { \prime } \in ( S , A , S _ { 2 } ) } p ^ { \pi } ( \boldsymbol { s } , \boldsymbol { a } , \boldsymbol { s } _ { 2 } ^ { \prime } ) \nabla _ { \theta _ { 1 } } \log \frac { p ( s _ { 2 } ^ { \prime } | \boldsymbol { s } , \boldsymbol { a } ) } { p ^ { \pi } ( s _ { 2 } ^ { \prime } | \boldsymbol { s } _ { 2 } , \boldsymbol { a } _ { 2 } ) } . } \end{array}
+$$
+
+While the second term is an expectation over the trajectory and can be shown to be zero (see Appendix B.1), it is unwieldy to deal with the first term because it requires the gradient of the stationary distribution, which depends on the policies and the dynamics of the environment. Fortunately, the gradient can still be estimated purely from sampled trajectories by drawing inspiration from the proof of the policy gradient theorem (Sutton et al., 2000).
+
+The resulting policy gradient update is:
+
+$$
+\nabla _ { \boldsymbol { \theta } _ { 1 } } J _ { \boldsymbol { \theta } _ { 1 } } ( t ) = \left( \hat { R } _ { 1 } ^ { t } - \hat { V } _ { 1 } ^ { \pi } ( s _ { t } ) \right) \nabla _ { \boldsymbol { \theta } _ { 1 } } \log \pi _ { \boldsymbol { \theta } _ { 1 } } ( a _ { 1 } ^ { t } | s _ { 1 } ^ { t } )
+$$
+
+where $\hat { V } _ { 1 } ^ { \pi } ( s _ { t } )$ is an augmented value function of $\begin{array} { r } { \hat { R } _ { 1 } ^ { t } = \sum _ { t ^ { \prime } = t } ^ { h } \hat { r } _ { 1 } ^ { t ^ { \prime } } } \end{array}$ and
+
+$$
+\begin{array} { r } { \hat { r } _ { 1 } ^ { t } = r ^ { t } + u _ { 1 } ^ { t } + \beta \log \frac { p ( s _ { 2 } ^ { t + 1 } | s _ { 1 } ^ { t } , s _ { 2 } ^ { t } , a _ { 1 } ^ { t } , a _ { 2 } ^ { t } ) } { p ( s _ { 2 } ^ { t + 1 } | s _ { 2 } ^ { t } , a _ { 2 } ^ { t } ) } . } \end{array}
+$$
+
+The third term, which we call EITI reward, is 0 when the agents are transition-independent, i.e., when $p ( s _ { 2 } ^ { t + 1 } | s _ { 1 } ^ { t } , s _ { 2 } ^ { t } , a _ { 1 } ^ { t } , a _ { 2 } ^ { t } ) \stackrel { } { = } p ( s _ { 2 } ^ { t + 1 } | s _ { 2 } ^ { t } , a _ { 2 } ^ { t } )$ , and is positive when $s _ { 1 } ^ { t } , a _ { 1 } ^ { t }$ increase the probability 2   of agent 2 translating to $s _ { 2 } ^ { t + 1 }$ 2    . Therefore, the EITI reward is an intrinsic motivation that encourages agent 1 to visit more frequently the state-action pairs where it can influence the trajectory of agent 2. The estimation of $p ( s _ { 2 } ^ { t + 1 } | s _ { 1 } ^ { t } , s _ { 2 } ^ { \dot { t } } , a _ { 1 } ^ { t } , a _ { 2 } ^ { t } )$ and $p ( s _ { 2 } ^ { t + 1 } | s _ { 2 } ^ { t } , a _ { 2 } ^ { t } )$ are discussed in Appendix C. We assume that agents know the states and actions of other agents, but this information is only available during centralized training. When execution, agents only have access to their local observations.
+
+# 3.2 EXPLORATION VIA DECISION-THEORETIC INFLUENCE
+
+Mutual information characterizes the influence of one agent’s trajectory on that of the other and captures interactions between the transition functions of the agents. However, it does not provide the value of these interactions to identify interactions related to more internal and external rewards $( \tilde { r } )$ . To address this issue, we propose exploration via decision-theoretic influence (EDTI) based on a decision-theoretic measure of $I$ , called Value of Interaction (VoI), which disentangles both transition and reward influences. VoI is defined as the expected difference between the action-value function of one agent (e.g., agent 2) and its counterfactual action-value function without considering the state and action of the other agent (e.g., agent 1):
+
+$$
+{ V o I } _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \sum _ { s , a , s _ { 2 } ^ { \prime } \in ( S , A , S _ { 2 } ) } p ^ { \pi } ( s , a , s _ { 2 } ^ { \prime } ) \left[ Q _ { 2 } ^ { \pi } ( s , a , s _ { 2 } ^ { \prime } ) - Q _ { 2 | 1 } ^ { \pi , * } ( s _ { 2 } , a _ { 2 } , s _ { 2 } ^ { \prime } ) \right] ,
+$$
+
+where $Q _ { 2 } ^ { \pi } ( s , { \pmb a } , s _ { 2 } ^ { \prime } )$ is the expected rewards (including intrinsic rewards) of agent 2 defined as:
+
+$$
+Q _ { 2 } ^ { \pi } ( \pmb { s } , \pmb { a } , \pmb { s } _ { 2 } ^ { \prime } ) = \tilde { r } _ { 2 } ( \pmb { s } , \pmb { a } ) + \gamma \sum _ { \pmb { s } _ { 1 } ^ { \prime } } p ( \pmb { s } _ { 1 } ^ { \prime } | \pmb { s } , \pmb { a } , \pmb { s } _ { 2 } ^ { \prime } ) V _ { 2 } ^ { \pi } ( \pmb { s } ^ { \prime } ) ,
+$$
+
+and the counterfactual action-value function $Q _ { 2 } ^ { \pi , * }$ (also includes intrinsic and extrinsic rewards) can be obtained by marginalizing out the state and action of agent 1:
+
+$$
+\partial _ { 2 | 1 } ^ { \pi , * } ( s _ { 2 } , a _ { 2 } , s _ { 2 } ^ { \prime } ) = \sum _ { s _ { 1 } ^ { * } , a _ { 1 } ^ { * } } p ^ { \pi } ( s _ { 1 } ^ { * } , a _ { 1 } ^ { * } | s _ { 2 } , a _ { 2 } ) [ \tilde { r } _ { 2 } ( s _ { 1 } ^ { * } , s _ { 2 } , a _ { 1 } ^ { * } , a _ { 2 } ) + \gamma \sum _ { s _ { 1 } ^ { \prime } } p ( s _ { 1 } ^ { \prime } | s _ { 1 } ^ { * } , s _ { 2 } , a _ { 1 } ^ { * } , a _ { 2 } , s _ { 2 } ^ { \prime } ) V _ { 2 } ^ { \pi } ( s ^ { \prime } ) ] .
+$$
+
+Note that the definition of VoI is analogous to that of MI and the difference lies in that $\log p ( \cdot )$ measures the amount of information while $Q$ measures the action value. Although VoI can be obtained by learning $Q _ { 2 } ^ { \pi } ( s , \pmb { a } )$ and $Q _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } )$ and calculating the difference, we propose to explicitly marginalize out $s _ { 1 } ^ { * }$ and $a _ { 1 } ^ { * }$ utilizing the estimated model transition probability $p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } )$ and $p ( s _ { 2 } ^ { \prime } | s , \mathbf { a } )$ to get a more accurate value estimate (Feinberg et al., 2018). The performance of these two formulations are compared in the experiments.
+
+Value functions $Q$ and $V$ used in VoI contains both expected external rewards and internal rewards, which will not only encourage coordinated exploration by the influence between intrinsic rewards but also filter out meaningless interactions which can not lead to extrinsic reward after intrinsic reward diminishes. To facilitate the optimization of VoI, we rewrite it as an expectation over stateaction trajectories.
+
+$$
+V o I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \mathbb { E } _ { \tau } \left[ \tilde { r } _ { 2 } ( s , a ) - \tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } ) + \gamma \left( 1 - \frac { p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s , a ) } \right) V _ { 2 } ^ { \pi } ( s ^ { \prime } ) \right] ,
+$$
+
+where $\tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } )$ is the counterfactual immediate reward. The detailed proof is deferred to Appendix B.2. From this definition, we can intuitively see how VoI reflects the value of interactions. $\tilde { r } _ { 2 } ( s , a ) -$ $\tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } )$ and $1 - p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) / p ( s _ { 2 } ^ { \prime } | \bar { s , } a )$ measure the influence of agent 1 on the immediate reward and the transition function of agent 2, and $V _ { 2 } ^ { \pi } ( s ^ { \prime } )$ serves as a scale factor in terms of future value. Only when agent 1 and agent 2 are both transition- and reward-independent, i.e., when $p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , \bar { a _ { 2 } } ) = p ( s _ { 2 } ^ { \prime } | s , \pmb { a } )$ and $r _ { 2 } ^ { \pi } \bar { ( } s _ { 2 } , a _ { 2 } ) = r _ { 2 } ( s , a )$ will VoI equal to 0. In particular, maximizing
+
+VoI with respect to policy parameters $\theta _ { 1 }$ will lead agent 1 to meaningful interaction points, where $V _ { 2 } ^ { \pi } ( s ^ { \prime } )$ is high and $s _ { 1 } , a _ { 1 }$ can increase the probability that $\scriptstyle { \boldsymbol { s } } ^ { \prime }$ is reached.
+
+In this learning framework, agents initially explore the environment individually driven by its own curiosity, during which process they will discover potentially valuable interaction points where they can influence the transition function and (intrinsic) rewarding structure of each other. VoI highlights these points and encourages agents to visit these configurations more frequently. As intrinsic reward diminishes, VoI can gradually distinguish those interaction points which are necessary to get extrinsic rewards.
+
+# 3.2.1 POLICY OPTIMIZATION WITH VOI
+
+We want to optimize $J _ { \theta _ { i } }$ with respect to the policy parameters $\theta _ { i }$ , where the most cumbrous term is $\nabla _ { \theta _ { i } } V o I _ { - i | i }$ . For brevity, we can consider a two-agent case, e.g., optimizing $V o I _ { 2 | 1 }$ with respect to the policy parameters $\theta _ { 1 }$ . Directly computing the gradient $\nabla _ { \theta _ { 1 } } V o I _ { 2 | 1 }$ is not stable, because $V o I _ { 2 | 1 }$ contains policy-dependent functions $\tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } ) , p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } )$ , and $V _ { 2 } ^ { \pi } ( s ^ { \prime } )$ (see Eq. 12). To stabilize training , we use target functions to approximate these policy-dependent functions, which is a commonly used technique in deep RL (Mnih et al., 2015). With this approximation, we denote
+
+$$
+g _ { 2 } ( s , a ) = \tilde { r } _ { 2 } ( s , a ) - \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) + \gamma \sum _ { s ^ { \prime } } T ( s ^ { \prime } | s , a ) \left( 1 - \frac { p ^ { - } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s , a ) } \right) V _ { 2 } ^ { - } ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } ) .
+$$
+
+where $r _ { 2 } ^ { - } , p ^ { - }$ , and $V _ { 2 } ^ { - }$ are corresponding target functions. As these target functions are only periodically updated during the learning, their gradients over $\theta _ { 1 }$ can be approximately ignored. Therefore, from Eq. 12, we have
+
+$$
+\nabla _ { \theta _ { 1 } } V o I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) \approx \sum _ { s , a \in ( S , A ) } \left( \nabla _ { \theta _ { 1 } } p ^ { \pi } ( s , a ) \right) g _ { 2 } ( s , a ) .
+$$
+
+Similar to the calculation of $\nabla _ { \boldsymbol { \theta } _ { i } } M \boldsymbol { I }$ , we get the gradient at every step (see Appendix B.3 for proof):
+
+$$
+\begin{array} { r } { \nabla _ { \theta _ { 1 } } J _ { \theta _ { 1 } } ( t ) \approx \left( \hat { R } _ { 1 } ^ { t } - \hat { V } _ { 1 } ^ { \pi } ( s _ { t } ) \right) \nabla _ { \theta _ { 1 } } \log \pi _ { \theta _ { 1 } } ( a _ { 1 } ^ { t } | s _ { 1 } ^ { t } ) , } \end{array}
+$$
+
+where $\hat { V } _ { 1 } ^ { \pi } ( s _ { t } )$ is an augmented value function regressed towards $\begin{array} { r } { \hat { R } _ { 1 } ^ { t } = \sum _ { t ^ { \prime } = t } ^ { h } \hat { r } _ { 1 } ^ { t ^ { \prime } } } \end{array}$ and
+
+$$
+\hat { r } _ { 1 } ^ { t } = r ^ { t } + u _ { 1 } ^ { t } + \beta \left[ u _ { 2 } ^ { t } + \gamma \left( 1 - \frac { p ^ { - } ( s _ { 2 } ^ { t + 1 } | s _ { 2 } ^ { t } , a _ { 2 } ^ { t } ) } { p ( s _ { 2 } ^ { t + 1 } | s _ { 1 } ^ { t } , s _ { 2 } ^ { t } , a _ { 1 } ^ { t } , a _ { 2 } ^ { t } ) } \right) V _ { 2 } ^ { - } ( s _ { 1 } ^ { t + 1 } , s _ { 2 } ^ { t + 1 } ) \right] .
+$$
+
+We call ut2 + γ 1 − p−(st+12 |st2,at2)p(st+1|st ,st ,at ,at ) $\begin{array} { r } { u _ { 2 } ^ { t } + \gamma \left( 1 - \frac { p ^ { - } ( s _ { 2 } ^ { t + 1 } | s _ { 2 } ^ { t } , a _ { 2 } ^ { t } ) } { p ( s _ { 2 } ^ { t + 1 } | s _ { 1 } ^ { t } , s _ { 2 } ^ { t } , a _ { 1 } ^ { t } , a _ { 2 } ^ { t } ) } \right) V _ { 2 } ^ { - } ( s _ { 1 } ^ { t + 1 } , s _ { 2 } ^ { t + 1 } ) } \end{array}$ the EDTI reward.
+
+# 3.3 DISCUSSIONS
+
+Scale to Large Settings: For cases with more than two agents, the VoI of agent $i$ on other agents can be defined similarly to Eq. 9, which is annotated with $V o I _ { - i | i } ^ { \pi } ( S _ { - i } ^ { \prime } ; S _ { i } , A _ { i } | S _ { - i } , A _ { - i } )$ , where $S _ { - i }$ and $A _ { - i }$ are the state and action sets of all agents other than agent $i$ . In practice, agents interaction can often be decomposed to pairwise interaction so $V o I _ { - i | i } ^ { \pi } ( S _ { - i } ^ { \prime } ; S _ { i } , A _ { i } | S _ { - i } , A _ { - i } )$ is well approximated by the sum of values of pairwise value of interaction:
+
+$$
+V o I _ { - i | i } ^ { \pi } ( S _ { - i } ^ { \prime } ; S _ { i } , A _ { i } | S _ { - i } , A _ { - i } ) \approx \sum _ { j \in N , j \ne i } V o I _ { j | i } ^ { \pi } ( S _ { j } ^ { \prime } ; S _ { i } , A _ { i } | S _ { - i } , A _ { - i } ) .
+$$
+
+Relationship between EITI and EDTI: EITI and EDTI gradient updates are obtained by information- and decision-theoretical influence respectively. Therefore, it is nontrivial to derive that part of the EDTI reward is a lower bound of the EITI reward:
+
+$$
+1 - \frac { p ( s _ { - i } ^ { \prime } | s _ { - i } , a _ { - i } ) } { p ( s _ { - i } ^ { \prime } | s , a ) } \leq \log \frac { p ( s _ { - i } ^ { \prime } | s , a ) } { p ( s _ { - i } ^ { \prime } | s _ { - i } , a _ { - i } ) } , \forall s , a , s _ { - i } ^ { \prime }
+$$
+
+which easily follows given that $\log x \geq 1 - 1 / x$ for $\forall x > 0$ . This draws a connection between EITI and EDTI reward.
+
+Table 1: Baseline algorithms. The third column is the reward used to train the value function of PPO. $u _ { i }$ and $u _ { c e n }$ are curiosity about individual state $s _ { i }$ and global state $\pmb { s }$ , $\begin{array} { r l } { T _ { 1 } } & { { } = } \end{array}$ $\log \left( p ( s _ { - i } ^ { \prime } | \pmb { s } , \pmb { a } ) / p ( s _ { - i } ^ { \prime } | s _ { - i } , a _ { - i } ) \right)$ , $T _ { 2 } = 1 - p ( s _ { - i } ^ { \prime } | s _ { - i } , a _ { - i } ) / p ( s _ { - i } ^ { \prime } | s , a )$ , and $\begin{array} { r l } { \Delta Q _ { - i } ( s , \pmb { a } ) } & { { } = } \end{array}$ $Q _ { - i } ( s , { \pmb a } ) - Q _ { - i } ( s _ { - i } , a _ { - i } )$ . Social influence (Jaques et al., 2018) and COMA (Foerster et al., 2018) are augmented with curiosity.   
+
+<table><tr><td></td><td>Alg.</td><td>Reward</td><td>Description</td></tr><tr><td>Ours</td><td>EITI EDTI</td><td>r+ui+βTi r+ui+β(u-i+γTV-i)</td><td>Influence-theoretic influence Decision-theoretic influence</td></tr><tr><td>Other Exploration Methods</td><td>random cen dec cen_control</td><td>r r+ucen r+ui r+ucen</td><td>Pure PPO Decentralized PPO with cen curiosity Decentralized PPO with dec curiosity Centralized PPO with cen curiosity</td></tr><tr><td>Ablations</td><td>r_influence plusV shared_critic Q-Q</td><td>r+ui+βu-i r+ui+βV-i r+ucen r+ui + β△Q-i(s,a)</td><td>Disentangle reward interaction Use other agents’value functions PPO with shared V and cen curiosity EDTI without explicit counterfactual</td></tr><tr><td>Related Works</td><td>social COMA Multi</td><td></td><td>By Jaques et al. (2018) By Foerster et al. (2018) By Iqbal &amp; Sha (2019b)</td></tr></table>
+
+Comparing EDTI to Centralized Methods: Different from a centralized method which directly includes value functions of other agents in the optimization objective, (i.e., by setting total reward $\hat { r } _ { i } = r + u _ { i } + \beta ( u _ { - i } + \gamma V _ { - i } )$ , which is called plusV henceforth), the EDTI reward for agent $i$ disentangles its contributions to values of another agents using a counterfactual formulation. This difference is important for quantifying influence because the value of another agent does not just contain the contributions from agent $i$ , but also those of itself and third-party agents. Therefore, EDTI is a kind of intrinsic reward assignment. Our experiments in the next section will compare the performance of plusV against our methods, which verify the importance of the intrinsic reward assignment.
+
+# 4 EXPERIMENTAL RESULTS
+
+Our experiments aim to answer the following questions: (1) Can EITI and EDTI rewards capture interaction points? If they can, how do these points change throughout exploration? (2) Can exploiting these interaction points facilitate exploration and learning performance? (3) Can EDTI filter out interaction points that are not related to environmental rewards? (4) What if only reward influence between agents are disentangled? We evaluate our approach on a set of multi-agent tasks with sparse rewards based on a discrete version of multi-agent particle world environment (Lowe et al., 2017). PPO (Schulman et al., 2017) is used as the underlying algorithm. For evaluation, all experiments are carried out with 5 different random seeds and results are shown with $9 5 \%$ confidence interval. Demonstrative videos1 are available online.
+
+Baselines We compare our methods with various baselines shown in Table 1. In particular, we carry out the following ablation studies: i) r influence disentangles immediate reward influence between agents, (derivation of the associated augmented reward can be found in Appendix B.4. Reward influence in long term is not considered because it inevitably involves transition interactions) ii) PlusV as described in Sec. 3.3. iii) Shared critic uses decentralized PPO agents with shared centralized value function and thus is a cooperative version of MADDPG (Lowe et al., 2017) augmented with intrinsic reward of curiosity. iv) Q-Q is similar to EDTI but without explicit counterfactual formulation, as described in Sec. 3.2. We also note that EITI is an ablation of EDTI which considers transition interactions. PlusV, shared critic, Q-Q, and cen control have access to global or other agents’ value functions during training. When execution, all the methods except cen control only require local state.
+
+![](images/d4cd7361d2c5f14c3d0fb9d6ca0617e77f21d9c42e6d5847123a48086a4188cf.jpg)  
+Figure 1: Didactic examples. Left: task Pass. Two agents starting at the upper-left corner are only rewarded when both of them reach the other room through the door, which will open only when at least one of the switches is occupied by one or more agents. Middle: Secret-Room. An extension of Pass with 4 rooms and switches. When the switch 1 is occupied, all the three doors turn open. And the three switches on the right only control the door of its room. The agents need to reach the upper right room to achieve any reward. Right: comparison of our methods with ablations on Pass.
+
+We present two didactic examples of multi-agent cooperation tasks with sparse reward to explain how EITI and EDTI work. The first didactic example consists of a $3 0 \times 3 0$ maze with two rooms and a door with two switches (Fig. 1 left). In the optimal strategy, one agent should first step on switch 1 to help the other agent pass the door, and then the agent that has already reached the right half should further go to switch 2 to bring the remaining agent in. There are two pairs of interaction points in this task: (switch 1, door) and (switch 2, door), i.e., transition probability of the agent near door is determined by whether another agent is on one of the switch.
+
+Fig. 1-right and Fig. 2-top show the learning curves of our methods and all the baselines, among which EITI, EDTI, r influence, Multi, and centralized control can learn the winning strategy and ours learn much more efficiently. Fig. 2-bottom gives a possible explanation why our methods work. EITI and EDTI rewards successfully highlight the interaction points (before 100 and 2100 updates, respectively). Agents are encouraged to explore these configurations more frequently and thus have better chance to learn the goal strategy. EDTI reward considers the value function of the other agent, so it converges slower than the EITI reward. In contrast, directly adding the other agent’s intrinsic rewards and value functions is noisy (see ”plusV reward”) and confuses the agent because these contain the effect of the other agent’s exploration. As for centralized control, global curiosity encourages agents to try all possible configurations, so it can find environmental rewards in most tasks. However, visiting all configurations without bias renders it inefficient – external rewards begin to dominate the behaviors of agents after 7000 updates even with the help of centralized learning algorithm. Our methods use the same information as centralized exploration but take advantages of agents’ interactions to accelerate exploration.
+
+In order to evaluate whether EDTI can help filter out noisy interaction points and accelerate exploration, we conduct experiments in a second didactic task (see Fig. 1 middle). It is also a navigation task in a $2 5 \times 2 5$ maze where agents are rewarded for being in a goal room. However, in this experiment, we consider a case where there are four rooms and the upper right one is attached to reward. This task contains 6 pairs of interaction points (switch 1 with each of the doors, each switch with the door of the same room), but only two of them are related to external rewards, i.e., (switch 1, door 1) and (switch 2, door 1). As Fig. 3-right shows, EITI agents treat three doors equally even after 7400 updates (see Fig. 3 right, 7400 updates, top row). In comparison, although EDTI reward suffers from noise in the beginning, it clearly highlight two pairs of valuable interaction points (see Fig. 3 right, 7400 updates, bottom row) as intrinsic reward diminishes. This can explain why EDTI outperforms EITI (Fig. 3 left).
+
+![](images/481a75fa7afe93608ea74ad7f887a6b4d8d62ed0b0b57a4ab78862de137af13a.jpg)  
+Figure 2: Development of performance of our methods compared to baselines and intrinsic reward terms of EITI, EDTI, and plusV over the training period of 9000 PPO updates segmented into three phases. ”Team Reward” shows averaged team reward gained in a episode, with a maximum of 1000. It shows that only EITI, EDTI, and centralized control and Multi can learn the strategy during this stage. ”EITI reward”, ”EDTI reward”, and ”plusV reward” demonstrate the evolving of corresponding intrinsic rewards.
+
+![](images/8b5459f41659ea240e58c5b268e5e06ac9cde7dfc6ca53a266ce8fab093c04dd.jpg)
+
+![](images/ad5d67074689589b545ef750ccb90f18378f9f0c2ee0651d32c85600d59a7136.jpg)  
+Figure 3: Left: performance comparison between EDTI and EITI on Secret-Room over 7400 PPO updates. Right: EITI and EDTI terms of two agents after 100, 2900, and 7400 updates.   
+Figure 4: Comparison of our methods against ablations for Push-Box, Island, and Large-Island. Comparison with baselines is shown in Fig. 8 in Appendix D.
+
+# 4.2 EXPLORATION IN COMPLEX TASKS
+
+Next, we evaluate the performance of our methods on more complex tasks. To this end, we use three sparse reward cooperative multi-agent tasks depicted in Fig. 7 of Appendix D and analyzed below. Details of implementation and experiment settings are also described in Appendix D.
+
+Push-Box: A $1 5 \times 1 5$ room is populated with 2 agents and 1 box. Agents need to push the box to the wall in 300 environment steps to get a reward of 1000. However, the box is so heavy that only when two agents push it in the same direction at the same time can it be moved a grid. Agents need to coordinate their positions and actions for multiple steps to earn a reward. The purpose of this task is to demonstrate that EITI and EDTI can explore long-term cooperative strategy.
+
+Island: This task is a modified version of the classic Stag Hunt game (Peysakhovich & Lerer, 2018) where two agents roam a $1 0 \times 1 0$ island populated with 9 treasures and a random walking beast for 300 environment steps. Agents can collect a treasure by stepping on it to get a team reward of 10 or, by attacking the beast within their attack range, capture it for a reward of 300. The beast would also attack the agents when they are too close. The beast and agent have a maximum energy of 8 and 5 respectively, which will be subtracted by 1 every time attacked. Therefore, an agent is too weak to beat the beast alone and they have to cooperate. In order to learn optimal strategy in this task, one method has to keep exploring after sub-optimal external rewards are found.
+
+Large-Island: Similar to Island but with more agents (4), more treasures (16), and a beast with more energy (16) and a higher reward (600) for being caught. This task aims to demonstrate feasibility of our methods in cases with more than 2 agents.
+
+Push-Box requires agents to take coordinated actions at certain positions for multiple steps to get rewarded. Therefore, this task is particularly challenging and all the baselines struggle to earn any reward (Fig. 4 left and Fig. 8 left). Our methods are considerably more successful because interaction happens when the box is moved – agents remain unmoved when they push the box alone but will move by a grid if push it together. In this way, EITI and EDTI agents are rewarded intrinsically to move the box and thus are able to quickly find the optimal policy.
+
+In the Island task, collecting treasures is a easily-attainable local optimal. However, efficient treasures collecting requires the agents to spread on the island. This leads to a situation where attempting to attack the beast seems a bad choice since it is highly possible that agents will be exposed to the beast’s attack alone. They have to give up profitable spreading strategy and take the risk of being killed to discover that if they attack the beast collectively for several timesteps, they will get much more rewards. Our methods help solve this challenge by giving agents intrinsic incentives to appear together in the attack range of the beast, where they have indirect interactions (health is part of the state and it decreases slower when the two are attacked alternatively). Fig. 9 in Appendix D demonstrates that our methods learn to catch the beast quickly, and thus have better performance (Fig. 8 right).
+
+Finally, outperformance of our methods on Large-Island proves that they can successfully handle cases with more than two agents.
+
+In summary, both of our methods are able to facilitate effective exploration on all the tasks by exploiting interactions. EITI outperforms EDTI in scenarios where all interaction points align with extrinsic rewards. On other tasks, EDTI performs better than EITI due to its ability to filter out interaction points that can not lead to more values.
+
+We also study EDTI with only intrinsic rewards, discussion and results are included in Appendix A.
+
+# 5 RELATED WORKS
+
+Single-agent exploration achieves conspicuous success recently. Provably efficient methods are proposed, such as upper confidence bound (UCB) (Jaksch et al., 2010; Azar et al., 2017; Jin et al., 2018) and posterior sampling for reinforcement learning (PSRL) (Strens, 2000; Osband et al., 2013; Osband & Van Roy, 2016; Agrawal & Jia, 2017). Given that these methods do not scale well to large or continuous settings, another line of research has been focusing on curiosity-driven exploration (Schmidhuber, 1991; Chentanez et al., 2005; Oudeyer et al., 2007; Barto, 2013; Bellemare et al., 2016; Pathak et al., 2017; Ostrovski et al., 2017), and have shown impressive results (Burda et al., 2019; 2018; Hyoungseok Kim, 2019). In addition, methods based on variational information maximization (Houthooft et al., 2016; Barron et al., 2018) and mutual information (Rubin et al., 2012; Still & Precup, 2012; Salge et al., 2014; Mohamed & Rezende, 2015; Hyoungseok Kim, 2019) have been proposed for single-agent intrinsically motivated exploration.
+
+Although multi-agent reinforcement learning (MARL) has been making significant progresses in recent years (Foerster et al., 2018; Lowe et al., 2017; Wen et al., 2019; Iqbal & Sha, 2019a; Sunehag et al., 2018; Son et al., 2019; Rashid et al., 2018), less attention has been drawn to multi-agent exploration. Dimakopoulou & Van Roy (2018) and Dimakopoulou et al. (2018) propose posterior sampling methods for exploration of concurrent reinforcement learning in coverage problems, Bargiacchi et al. (2018) presents a multi-agent upper confidence exploration method for repeated single-stage problems, and Iqbal & Sha (2019b) investigates methods to combine several decentralized curiosity-driven exploration strategies. All these works focus on transition-independent settings. Another Bayesian exploration approach has been proposed for learning in stateless repeated games (Chalkiadakis & Boutilier, 2003). In contrast, this paper focuses on more general multi-agent sequential decision making problems with complex reward dependencies and transition interactions among agents.
+
+In the literature of MARL, COMA (Foerster et al., 2018) shares some similarity with our decisiontheoretic EDTI approach in that both of them use the idea of counterfactual formulations. However, they are quite different in terms of definition and optimization: (1) conceptually, EDTI measures the influence of one agent on the value functions of other agents, while COMA quantifies individual contribution to the team value; (2) EDTI is defined on counterfactual Q-value over state-action pairs of other agents given its own state-action pair, while COMA uses the counterfactual Q-value just over its own action without considering state information, which is critical for exploration; (3) we explicitly derive the gradients for optimizing EDTI influence for coordinated exploration in the policy gradient framework, which provides more accurate feedback, while COMA uses the counterfactual Q value as a critic. Another line of relevant works (Oliehoek et al., 2012; de Castro et al., 2019) propose influence-based abstraction to predict influence sources to help local decision making of agents. In contrast, this paper presents two novel approaches that quantify and maximize the influence between agents for enabling coordinated multi-agent exploration.
+
+In addition, some previous MARL work has also studied intrinsic rewards. One notably relevant work is Jaques et al. (2018), which models the social influence of one agent on other agents’ policies. In contrast, EITI measures the influence of one agent on the transition dynamics of other agents. Accompanying this distinction, EITI includes states of agents in the calculation of influence while social influence dos not. Apart from that, the optimization methods are also different – we directly derive the gradients of mutual information and incorporate its optimization in the policy gradient framework, while Jaques et al. (2018) adds social influence reward to the immediate environmental reward for training policies. Hughes et al. (2018) proposes an inequality aversion reward for learning in intertemporal social dilemmas. Strouse et al. (2018) uses mutual information between goal and states or actions as an intrinsic reward to train the agent to share or hide their intentions.
+
+# 6 CLOSING REMARKS
+
+In this paper, we study the multi-agent exploration problem and propose two influence-based methods that exploits the interaction structure. These methods are based on two interaction measures, MI and Value of Interaction (VoI), which respectively measure the amount and value of one agent’s influence on the other agents’ exploration processes. These two measures can be best regraded as exploration bonus distribution. We also propose an optimization method in the policy gradient framework, which enables agents to achieve coordinated exploration in a decentralized manner and optimize team performance.
+
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+# APPENDIX
+
+# A INTRINSIC EDTI
+
+Value of interaction (VoI) captures both transition and reward influence among agents, and it facilitates coordinated exploration by encouraging interactions. VoI contains influence of both intrinsic and extrinsic rewards. Since single-agent literature has studied purely curiosity-driven learning and gets cutting-edge performance (Burda et al., 2019), it is interesting to investigate the performance of VoI given only intrinsic rewards.
+
+Intuitively, intrinsic VoI distributes individual curiosity among team members and facilitates exploration by encouraging agents to help each other to reach under-explored states. Specifically, we use the following objective:
+
+$$
+J _ { \theta _ { i } } [ \pi _ { i } | \pi _ { - i } , p _ { 0 } ] \equiv V ^ { e x t , \pi } ( s _ { 0 } ) + V _ { i } ^ { i n t , \pi } ( s _ { 0 } ) + \beta \cdot V o I _ { - i | i } ^ { i n t , \pi } .
+$$
+
+The corresponding augmented reward is:
+
+$$
+\hat { r } _ { 1 } ^ { t } = r _ { t } + u _ { 1 } ^ { t } + \beta \left[ u _ { 2 } ^ { t } + \gamma \left( 1 - \frac { p ^ { - } ( s _ { 2 } ^ { t + 1 } | s _ { 2 } ^ { t } , a _ { 2 } ^ { t } ) } { p ( s _ { 2 } ^ { t + 1 } | s _ { 1 } ^ { t } , s _ { 2 } ^ { t } , a _ { 1 } ^ { t } , a _ { 2 } ^ { t } ) } \right) V _ { 2 } ^ { i n t , - } ( s _ { 1 } ^ { t + 1 } , s _ { 2 } ^ { t + 1 } ) \right]
+$$
+
+We use this method (intrinsic EDTI) to train the agents on Pass, Secret-Room, Push-Box, and Island and show the results in Fig. 5.
+
+# B MATHEMATICAL DETAILS
+
+# B.1 GRADIENT OF MUTUAL INFORMATION
+
+To encourage agents to exert influence on transitions of other agents, we optimize mutual information between agent’s trajectories. In particular, in the following, we show that term 2 in Eq. 6 is always zero.
+
+$$
+\begin{array} { r c l } { \tau _ { \mathrm { S } } } & { - } & { \displaystyle \sum _ { \alpha \in \mathbb { Z } _ { h } } \frac { \tau _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } } \\ & { - } & { \displaystyle \sum _ { \alpha \in \mathbb { Z } _ { h } } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } } \\ & { = } & { \displaystyle \sum _ { \alpha \in \mathbb { Z } _ { h } } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } } \\ & { - } &  \displaystyle \sum _ { \alpha \in \mathbb { Z } _ { h } } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \frac { \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ { h } ) } { \sigma ( \Delta \alpha _ { h } , \zeta _ { h } ) } \\ & { - } &  \displaystyle \sum _ { \alpha \in \mathbb { Z } _ { h } } \frac  \epsilon _ { \alpha } ^ { \alpha } ( \Delta \alpha _ { h } , \zeta _ \end{array}
+$$
+
+![](images/5f8718e5e102d7ac4149eae8d4e67da63837b1c9dd7986f23dea24c38b058917.jpg)  
+Figure 5: Performance of intrinsic EDTI in comparison with EITI and EDTI on Pass, Secret-Room, Push-Box, and Island.
+
+$$
+\begin{array} { r l } { = } & { { } - \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } ^ { * } } p ( s _ { 1 } ^ { * } | s _ { 2 } , a _ { 2 } ) \nabla _ { \theta _ { 1 } } \sum _ { a _ { 1 } ^ { * } } \pi _ { \theta _ { 1 } } ( a _ { 1 } ^ { * } | s _ { 1 } ^ { * } ) } \\ { = } & { { } - \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } ^ { * } } p ( s _ { 1 } ^ { * } | s _ { 2 } , a _ { 2 } ) \nabla _ { \theta _ { 1 } } 1 } \\ { = } & { { } 0 } \end{array}
+$$
+
+# B.2 DEFINITION OF Value of Interaction
+
+To capture both transition and reward interactions between agents and thereby achieve intrinsic reward distribution, we propose a decision-theoretic measure called Value of Interaction. We start from 2-agent cases and the following theorem gives the definition of $V o I _ { 2 | 1 }$ in the form of an expectation over trajectories, which is especially helpful in the derivation of the EDTI policy gradient update shown Eq. 15.
+
+Theorem 1. Value of Interaction of agent 1 on agent 2 is:
+
+$$
+V o I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \mathbb { E } _ { \tau } \left[ \tilde { r } _ { 2 } ( s , a ) - \tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } ) + \gamma \left( 1 - \frac { p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s , a ) } \right) V _ { 2 } ^ { \pi } ( s ^ { \prime } ) \right] ,
+$$
+
+where $\tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } )$ is the counterfactual immediate reward.
+
+$V o I _ { 2 | 1 }$ can be defined similarly. To lighten notation in the proof, we define
+
+$$
+V _ { 2 } ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) = \sum _ { s _ { 1 } ^ { \prime } } p ( s _ { 1 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } , s _ { 2 } ^ { \prime } ) V _ { 2 } ^ { \pi } ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } )
+$$
+
+$$
+\begin{array} { c } { { \displaystyle { \tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } ) = \sum _ { s _ { 1 } ^ { * } , a _ { 1 } ^ { * } } p ^ { \pi } ( s _ { 1 } ^ { * } , a _ { 1 } ^ { * } | s _ { 2 } , a _ { 2 } ) \tilde { r } _ { 2 } ( s _ { 1 } ^ { * } , s _ { 2 } , a _ { 1 } ^ { * } , a _ { 2 } ) , } } } \\ { { \displaystyle { V _ { 2 } ^ { \pi , * } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) = \sum _ { s _ { 1 } ^ { * } , a _ { 1 } ^ { * } } p ^ { \pi } ( s _ { 1 } ^ { * } , a _ { 1 } ^ { * } | s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } ^ { \prime } } p ( s _ { 1 } ^ { \prime } | s _ { 1 } ^ { * } , s _ { 2 } , a _ { 1 } ^ { * } , a _ { 2 } , s _ { 2 } ^ { \prime } ) V _ { 2 } ^ { \pi } ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } ) . } } } \end{array}
+$$
+
+We first prove Lemma 1, which is used in the proof of Theorem 1.
+
+# Lemma 1.
+
+$$
+\begin{array} { r l r } {  { \sum _ { s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } } \atop 0 } p ^ { \pi } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \gamma \sum _ { s _ { 2 } ^ { \prime } } p ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) V _ { 2 } ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) }  & { ( 3 7 ) } \\ & { = } & { \sum _ { s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } } p ^ { \pi } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \gamma \sum _ { s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } } T ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \cdot \frac { p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) } V _ { 2 } ^ { \pi } ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } ) . } \end{array}
+$$
+
+Proof.
+
+$$
+\begin{array} { r l } & { \quad \sum _ { j = 1 } ^ { N } \frac { \partial ^ { j } } { \partial x _ { j } } g ^ { ( j ) } ( x _ { j + 1 } , y _ { j + 1 } , x _ { j } ) \geq \frac { \gamma _ { j } ^ { j } } { N } g ^ { ( j ) } ( x _ { j + 1 } , y _ { j } , x _ { j } ) < \beta _ { j } ^ { j } < \zeta _ { j } , } \\ { = } & { \quad \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } \left( g ^ { ( j ) } ( x _ { j + 1 } , y _ { j } , x _ { j } ) + \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } g ^ { ( j ) } ( x _ { j } , y _ { j } , x _ { j } ) \right) , } \\ { = } & { \quad \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } \left( g ^ { ( j ) } ( x _ { j } , y _ { j } , x _ { j } ) + \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } \right) , } \\ { = } & { \quad \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } \left( g ^ { ( j ) } ( x _ { j } , y _ { j } , x _ { j } ) \right) > \frac { \gamma _ { j } ^ { j } } { N } g ^ { ( j ) } ( x _ { j } , y _ { j } , x _ { j } ) < \beta _ { j } ^ { j } , } \\ { = } & { \quad \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } g ^ { ( j ) } ( x _ { j } , y _ { j + 1 } , x _ { j } ) > \frac { \gamma _ { j } ^ { j } } { N } g ^ { ( j ) } ( x _ { j } , y _ { j + 1 } , x _ { j } ) < \beta _ { j } ^ { j } , } \\ { = } & { \quad \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } g ^ { ( j ) } ( x _ { j } , y _ { j + 1 } , x _ { j } ) > \frac { \gamma _ { j } ^ { j } } { N } g ^ { ( j ) } ( x _ { j } , y _ { j + 1 } , x _ { j } ) , } \\ &  \quad \sum _ { j = 1 } ^ { N } \frac { \beta ^ { j } } { N } \left( g ^ { ( j ) } ( x _ { j } , y _ { j + 1 } , x _ { j } ) \right) > \frac { \gamma _ { j } ^ { j } } \end{array}
+$$
+
+We now give the proof of Theorem 1:
+
+Proof.
+
+$$
+\begin{array} { r l } & { V o I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) } \\ { = } & { \underset { s , a , s _ { 2 } \in ( S , A , S _ { 2 } ) } { \sum } p ^ { \pi } ( s , a , s _ { 2 } ^ { \prime } ) \left[ Q _ { 2 } ^ { \pi } ( s , a , s _ { 2 } ^ { \prime } ) - Q _ { 2 | 1 } ^ { \pi , * } ( s _ { 2 } , a _ { 2 } , s _ { 2 } ^ { \prime } ) \right] } \\ { = } & { \underset { s _ { 1 } , s _ { 2 } , a , 1 , a _ { 2 } } { \sum } p ^ { \pi } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \big ( \widetilde { r } _ { 2 } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) - \widetilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } ) + } \\ & { \underset { s _ { 1 } , s _ { 2 } , a , 1 , a _ { 2 } } { \sum } p ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \big ( \widetilde { V } _ { 2 } ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) - \widetilde { V } _ { 2 } ^ { \pi , * } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) \big ) } \\ & { \overset { } { \underset { s _ { 2 } ^ { \prime } } { \sum } } p ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \big ( \widetilde { V } _ { 2 } ^ { \pi } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) - \widetilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) \big ) } \\ { = } &  \underset { s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } } { \sum } p ^ { \pi } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \big ( \widetilde { r } _ { 2 } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } )  \end{array}
+$$
+
+$$
+\begin{array} { r l } & { \gamma \displaystyle \sum _ { s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } } T ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) ( 1 - \frac { p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) } ) V _ { 2 } ^ { \pi } ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } ) ) \ ( { \bf L e m m a \ 1 } ) } \\ { = } & { \mathbb { E } _ { \tau } \left[ \tilde { r } _ { 2 } ( s , a ) - \tilde { r } _ { 2 } ^ { \pi } ( s _ { 2 } , a _ { 2 } ) + \gamma \left( 1 - \frac { p ^ { \pi } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s , a ) } \right) V _ { 2 } ^ { \pi } ( s ^ { \prime } ) \right] . } \end{array}
+$$
+
+# B.3 CALCULATING GRADIENT OF VOI
+
+In order to optimize $V o I$ with respect to the parameters of agent policy, in Sec. 3.2.1, we propose to use target function and get:
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta _ { 1 } } V o I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) \approx \displaystyle \sum _ { s , a \in ( S , A ) } \left( \nabla _ { \theta _ { 1 } } p ^ { \pi } ( s , a ) \right) [ \tilde { r } _ { 2 } ( s , a ) - \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) + } \\ & { \qquad \quad \qquad \ \gamma \displaystyle \sum _ { s ^ { \prime } } T ( s ^ { \prime } | s , a ) \left( 1 - \frac { p ^ { - } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) } { p ( s _ { 2 } ^ { \prime } | s , a ) } \right) V _ { 2 } ^ { - } ( s _ { 1 } ^ { \prime } , s _ { 2 } ^ { \prime } ) ] . } \end{array}
+$$
+
+We prove that $\begin{array} { r l } { { } } & { { } \sum _ { s , a } \left( \nabla _ { \theta _ { 1 } } p ^ { \pi } ( s , a ) \right) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) } \end{array}$ is $0$ in the following lemma.
+
+Lemma 2.
+
+$$
+\sum _ { s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } } \left( \nabla _ { \theta _ { 1 } } p ^ { \boldsymbol { \pi } } ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) \right) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) = 0 .
+$$
+
+Proof. Similar to the way that policy gradient theorem was proved by Sutton et al. (2000),
+
+$$
+\begin{array} { r l }  \underset  \{ \mathbf { x } _ { h } ^ { 2 } , \mathbf { x } _ { h } ^ { 2 } \geq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf { x } _ { h } ^ { 2 } \leq 0 , \mathbf  x  \end{array}
+$$
+
+$$
+\begin{array} { r l } { = } & { { } \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } , a _ { 1 } } p ^ { \pi } ( s _ { 1 } | s _ { 2 } , a _ { 2 } ) \left( \nabla _ { \theta _ { 1 } } \pi ( a _ { 1 } | s _ { 1 } , s _ { 2 } ) \right) } \\ { = } & { { } \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } } p ^ { \pi } ( s _ { 1 } | s _ { 2 } , a _ { 2 } ) \sum _ { a _ { 1 } } \left( \nabla _ { \theta _ { 1 } } \pi ( a _ { 1 } | s _ { 1 } , s _ { 2 } ) \right) } \\ { = } & { { } \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } } p ^ { \pi } ( s _ { 1 } | s _ { 2 } , a _ { 2 } ) \left( \nabla _ { \theta _ { 1 } } \sum _ { a _ { 1 } } \pi ( a _ { 1 } | s _ { 1 } , s _ { 2 } ) \right) } \\ { = } & { { } \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } } p ^ { \pi } ( s _ { 1 } | s _ { 2 } , a _ { 2 } ) \left( \nabla _ { \theta _ { 1 } } \sum _ { a _ { 1 } } \pi ( a _ { 1 } | s _ { 1 } , s _ { 2 } ) \right) } \\ { = } & { { } \displaystyle \sum _ { s _ { 2 } , a _ { 2 } } p ^ { \pi } ( s _ { 2 } , a _ { 2 } ) \tilde { r } _ { 2 } ^ { - } ( s _ { 2 } , a _ { 2 } ) \sum _ { s _ { 1 } } p ^ { \pi } ( s _ { 1 } | s _ { 2 } , a _ { 2 } ) \left( \nabla _ { \theta _ { 1 } } 1 \right) } \end{array}
+$$
+
+# B.4 IMMEDIATE REWARD INFLUENCE
+
+Similar to MI and $V o I$ , we can define influence of agent 1 on the immediate rewards of agent 2 as:
+
+$$
+R I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \sum _ { s , a \in ( S , A ) } p ^ { \pi } ( s , a ) [ \tilde { r } _ { 2 } ( s , a ) - \tilde { r } _ { 2 } ( s _ { 2 } , a _ { 2 } ) ] .
+$$
+
+Use Lemma 2, we can get:
+
+$$
+\nabla _ { \theta _ { 1 } } R I _ { 2 | 1 } ^ { \pi } ( S _ { 2 } ^ { \prime } ; S _ { 1 } , A _ { 1 } | S _ { 2 } , A _ { 2 } ) = \sum _ { s , a \in ( S , A ) } \nabla _ { \theta _ { 1 } } ( p ^ { \pi } ( s , a ) ) \tilde { r } _ { 2 } ( s , a ) .
+$$
+
+Now we have
+
+$$
+\begin{array} { r } { \nabla _ { \boldsymbol { \theta } _ { 1 } } J _ { \boldsymbol { \theta } _ { 1 } } ( t ) \approx \left( \hat { R } _ { 1 } ^ { t } - \hat { V } _ { 1 } ^ { \pi } ( \boldsymbol { s } _ { t } ) \right) \nabla _ { \boldsymbol { \theta } _ { 1 } } \log \pi _ { \boldsymbol { \theta } _ { 1 } } ( a _ { 1 } ^ { t } | \boldsymbol { s } _ { 1 } ^ { t } ) , } \end{array}
+$$
+
+where $\hat { V } _ { 1 } ^ { \pi } ( s _ { t } )$ is an augmented value function of $\begin{array} { r } { \hat { R } _ { 1 } ^ { t } = \sum _ { t ^ { \prime } = t } ^ { h } \hat { r } _ { 1 } ^ { t ^ { \prime } } } \end{array}$ and
+
+$$
+\hat { r } _ { 1 } ^ { t } = r ^ { t } + u _ { 1 } ^ { t } + \beta u _ { 2 } ^ { t } .
+$$
+
+# C ESTIMATION OF CONDITIONAL PROBABILITIES
+
+To quantify interdependence among exploration processes of agents, we use mutual information and value of interaction. Calculations of MI and VoI need estimation of $p ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } )$ and $p ( s _ { 2 } ^ { \prime } | s , \mathbf { a } )$ . In practice, we track the empirical frequencies $p _ { e m p } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } )$ and $p _ { e m p } ( s _ { 2 } ^ { \prime } | s , \pmb { a } )$ and substitute them for the corresponding terms in Eq. 8 and 16.
+
+Estimating $p _ { e m p } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } )$ and $p _ { e m p } ( s _ { 2 } ^ { \prime } | s , \pmb { a } )$ is one obstacle to the scalability of our method, we now discuss how to solve this problem. When the state and action space is small, we can use hash table to implement Monte Carlo method (MC) for estimating the distributions accurately. In the MC sampling, we count from the samples the state frequencies $\begin{array} { r } { p ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } ) \equiv \frac { N ( s _ { 2 } ^ { \prime } , s _ { 2 } , a _ { 2 } ) } { N ( s _ { 2 } , a _ { 2 } ) } } \end{array}$ and $\begin{array} { r } { p ( s _ { 2 } ^ { \prime } | \pmb { s } , \pmb { a } ) \equiv \frac { N ( s _ { 2 } ^ { \prime } , s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) } { N ( s _ { 1 } , s _ { 2 } , a _ { 1 } , a _ { 2 } ) } } \end{array}$ , where $N ( \cdot )$ is the number of times each state-action pair was visited during the learning process. When the problem space becomes large, MC consumes large memory in practice. As an alternative, we adopt variational inference (Fox & Roberts, 2012) to learn variational distributions $q _ { \xi _ { 1 } } ( s _ { 2 } ^ { \prime } | s _ { 2 } , a _ { 2 } )$ and $q _ { \xi _ { 2 } } ( s _ { 2 } ^ { \prime } | s , \mathbf { a } )$ , parameterized via neural networks with parameters $\xi _ { 1 }$ and $\xi _ { 2 }$ , by optimizing the evidence lower bound. In Fig. 6, we show the performance of EDTI estimated using variational inference and the changing of associated EDTI rewards on Pass during 9000 PPO updates. Variational inference introduces some noise in EDTI rewards estimation and thus requires slightly more steps to learn the true probability and the strategy. However, estimating using MC sampling consumes 1.6G memory to save the hash table with 100M items each agent while variational inference needs a three-layer fully connected network with 74800 parameters occupying about 0.60M memory. This results highlights the feasibility of estimating EITI and EDTI rewards using variational inference in problem with large state-action space.
+
+![](images/e42a743cf675536d9ebc5b6748ee5414965394c6bec6ac5cf94f2b4c2e5dc060.jpg)  
+Figure 6: Left: Performance of EDTI (vi) (EDIT estimated using variational inference) compared with EITI and EDTI estimated using MC sampling. Others: Development of EDTI (vi) rewards during exploration process. Top row: EDTI (vi) rewards of agent 1; bottom row: EDTI (vi) rewards of agent 2.
+
+Table 2: The scaling weights for different intrinsic reward terms in various tasks. $\beta _ { \mathrm { T } }$ is the weight of term $T _ { 1 }$ (see Table 1). $\beta _ { \mathrm { i n t } }$ and $\beta _ { \mathrm { e x t } }$ are scaling factors to combine $r$ and $u _ { i }$ in $\tilde { r }$ . $u _ { - i }$ in r influence is scaled by $\beta _ { \mathrm { r } }$ while $V _ { - i } ^ { i n t }$ and $V _ { - i } ^ { e x t }$ in plusV are respectively scaled by $\beta _ { \mathrm { i n t } } ^ { \mathrm { p l u s V } }$ and $\beta _ { \mathrm { e x t } } ^ { \mathrm { p l u s V } }$   
+
+<table><tr><td>Task</td><td>n</td><td>β</td><td>βint</td><td>βext</td><td>β</td><td></td><td>pouV</td></tr><tr><td>Pass</td><td>10.</td><td>10</td><td>1.</td><td>0.1</td><td>1.</td><td>0.1</td><td>0.01</td></tr><tr><td>Secret-Room</td><td>10.</td><td>10</td><td>1.</td><td>0.1</td><td></td><td></td><td>一</td></tr><tr><td>Push-Box</td><td>1.</td><td>100.</td><td>100.</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.01</td></tr><tr><td>Island</td><td>1.</td><td>10</td><td>10.</td><td>0.5</td><td>0.1</td><td>0.1</td><td>0.01</td></tr><tr><td>Large-Island</td><td>1.</td><td>10</td><td>1.</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.01</td></tr></table>
+
+# D IMPLEMENTATION DETAILS
+
+D.1 NETWORK ARCHITECTURE, HYPERPARAMETERS, AND INFRASTRUCTURE
+
+We base our framework on OpenAI implementation of PPO2 (Dhariwal et al., 2017) and use its default parameters to carry out all the experiments. We train our models on an NVIDIA RTX 2080TI GPU using experience sampled from 32 parallel environments. We use visitation count to calculate the intrinsic reward, for its provable effectiveness (Azar et al., 2017; Jin et al., 2018). For all our methods and baselines, we use $\eta / \sqrt { N ( s ) }$ as the exploration bonus for $N ( s )$ -th visit to state s. Specific values of $\eta$ and scaling weights can be found in Table 2.
+
+As for variational inference, the inference network is a 3-layer fully-connected network coupled with a 64-dimensional reparameterization estimator. ReLU is used as the activation function for the first two layers and the sum of negative log-likelihood and negative Evidence Lower Bound is used as loss. We use Adam optimizer (Kingma & Ba, 2014) with learning rate $1 \times 1 0 ^ { - 3 }$ and batchsize 2048. To speed up the learning of variational distributions estimation, we equip the learning with proportional prioritized experience replay (Schaul et al., 2015).
+
+# D.2 TASK STRUCTURE
+
+In this section, we describe the detailed settings of the experimental tasks.
+
+Pass: There are two agents and two switches to open the door in a $3 0 \times 3 0$ grid. Only when at least one of the switches are occupied will the door open. The agents need navigate from left to right and the team reward, which is 1000, is only provided when all agents reach the target zone. Agents can observe the position of another agents.
+
+Secret-Room: This is an extension of the Pass task with 4 rooms and 4 switches locating in different rooms. The size of the grid is $2 5 \times 2 5$ . When the left switch is occupied, all the three doors are open. And the three switches in each room on the right only control the door of its room. The agents need to navigate towards the desired room (in light red of Fig. 1 middle) to achieve the extrinsic team reward 1000. Agents can observe the position of the other agents.
+
+![](images/29e288b51665942791b82c324b076195f75706bbdbae718a605f9210149a83bf.jpg)  
+Figure 7: Task Push-Box, Island, and Large-Island
+
+![](images/40079d75eb6a27c26c9560c4e85cc342643565f83b6d17c57563be780530b3b9.jpg)  
+Figure 8: Comparison of our methods against baselines on Push-Box (left), Island (right).
+
+Push-Box: There are two agents and one box in a $1 5 \times 1 5$ grid. Agents need to push the box to the wall. However, the box is so heavy that only when two agents push it in the same direction at the same time can it be moved a grid. The only team reward, 1000, is given when the box is placed right against the wall. Agents can observe the coordinates of their teammate and the location of the box.
+
+Island: A group of two agents are hunting for treasure on an island. However, a random walking beast may attack the agents when they are too close. The agents can also attack the beast within their attack range. This hurt doubles when more than one agent attack at the same time. Each agent has a maximum health of 5 and will lose $1 / n$ health per step when there are $n$ agents within the attack range of the beast. Island is a modified version of the classic coordination scenario Stag-Hunt with local optimal, because finding each treasure (9 in total) will trigger a team reward of 10 but catching the beast gives a higher team reward of 300. Agents can observe the position and health of each other, and the coordinates of the beast. Fig. 9 shows the development of the probability of catching the beast and the averaged number of treasures found in an episode during 9000 PPO updates.
+
+Large-Island: Settings are similar to that of Island but with more agents (4), more treasures (16), and a beast with more energy (16) and a higher reward (600) for being caught.
+
+The horizon of one episode is set to 300 timesteps in all these tasks.
+
+# E COMPARISON WITH SINGLE-AGENT EXPLORATION METHODS
+
+In this paper, we study the exploration problem in multi-agent settings from a decentralized perspective. Alternatively, exploration can be carried out in a centralized manner – treating agents as a joint one and using single-agent exploration algorithms. In this section, we compare our methods with centralized exploration strategies using RND (Burda et al., 2018) and EMI (Hyoungseok Kim, 2019), which are among the most cutting-edge exploration algorithms driven by curiosity and based on mutual information, respectively. We use codes published by their authors and carry out a modest grid search over hyperparameters. For RND, we search intrinsic reward coefficient in the range of [0.005, 1.0] and extrinsic reward coefficient in range [0.05, 2.0]. For EMI, we test difference combinations of loss weights. Results averaged over four random seeds with the best found parameters are shown below.
+
+![](images/826ff5810bdd8edbe24f0c390152e5d05618e7df1bd358fc3bde3e853b230b3a.jpg)  
+Figure 9: Comparison of our methods against baselines and ablations on Island in terms of the probability of catching the beast and the averaged treasures collected in an episode.
+
+![](images/a2cf21a017a7442ca8047fa34dd882ebd7696cb7188570c59f28541e9c041b95.jpg)  
+Figure 10: Comparison of our methods against centralized single-agent exploration algorithms on Pass (left), Secret-Room (middle), and Push-Box (right).
+
+Performance comparisons on problems of Pass, Secret-Room, and Push-Box are illustrated in Fig. 10. We can observe that our methods significantly outperform centralized exploration strategies using RND or EMI. To better understand this observation, we plot visitation heatmaps over time for RND and EMI, respectively, in Fig. 11 and 12.
+
+Fig. 11 shows visitation heatmaps of RND on the Pass problem. From Fig. 11 (b), we can see that RND seems finding good policies for agents to pass the door in the first 4671 updates. However, agents’ policies seem to collapse quickly after that and their visits scatter around rooms again, which explains its learning curve in Fig. 10. From the evolution of its visitation heatmaps, we hypothesize that after visiting the center of the room for many times, agents’ curiosity models overfit on a particular set of states and they start to be curious about the relatively unfamiliar transition dynamics around the wall. As the result, the RND intrinsic reward drags the agents to the walls, as shown in Fig. 11(c) and (d), and their performance quickly drops within several updates (i.e., update 4671- 4677 shown by Fig. 11(b-d)). After a while, agents then leave from the walls and visit around in the room again, as shown in Fig. 11(e). The whole exploration process repeated. Similar behaviors are also observed on the Secret-Room problem.
+
+![](images/159cb137c5e51c6804ade198a624d21bf353f27b134c8bc574e46dd663b6d5a9.jpg)  
+Figure 11: Visitation heatmap of RND agents on Pass of most recent $1 k$ episodes. The brighter the yellow color, the higher the visitation frequency. Top: agent 1, bottom: agent 2.
+
+![](images/2d1ff41d473c11e3c2c9ff838b320e4304721e71938ebbb5d9ca2d7c40dfcbbf.jpg)  
+Figure 12: Visitation heatmap in most recent $1 k$ episodes of EMI agents on Pass. The brighter the yellow color, the higher the visitation frequency. Top: agent 1, bottom: agent 2.
+
+We also analyze the exploration behaviors of EMI agents on Pass, as illustrated by visitation heatmaps in Fig. 12. EMI tends to explore the state-action pairs where the transition dynamics is relatively complex, such as the edges and corners of the room (Fig. 12(a-c)). For problems where these state-action pairs do not lead to goals, EMI is not very effective. As the (centralized) transition dynamics of the Pass problem is relatively simple, EMI intrinsic reward quickly diminishes, which results in the behaviors of agents keeping unchanged after 500 updates (Fig. 12(d-e)).
+
+In summary, centralized single-agent exploration methods encode some heuristics to facilitate exploration, but they typically do not place a great emphasis on interactions among agents and are thus not very efficient for multi-agent exploration with sparse interactions.

parse/train/H1VGkIxRZ/H1VGkIxRZ.md

@@ -0,0 +1,504 @@
+# ENHANCING THE RELIABILITY OF OUT-OF-DISTRIBUTION IMAGE DETECTION IN NEURAL NETWORKS
+
+Shiyu Liang Coordinated Science Lab, Department of ECE University of Illinois at Urbana-Champaign sliang26@illinois.edu
+
+# R. Srikant
+
+Coordinated Science Lab, Department of ECE University of Illinois at Urbana-Champaign rsrikant@illinois.edu
+
+# ABSTRACT
+
+We consider the problem of detecting out-of-distribution images in neural networks. We propose ODIN, a simple and effective method that does not require any change to a pre-trained neural network. Our method is based on the observation that using temperature scaling and adding small perturbations to the input can separate the softmax score distributions between in- and out-of-distribution images, allowing for more effective detection. We show in a series of experiments that ODIN is compatible with diverse network architectures and datasets. It consistently outperforms the baseline approach (Hendrycks & Gimpel, 2017) by a large margin, establishing a new state-of-the-art performance on this task. For example, ODIN reduces the false positive rate from the baseline $3 4 . 7 \%$ to $4 . 3 \%$ on the DenseNet (applied to CIFAR-10 and Tiny-ImageNet) when the true positive rate is $9 5 \%$ .
+
+# 1 INTRODUCTION
+
+Modern neural networks are known to generalize well when the training and testing data are sampled from the same distribution (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016; Cho et al., 2014; Zhang et al., 2017). However, when deploying neural networks in real-world applications, there is often very little control over the testing data distribution. Recent works have shown that neural networks tend to make high confidence predictions even for completely unrecognizable (Nguyen et al., 2015) or irrelevant inputs (Hendrycks & Gimpel, 2017; Szegedy et al., 2014; Moosavi-Dezfooli et al., 2017). It has been well documented (Amodei et al., 2016) that it is important for classifiers to be aware of uncertainty when shown new kinds of inputs, i.e., out-ofdistribution examples. Therefore, being able to accurately detect out-of-distribution examples can be practically important for visual recognition tasks (Krizhevsky et al., 2012; Farabet et al., 2013; Ji et al., 2013).
+
+A seemingly straightforward approach of detecting out-of-distribution images is to enlarge the training set of both in- and out-of-distribution examples. However, the number of out-of-distribution examples can be infinitely many, making the re-training approach computationally expensive and intractable. Moreover, to ensure that a neural network accurately classifies in-distribution samples into correct classes while correctly detecting out-of-distribution samples, one might need to employ exceedingly large neural network architectures, which further complicates the training process.
+
+Hendrycks & Gimpel (2017) proposed a baseline method to detect out-of-distribution examples without further re-training networks. The method is based on an observation that a well-trained neural network tends to assign higher softmax scores to in-distribution examples than out-of-distribution examples. In this paper, we go further. We observe that after using temperature scaling in the softmax function (Hinton et al., 2015; Pereyra et al., 2017) and adding small controlled perturbations to inputs, the softmax score gap between in - and out-of-distribution examples is further enlarged. We will show that the combination of these two techniques (temperature scaling and input perturbation) can lead to better detection performance. For example, provided with a pre-trained DenseNet (Huang et al., 2016) on CIFAR-10 dataset (positive samples), we test against images from TinyImageNet dataset (negative samples). Our method reduces the False Positive Rate (FPR), i.e., the fraction of misclassified out-of-distribution samples, from $3 4 . 7 \%$ to $4 . 3 \%$ , when $9 5 \%$ of in-distribution images are correctly classified. We summarize the main contributions of this paper as the following:
+
+• We propose a simple and effective method, ODIN (Out-of-DIstribution detector for Neural networks), for detecting out-of-distribution examples in neural networks. Our method does not require re-training the neural network and is easily implementable on any modern neural architecture. • We test ODIN on state-of-the-art network architectures (e.g., DenseNet (Huang et al., 2016) and Wide ResNet (Zagoruyko & Komodakis, 2016)) under a diverse set of in- and out-distribution dataset pairs. We show ODIN can significantly improve the detection performance, and consistently outperforms the state-of-the-art method (Hendrycks & Gimpel, 2017) by a large margin. • We empirically analyze how parameter settings affect the performance, and further provide simple analysis that provides some intuition behind our method.
+
+The outline of this paper is as follows. In Section 2, we present the necessary definitions and the problem statement. In Section 3, we introduce ODIN and present performance results in Section 4. We experimentally analyze the proposed method and provide some justification for our method in Section 5. We summarize the related works and future directions in Section 6 and conclude the paper in Section 7.
+
+# 2 PROBLEM STATEMENT
+
+In this paper, we consider the problem of distinguishing in- and out-of-distribution images on a pretrained neural network. Let $P _ { X }$ and $Q _ { X }$ denote two distinct data distributions defined on the image space $\mathcal { X }$ . Assume that a neural network $f$ is trained on a dataset drawn from the distribution $P _ { X }$ . Thus, we call $P _ { X }$ the in-distribution and $Q _ { X }$ the out-distribution, respectively. In testing, we draw new images from a mixture distribution $\mathbb { P } _ { X \times Z }$ defined on $\mathcal { X } \times \{ 0 , 1 \}$ , where the conditional probability distributions $\mathbb { P } _ { X | Z = 0 } = P _ { X }$ and $\mathbb { P } _ { X | Z = 1 } = Q _ { X }$ denote in- and out-distribution respectively. Now we focus on the following problem: Given an image $\boldsymbol { X }$ drawn from the mixture distribution $\mathbb { P } _ { X \times Z }$ can we distinguish whether the image is from in-distribution $P _ { X }$ or not?
+
+In this paper, we focus on detecting out-of-distribution images. However, it is equally important to correctly classify an image into the right class if it is an in-distribution image. But this can be easily done: once it has been detected that an image is in-distribution, we can simply use the original image and run it through the neural network to classify it. Thus, we do not change the predictions of the neural network for in-distribution images and only focus on improving the detection performance for out-of-distribution images.
+
+# 3 ODIN: OUT-OF-DISTRIBUTION DETECTOR
+
+In this section, we present our method, ODIN, for detecting out-of-distribution samples. The detector is built on two components: temperature scaling and input preprocessing. We describe the details of both components below.
+
+Temperature Scaling. Assume that the neural network $\pmb { f } = ( f _ { 1 } , . . . , f _ { N } )$ is trained to classify $N$ classes. For each input $_ { \textbf { \em x } }$ , the neural network assigns a label ${ \hat { y } } ( \mathbf { x } ) = \arg \operatorname* { m a x } _ { i } S _ { i } ( \mathbf { x } ; T )$ by computing the softmax output for each class. Specifically,
+
+$$
+S _ { i } ( { \pmb x } ; T ) = \frac { \exp \left( f _ { i } ( { \pmb x } ) / T \right) } { \sum _ { j = 1 } ^ { N } \exp \left( f _ { j } ( { \pmb x } ) / T \right) } ,
+$$
+
+where $T \in \mathbb { R } ^ { + }$ is the temperature scaling parameter and set to 1 during the training. For a given input $_ { \textbf { \em x } }$ , we call the maximum softmax probability, i.e., $S _ { \hat { y } } ( \pmb { x } ; T ) = \operatorname* { m a x } _ { i } S _ { i } ( \pmb { x } ; T )$ the softmax score. In this paper, we use notations $S _ { \hat { y } } ( \pmb { x } ; T )$ and $S ( \pmb { x } ; T )$ interchangeably. Prior works have established the use of temperature scaling to distill the knowledge in neural networks (Hinton et al., 2015) and calibrate the prediction confidence in classification tasks (Guo et al., 2017). As we shall see later, a good manipulation of temperature $T$ can push the softmax scores of in- and out-of-distribution images further apart from each other, making the out-of-distribution images distinguishable.
+
+Input Preprocessing. Before feeding the image $_ { \textbf { \em x } }$ into the neural network, we preprocess the input by adding small perturbations to it. The preprocessed image is given by
+
+$$
+\tilde { \pmb { x } } = \pmb { x } - \varepsilon \mathrm { s i g n } ( - \nabla _ { \pmb { x } } \log S _ { \hat { \pmb { y } } } ( \pmb { x } ; T ) ) ,
+$$
+
+where the parameter $\varepsilon$ can be interpreted as the perturbation magnitude. The method is inspired by the idea in the reference (Goodfellow et al., 2015), where small perturbations are added to decrease the softmax score for the true label and force the neural network to make a wrong prediction. Here, our goal and setting are rather different: we aim to increase the softmax score of any given input, without the need for a class label at all. As we shall see later, the perturbation can have stronger effect on the in- distribution images than that on out-of-distribution images, making them more separable. Note that the perturbations can be easily computed by back-propagating the gradient of the cross-entropy loss w.r.t the input.
+
+Out-of-distribution Detector. The proposed approach works as follows. For each image $_ { \textbf { \em x } }$ , we first calculate the preprocessed image $\tilde { \pmb x }$ according to the equation (2). Next, we feed the preprocessed image $\tilde { \pmb x }$ into the neural network, calculate its softmax score $S ( \tilde { \mathbf { } x } ; T )$ and compare the score to the threshold $\delta$ . We say that the image $_ { \textbf { \em x } }$ is an in-distribution example if the softmax score is above the threshold and that the image $_ { \textbf { \em x } }$ is an out-of-distribution example, otherwise. Therefore, the out-of-distribution detector is given by
+
+$$
+\begin{array} { r } { \overline { { g ( { \pmb x } ; \delta , T , \varepsilon ) } } = \displaystyle \left. \begin{array} { l l } { 1 } & { \mathrm { i f } \mathrm { m a x } _ { i } p ( \tilde { { \pmb x } } ; T ) \leq \delta , } \\ { 0 } & { \mathrm { i f } \mathrm { m a x } _ { i } p ( \tilde { { \pmb x } } ; T ) > \delta . } \end{array} \right. } \end{array}
+$$
+
+The parameters $T , \varepsilon$ and $\delta$ are chosen so that the true positive rate (i.e., the fraction of in-distribution images correctly classified as in-distribution images) under some out-of-distribution image data set is $9 \hat { 5 } \%$ . (The choice of the out-of-distribution images to tune the parameters $T , \varepsilon$ and $\delta$ appears to be unimportant, as demonstrated in the appendix H.) Having chosen the parameters as above, we evaluate the performance of our algorithm using various metrics in the next section.
+
+# 4 EXPERIMENTS
+
+In this section, we demonstrate the effectiveness of ODIN on several computer vision benchmark datasets. We run all experiments with PyTorch1 and we will release the code to reproduce all experimental results2.
+
+# 4.1 TRAINING SETUP
+
+Architectures and training configurations. We adopt two state-of-the-art neural network architectures, including DenseNet (Huang et al., 2016) and Wide ResNet (Zagoruyko & Komodakis, 2016). For DenseNet, our model follows the same setup as in (Huang et al., 2016), with depth $L = 1 0 0$ , growth rate $k = 1 2$ (Dense-BC) and dropout rate 0. In addition, we evaluate the method on a Wide ResNet, with depth 28, width 10 (WRN-28-10) and dropout rate 0. Furthermore, in Appendix A.1, we provide additional experimental results on another Wide ResNet with depth 40, width 4 (WRN-40-4). The hyper-parameters of neural networks are set identical to the original Wide ResNet (Zagoruyko & Komodakis, 2016) and DenseNet (Huang et al., 2016) implementations. All neural networks are trained with stochastic gradient descent with Nesterov momentum (Duchi et al., 2011; Kingma & Ba, 2014). Specifically, we train Dense-BC for 300 epochs with batch size 64 and momentum 0.9; and Wide ResNet for 200 epochs with batch size 128 and momentum 0.9. The learning rate starts at 0.1, and is dropped by a factor of 10 at $5 0 \%$ and $7 5 \%$ of the training progress, respectively.
+
+Accuracy. Each neural network architecture is trained on CIFAR-10 (C-10) and CIFAR-100 (C-100) datasets (Krizhevsky & Hinton, 2009), respectively. CIFAR-10 and CIFAR-100 images are drawn from 10 and 100 classes, respectively. Both datasets consist of
+
+Table 1: Test error rates on CIFAR-10 and CIFAR-100 datasets.   
+
+<table><tr><td>Architecture</td><td>C-10</td><td>C-100</td></tr><tr><td>Dense-BC</td><td>4.81</td><td>22.37</td></tr><tr><td>WRN-28-10</td><td>3.71</td><td>19.86</td></tr></table>
+
+50,000 training images and 10,000 test images. The test error on CIFAR datasets are summarized in Table 1.
+
+# 4.2 OUT-OF-DISTRIBUTION DATASETS
+
+At test time, the test images from CIFAR-10 (CIFAR-100) datasets can be viewed as the in-distribution (positive) examples. For out-of-distribution (negative) examples, we follow the setting in (Hendrycks & Gimpel, 2017) and test on several different natural image datasets and synthetic noise datasets. All the datasets considered are listed below.
+
+(1) TinyImageNet. The Tiny ImageNet dataset3 consists of a subset of ImageNet images (Deng et al., 2009). It contains 10,000 test images from 200 different classes. We construct two datasets, TinyImageNet (crop) and TinyImageNet (resize), by either randomly cropping image patches of size $3 2 \times 3 2$ or downsampling each image to size $3 2 \times 3 2$ .   
+(2) LSUN. The Large-scale Scene UNderstanding dataset (LSUN) has a testing set of 10,000 images of 10 different scenes (Yu et al., 2015). Similar to TinyImageNet, we construct two datasets, LSUN (crop) and LSUN (resize), by randomly cropping and downsampling the LSUN testing set, respectively.   
+(3) iSUN. The iSUN (Xu et al., 2015) consists of a subset of SUN images. We include the entire collection of 8925 images in iSUN and downsample each image to size 32 by 32.   
+(4) Gaussian Noise. The synthetic Gaussian noise dataset consists of 10,000 random 2D Gaussian noise images, where each RGB value of every pixel is sampled from an i.i.d Gaussian distribution with mean 0.5 and unit variance. We further clip each pixel value into the range $[ 0 , 1 ]$ .   
+(5) Uniform Noise. The synthetic uniform noise dataset consists of 10,000 images where each RGB value of every pixel is independently and identically sampled from a uniform distribution on $[ 0 , 1 ]$ .
+
+# 4.3 EVALUATION METRICS
+
+We adopt the following four different metrics to measure the effectiveness of a neural network in distinguishing in- and out-of-distribution images.
+
+(1) FPR at $9 5 \%$ TPR can be interpreted as the probability that a negative (out-of-distribution) example is misclassified as positive (in-distribution) when the true positive rate (TPR) is as high as $9 5 \%$ . True positive rate can be computed by $\mathrm { T P R } = \mathrm { T P } / \left( \mathrm { T P } { + } \mathrm { F N } \right)$ , where TP and FN denote true positives and false negatives respectively. The false positive rate (FPR) can be computed by $\mathrm { F P R = }$ FP / $\left( \mathrm { F P + T N } \right)$ , where FP and TN denote false positives and true negatives respectively.   
+(2) Detection Error, i.e., $P _ { e }$ measures the misclassification probability when TPR is $9 5 \%$ . The definition of $P _ { e }$ is given by $P _ { e } = 0 . 5 ( 1 - \mathrm { T P R } ) + 0 . 5 \mathrm { F P R }$ , where we assume that both positive and negative examples have the equal probability of appearing in the test set.   
+(3) AUROC is the Area Under the Receiver Operating Characteristic curve, which is also a thresholdindependent metric (Davis & Goadrich, 2006). The ROC curve depicts the relationship between TPR and FPR. The AUROC can be interpreted as the probability that a positive example is assigned a higher detection score than a negative example (Fawcett, 2006). A perfect detector corresponds to an AUROC score of $1 0 0 \%$ .   
+(4) AUPR is the Area under the Precision-Recall curve, which is another threshold independent metric (Manning et al., 1999; Saito & Rehmsmeier, 2015). The PR curve is a graph showing the precision $\ c =$ TP/ $\mathrm { T P + F P ) }$ and recall=TP/ $\mathrm { \mathrm { T P } } { + } \mathrm { F N } )$ against each other. The metric AUPR-In and AUPR-Out in Table 2 denote the area under the precision-recall curve where in-distribution and out-of-distribution images are specified as positives, respectively.
+
+# 4.4 EXPERIMENTAL RESULTS
+
+Comparison with baseline. In Figure 1, we show the ROC curves when DenseNet-BC-100 is evaluated on CIFAR-10 (positive) images against TinyImageNet (negative) test examples. The red curve corresponds to the ROC curve when using baseline method (Hendrycks & Gimpel, 2017), whereas the blue curve corresponds to our method with temperature $T = 1 0 0 0$ and perturbation magnitude $\varepsilon = 0 . 0 0 1 2$ . We observe a strikingly large gap between the blue and red ROC curves. For example, when $\mathrm { T P R } { = } 9 5 \%$ , the FPR can be reduced from $3 4 \%$ to $4 . 2 \%$ by using our approach.
+
+Table 2: Distinguishing in- and out-of-distribution test set data for image classification. All values are percentages. $\uparrow$ indicates larger value is better, and $\downarrow$ indicates lower value is better. All parameter settings are shown in Appendix A.2. Additional results on WRN-40-4 and MNIST dataset are reported in Appendix A.1.   
+
+<table><tr><td>dataset</td><td>Out-of-distribution</td><td>FPR (95% TPR) √</td><td>Detection Error √</td><td>AUROC 个</td><td>AUPR In</td><td>AUPR Out</td></tr><tr><td rowspan="7">Dense-BC CIFAR-10</td><td></td><td colspan="3">Baseline (Hendrycks &amp; Gimpel,2017) /ODIN</td><td>→</td><td>→</td></tr><tr><td>TinyImageNet (crop)</td><td>34.7/4.3</td><td>19.9/4.7</td><td>95.3/99.1</td><td>96.4/99.1</td><td>93.8/99.1</td></tr><tr><td>TinyImageNet (resize)</td><td>40.8/7.5</td><td>22.9/6.3</td><td>94.1/98.5</td><td>95.1/98.6</td><td>92.4/98.5</td></tr><tr><td>LSUN (crop)</td><td>39.3/8.7</td><td>22.2/6.9</td><td>94.8/98.2</td><td>96.0/98.5</td><td>93.1/97.8</td></tr><tr><td>LSUN (resize)</td><td>33.6/3.8</td><td>19.3/4.4</td><td>95.4/99.2</td><td>96.4/99.3</td><td>94.0/99.2</td></tr><tr><td>iSUN</td><td>37.2/6.3</td><td>21.1/5.7</td><td>94.8/98.8</td><td>95.9/98.9</td><td>93.1/98.8</td></tr><tr><td>Uniform</td><td>23.5/0.0</td><td>14.3/2.5</td><td>96.5/99.9</td><td>97.8/100.0</td><td>93.0/99.9</td></tr><tr><td rowspan="7">Dense-BC CIFAR-100</td><td>Gaussian</td><td>12.3/0.0</td><td>8.7/2.5</td><td>97.5/100.0</td><td>98.3/100.0</td><td>95.9/100.0</td></tr><tr><td>TinyImageNet (crop)</td><td>67.8/17.3</td><td>36.4/11.2</td><td>83.0/97.1</td><td>85.3/97.4</td><td>80.8/96.8</td></tr><tr><td>TinyImageNet (resize)</td><td>82.2/44.3</td><td>43.6/24.6</td><td>70.4/90.7</td><td>71.4/91.4</td><td>68.6/90.1</td></tr><tr><td>LSUN (crop)</td><td>69.4/17.6</td><td>37.2/11.3</td><td>83.7/96.8</td><td>86.2/97.1</td><td>80.9/96.5</td></tr><tr><td>LSUN (resize)</td><td>83.3/44.0</td><td>44.1/24.5</td><td>70.6/91.5</td><td>72.5/92.4</td><td>68.0/90.6</td></tr><tr><td>iSUN</td><td>84.8/49.5</td><td>44.7/27.2</td><td>69.9/90.1</td><td>71.9/91.1</td><td>67.0/88.9</td></tr><tr><td>Uniform Gaussian</td><td>88.3/0.5 95.4/0.2</td><td>46.6/2.8</td><td>83.2/99.5</td><td>88.1/99.6</td><td>73.1/99.0</td></tr><tr><td rowspan="7">WRN-28-10 CIFAR-10</td><td></td><td></td><td>50.2/2.6</td><td>81.8/99.6</td><td>87.6/99.7</td><td>70.1/99.1</td></tr><tr><td>TinyImageNet (crop) TinyImageNet (resize)</td><td>38.9/23.4</td><td>21.9/14.2</td><td>92.9/94.2</td><td>92.5/92.8</td><td>91.9/94.7</td></tr><tr><td></td><td>45.6/25.5</td><td>25.3/15.2</td><td>91.0/92.1</td><td>89.7/89.0</td><td>89.9/93.6</td></tr><tr><td>LSUN (crop)</td><td>35.0/21.8</td><td>20.0/13.4</td><td>94.5/95.9</td><td>95.1/95.8</td><td>93.1/95.5</td></tr><tr><td>LSUN (resize)</td><td>35.0/17.6</td><td>20.0/11.3</td><td>93.9/95.4</td><td>93.8/93.8</td><td>92.8/96.1</td></tr><tr><td>iSUN</td><td>40.6/21.3</td><td>22.8/13.2</td><td>92.5/93.7</td><td>91.7/91.2</td><td>91.5/94.9</td></tr><tr><td>Uniform Gaussian</td><td>1.6/0.0 0.3/0.0</td><td>3.3/2.5</td><td>99.2/100.0</td><td>99.3/100.0</td><td>98.9/100.0</td></tr><tr><td rowspan="7">WRN-28-10 CIFAR-100</td><td></td><td></td><td>2.6/2.5</td><td>99.5/100.0</td><td>99.6/100.0</td><td>99.3/100.0</td></tr><tr><td>TinyImageNet (crop)</td><td>66.6/43.9</td><td>35.8/24.4</td><td>82.0/90.8</td><td>83.3/91.4</td><td>80.2/90.0</td></tr><tr><td>TinyImageNet (resize)</td><td>79.2/55.9</td><td>42.1/30.4</td><td>72.2/84.0</td><td>70.4/82.8</td><td>70.8/84.4</td></tr><tr><td>LSUN (crop)</td><td>74.0/39.6</td><td>39.5/22.3</td><td>80.3/92.0</td><td>83.4/92.4</td><td>77.0/91.6</td></tr><tr><td>LSUN (resize)</td><td>82.2/56.5</td><td>43.6/30.8</td><td>73.9/86.0</td><td>75.7/86.2</td><td>70.1/84.9</td></tr><tr><td>iSUN</td><td>82.7/57.3</td><td>43.9/31.1</td><td>72.8/85.6</td><td>74.2/85.9</td><td>69.2/84.8</td></tr><tr><td>Uniform Gaussian</td><td>98.2/0.1 99.2/1.0</td><td>51.6/2.5 52.1/3.0</td><td>84.1/99.1 84.3/98.5</td><td>89.9/99.4 90.2/99.1</td><td>71.0/97.5 70.9/95.9</td></tr></table>
+
+Choosing parameters. For each out-of-distribution dataset, we randomly hold out 1,000 images for tuning the parameters $T$ and $\varepsilon$ . For temperature $T$ , we select among 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000; and for perturbation magnitude $\varepsilon$ we choose from 21 evenly spaced numbers starting from 0 and ending at 0.004. The optimal parameters are chosen to minimize the FPR at TPR $9 5 \%$ on the holdout set. We evaluate the our approach on the remaining test images. All parameter settings are reported in the Appendix A. We provide additional details on the effect of parameters in Section 5.
+
+Main results. The main results are summarized in Table 2. For each in- and out-of-distribution dataset pair, we report both the performance of the baseline (Hendrycks & Gimpel, 2017) and our approach using temperature scaling and input preprocessing. In Table 2, we observe improved performance across all neural architectures and all dataset pairs. Noticeably, our method consistently outperforms the baseline by a large margin when measured by FPR at $9 5 \%$ TPR and detection error.
+
+![](images/d84810ff40d5d49d8e9ca22e56b3cf747848bafef569d357b8658fb90c8b775c.jpg)  
+Figure 1: (a) ROC curves of baseline (red) and our method (blue) on DenseNet-BC-100 network, where CIFAR-10 and TinyImageNet (crop) are in- and out-of-distribution dataset, respectively.
+
+![](images/c6f305faa3915e41faab2650f74a706dbf8ab8bb85d214e3b0dc67dfad04ad27.jpg)  
+Figure 2: (a)-(d) Performance of our method vs. MMD between in- and out-of-distribution datasets. Neural networks are trained on CIFAR-100 and CIFAR-80, respectively. The out-of-distribution datasets are 1: LSUN (cop), 2: TinyImageNet (crop), 3: LSUN (resize), 4: is iSUN (resize), 5: TinyImageNet (resize) and 6: CIFAR-20.
+
+# 4.5 EXTENSIONS
+
+In this subsection, we analyze how the statistical distance be  
+tween in- and out-of-distribution natural image dataset affects the detection performance of the proposed method.
+
+Data distribution distance vs. Detection performance. To measure the statistical distance between in- and out-of-distribution datasets, we adopt a commonly used metric, maximum mean discrepancy (MMD) with Gaussian RBF kernel (Sriperumbudur et al., 2010; Gretton et al., 2012; Sutherland et al., 2016). Specifically, given two image sets, $V = \{ v _ { 1 } , . . . , v _ { m } \}$ and $W = \{ w _ { 1 } , . . . , w _ { m } \}$ , the maximum mean discrepancy between $V$ and $Q$ is defined as
+
+$$
+\widehat { \overline { { { \bf { M M D } } } } } ^ { 2 } ( V , W ) = \frac { 1 } { { \binom { m } { 2 } } } \sum _ { i \neq j } k ( v _ { i } , v _ { j } ) + \frac { 1 } { { \binom { m } { 2 } } } \sum _ { i \neq j } k ( w _ { i } , w _ { j } ) - \frac { 2 } { { \binom { m } { 2 } } } \sum _ { i \neq j } k ( v _ { i } , w _ { j } ) ,
+$$
+
+where $k ( \cdot , \cdot )$ is the Gaussian RBF kernel, i.e., $\begin{array} { r } { k ( x , x ^ { \prime } ) = \exp { \left( - \frac { \| x - x ^ { \prime } \| _ { 2 } ^ { 2 } } { 2 \sigma ^ { 2 } } \right) } } \end{array}$ . We use the same method used by Sutherland et al. (2016) to choose $\sigma$ , where $2 \sigma ^ { 2 }$ is set to the median of all Euclidean distances between all images in the aggregate set $V \cup W$ .
+
+In Figure 2 (a)(b), we show how the performance of ODIN varies against the MMD distances between in- and out-of-distribution datasets4. The datasets (on $\mathbf { X }$ -axis) are ranked in the descending order of MMD distances with CIFAR-100. There are two interesting observations can be drawn from these figures. First, we find that the MMD distances between the cropped datasets and CIFAR-100 tend to be larger. This is likely due to the fact that cropped images only contain local image context and are therefore more distinct from CIFAR-100 images, while resized images contain global patterns and are thus similar to images in CIFAR-100. Second, we observe that the MMD distance tends to be negatively correlated with the detection performance. This suggests, not surprisingly, that the detection task becomes harder as in and out-of-distribution images are more similar to each other.
+
+Same-manifold datasets. Furthermore, we investigate the extreme scenario when in- and out-ofdistribution datasets are on the same manifold. In experiment, we randomly split CIFAR-100 into two disjoint datasets containing 80 and 20 classes each. We name them CIFAR-80 and CIFAR20, respectively. We train both DenseNet and Wide ResNet-28-10 on the CIFAR-80 dataset (indistribution) and evaluate the detection performance on the CIFAR-20 dataset (out-distribution). All hyperparameters used here are exactly the same as in Section 4.1. The MMD distance between CIFAR-20 and CIFAR-80 is much smaller than other dataset pairs. In Figure 2 (c)(d), we observe that both FPR at TPR $9 5 \%$ and detection error become larger on the CIFAR-20 dataset. This coincides with our expectation that the detection task becomes extremely hard when in- and out-of-distribution dataset locate on the same manifold. We provide additional experimental results in Appendix A.1 and Appendix G.
+
+# 5.1 EFFECTS OF PARAMETERS
+
+# 5 DISCUSSIONS
+
+In this subsection, we empirically show how temperature $T$ and perturbation magnitude $\varepsilon$ affect FPR at TPR $9 5 \%$ and AUROC on DenseNet and Wide ResNet-28-10. Additional results on other metrics and architectures are provided in Appendix B. We show the detection performance when using only the temperature scaling method (see Figure 3(a)(b), $\varepsilon = 0$ ), or the input preprocessing method (see Figure 3(c)(d), $T = 1$ ). In Figure 4, we show the detection performance w.r.t $\varepsilon$ when $T$ is optimal (e.g., $\scriptstyle { T = 1 0 0 0 }$ ). First, from Figure 3 (a)(b), we observe that increasing the temperature can improve the detection performance, although the effects diminish when $T$ is sufficiently large (e.g., $T > 1 0 0 $ ). Next, from Figure 3(c)(d) and Figure 4, we observe that we can further improve the detection performance by appropriately choosing the perturbation magnitudes. We can achieve overall better performance by combining both (1) temperature scaling and (2) input preprocessing.
+
+![](images/a65e17e18f8c7c4ad30da3bdde2fc91b1d8b066ce76252936f1de351b9a4b636.jpg)  
+Figure 3: (a)(b) Effects of temperature $T$ when $\varepsilon = 0$ . (c)(d) Effects of perturbation magnitude $\varepsilon$ when $T = 1$ . All networks are trained on CIFAR-10 (in-distribution). Additional results on other metrics and Wide ResNet-40 are provided in Appendix B.
+
+![](images/499c75f5bd3fb91211c28ff8081ba9bbee7ac9d9881dc9d86e5e807362521422.jpg)  
+Figure 4: (a)(b) Effects of perturbation magnitude $\varepsilon$ on DenseNet when $T$ is large (e.g., $T = 1 0 0 0$ ). (c)(d) Effects of perturbation magnitude of $\varepsilon$ on Wide-ResNet-28-10 when $T$ is large (e.g., $T = 1 0 0 0$ ). All networks are trained on CIFAR-10. Additional results on other metrics and Wide ResNet-40 are provided in Appendix B.
+
+# 5.2 ANALYSIS ON TEMPERATURE SCALING
+
+In this subsection, we analyze the effectiveness of the temperature scaling method. As shown in Figure 3 (a) and (b), we observe that a sufficiently large temperature yields better detection performance although the effects diminish when $T$ is too large. To gain insight, we can use the Taylor expansion of the softmax score (details provided in Appendix D). When $T$ is sufficiently large, we have
+
+$$
+S _ { \hat { y } } ( { \pmb x } ; T ) \approx \frac { \hat { \textbf { \textit { 1 } } } } { N - \frac { 1 } { T } \sum _ { i } [ f _ { \hat { y } } ( { \pmb x } ) - f _ { i } ( { \pmb x } ) ] + \frac { 1 } { 2 T ^ { 2 } } \sum _ { i } [ f _ { \hat { y } } ( { \pmb x } ) - f _ { i } ( { \pmb x } ) ] ^ { 2 } } ,
+$$
+
+by omitting the third and higher orders. For simplicity of notation, we define
+
+$$
+U _ { 1 } ( \pmb { x } ) = \frac { 1 } { N - 1 } \sum _ { i \neq \hat { \pmb { y } } } [ f _ { \hat { \pmb { y } } } ( \pmb { x } ) - f _ { i } ( \pmb { x } ) ] \quad \mathrm { a n d } \quad U _ { 2 } ( \pmb { x } ) = \frac { 1 } { N - 1 } \sum _ { i \neq \hat { \pmb { y } } } [ f _ { \hat { \pmb { y } } } ( \pmb { x } ) - f _ { i } ( \pmb { x } ) ] ^ { 2 } .
+$$
+
+Interpretations of $U _ { 1 }$ and $U _ { 2 }$ . By definition, $U _ { 1 }$ measures the extent to which the largest unnormalized output of the neural network deviates from the remaining outputs; while $U _ { 2 }$ measures the extent to which the remaining smaller outputs deviate from each other. We provide formal mathematical derivations in Appendix F. In Figure 5(a), we show the distribution of $U _ { 1 }$ for each out-of-distribution dataset vs. the in-distribution dataset (in red). We observe that the largest outputs of the neural network on in-distribution images deviate more from the remaining outputs. This is likely due to the fact that neural networks tend to make more confident predictions on in-distribution images.
+
+Further, we show in Figure 5(b) the expectation of $U _ { 2 }$ conditioned on $U _ { 1 }$ , i.e., $E [ U _ { 2 } | U _ { 1 } ]$ , for each dataset. The red curve (in-distribution images) has overall higher expectation. This indicates that, when two images have similar values on $U _ { 1 }$ , the in-distribution image tends to have a much higher value of $U _ { 2 }$ than the out-of-distribution image. In other words, for in-distribution images, the remaining outputs (excluding the largest output) tend to be more separated from each other compared to out-of-distribution datasets. This may happen when some classes in the in-distribution dataset share common features while others differ significantly. To illustrate this, in Figure 5 (f)(g), we show the outputs of each class using a DenseNet (trained on CIFAR-10) on a dog image from CIFAR-10, and another image from TinyImageNet (crop). For the image of dog, we can observe that the largest output for the label dog is close to the output for the label cat but is quite separated from the outputs for the label car and truck. This is likely due to the fact that, in CIFAR-10, images of dogs are very similar to the images of cats but are quite distinct from images of car and truck. For the image from TinyImageNet (crop), despite having one large output, the remaining outputs are close to each other and thus have a smaller deviation.
+
+![](images/25aa25ed3f4a8ee8f57be1980afd515288dce7137a9ae0f1f0c46ac695a6d381.jpg)  
+Figure 5: (a) Probability density of $U _ { 1 }$ under different datasets on DenseNet. (b) Expectations of $U _ { 2 }$ conditioned on $U _ { 1 }$ on DenseNet. (c) Probability density of the norm of gradient on DenseNet under temperature 1, 000. (c)(d) Expectation of the norm of gradient conditioned on the softmax scores on DenseNet under temperature $T = 1 0 \bar { 0 } 0$ and $T = 1$ , respectively. (f)(g) Outputs of DenseNet on each class for an image of dog from CIFAR-10 and an image from TinyImageNet (crop). The DenseNet is trained on CIFAR-10. Additional results on other architectures are provided in Appendix C.
+
+The effects of $T$ . To see the usefulness of adopting a large $T$ , we can first rewrite the softmax score function in Equation (3) as $S \propto ( U _ { 1 } - U _ { 2 } / 2 \bar { T } ) / \bar { T }$ . Hence the softmax score is largely determined by $U _ { 1 }$ and $U _ { 2 } / 2 T$ . As noted earlier, $U _ { 1 }$ makes in-distribution images produce larger softmax scores than out-of-distribution images since $S \propto U _ { 1 }$ , while $U _ { 2 }$ has the exact opposite effect since $S \propto - U _ { 2 }$ . Therefore, by choosing a sufficiently large temperature, we can compensate the negative impacts of $U _ { 2 } / 2 T$ on the detection performance, making the softmax scores between in- and out-of-distribution images more separable. Eventually, when $T$ is sufficiently large, the distribution of softmax score is almost dominated by the distribution of $U _ { 1 }$ and thus increasing the temperature further is no longer effective. This explains why we see in Figure 3 (a)(b) that the performance does not change when $T$ is too large (e.g., $T > 1 0 0$ ). In Appendix $\mathrm { E }$ , we provide a formal proof showing that the detection error eventually converges to a constant number when $T$ goes to infinity.
+
+# 5.3 ANALYSIS ON INPUT PREPROCESSING
+
+As noted previously, using the temperature scaling method by itself can be effective in improving the detection performance. However, the effectiveness quickly diminishes as $T$ becomes very large. In order to make further improvement, we complement temperature scaling with input preprocessing. This has already been seen in Figure 4, where the detection performance is improved by a large margin on most datasets when $T = 1 0 0 0$ , provided with an appropriate perturbation magnitude $\varepsilon$ is chosen. In this subsection, we provide some intuition behind this.
+
+To explain, we can look into the first order Taylor expansion of the log-softmax function for the perturbed image $\tilde { \pmb x }$ , which is given by
+
+$$
+\log S _ { \hat { y } } ( \tilde { x } ; T ) = \log S _ { \hat { y } } ( x ; T ) + \varepsilon \left. \nabla _ { x } \log S _ { \hat { y } } ( x ; T ) \right. _ { 1 } + o ( \varepsilon ) ,
+$$
+
+where $_ { \textbf { \em x } }$ is the original input.
+
+The effects of gradient. In Figure 5 (c), we present the distribution of $\| \nabla _ { \pmb { x } } \log S ( \pmb { x } ; T ) \| _ { 1 }$ — the 1-norm of gradient of log-softmax with respect to the input $_ { \textbf { \em x } }$ — for all datasets. A salient observation is that CIFAR-10 images (in-distribution) tend to have larger values on the norm of gradient than most out-of-distribution images. To further see the effects of the norm of gradient on the softmax score, we provide in Figures 5 (d) the conditional expectation $E [ | | \nabla _ { \pmb { x } } \log \bar { S } ( { \pmb x } ; T ) | | _ { 1 } | S ]$ . We can observe that, when an in-distribution image and an out-of-distribution image have the same softmax score, the value of $\| \nabla _ { x } \log S ( { \pmb x } ; T ) \| _ { 1 }$ for in-distribution image tends to be larger.
+
+We illustrate the effects of the norm of gradient in Figure 6. Suppose that an in-distribution image $\scriptstyle { \mathbf { \mathscr { x } } } _ { 1 }$ (blue) and an out-of-distribution image $\mathbf { \boldsymbol { x } } _ { 2 }$ (red) have similar softmax scores, i.e., $S ( \pmb { x } _ { 1 } ) \approx S ( \pmb { x } _ { 2 } )$ . After input processing, the in-distribution image can have a much larger softmax score than the out-of-distribution image $\mathbf { x } _ { 2 }$ since $\scriptstyle { \mathbf { \mathscr { x } } } _ { 1 }$ results in a much larger value on the norm of softmax gradient than that of $\mathbf { x } _ { 2 }$ . Therefore, in- and out-of-distribution images are more separable from each other after input preprocessing5.
+
+The effect of $\varepsilon$ . When the magnitude $\varepsilon$ is sufficiently small, adding perturbations does not change the predictions of the neural network, i.e., ${ \hat { y } } ( { \tilde { \mathbf { x } } } ) = { \hat { y } } ( \mathbf { x } )$ . However, when $\varepsilon$ is not negligible, the gap of softmax scores between in- and out-of-distribution images can be affected by $\| \nabla _ { \pmb { x } } \log S ( \pmb { x } ; T ) \| _ { 1 }$ . Our observation is consistent with that in (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi
+
+![](images/c850b024deebdefc6a42320e2701738117f4f0670d6531ad63a402c3a355cd34.jpg)  
+Figure 6: Illustration of effects of the input preprocessing.
+
+Dezfooli et al., 2017), which show that the softmax scores tend to change significantly if small perturbations are added to the in-distribution images. It is also worth noting that using a very large $\varepsilon$ can lead to performance degradation, as seen in Figure 4. This is likely due to the fact that the second and higher order terms in the Taylor expansion are no longer insignificant when the perturbation magnitude is too large.
+
+# 6 RELATED WORKS AND FUTURE DIRECTIONS
+
+The problem of detecting out-of-distribution examples in low-dimensional space has been well-studied in various contexts (see the survey by Pimentel et al. (2014)). Conventional methods such as density estimation, nearest neighbor and clustering analysis are widely used in detecting low-dimensional outof-distribution examples (Chow, 1970; Vincent & Bengio, 2003; Ghoting et al., 2008; Devroye et al., 2013), . The density estimation approach uses probabilistic models to estimate the in-distribution density and declares a test example to be out-of-distribution if it locates in the low-density areas. The clustering method is based on the statistical distance, and declares an example to be out-ofdistribution if it locates far from its neighborhood. Despite various applications in low-dimensional spaces, unfortunately, these methods are known to be unreliable in high-dimensional space such as image space (Wasserman, 2006; Theis et al., 2015). In recent years, out-of-distribution detectors based on deep models have been proposed. Schlegl et al. (2017) train a generative adversarial networks to detect out-of-distribution examples in clinical scenario. Sabokrou et al. (2016) train a convolutional network to detect anomaly in scenes. Andrews et al. (2016) adopt transfer representation-learning for anomaly detection. All these works require enlarging or modifying the neural networks. In a more recent work, Hendrycks & Gimpel (2017) found that pre-trained neural networks can be overconfident to out-of-distribution example, limiting the effectiveness of detection. Our paper aims to improve the performance of detecting out-of-distribution examples, without requiring any change to an existing well-trained model.
+
+Our approach leverages the following two interesting observations to help better distinguish between in- and out-of-distribution examples: (1) On in-distribution images, modern neural networks tend to produce outputs with larger variance across class labels, and (2) neural networks have larger norm of gradient of log-softmax scores when applied on in-distribution images. We believe that having a better understanding of these phenomenon can lead to further insights into this problem.
+
+# 7 CONCLUSIONS
+
+In this paper, we propose a simple and effective method to detect out-of-distribution data samples in neural networks. Our method does not require retraining the neural network and significantly improves on the baseline (state-of-the-art) on different neural architectures across various in and out-distribution dataset pairs. We empirically analyze the method under different parameter settings, and provide some insights behind the approach. Future work involves exploring our method in other applications such as speech recognition and natural language processing.
+
+# ACKNOWLEDGMENTS
+
+The research reported here was supported by NSF Grant CPS ECCS 1739189.
+
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+
+# A SUPPLEMENTARY RESULTS IN SECTION 4.4
+
+A.1 EXPERIMENTAL RESULTS   
+Table 3: Distinguishing in- and out-of-distribution test set data for image classification. All values are percentages. $\uparrow$ indicates larger value is better, and $\downarrow$ indicates lower value is better.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Out-of-distribution dataset</td><td rowspan="2">FPR (95% TPR) ←</td><td rowspan="2">Detection Error √</td><td>AUROC</td><td>AUPR In</td><td>AUPR Out</td></tr><tr><td>→</td><td>个</td><td>→</td></tr><tr><td rowspan="8">WRN-40-4 CIFAR-10</td><td colspan="6">Baseline (Hendrycks &amp; Gimpel, 2017) /Ours</td></tr><tr><td>TinyImageNet (crop)</td><td>49.8/36.7</td><td>27.4/20.9</td><td>87.3/89.3</td><td>85.1/86.7</td><td>87.2/90.7</td></tr><tr><td>TinyImageNet (resize)</td><td>62.3/49.1</td><td>33.6/27.1</td><td>79.3/81.6</td><td>73.5/76.9</td><td>80.6/84.8</td></tr><tr><td>LSUN (crop)</td><td>34.6/23.0</td><td>19.8/14.0</td><td>93.4/95.1</td><td>93.1/94.3</td><td>92.4/95.2 85.3/89.7</td></tr><tr><td>LSUN (resize)</td><td>54.5/35.1</td><td>29.8/20.1</td><td>84.7/87.0</td><td>79.8/82.6</td><td></td></tr><tr><td>iSUN Uniform</td><td>58.6/41.0 26.6/3.2</td><td>31.8/23.0 15.8/4.1</td><td>82.1/84.9 96.1/99.2</td><td>76.4/80.2</td><td>83.2/87.8</td></tr><tr><td>Gaussian</td><td>21.8/0.9</td><td>13.4/3.0</td><td>96.5/99.7</td><td>97.0/99.2 97.5/99.7</td><td>94.8/99.2 94.7/99.7</td></tr><tr><td rowspan="7">WRN-40-4 CIFAR-100</td><td>TinyImageNet (crop)</td><td>66.9/43.3</td><td>36.0/24.1</td><td>81.3/88.5</td><td>80.6/87.2</td><td>80.1/89.1</td></tr><tr><td>TinyImageNet (resize)</td><td>78.1/55.1</td><td>41.5/30.1</td><td>72.6/81.6</td><td>69.4/78.0</td><td>71.6/83.4</td></tr><tr><td>LSUN (crop)</td><td>74.9/35.9</td><td>40.0/20.4</td><td>79.1/90.8</td><td>81.4/89.9</td><td>76.3/91.5</td></tr><tr><td>LSUN (resize)</td><td>77.9/50.0</td><td>41.5/27.5</td><td>75.2/85.6</td><td>73.1/83.5</td><td>73.3/86.4</td></tr><tr><td>iSUN</td><td>79.5/52.9</td><td>42.2/28.9</td><td>74.3/84.3</td><td>72.9/81.9</td><td>71.9/85.1</td></tr><tr><td>Uniform</td><td>84.7/3.3</td><td>44.9/4.2</td><td>86.3/98.8</td><td>90.5/99.1</td><td>77.0/97.9</td></tr><tr><td>Gaussian</td><td>77.2/3.1</td><td>41.1/4.0</td><td>86.4/99.0</td><td>90.2/99.2</td><td>78.6/98.6</td></tr><tr><td rowspan="8">Dense-BC CIFAR-80</td><td>CIFAR-20</td><td>84.1/81.1</td><td>44.9/43.0</td><td>76.6/77.8</td><td>79.4/80.6</td><td>71.6/73.6</td></tr><tr><td>TinyImageNet (crop)</td><td>72.9/22.7</td><td>39.0/13.8</td><td>83.4/96.2</td><td>86.3/96.6</td><td>79.9/95.8</td></tr><tr><td>TinyImageNet (resize)</td><td>84.4/46.3</td><td>44.7/25.6</td><td>76.8/91.7</td><td>80.3/92.7</td><td>71.5/90.4</td></tr><tr><td>LSUN (crop)</td><td>67.1/20.9</td><td>36.0/12.9</td><td>84.6/96.2</td><td>86.9/96.4</td><td>82.1/96.0</td></tr><tr><td>LSUN (resize)</td><td>84.9/45.9</td><td>45.0/25.4</td><td>77.5/91.8</td><td>81.4/92.9</td><td>71.6/90.2</td></tr><tr><td>iSUN</td><td>86.1/50.2</td><td>50.5/27.6</td><td>76.1/90.5</td><td>79.8/91.3</td><td>69.9/88.8</td></tr><tr><td>Uniform</td><td>100.0/0.9</td><td>52.5/3.0</td><td>64.3/98.6</td><td>78.4/99.1</td><td>52.2/96.6</td></tr><tr><td>Gaussian</td><td>98.5/1.2</td><td>51.8/3.1</td><td>80.4/99.6</td><td>86.7/99.6</td><td>68.0/99.1</td></tr><tr><td rowspan="8">WRN-28-10 CIFAR-80</td><td>CIFAR-20</td><td>80.4/78.3</td><td>42.7/41.6</td><td>79.2/80.4</td><td>81.5/82.2</td><td>74.2/76.2</td></tr><tr><td>TinyImageNet (crop)</td><td>71.3/46.7</td><td>38.1/25.9</td><td>83.1/91.9</td><td>85.9/92.6</td><td>79.7/90.7</td></tr><tr><td>TinyImageNet (resize)</td><td>81.0/48.8</td><td>43.0/26.9</td><td>77.1/89.2</td><td>80.0/89.5</td><td>72.6/88.5</td></tr><tr><td>LSUN (crop)</td><td>74.4/45.5</td><td>39.7/25.2</td><td>82.0/92.9</td><td>84.4/93.0</td><td>78.2/91.5</td></tr><tr><td>LSUN (resize)</td><td>81.9/49.0</td><td>43.5/27.0</td><td>78.8/90.1</td><td>82.2/90.8</td><td>73.4/88.8</td></tr><tr><td>iSUN</td><td>82.7/51.1</td><td>43.9/28.1</td><td>78.3/89.4</td><td>81.5/90.0</td><td>72.6/88.0</td></tr><tr><td>Uniform</td><td>99.6/1.4</td><td>52.3/3.2</td><td>80.6/98.9</td><td>87.7/99.2</td><td>66.8/97.6</td></tr><tr><td>Gaussian</td><td>100.0/0.4</td><td>52.5/2.7</td><td>79.7/99.1</td><td>87.4/99.4</td><td>65.5/98.0</td></tr><tr><td rowspan="8">WRN-40-4 CIFAR-80</td><td>CIFAR-20</td><td>82.4/78.4</td><td>43.7/41.7</td><td>76.8/78.1</td><td>78.9/79.1</td><td>72.2/75.0</td></tr><tr><td>TinyImageNet (crop)</td><td>68.3/34.3</td><td>36.6/19.6</td><td>83.6/93.4</td><td>85.9/94.0</td><td>81.2/92.5</td></tr><tr><td>TinyImageNet (resize)</td><td>80.6/53.5</td><td>42.8/29.2</td><td>76.2/87.7</td><td>78.5/88.3</td><td>72.5/86.1</td></tr><tr><td>LSUN (crop)</td><td>72.2/33.2</td><td>38.6/19.1</td><td>83.1/93.4</td><td>86.3/93.7</td><td>79.7/93.1</td></tr><tr><td>LSUN (resize)</td><td>79.1/51.2</td><td>42.0/28.1</td><td>77.6/88.8</td><td>80.0/89.4</td><td>73.9/87.3</td></tr><tr><td>iSUN</td><td>81.2/53.2</td><td>43.1/29.1</td><td>76.2/87.7</td><td>78.7/88.3</td><td>72.2/86.1</td></tr><tr><td>Uniform</td><td>99.7/48.6</td><td>52.4/26.8</td><td>65.6/93.8</td><td>77.3/95.7</td><td>53.7/88.7</td></tr><tr><td>Gaussian</td><td>99.7/10.7</td><td>52.4/7.8</td><td>74.3/97.7</td><td>83.0/98.4</td><td>61.2/95.2</td></tr><tr><td rowspan="5">MNIST</td><td>Omniglot</td><td>0.2/0.0</td><td>2.6/2.5</td><td>99.6/100.0</td><td>99.7/100.0</td><td>99.5/100.0</td></tr><tr><td>notMNIST</td><td>10.3/8.7</td><td>7.7/6.8</td><td>97.2/98.2</td><td>97.5/98.4</td><td>97.4/98.0</td></tr><tr><td>CIFAR-10bw</td><td>0.1/0.0</td><td>2.5/2.5</td><td>99.7/100.0</td><td>99.8/100.0</td><td>99.7/100.0</td></tr><tr><td>Gaussian</td><td>0.0/0.0</td><td>2.5/2.5</td><td>99.7/100.0</td><td>99.8/100.0</td><td>99.7/100.0</td></tr><tr><td>Uniform</td><td>0.0/0.0</td><td>2.5/2.5</td><td>99.9/100.0</td><td>99.9/100.0</td><td>99.9/100.0</td></tr></table>
+
+MNIST: We used the same MNIST classifier used by Hendrycks & Gimpel (2017), which is a three-layer, 256 neuron-wide, fully connected network trained for 30 epochs with Adam (Kingma & Ba, 2014). The classifier achieve $9 9 . 3 4 \%$ test accuracy on the MNIST test set. We compare our method with the baseline (Hendrycks & Gimpel, 2017) on five different out-of-distribution datasets: (1) Omniglot dataset (Lake et al., 2015) contains images of handwritten characters in stead of the handwritten digits in MNIST; (2) notMNIST (Bulatov, 2011) dataset contains typeface characters;
+
+(3) CIFAR-10bw contains black and white rescaled CIFAR-10 images; (4)(5) Gaussian and Uniform image set contains the synthetic Gaussian and Uniform noise images used in Section 4.2.
+
+Wide ResNet-40-4: We use the same architecture used by Hendrycks & Gimpel (2017) to evaluate the baseline and our method. The Wide ResNet-40-4 achieves $9 5 . 7 \%$ test accuracy on CIFAR-10 dataset and achieve $7 9 . 2 7 \%$ test accuracy on CIFAR-100.
+
+CIFAR-80: DenseNet-BC-100 achieves $7 8 . 9 4 \%$ test accuracy on CIFAR-80, while Wide ResNet28-10 achieves $8 1 . 7 1 \%$ test accuracy and Wide ResNet-40-4 achieves $7 9 . 5 3 \%$ test accuracy on CIFAR-80.
+
+# A.2 PARAMETER SETTINGS
+
+For MNIST, we set $T = 1 0 0 0$ and $\varepsilon = 0$ . The parameter settings for other structures are shown as follows.   
+
+<table><tr><td colspan="4">DenseNet-BC-100</td></tr><tr><td>Out-of-distribution datasets</td><td>CIFAR-10</td><td>CIFAR-80</td><td>CIFAR-100</td></tr><tr><td>TinyImageNet (crop)</td><td>0.0014</td><td>0.002</td><td>0.002</td></tr><tr><td>TinyImageNet (resize)</td><td>0.0014</td><td>0.0022</td><td>0.0022</td></tr><tr><td>LSUN (crop)</td><td>0</td><td>0.0036</td><td>0.0038</td></tr><tr><td>LSUN (resize)</td><td>0.0014</td><td>0.002</td><td>0.0018</td></tr><tr><td>iSUN</td><td>0.0014</td><td>0.002</td><td>0.002</td></tr><tr><td>Uniform</td><td>0.0014</td><td>0.0028</td><td>0.0024</td></tr><tr><td>Gaussian</td><td>0.0014</td><td>0.0026</td><td>0.0028</td></tr><tr><td>CIFAR-20</td><td>-</td><td>0.0002</td><td>-</td></tr></table>
+
+Table 4: Optimal perturbation magnitude $\varepsilon$ for reproducing main results in Table 2 and 3.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">DenseNet-BC-100</td></tr><tr><td>Out-of-distribution datasets CIFAR-10</td><td>CIFAR-80</td><td>CIFAR-100</td></tr><tr><td>TinyImageNet (crop)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>TinyImageNet (resize)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>LSUN (crop)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>LSUN (resize)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>iSUN</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>Uniform</td><td>1000</td><td>1</td><td>1</td></tr><tr><td>Gaussian</td><td>1000</td><td>1</td><td>1</td></tr><tr><td>CIFAR-20</td><td>-</td><td>1</td><td>-</td></tr></table>
+
+Table 5: Optimal Temperature $T$ for reproducing main results in Table 2 and 3.   
+
+<table><tr><td colspan="4">Wide-ResNet-28-10</td></tr><tr><td>Out-of-distribution datasets</td><td>CIFAR-10</td><td>CIFAR-80</td><td>CIFAR-100</td></tr><tr><td>TinyImageNet (crop)</td><td>0.0005</td><td>0.0002</td><td>0.0026</td></tr><tr><td>TinyImageNet (resize)</td><td>0.0011</td><td>0.0004</td><td>0.0024</td></tr><tr><td>LSUN (crop)</td><td>0</td><td>0.0002</td><td>0.0038</td></tr><tr><td>LSUN (resize)</td><td>0.0006</td><td>0.0002</td><td>0.0026</td></tr><tr><td>iSUN</td><td>0.0008</td><td>0.0002</td><td>0.0026</td></tr><tr><td>Uniform</td><td>0.0014</td><td>0.0002</td><td>0.0032</td></tr><tr><td>Gaussian</td><td>0.0014</td><td>0.0002</td><td>0.0032</td></tr><tr><td>CIFAR-20</td><td>-</td><td>5e-05</td><td>-</td></tr></table>
+
+Table 6: Optimal perturbation magnitude $\varepsilon$ for reproducing main results in Table 2 and 3.
+
+Table 7: Optimal Temperature $T$ for reproducing main results in Table 2 and 3.   
+
+<table><tr><td colspan="4">Wide-ResNet-28-10</td></tr><tr><td>Out-of-distribution datasets</td><td>CIFAR-10</td><td>CIFAR-80</td><td>CIFAR-100</td></tr><tr><td>TinyImageNet (crop)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>TinyImageNet (resize)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>LSUN (crop)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>LSUN (resize)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>iSUN</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>Uniform</td><td>1000</td><td>1</td><td>1000</td></tr><tr><td>Gaussian</td><td>1000</td><td>1</td><td>1000</td></tr><tr><td>CIFAR-20</td><td>1</td><td>1</td><td>-</td></tr></table>
+
+<table><tr><td colspan="4">Wide-ResNet-40-4</td></tr><tr><td>Out-of-distribution datasets</td><td>CIFAR-10</td><td>CIFAR-80</td><td>CIFAR-100</td></tr><tr><td>TinyImageNet (crop)</td><td>0.0004</td><td>0.0002</td><td>0.0014</td></tr><tr><td>TinyImageNet (resize)</td><td>0.0008</td><td>0.0004</td><td>0.0016</td></tr><tr><td>LSUN (crop)</td><td>0</td><td>0.0002</td><td>0.0038</td></tr><tr><td>LSUN (resize)</td><td>0.001</td><td>0.0002</td><td>0.0014</td></tr><tr><td>iSUN</td><td>0.0008</td><td>0.0002</td><td>0.0016</td></tr><tr><td>Uniform</td><td>0.0016</td><td>0.0002</td><td>0.0024</td></tr><tr><td>Gaussian</td><td>0.0016</td><td>0.0002</td><td>0.0026</td></tr><tr><td>CIFAR-20</td><td>-</td><td>0.0002</td><td>-</td></tr></table>
+
+Table 8: Optimal perturbation magnitude $\varepsilon$ for reproducing main results in Table 2 and 3.
+
+Table 9: Optimal Temperature $T$ for reproducing main results in Table 2 and 3.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Wide-ResNet-40-4</td></tr><tr><td>Out-of-distribution datasets CIFAR-10</td><td>CIFAR-80</td><td>CIFAR-100</td></tr><tr><td>TinyImageNet (crop)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>TinyImageNet (resize)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>LSUN (crop)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>LSUN (resize)</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>iSUN</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>Uniform</td><td>1000</td><td>1</td><td>1000</td></tr><tr><td>Gaussian</td><td>1000</td><td>1</td><td>1000</td></tr><tr><td>CIFAR-20</td><td>-</td><td>1</td><td>-</td></tr></table>
+
+![](images/ea42fd1bd24fe913c28758afd94e879b17b3646a75d61ceec7dab1601dec47ae.jpg)  
+Figure 7: Detection performance on DenseNet, Wide ResNet-28-10 and Wide ResNet-40-4 under different temperature, when input preprocessing is not used, i.e., $\varepsilon = 0$ . All networks are trained on CIFAR-10.
+
+![](images/da6fbf7d380a009f288b75da5f98ca92fbe61512fa4d08d0c35c87d1449de2fa.jpg)  
+Figure 8: Detection performance on DenseNet, Wide ResNet-28-10 and Wide ResNet-40-4 under different perturbation magnitude, when temperature scaling is not used, i.e., $T = 1$ . All networks are trained on CIFAR-10.
+
+![](images/9284ea875fa9c88ac34d6dddf279322ca86daad61635f6f3b3060ada44d9e10e.jpg)  
+Figure 9: Detection performance on DenseNet, Wide ResNet-28-10 and Wide ResNet-40-4 under different perturbation magnitude, when the optimal temperature is used, i.e., $T = 1 0 0 0$ . All networks are trained on CIFAR-10.
+
+![](images/c39ecd2fcf00ebff9a84cf1dd27e3075b0254e8ce522b36863629db122b12d98.jpg)  
+Figure 10: Expectation of the second order term $U _ { 2 }$ conditioned on the first order term $U _ { 1 }$ under DenseNet, Wide-ResNet-28-10 and Wide ResNet-40-4. All networks are trained on CIFAR-10.
+
+![](images/10d294280cf43184e465f1c90e8570af55e9cb5483487a22aa27c00a267faef6.jpg)  
+Figure 11: Expectation of gradient norms conditioned on the softmax scores under DenseNet, Wide-ResNet-28- 10 and Wide ResNet-40-4, where the temperature scaling is not used. All networks are trained on CIFAR-10.
+
+![](images/677f34f8c3f5b4825a72ba43ccc2417d40de1307a872ac52f8c113dcf021ec94.jpg)  
+Figure 12: Expectation of gradient norms conditioned on the softmax scores under DenseNet, Wide-ResNet-28- 10 and Wide ResNet-40-4, where the optimal temperature is used, i.e., $T = 1 0 0 0$ . All networks are trained on CIFAR-10.
+
+# D TAYLOR EXPANSION
+
+In this section, we present the Taylor expansion of the soft-max score function:
+
+$$
+\begin{array} { r l } { S _ { \hat { y } } ( { \boldsymbol x } ; T ) = \frac { \exp { \left( f _ { \hat { y } } ( { \boldsymbol x } ) / T \right) } } { \sum _ { i = 1 } ^ { N } \exp ( f _ { i } ( { \boldsymbol x } ) / T ) } } \\ & { = \frac { 1 } { \sum _ { i = 1 } ^ { N } \exp { \left( \frac { f _ { i } ( { \boldsymbol x } ) - f _ { i } ( { \boldsymbol x } ) } { T } \right) } } } \\ & { = \frac { 1 } { \sum _ { i = 1 } ^ { N } \left[ 1 + \frac { f _ { i } ( { \boldsymbol x } ) - f _ { \hat { y } } ( { \boldsymbol x } ) } { T } + \frac { 1 } { 2 ! } \frac { ( f _ { i } ( { \boldsymbol x } ) - f _ { \hat { y } } ( { \boldsymbol x } ) ) ^ { 2 } } { T ^ { 2 } } + o \left( \frac { 1 } { T ^ { 2 } } \right) \right] } } \\ & { \approx \frac { 1 } { N - \frac { 1 } { T } \sum _ { i = 1 } ^ { N } [ f _ { \hat { y } } ( { \boldsymbol x } ) - f _ { i } ( { \boldsymbol x } ) ] + \frac { 1 } { 2 ^ { \frac { 1 } { 2 } } T ^ { 2 } } \sum _ { i = 1 } ^ { N } [ f _ { i } ( { \boldsymbol x } ) - f _ { \hat { y } } ( { \boldsymbol x } ) ] ^ { 2 } } } \end{array}
+$$
+
+by Taylor expansion
+
+# E PROPOSITION 1
+
+The following proposition 1 shows that the detection error $P _ { e } ( T , 0 ) \approx c$ if $T$ is sufficiently large.   
+Thus, increasing the temperature further can only slightly improve the detection performance.
+
+Proposition 1. There exists a constant c only depending on function $U _ { 1 }$ , in-distribution $P _ { X }$ and out-of-distribution $Q _ { X }$ such that $\operatorname* { l i m } _ { T \to \infty }$ $P _ { e } ( T , \varepsilon ) = c$ , when $\varepsilon = 0$ (i.e., no input preprocessing).
+
+Proof. Since
+
+$$
+S _ { \hat { y } } ( X ; T ) = { \frac { \exp ( f _ { \hat { y } } ( X ) / T ) } { \sum _ { i = 1 } ^ { N } \exp ( f _ { i } ( X ) / T ) } } = { \frac { 1 } { 1 + \sum _ { i \neq \hat { y } } \exp ( [ f _ { i } ( X ) - f _ { \hat { y } } ( X ) ] / T ) } }
+$$
+
+Therefore, for any $\boldsymbol { X }$ ,
+
+$$
+\begin{array} { l } { { \displaystyle \operatorname* { l i m } _ { T \to \infty } T \left( - \frac { 1 } { S _ { \hat { y } } ( X ; T ) } + N \right) = \operatorname* { l i m } _ { T \to \infty } \sum _ { i \neq \hat { y } } T \left[ 1 - \exp \left( \frac { f _ { i } ( X ) - f _ { \hat { y } } ( X ) } { T } \right) \right] } } \\ { { \displaystyle \qquad = \sum _ { i \neq \hat { y } } [ f _ { \hat { y } } ( X ) - f _ { i } ( X ) ] = ( N - 1 ) U _ { 1 } ( X ) } } \end{array}
+$$
+
+This indicates that the random variable
+
+$$
+T \left( - \frac { 1 } { S _ { \hat { y } } ( X ; T ) } + N \right) \to ( N - 1 ) U _ { 1 } ( X ) \quad a . s .
+$$
+
+as $T \to \infty$ . This means that for a specific $\alpha > 0$ , choosing the threshold $\delta _ { T } = 1 / ( N - \alpha / T )$ , then the false positive rate
+
+$$
+\begin{array} { r l r } & { } & { \mathrm { F P R } ( T ) = Q _ { X } ( S _ { \hat { y } } ( X ; T ) > 1 / ( N - \alpha / T ) ) = Q _ { X } ( T ( N - \displaystyle \frac { 1 } { S _ { \hat { y } } ( X ; T ) } ) > \alpha ) } \\ & { } & { \xrightarrow { T  \infty } Q _ { X } ( ( N - 1 ) U _ { 1 } ( X ) > \alpha ) , } \end{array}
+$$
+
+and the true positive rate
+
+$$
+\begin{array} { r } { \mathrm { T P R } ( T ) = P _ { X } ( S _ { \hat { y } } ( { X } ; T ) > 1 / ( N - \alpha / T ) ) = P _ { X } ( T ( N - \displaystyle \frac { 1 } { S _ { \hat { y } } ( { X } ; T ) } ) > \alpha ) } \\ { \xrightarrow { T  \infty } P _ { X } ( ( N - 1 ) U _ { 1 } ( X ) > \alpha ) . } \end{array}
+$$
+
+Choosing $\alpha ^ { * }$ such that $P _ { X } \left( ( N - 1 ) U _ { 1 } ( X ) > \alpha ^ { * } \right) = 0 . 9 5 ,$ , then $\mathrm { T P R } ( T )  0 . 9 5$ as $T \to \infty$ and at the same time $\mathrm { F P R } ( T ) \to Q _ { X } \left( ( N - 1 ) U _ { 1 } ( X ) > \alpha ^ { * } \right)$ as $T \to \infty$ . There exists a constant $c$ depending on $U _ { 1 } , P _ { X } , Q _ { X }$ and $P _ { Z }$ , such that
+
+$$
+\operatorname* { l i m } _ { T \to \infty } P _ { e } ( T , 0 ) = 0 . 0 5 P ( Z = 0 ) + P ( Z = 1 ) Q _ { X } \left( ( N - 1 ) U _ { 1 } ( X ) > \alpha ^ { * } \right) = c .
+$$
+
+# F ANALYSIS OF TEMPERATURE
+
+For simplicity of the notations, let $\Delta _ { i } = f _ { \hat { y } } - f _ { i }$ and thus $\Delta = \{ \Delta _ { i } \} _ { i \neq \hat { y } }$ . Besides, let $\bar { \Delta }$ denote the mean of the set $\Delta$ . Therefore,
+
+$$
+\bar { \Delta } = \frac { 1 } { N - 1 } \sum _ { i \neq \hat { y } } \Delta _ { i } = \frac { 1 } { N - 1 } \sum _ { i \neq \hat { y } } [ f _ { \hat { y } } - f _ { i } ] = U _ { 1 } .
+$$
+
+Equivalently,
+
+$$
+U _ { 1 } = \mathbf { M e a n } ( \Delta ) .
+$$
+
+Next, we will show
+
+$$
+U _ { 2 } = \frac { 1 } { N - 1 } \sum _ { i \neq \hat { y } } [ f _ { \hat { y } } - f _ { i } ] ^ { 2 } = \overbrace { \frac { 1 } { N - 1 } \sum _ { i \neq \hat { y } } [ \Delta _ { i } - \bar { \Delta } ] ^ { 2 } } ^ { \mathrm { V a r i a n c e } ^ { 2 } ( \Delta ) } + \overbrace { \bar { \Delta } ^ { 2 } } ^ { \mathrm { M e a n } ^ { 2 } ( \Delta ) } .
+$$
+
+Since
+
+$$
+\mathsf { b y } \Delta _ { i } = f _ { \hat { y } } - f _ { i }
+$$
+
+$$
+\begin{array} { l } { { { \displaystyle U _ { 2 } = \frac { 1 } { N - 1 } \sum _ { i \neq \bar { \rho } } \Delta _ { i } ^ { 2 } } } } \\ { { { \displaystyle \quad = \frac { 1 } { N - 1 } \sum _ { i \neq \bar { \rho } } ( \Delta _ { i } - \bar { \Delta } + \bar { \Delta } ) ^ { 2 } } } } \\ { { { \displaystyle \quad = \frac { 1 } { N - 1 } \sum _ { i \neq \bar { \rho } } [ ( \Delta _ { i } - \bar { \Delta } ) ^ { 2 } - 2 ( \Delta _ { i } - \bar { \Delta } ) \bar { \Delta } + \bar { \Delta } ^ { 2 } ] } } } \\ { { { \displaystyle \quad = \frac { 1 } { N - 1 } \sum _ { i \neq \bar { \rho } } [ \Delta _ { i } - \bar { \Delta } ] ^ { 2 } - \underbrace { \frac { 2 \bar { \Delta } } { N - 1 } \sum _ { i \neq \bar { \rho } } ( \Delta _ { i } - \bar { \Delta } ) } _ { \mathrm { V o l u a r e } ^ { 2 } ( \Delta ) } + \underbrace { \bar { \Delta } ^ { 2 } } _ { \mathrm { M a r } ^ { 2 } ( \Delta ) } } } } \end{array}
+$$
+
+then
+
+$$
+U _ { 2 } = \mathrm { V a r i a n c e } ^ { 2 } ( \Delta ) + \mathrm { M e a n } ^ { 2 } ( \Delta )
+$$
+
+# G ADDITIONAL RESULTS IN SECTION 4.5
+
+Apart from the Maximum Mean Discrepancy, we as well calculate the Energy distance between inand out-of-distribution datasets. Let $P$ and $Q$ denote two different distributions. Then the energy distance between distributions $P$ and $Q$ is defined as
+
+$$
+D _ { \mathrm { e n e r g y } } ^ { 2 } ( P , Q ) = 2 \mathbb { E } _ { V \sim P , W \sim Q } \| X - Y \| - \mathbb { E } _ { V , V ^ { \prime } \sim P } \| X - X ^ { \prime } \| - \mathbb { E } _ { W , W ^ { \prime } \sim Q } \| Y - Y ^ { \prime } \| .
+$$
+
+Therefore, the energy distance between two datasets $V ~ = ~ \{ V _ { 1 } , . . . , V _ { m } \} ~ \stackrel { i i d } { \sim } ~ P$ and $W =$ $\{ W _ { 1 } , . . . , W _ { m } \} \stackrel { i i d } { \sim } Q$ is defined as
+
+$$
+\widehat { D _ { \mathrm { e n e r g y } } } ^ { 2 } ( P , Q ) = \frac { 2 } { m ^ { 2 } } \sum _ { i = 1 } ^ { m } \sum _ { j = 1 } ^ { m } \| V _ { i } - W _ { j } \| - \frac { 1 } { \binom { m } { 2 } } \sum _ { i \neq j } \| V _ { i } - V _ { j } \| - \frac { 1 } { \binom { m } { 2 } } \sum _ { i \neq j } \| W _ { i } - W _ { j } \| .
+$$
+
+In the experiment, we use the 2-norm $\| \cdot \| _ { 2 }$ .
+
+<table><tr><td>In-distribution datasets</td><td>Out-of-distribution Datasets</td><td>MMD Distance</td><td>Energy Distance</td></tr><tr><td rowspan="5">CIFAR-100</td><td>Tiny-ImageNet (crop)</td><td>0.41</td><td>2.25</td></tr><tr><td>LSUN (crop)</td><td>0.43</td><td>2.31</td></tr><tr><td>Tiny-ImageNet (resize)</td><td>0.088</td><td>0.54</td></tr><tr><td>LSUN (resize)</td><td>0.12</td><td>0.63</td></tr><tr><td>iSUN (resize)</td><td>0.11</td><td>0.56</td></tr><tr><td rowspan="6">CIFAR-80</td><td>Tiny-ImageNet (crop)</td><td>0.4</td><td>2.22</td></tr><tr><td>LSUN (crop)</td><td>0.43</td><td>2.29</td></tr><tr><td>Tiny-ImageNet (resize)</td><td>0.095</td><td>0.57</td></tr><tr><td>LSUN (resize)</td><td>0.120</td><td>0.62</td></tr><tr><td>iSUN (resize)</td><td>0.116</td><td>0.61</td></tr><tr><td>CIFAR-20</td><td>0.057</td><td>0.35</td></tr></table>
+
+![](images/6fe8653b8e35460d9830168f4ed71f6c9c069a8058fb147a876de465263b10ab.jpg)  
+Figure 13: False positive rate (FPR) and true positive rate (TPR) under different thresholds $( \delta )$ when the temperature $( T )$ is set to $1 , 0 0 0$ and the perturbation magnitude (ε) is set to 0.0014. The DenseNet is trained on CIFAR-10.
+
+![](images/3fa62a6e8a6ffc888e928740184f2ca4896529f91d48edb8bdfdcc61dd3d3b28.jpg)  
+Figure 14: Detection performance on Tiny-ImageNet (resize), LSUN (resize) and iSUN (resize) when parameters are tuned on six different out-of-distribution datasets. Each tuning set contains 1,000 images and each test set contains 9,000 images. Both DenseNet and Wide-ResNet are trained on CIFAR-10. Additional results on other datasets are provide in Table 10 and 11.
+
+# H ADDITIONAL DISCUSSIONS
+
+In this section, we present additional discussion on the proposed method. We first empirically show how the threshold $\delta$ affects the detection performance. We next show how the proposed method performs when the parameters are tuned on a certain out-of-distribution dataset and are evaluated on other out-of-distribution datasets. Finally, we show how the size of dataset for choosing parameters affects the detection performance.
+
+Effects of the threshold. We analyze how the threshold affects the following metrics: (1) FPR, i.e., the fraction of out-of-distribution images misclassified as in-distribution images; (2) TPR, i.e, the fraction of in-distribution images correctly classified as in-distribution images. In Figure 13, we show how the thresholds affect FPR and TPR when the temperature and perturbation magnitude are chosen optimally (i.e., $T = 1 , 0 0 0$ , $\varepsilon = 0 . 0 0 1 4$ ). From the figure, we can observe that the threshold corresponding to $9 5 \%$ TPR can produce small FPRs on all out-of-distribution datasets.
+
+Performance across datasets. To investigate how the parameters generalize across datasets, we tune the parameters using one out-of-distribution dataset and then evaluate on a different one. Given an out-of-distribution dataset, we first split the dataset into two disjoint subsets: tuning set and test set. The tuning set contains 1,000 images and the test set contains 9,000 images. We tune the parameters on the tuning set and evaluate the detection performance on the test set. We first choose the temperature $T$ and the perturbation magnitude $\varepsilon$ such that the FPR at TPR $9 5 \%$ is minimized on the tuning set of one out-of-distribution dataset. Next, we set $\delta$ to the threshold corresponding to $9 5 \%$ TPR and calculate the false positive rates on the test sets of other out-of-distribution datasets.
+
+In Figure 14, we show the detection performance on three out-of-distribution datasets when the parameters are tuned on six different datasets. From Figure 14, we can observe that the parameters tuned on different tuning sets can have quite similar detection FPRs on all of three out-of-distribution image sets. This may be due to the fact, shown in Figure 13, that the threshold corresponding to $9 5 \%$ TPR can produce small FPRs on all datasets.
+
+![](images/753533d77dc18af42ccc6c8fc5f69d009acf71c4015a87df28f24f09659ff8b3.jpg)  
+Figure 15: FPR at TPR $9 5 \%$ under different tuning set sizes. The DenseNet is trained on CIFAR-10 and each test set contains 8,000 out-of-distribution images.
+
+Performance vs. tuning set size. To show the effects of the tuning set size on the detection performance, we devise the following experiment. For each out-of-distribution dataset, we choose the tuning set size from 200, 400, 600, 800, 1000, 1200, 1400, 1600, 1800, 2000. For each set size, we tune the temperature $T$ and perturbation magnitude $\varepsilon$ to minimize the FPR at TPR $9 5 \%$ and calculate the FPR. In Figure 15, we show the detection performance of ODIN under different tuning set size. From Figure 15, we can observe that the FPR at TPR $9 5 \%$ tends to stabilize when the set size grows above 1,000.
+
+<table><tr><td colspan="8">DenseNet-BC-100</td></tr><tr><td>Test set</td><td>ImgNet (c)</td><td>ImgNet (r)</td><td>LSUN (c)</td><td>LSUN (c)</td><td>iSUN</td><td>Gaussian</td><td>Uniform</td></tr><tr><td colspan="8">Baseline (Hendrycks &amp; Gimpel, 2017) / Ours</td></tr><tr><td>ImgNet (c)</td><td>34.7/4.3</td><td>34.7/4.3</td><td>34.7/6.6</td><td>34.7/4.3</td><td>34.7/4.3</td><td>34.7/4.3</td><td>34.7/4.3</td></tr><tr><td>ImgNet (r)</td><td>40.7/7.5</td><td>40.7/7.5</td><td>40.7/14.9</td><td>40.7/7.5</td><td>40.7/7.5</td><td>40.7/7.5</td><td>40.7/7.5</td></tr><tr><td>LSUN (c)</td><td>39.3/13.8</td><td>39.3/13.8</td><td>39.3/8.1</td><td>39.3/13.8</td><td>39.3/13.8</td><td>39.3/13.8</td><td>39.3/13.8</td></tr><tr><td>LSUN (r)</td><td>33.6/4.8</td><td>33.6/4.8</td><td>33.6/10.4</td><td>33.6/4.8</td><td>33.6/4.8</td><td>33.6/4.8</td><td>33.6/4.8</td></tr><tr><td>iSUN</td><td>37.2/6.3</td><td>37.2/6.3</td><td>37.2/12.6</td><td>37.2/6.3</td><td>37.2/6.3</td><td>37.2/6.3</td><td>37.2/6.3</td></tr><tr><td>Gaussian</td><td>23.5/0.0</td><td>23.5/0.0</td><td>23.5/0.4</td><td>23.5/0.0</td><td>23.5/0.0</td><td>23.5/0.0</td><td>23.5/0.0</td></tr><tr><td>Uniform</td><td>12.3/0.0</td><td>12.3/0.0</td><td>12.3/4.5</td><td>12.3/0.0</td><td>12.3/0.0</td><td>12.3/0.0</td><td>12.3/0.0</td></tr></table>
+
+Table 10: Detection performance across different datasets. Each row corresponds to the FPR at TPR $9 5 \%$ on the same test set where parameters are tuned under different tuning sets. Each column corresponds to the FPR at TPR $9 5 \%$ on different test sets where parameters are tuned under the same tuning set. The DenseNet is trained on CIFAR-10.
+
+<table><tr><td colspan="8">Wide-ResNet- 28-10</td></tr><tr><td>Test set</td><td>ImgNet (c)</td><td>ImgNet (r)</td><td>LSUN (c)</td><td>LSUN (c)</td><td>iSUN</td><td>Gaussian</td><td>Uniform</td></tr><tr><td colspan="8">Baseline (Hendrycks &amp; Gimpel, ,2017) /Ours</td></tr><tr><td>ImgNet (c)</td><td>38.9/23.4</td><td>38.9/24.5</td><td>38.9/27.1</td><td>38.9/23.4</td><td>38.9/24.1</td><td>38.9/26.5</td><td>38.9/26.5</td></tr><tr><td>ImgNet (r)</td><td>45.6/25.5</td><td>45.6/25.5</td><td>45.6/32.9</td><td>45.6/25.8</td><td>45.6/25.8</td><td>45.6/27.9</td><td>45.6/27.9</td></tr><tr><td>LSUN (c)</td><td>35.0/28.1</td><td>35.0/31.27</td><td>35.0/21.8</td><td>35.0/28.2</td><td>35.0/28.9</td><td>35.0/29.7</td><td>35.0/29.7</td></tr><tr><td>LSUN (r)</td><td>35.0/18.9</td><td>35.0/18.0</td><td>35.0/25.6</td><td>35.0/17.6</td><td>35.0/18.0</td><td>35.0/19.1</td><td>35.0/19.1</td></tr><tr><td>iSUN</td><td>40.6/21.8</td><td>40.6/21.7</td><td>40.6/28.9</td><td>40.6/ 21.9</td><td>40.6/21.3</td><td>40.6/22.8</td><td>40.6/22.8</td></tr><tr><td>Gaussian</td><td>1.6/0.0</td><td>1.6/0.0</td><td>1.6/0.4</td><td>1.6/0.0</td><td>1.6/0.0</td><td>1.6/0.0</td><td>1.6/0.0</td></tr><tr><td>Uniform</td><td>0.3/0.0</td><td>0.3/0.0</td><td>0.3/0.0</td><td>0.3/0.0</td><td>0.3/0.0</td><td>0.3/0.0</td><td>0.3/0.0</td></tr></table>
+
+Table 11: Detection performance across different datasets. Each row corresponds to the FPR at TPR $9 5 \%$ on the same test set where parameters are tuned under different tuning sets. Each column corresponds to the FPR at TPR $9 5 \%$ on different test sets where parameters are tuned under the same tuning set. The Wide-ResNet is trained on CIFAR-10.
+
+![](images/ca62f28095afb45ac95cc6b63d2cf6e5da5918b077eaa2278439e09e325c0d61.jpg)  
+Figure 16: (a) The test accuracy on the images having softmax scores above the threshold corresponding to a certain true positive rate. (b) The test accuracy on the images having softmax scores below the threshold corresponding to a certain true positive rate. All networks are trained on CIFAR-10.
+
+![](images/79ac9013944515920bb62f247f16785ff2ed034895842149971b4dec19911fc5.jpg)  
+Figure 17: Outputs of DenseNet on thirty classes for an image of apple from CIFAR-80 and an image of red pepper from CIFAR-20. The label $\ddot { } \mathrm { ~ 0 ~ } ^ { 5 }$ denotes the class “apple” and the label “49" denotes the class “orange".
+
+# I ADDITIONAL ANALYSIS
+
+Difficult-to-classify images and difficult-to-detect images. We analyze the correlation between the images that tend to be out-of-distribution and images on which the neural network tend to make incorrect predictions. To understand the correlation, we devise the following experiment. For the fixed temperature $T$ and perturbation magnitude $\varepsilon$ , we first set $\delta$ to the softmax score threshold corresponding to a certain true positive rate. Next, we calculate the test accuracy on the images with softmax scores above $\delta$ and the test accuracy on the images with softmax score below $\delta$ , respectively. We report the results in Figure 16(a) and (b). From these two figures, we can observe that the images that are difficult to detect are more likely to be the images that are difficult to classify. For example, the DenseNet can achieve up to $9 8 . 5 \%$ test accuracy on the images having softmax scores above the threshold corresponding to $80 \%$ TPR, but can only achieve around $82 \%$ test accuracy on the images having softmax scores below the threshold corresponding to $80 \%$ TPR.
+
+Same manifold datasets. We provide additional empirical results showing how the term $E [ U _ { 2 } | U _ { 1 } ]$ affects the detection performance when in- and out-of-distribution datasets locate on the same manifold. In Figure 17, we show the outputs of DenseNet on thirty classes for an image of apple from CIFAR-80 (in-distribution) and an image of red pepper of CIFAR-20 (out-distribution). We can observe that the outputs of DenseNet for both images are quite similar to each other. In addition, we can observe that for both images, the second and third largest output are quite close to the largest output. This may be due the fact the image of red pepper shares some common features with the images in CIFAR-80. Furthermore, the similarity between the outputs for the images from
+
+Table 12: Distinguishing in- and out-of-distribution test set data for image classification. All values are percentages. $\uparrow$ indicates larger value is better, and $\downarrow$ indicates lower value is better. The architecture is DenseNet.   
+
+<table><tr><td></td><td>Out-of-distribution dataset</td><td>FPR (95% TPR) √</td><td>Detection Error √</td><td>AUROC</td><td>AUPR In</td><td>AUPR Out →</td></tr><tr><td></td><td></td><td colspan="5">Baseline (Hendrycks &amp; Gimpel, 2017)/Ours</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>57.1/47.2</td><td>31.1/26.1</td><td>89.0/89.8</td><td>91.2/91.4</td><td>86.8/88.7</td></tr><tr><td>CIFAR-100</td><td>CIFAR-10</td><td>81.8/81.4</td><td>43.4/43.2</td><td>76.1/76.7</td><td>79.9/80.4</td><td>71.3/72.6</td></tr></table>
+
+CIFAR-80 and CIFAR-20 can help explain that the detection task becomes harder when in- and out-of-distribution datasets locate on the same manifold.
+
+Reciprocal results between datasets. In Table 12, we show the reciprocal results between datasets. First, we train the DenseNet on the CIFAR-10 dataset (in-distribution) and evaluate the detection performance on the CIFAR-100 dataset (out-distribution). Next, we train the DenseNet on the CIFAR-100 dataset (in-distribution) and evaluate the detection performance on the CIFAR-10 dataset (out-distribution). From Table 12, we can observe that the performance of the DenseNet trained on CIFAR-10 is better than the performance of the DenseNet trained on CIFAR-100. This may be due to the fact that the DenseNet has a higher test accuracy on CIFAR-10 (around $9 5 \%$ ) compared to the test accuracy on CIFAR-100 (around $7 7 \%$ ).

parse/train/O6LPudowNQm/O6LPudowNQm.md

@@ -0,0 +1,806 @@
+# INT: AN INEQUALITY BENCHMARK FOR EVALUATING GENERALIZATION IN THEOREM PROVING
+
+Yuhuai $\mathbf { W } \mathbf { u } ^ { * } ,$ , Albert Qiaochu Jiang∗, Jimmy Ba & Roger Grosse University of Toronto & Vector Institute {ywu, ajiang, jba, rgrosse}@cs.toronto.edu
+
+# ABSTRACT
+
+In learning-assisted theorem proving, one of the most critical challenges is to generalize to theorems unlike those seen at training time. In this paper, we introduce INT, an INequality Theorem proving benchmark designed to test agents’ generalization ability. INT is based on a theorem generator, which provides theoretically infinite data and allows us to measure 6 different types of generalization, each reflecting a distinct challenge, characteristic of automated theorem proving. In addition, INT provides a fast theorem proving environment with sequence-based and graph-based interfaces, conducive to performing learning-based research. We introduce baselines with architectures including transformers and graph neural networks (GNNs) for INT. Using INT, we find that transformer-based agents achieve stronger test performance for most of the generalization tasks, despite having much larger outof-distribution generalization gaps than GNNs. We further find that the addition of Monte Carlo Tree Search (MCTS) at test time helps to prove new theorems.
+
+# 1 INTRODUCTION
+
+Advances in theorem proving can catalyze developments in fields including formal mathematics (McCune, 1997), software verification (Darvas et al., 2005), and hardware design (Kern and Greenstreet, 1999). Following its recent success across other application domains, machine learning has significantly improved the performance of theorem provers (Bansal et al., 2019; Bridge et al., 2014; Gauthier et al., 2018; Huang et al., 2019; Irving et al., 2016; Kaliszyk et al., 2018; Lee et al., 2020; Loos et al., 2017; Urban et al., 2011; Wang and Deng, 2020; Yang and Deng, 2019; Li et al., 2020; Rabe et al., 2020; Polu and Sutskever, 2020). Two key factors that make theorem proving particularly challenging for ML are data sparsity and that it requires out-of-distribution generalization.
+
+Firstly, due to the difficulty of formalizing mathematics for humans, manually generated formal proofs are necessarily expensive. Typical formal mathematics datasets contain thousands (Huang et al., 2019) to tens-of-thousands (Yang and Deng, 2019) of theorems — orders of magnitude smaller than datasets that enabled breakthroughs in areas such as vision (Deng et al., 2009) and natural language processing (Rajpurkar et al., 2016). Secondly, the assumption frequently made in machine learning that each data point is identically and independently distributed does not hold in general for theorem proving: interesting problems we want to prove are non-trivially different from those we have proofs for. Hence, the out-of-distribution generalization ability is crucial.
+
+Synthetic datasets that rely on procedural generation provide a potentially unlimited amount of data. Well-designed synthetic datasets have been shown to help understand the capabilities of machine learning models (Johnson et al., 2017; Ros et al., 2016; Weston et al., 2016). With the goal of alleviating the data scarcity problem and understanding out-of-distribution generalization for theorem proving, we introduce INT. INT is a synthetic INequality Theorem proving benchmark designed for evaluating generalization. It can generate a theoretically unlimited number of theorems and proofs in the domain of algebraic equalities and inequalities. INT allows tweaking of its problem distribution along 6 dimensions, enabling us to probe multiple aspects of out-of-distribution generalization. It is accompanied by a fast proof assistant with sequence and graph-based interfaces. A common reservation to hold for synthetic datasets is one of realism: can synthetic data help to prove realistic theorems? Polu and Sutskever (2020) adopted our generation method and showed that augmentation of $1 \%$ of synthetic theorems in training helped to complete $2 . 3 \%$ more proofs on Metamath (Megill and Wheeler, 2019). This demonstrates the usefulness of INT in real mathematics.
+
+Time and memory requirements for the proof assistant have often been an obstacle for using theorem provers as RL environments. Most existing proof assistants require a large software library to define numerous mathematical theorems, leading to slow simulation. Therefore, a key design objective for INT was to be lightweight and swift. Taking advantage of the limited scope of inequality theorems, we load a minimal library and achieve fast simulation. Reducing the simulation overhead allows for experimentation with planning methods such as MCTS which requires many calls to a simulator.
+
+We summarize the contributions of this paper as follows:
+
+1. We make, to the best of our knowledge, the first attempt to investigate an important question in learning-assisted theorem proving research, i.e., can theorem provers generalize to different problem distributions? We introduce INT for evaluating six dimensions of generalization. 2. We introduce and benchmark baseline agents for the six types of generalization tasks in INT. We find that transformer-based agents’ generalization abilities are superior when training and test data are drawn from the same distribution and inferior in out-of-distribution tasks in INT, compared to GNN-based agents. Surprisingly, despite larger generalization gaps, transformer-based agents have favorable test success rates over GNN-based ones in most cases. 3. We find that searching with MCTS at test time greatly improves generalization.
+
+# 2 RELATED WORKS
+
+Automatic and Interactive Theorem Proving. Modern Automatic Theorem Provers (ATPs) such as E (Schulz, 2013) and Vampire (Kovács and Voronkov, 2013) represent mathematical theorems in first-order logic and prove them with resolution-based proof calculi. On the other hand, Interactive Theorem Provers (ITPs) allow human formalization of proofs. This perhaps makes them more suitable for biologically inspired methods such as machine learning. Famous ITPs include Isabelle (Paulson, 1986), Coq (Barras et al., 1999), LEAN (de Moura et al., 2015), and HOL Light (Harrison, 1996).
+
+Learning-assisted Theorem Proving. Theorem provers have been improved by supervised learning (Urban et al., 2011; Bridge et al., 2014; Irving et al., 2016; Loos et al., 2017; Wang et al., 2017; Rocktäschel and Riedel, 2017; Bansal et al., 2019; Gauthier et al., 2018; Huang et al., 2019; Yang and Deng, 2019; Kaliszyk and Urban, 2015; Polu and Sutskever, 2020; Li et al., 2020; Rabe et al., 2020; Jakubuv and Urban, 2019; Olsák et al., 2020; Jakubuv et al., 2020; Kaliszyk et al., 2015; Gauthier and Kaliszyk, 2015). Wang et al. (2017) used graph embeddings to represent logic formulas and achieved state-of-the-art classification accuracy on the HolStep dataset (Kaliszyk et al., 2017). Reinforcement learning (RL) was employed in (Zombori et al., 2019; Gauthier, 2019; 2020). Kaliszyk et al. (2018) combined MCTS with RL to prove theorems with connection tableau. Notably, GPT-f (Polu and Sutskever, 2020) adopts our INT generation method for dataset augmentation.
+
+Datasets for Theorem Proving. There have been many formal mathematical libraries (Megill and Wheeler, 2019; Rudnicki, 1992; Gauthier, 2019). Formalized mathematical theorems include the Feit-Thompson theorem (Gonthier et al., 2013) and the Kepler Conjecture (Hales et al., 2017). The largest human formal reasoning dataset is IsarStep (Li et al., 2020), where they mined the archive of formal proofs and brought together 143K theorems in total. These works rely on human efforts to formalize theorems, which leads to small to moderate-sized datasets. There have been studies on synthesizing theorems (Urban, 2007; Urban et al., 2008; Piotrowski and Urban, 2018; Gauthier et al., 2017; 2016; Chvalovsky et al., 2019; Lenat, 1976; Fajtlowicz, 1988; Colton, 2012; Johansson \` et al., 2014) It is worth mentioning that there have been a few approaches (Urban and Jakubv, 2020; Wang and Deng, 2020) on neural theorem synthesizers. Our theorem generator INT is designed to be capable of creating an infinite number of theorems, as well as benchmarking the generalization ability of learning-assisted theorem provers.
+
+# 3 THE INT BENCHMARK DATASET AND PROOF ASSISTANT
+
+Our INT benchmark dataset provides mathematical theorems and a means to study the generalization capability of theorem provers. For this purpose, we need control over the distribution of theorems:
+
+this is achieved by a highly customizable synthetic theorem generator. We used a set of ordered field axioms (Dummit and Foote, 2004) to generate inequality theorems and a subset of it to generate equality theorems. Details of the axiomization schemes can be found in Appendix A. The code for generating theorems and conducting experiments is available at https://github.com/ albertqjiang/INT.
+
+# 3.1 TERMINOLOGY
+
+The axiom combination of a proof refers to the set of axioms used in constructing it. The sequence of axioms applied in order in the proof is called the axiom order. For example, let $A , B , C$ denote three unique axioms, and their order of application in a proof be $[ B , B , A , C ]$ . In this case, the axiom combination is the set $\{ A , B , C \}$ and the axiom order is the sequence $\left[ B , \bar { B } , A , C \right]$ . An initial condition is a (usually trivial) logic statement (e.g. $a = a$ ) to initiate the theorem generation process. The degree of an expression is the number of arithmetic operators used to construct it. For example, degree $( a ) = 0$ while degree $( ( a * c ) * b ) ^ { 2 } ) = 3$ .
+
+# 3.2 INT ASSISTANT
+
+![](images/8afe46e08dddbcd9dcf0c5cfb84349e176cf3b9c5670955e0a56b15a120bbd1c.jpg)  
+Figure 1: A proof of $a + b + c = c + a + b$ in LEAN and INT, with seq2seq and graph interfaces.
+
+We built a lightweight proof assistant to interact with theorem provers. It has two interfaces, providing theorem provers with sequential and graph representations of the proof state, respectively.
+
+A problem in INT is represented by a goal and a set of premises (e.g. $a + 0 = a , \emptyset )$ , which are mathematical propositions. The INT assistant maintains a proof state composed of the goal and the proven facts. The proof state is initialized to be just the goal and premises of the theorem. A proof is a sequence of axiom-arguments tuples (e.g. [(AdditionZero, $[ a + 0 ] )$ ]). At each step of the proof, a tuple is used to produce a logical relation in the form of assumptions conclusions (e.g. $\emptyset  a + 0 = a _ { \mathrm { . } }$ ). Then, if the assumptions are in the proven facts, the conclusions are added to the proven facts; if the conclusions include the goal, the unproven assumptions will become the new goal. The assistant considers the theorem proven, if after all steps in the proof are applied, the goal is empty or trivial.
+
+In Figure 1, we present the same proof in LEAN (de Moura et al., 2015) and INT assistants. They both process proofs by simplifying the goal until it is trivial. The INT assistant’s seq2seq interface (Figure 1b) is very similar to that of LEAN (Figure 1a) with the rewrite tactic. An action is composed of an axiom followed by argument names and their positions in the proof state. in obj indicates that the arguments can be found in the objective. The graph interface (Figure 1c) of the INT assistant allows theorem provers to chose axiom arguments from the computation graphs of the proof state by node. We can view theorem proving with this interface as a graph manipulation task.
+
+INT assistant provides fast simulation. To demonstrate this, we produced 10,000 typical proof steps in both interfaces, 40-character-long on average. We executed them with HOL Light (Harrison, 1996) and INT assistant. The average time it takes per step is 7.96ms in HOL Light and $1 . 2 8 \mathrm { m s }$ in INT, resulting in a $6 . 2 \times$ speedup. The correctness of the proofs is ensured by a trusted core of fewer than 200 lines of code.
+
+# 3.3 THEOREM GENERATOR
+
+One of the main contributions of this paper is to provide a generation algorithm that is able to produce a distribution of non-trivial synthetic theorems given an axiom order. Generating theorems by randomly sampling axiom and argument applications will often yield theorems with short proofs. Instead, we write production rules for axioms in the form of transformation and extension rules. With these production rules, we can find arguments and new premises required for longer proofs.
+
+We provide the theorem generation algorithm in Algorithm 1. The general idea of the algorithm is to morph a trivial logic statement into one that requires a non-trivial proof; we call this statement the core logic statement. We initiate the core logic statement $C _ { 0 }$ to be one of the initial conditions. At step $t$ of the generation process, we are given an axiom $a _ { t }$ specified by the axiom order. We apply the MORPH function associated with the axiom $a _ { t }$ to $C _ { t - 1 }$ and derive a new logic statement $C _ { t }$ and corresponding premises $P _ { t }$ . The key design idea in the MORPH function is to ensure that the newly
+
+# Algorithm 1 Theorem Generator
+
+1: function GENERATE_THEOREM(initial conditions $\mathcal { T }$ , axiom order $A$ )   
+2: Axiom order length $L = \operatorname { l e n } ( A )$ .   
+3: Initialize core logic statement $C _ { 0 } \sim U n i f o r m ( \mathscr { T } )$ , and the set of premises $P = \{ C _ { 0 } \}$ .   
+4: for $t \gets 1$ to $L$ do   
+5: Get axiom $a _ { t }  A [ t ]$ .   
+6: Get new logic statement and premises: $C _ { t }$ $\Upsilon _ { t } , P _ { t } \gets \mathrm { M O R P H } \left( a _ { t } , C _ { t - 1 } \right) .$ .   
+7: Add new premises to the set of all premises: $P  P \cup P _ { t }$ .   
+8: end for   
+9: return $C _ { L } , P$   
+10: end function
+
+generated logic statement and the premises form the implication $C _ { t - 1 } , a _ { t } , P _ { t } \to C _ { t }$ (see Appendix B   
+for detailslength is statement herefore, weaxiom order: and its premi $\Upsilon _ { 0 } , \{ a _ { t } , P _ { t } \} _ { t = 1 } ^ { L } \dot { {  } } C _ { L }$ ns from, where urned as $L$ ll steps together to obtain a proof whosedenotes the length. The last core logiche theorem generated. Below we show a $C _ { L }$ $C _ { 0 }$ $\{ P _ { t } \} _ { t = 1 } ^ { L }$   
+step-by-step example of how a theorem is generated with our algorithm.
+
+# A worked example
+
+Use Algorithm 1 to generate a theorem with initial conditions $\mathcal { T }$ : { $a = a$ , $b = b$ , $c = c$ , $d = d$ , $e = e$ } and axiom order $A$ : [AdditionAssociativity (AA), AdditionCommutativity (AC), EquivalenceImpliesDoubleInequality (EIDI), FirstPrincipleOfInequality (FPI)].
+
+Core logic statement C0 ∼ Unifor $m ( { \mathcal { T } } ) : a = a$ .   
+Step 1: $a _ { 1 } = \mathtt { A } \mathtt { A }$ . $C _ { 1 }$ : $a + ( b + c ) = ( a + b ) + c , P _ { 1 } = \emptyset$ .   
+Step 2: $a _ { 2 } = \mathtt { A C }$ . $C _ { 2 }$ : $a + ( b + c ) = ( b + a ) + c ,$ $P _ { 2 } = \varnothing$ .   
+Step 3: $a _ { 3 } = \mathtt { E I D I }$ . $C _ { 3 }$ $\begin{array} { r } { \mathrm { ~ 3 : ~ } a + ( b + c ) \geq ( b + a ) + c , } \end{array}$ $P _ { 3 } = \varnothing$ .   
+Step 4: $a _ { 4 } = \mathrm { F P I } . C _ { 4 }$ : $( a + ( b + c ) ) + d \geq ( ( b + a ) + c ) + e , P _ { 4 } = \{ d \geq e \} .$
+
+Theorem generated: Given $d \geq e$ , prove $a + ( b + c ) + d \geq b + a + c + e .$
+
+With recorded axiom and argument applications, we can synthesize proofs to the theorems. The proofs can be used for behavior cloning. Appendix E shows statistics of the generated proofs, including the distribution of length of theorems in characters, the distribution of axioms, and the distribution of the number of nodes in proof state graphs.
+
+# 4 EXPERIMENTS
+
+Our experiments are intended to answer the following questions:
+
+1. Can neural agents generalize to theorems: 1) sampled from the same distribution as training data, 2) with different initial conditions, 3) with unseen axiom orders, 4) with unseen axiom combinations, 5) with different numbers of unique axioms, 6) with shorter or longer proofs?
+
+![](images/1a921ca31fcd7eb4896aac4e1593a6196c7e5aab49e53e9959f1ebd9ec62f8ea.jpg)  
+Figure 2: Proof success rates on problems generated with different $K$ and $L$ parameters. Left: When the IID assumption holds, the success rate decreases as the two generation parameters $K$ and $L$ are increased. Right: All agents are trained on degree-0 problems and evaluated against problems of degree 0, 1, and 2. We find that transformer-based agents deteriorate in performance as the test problems become more complex than training problems. For GNN-based agents, there are no obvious trends as to how the proof success rate changes as the degree of the initial entities is varied.
+
+![](images/b7a06dca13c599efe784413163137dad81c9b313a9b4d1a69a76c5fee0089940.jpg)
+
+2. How do different architectures (transformer vs. GNN) affect theorem provers’ in-distribution and out-of-distribution generalization?
+
+3. Can search at test time help generalization?
+
+# 4.1 EXPERIMENT DETAILS
+
+In the following experiments, we used the proofs generated by the INT generator to perform behavior cloning. We then evaluated the success rates of trained agents in a theorem proving environment. We denote the cardinality of an axiom combination as $K$ and the length of a proof as $L$ . In the worked example, $K = 4$ and $L = 4$ . For each theorem distribution, we first generated a fixed test set of 1000 problems, and then produced training problems in an online fashion, while making sure the training problems were different from the test ones. For each experiment, we generated 1000 problems and performed 10 epochs of training before generating the next 1000. We ran 1500 such iterations in total, with 1.5 million problems generated. We used the Adam optimizer (Kingma and Ba, 2015). We searched over the learning rates $\{ 1 0 ^ { - 5 } , 3 \cdot 1 0 ^ { - 5 } , 1 0 ^ { - 4 } , 3 \cdot 1 0 ^ { \frac { . } { - 4 } } \}$ in preliminary experiments and found $1 0 ^ { - 4 }$ to be the best choice, which was used for following experiments. We used one Nvidia P100 or Tesla T4 GPU with 4 CPU cores for training. For each experiment, we ran 2 random seeds, and picked the one with higher validation success rates for test evaluation. Since this paper focuses on inequalities, all figures and tables in the main text are based on results from the ordered-field axiomization. We also include results of GNN-based agents on equalities in Appendix G.
+
+# 4.2 NETWORK ARCHITECTURES
+
+In this section, we introduce four baselines built on commonly used architectures: Transformers (Vaswani et al., 2017), Graph Neural Networks (GNNs), TreeLSTMs (Tai et al., 2015) and Bag-of-Words (BoWs). In preliminary experiments, we found Graph Isomorphism Networks (GINs) (Xu et al., 2019) to have performed the best among several representative GNN architectures. So we used GIN as our GNN of choice. Transformers interact with the INT proof assistant through the seq2seq interface while the other baselines through the graph interface.
+
+![](images/1747adfa8923526e1b3cc58ce25424215edcb3a4852d3f63576c251e81565065.jpg)  
+Figure 3: Proof success rates on test problems generated with $K$ and $L$ settings. Transformer and GNN perform well; TreeLSTM has mediocre performance; and Bag-of-Words performs poorly: it cannot prove more than $5 \%$ of problems.
+
+For sequence-to-sequence training, we used a character-level transformer architecture with 6 encoding layers and 6 decoding layers. We used 512 embedding dimensions, 8 attention heads and 2048 hidden dimensions for position-wise feedforward layers. We used dropout with rate 0.1, label smoothing with coefficient 0.1, and a maximum 2048 tokens per batch. The library fairseq (Ott et al., 2019) was used for its implementation.
+
+Table 1: Left: Average success rates $( \mathrm { i n } \% )$ ) of agents trained on different numbers of axiom orders. Right: Average success rates (in $\%$ ) of agents trained on different numbers of axiom combinations.   
+
+<table><tr><td rowspan="2"># Axiom orders</td><td colspan="2">100</td><td colspan="2">500</td><td colspan="2">2000</td><td colspan="2">5000</td></tr><tr><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td></tr><tr><td>Transformer</td><td>93.2</td><td>10.0</td><td>93.4</td><td>62.8</td><td>93.6</td><td>87.9</td><td>93.7</td><td>91.8</td></tr><tr><td>GNN</td><td>87.6</td><td>21.1</td><td>86.6</td><td>53.6</td><td>79.0</td><td>70.4</td><td>75.7</td><td>74.7</td></tr></table>
+
+<table><tr><td rowspan="2">#Axiom combinations</td><td colspan="2">25</td><td colspan="2">100</td><td colspan="2">200</td><td colspan="2">300</td></tr><tr><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td></tr><tr><td>Transformer</td><td>96.1</td><td>29.3</td><td>96.0</td><td>71.8</td><td>95.4</td><td>88.4</td><td>94.4</td><td>91.3</td></tr><tr><td>GNN</td><td>79.1</td><td>47.5</td><td>76.6</td><td>68.0</td><td>72.6</td><td>72.4</td><td>72.8</td><td>71.9</td></tr></table>
+
+For data in the graph form, each node in computation graphs corresponds to a character in the formula. We first used a learnable word embedding of dimension 512 to represent each node. We then used 6 GIN layers to encode graph inputs into vector representations, each with 512 hidden dimensions. The graph representation was obtained by taking the sum of all the node embeddings. For the TreeLSTM and the BoW baselines, we used a bidirectional TreeLSTM with 512 hidden dimensions and a BoW architecture to compute the graph representation vectors from node embeddings. The hyper-parameters used were found to be optimal in preliminary experiments. We then proposed axioms conditioned on the graph representations, with a two-layer MLP of hidden dimension 256. Conditioning on the graph representation and axiom prediction, the arguments are selected in an autoregressive fashion. Namely, the prediction of the next node is conditioned on the previous ones. For each argument prediction, we used a one-layer MLP with a hidden size of 256. We used graph neural network libraries Pytorch Geometric (Fey and Lenssen, 2019) for the GIN implementation, and DGL (Wang et al., 2019) for the TreeLSTM implementation.
+
+We trained agents based on architectures mentioned above by behavior cloning on theorems of various length $( L )$ and number of axioms $( K )$ . The success rates for proving 1000 test theorems are plotted in Figure 3. As the BoW architecture did not utilize the structure of the state, it failed miserably at proving theorems, indicating the significance of the structural information. TreeLSTM performed worse than the graph neural network baseline. The transformer and the GNN baselines perform the best among the architectures chosen and they take inputs in sequential and graph forms, respectively. Thus, we used these two architectures in the following experiments to investigate generalization.
+
+# 4.3 BENCHMARKING SIX DIMENSIONS OF GENERALIZATION
+
+IID Generalization In this experiment, the training and test data are independently and identically distributed (IID). The performances of our transformer-based and GNN-based agents are displayed on the left in Figure 2. As can be seen, the performance of agents examined on train and test problems are very similar. The largest difference between train and test success rates is $2 \%$ $( K 3 L 7 )$ . Notably, transformer-based agents complete $1 5 . 3 \%$ more test proofs than GNN-based agents on average.
+
+Initial Condition Consider two theorems: (1) $( a + b ) ^ { 2 } = a ^ { 2 } + b ^ { 2 } + 2 a b$ and (2) $( a + ( b + c ) ) ^ { 2 } =$ $a ^ { 2 } + ( b + c ) ^ { 2 } + 2 a ( b + c )$ . The two problems take the same axioms and the same number of steps to prove. However, the axiom argument complexities are different, which can be seen as a result of varying initial conditions. Can agents trained on problems like (1) prove theorems like (2)?
+
+For an initial condition of the form $X = X$ , we use the degree of the entity $X$ to determine the complexity. In this experiment, we trained agents on problems with initial conditions made up of entities of degree 0, and evaluated them on ones of degrees 1 and 2. The results are presented in Figure 2 (b) with various $K$ and $L$ . For transformer-based agents, the success rate drops $2 5 . 6 \%$ on degree-1 problems and $3 1 . 5 \%$ on degree-2 problems on average. However, for GNN-based agents, the largest generalization gap between training and test success rates is $3 \%$ $X 3 L 5 )$ . This shows that GNN agents can generalize to problems of higher complexities while transformer agents struggle.
+
+Axiom Orders Let $A$ and $B$ represent two different axioms. There are multiple orders in which they can be applied in a $K 2 L 3$ problem. $O _ { 1 } = [ A , A , B ]$ and $O _ { 2 } = [ B , A , B ]$ are two examples. Can an agent trained on problems generated with $O _ { 1 }$ prove theorems generated with $O _ { 2 }$ ?
+
+For both architectures, we investigated how well agents can generalize to problems with different axiom orders than those in training. We generated 100, 500, 2000, and 5000 axiom orders to use in the training set for different $K$ and $L$ settings. We evaluated the test success rates on 1000 unseen axiom orders with the corresponding $K$ and $L$ settings and averaged them. The results averaged over different $K$ and $L$ settings are shown on the left of Table 1 (See Appendix G.5 for the full results).
+
+It can be observed in the table that the test success rates rise when we increase the number of axiom orders in the training set. We notice that transformer-based agents have worse generalization than
+
+![](images/6526a0ed7944380306284a8edca97e5c88bc72921ab66a356fa8fcde21db7d89.jpg)  
+Figure 4: Proof success rates on problems generated with different parameters. Left: We keep $L$ the same and vary $K$ . The success rate is likely to decrease when the test problems have different $K$ from the training problems. Right: We keep $K$ the same and vary $L$ . For all agents, the proof success rate is lower on theorems that require longer proofs.
+
+GNN-based ones, as their average generalization gap is larger. This is particularly true when the number of axiom orders in the training set is 100: transformer-based agents can prove only $1 0 . 0 \%$ of test theorems. Remarkably, they still manage to complete more proofs than GNNs when the number of axiom orders in the training set exceeds 500.
+
+Axiom Combinations Consider three problems provable in the ordered field axiomization (Appendix A): (1) $a ^ { 2 } \geq 0$ , (2) $a * ( b + c ) = b * a + a * c .$ , and (3) $a ^ { 2 } + b ^ { 2 } - 2 a b \geq 0 $ . Solving (1) requires axiom SquareGEQZero (SGEQZ). Solving (2) requires axiom AdditionMultiplicationDistribution (AMD) and axiom MultiplicationCommutativity (MC). Solving (3) requires axiom SGEQZ and axiom AMD. Notice that all axioms used to prove (3) appear in the proofs of (1) and (2). We ask: can an agent trained on theorems like (1) and (2) prove theorems like (3)?
+
+In this set of experiments, we investigated how well theorem provers can generalize to problems with different axiom combinations than those in training for both architectures. We used 25, 100, 200, and 300 axiom combinations to generate the training set with various $K$ and $L$ settings, and evaluated the agents on test sets generated with 300 unseen combinations. The results averaged over different $K$ and $L$ settings are displayed on the right in Table 1 (See Appendix G.5 for full results). As the number of axiom combinations in training set increases, the generalization gap decreases and test success rate improves. The transformer-based agents have larger generalization gaps than GNN-based ones. This is particularly obvious when there are 25 axiom combinations: the generalization gap is $6 6 . 8 \%$ for transformers and $3 1 . 6 \%$ for GNNs. The test success rate of transformers is $1 8 . 2 \%$ lower than that of GNNs in this setting. Yet when there are more than 100 axiom combinations in training, transformers always perform better on the test sets, completing $3 . 8 \% - 1 9 . 6 \%$ more proofs. When the data is diverse, transformers perform better; when it is insufficient, GNNs are better. This might be due to the difference in the inductive bias used by both structures and might explain the choice of neural architectures in deep learning practice.
+
+Number of Axioms Here we investigated how well theorem provers could generalize to test problems that were generated with a different number of axioms than at training time. For instance, let $A$ , $B$ and $C$ represent different axioms. Will agents trained on $K 2 L 3$ axiom orders like $[ A , B , A ]$ and $[ C , C , B ]$ be able to prove theorems generated with $K 3 L 3$ axiom orders like $[ A , B , C ] ^ { i }$ ?
+
+We trained the agents on problems that have the same proof length $ { \Sigma } =  { \mathsf { 7 } } ,$ ) and varying $K \mathbf { s } .$ . The results are on the left of Figure 4. It can be observed from the figure that in general, agents perform the best on the $K$ they were trained on and worse when $K$ shifts away. Transformer-based agents showed better performances in all $K$ and $L$ settings, completing $2 0 . 9 \%$ more proofs than GNN-based ones on average. The success rates of transformer-based agents drop $5 . 6 \%$ on average when the test $K$ is shifted away by 1 from the training $K$ . For GNN-based agents, this averages to $5 . 1 \%$ . This shows that their generalization abilities to different number of axioms are similar.
+
+Proof Length We tested the generalization ability of theorem provers over the dimension of proof length of the theorems. To do this, we kept the cardinality of the axiom set to be the same $K = 3$ ) and varied the evaluated problems’ proof length $\langle L = 3 , 5 , 7 \rangle$ ). The result is presented on the right of Figure 4. For all of the agents trained, the success rate decreases as the length of the proof increases. This is due to the natural difficulty of completing longer proofs. Observing the figure, we see that the longer the training problems, the less they deteriorate in performance when proofs becomes longer: agents trained on $K 3 L 3$ problems complete $1 8 . 8 \%$ fewer proofs when $L$ is increased by 1, while ones trained on $K 3 L 7$ complete $5 . 7 \%$ fewer. Furthermore, the performance of transformer-based agents decreases by $1 2 . 2 \%$ when the test proof length increases by 1, compared to $1 0 . 7 \%$ for GNN-based ones. This suggests that transformers have inferior proof length generalization abilities than GNNs.
+
+Table 2: The behavior cloning (BC) agents versus the MCTS-assisted (search) agents. Left: The average success rates (in $\%$ ) of agents with and without MCTS over 1000 test theorems. Right: The average length of successful proofs by agents with and without MCTS over 1000 test theorems. $K$ denotes the cardinality of the axiom combination of a proof, $L$ denotes the length of the proof.   
+
+<table><tr><td rowspan="2">Train Evaluation</td><td colspan="2">K3L3</td><td colspan="2">K3L5</td><td colspan="2">K3L7</td></tr><tr><td>BC</td><td>Search</td><td>BC</td><td>Search</td><td>BC</td><td>Search</td></tr><tr><td>K3L3</td><td>92</td><td>98</td><td>91</td><td>97</td><td>81</td><td>96</td></tr><tr><td>K3L5</td><td>50</td><td>64</td><td>80</td><td>92</td><td>70</td><td>92</td></tr><tr><td>K3L7</td><td>25</td><td>40</td><td>64</td><td>78</td><td>58</td><td>81</td></tr><tr><td>Average</td><td>56</td><td>67</td><td>78</td><td>89</td><td>69</td><td>90</td></tr></table>
+
+<table><tr><td rowspan="2">Train Evaluation</td><td colspan="2">K3L3</td><td colspan="2">K3L5</td><td colspan="2">K3L7</td></tr><tr><td>BC</td><td>Search</td><td>BC</td><td>Search</td><td>BC</td><td>Search</td></tr><tr><td>K3L3</td><td>3.83</td><td>3.33</td><td>4.00</td><td>3.52</td><td>5.00</td><td>3.67</td></tr><tr><td>K3L5</td><td>7.54</td><td>6.82</td><td>6.2</td><td>5.52</td><td>6.84</td><td>5.56</td></tr><tr><td>K3L7</td><td>9.05</td><td>8.54</td><td>8.01</td><td>7.53</td><td>8.39</td><td>7.50</td></tr><tr><td>Average</td><td>6.81</td><td>6.23</td><td>6.07</td><td>5.52</td><td>6.74</td><td>5.58</td></tr></table>
+
+# 4.4 GENERALIZING WITH SEARCH
+
+We investigated whether performing search at test time can help agents generalize. Specifically, we investigated the effectiveness of Monte-Carlo Tree Search (MCTS) in finding proofs for unseen theorems with GNN-based agents. We chose GNN-based agents because they are better at outof-distribution generalization than transformer-based ones. Straightforward application of MCTS is impractical: in our theorem proving environment, the action space can be as large as $1 . 3 { \bf M }$ in size (see Appendix H). Hence, it would be infeasible to expand all possible actions when constructing the MCTS trees. Thus, we only performed MCTS over the axiom space (18 distinct axioms in total), and the arguments were proposed by the behavior cloning agents. Following AlphaGo Zero/AlphaZero (Silver et al., 2017; 2018), we trained a value network to estimate the value of a state. The value network is an MLP with two hidden layers of size 256, taking the GNN global representations of graphs as input. It was trained on 1000 episodes of rollouts obtained by the behavior cloning agents, with a learning rate of $3 \cdot 1 0 ^ { - 6 }$ . We also followed AlphaZero for the choice of the upper confidence bound, and the way that actions are proposed using visit counts. We used 200 simulations for constructing MCTS trees. More details can be found in Appendix F. We took the agents trained on $" K 3 L 3 "$ , "K3L5", and "K3L7" from section 4.3, and evaluated the agents’ performance when boosted by MCTS.
+
+Generalization The average success rates on 1000 test theorems are presented on the left in Table 2. We can see that search greatly improved the generalization results. It helped to solve $2 1 \%$ more problems on average for the agent trained on theorem distribution $K 3 L 7$ . Remarkably, when evaluating on $K 3 L 7$ theorems, search helped the $K 3 L 3$ agent improve its success rate from $2 5 \%$ to $4 0 \%$ : a relative improvement of $6 0 \%$ . It is interesting to see the $K 3 L 7$ behavior cloning agent solved $9 \%$ fewer problems on average than the $K 3 L 5$ agent. But search brought about much larger improvement to the $K 3 L 7$ agent and helped it to solve the largest proportion of problems on average – $9 0 \%$ . This indicates that skills learned through behavior cloning can be better exploited by searching.
+
+The average proof length for 1000 problems is presented on the right in Table 2 (we count those unsolved problem as 15, the step limit of an episode). We can see that by performing search, we are able to discover proofs of length closer to the ground truth proof length. For test theorems requiring 3-step proofs, the $K 3 L 3$ agent was able to prove them in 3.33 steps on average, with a gap of 0.33 steps to the optimal value. Similarly, for test theorems requiring 5-step proofs, the $K 3 L 5$ agent was able to prove them in 5.52 steps on average, with a gap of 0.52 steps; and for theorems requiring 7-step proofs, $K 3 L 7$ agent achieved a gap of 0.5 steps.
+
+# 4.5 DISCUSSION
+
+Experimental results suggested that transformer-based agents can complete more proofs in the IID generalization scenario but have larger out-of-distribution generalization gaps than GNN-based ones. The larger gap may be due to the lack of constraints in the sequence-to-sequence framework, in which the model can propose sequences that are invalid actions, whereas the graph interface constrains the model to propose valid actions only. However, we still see that transformers are able to complete more proofs overall. This shows the superiority of transformers in model capacity when applied to theorem proving. This insight motivates us to explore the possibility of taking the best from both worlds, combining both graph structural information and the strong transformer architecture to improve learning-assisted theorem proving. We leave it for future work.
+
+# 5 CONCLUSION
+
+We addressed the problem of diagnosing the generalization weaknesses in learning-assisted theorem provers. We constructed INT, a synthetic benchmark of inequalities, to analyze the generalization of machine learning methods. We evaluated transformer-based and GNN-based agents and a variation of GNN-based agents with MCTS at test time. Experiments revealed that transformer-based agents generalize better when the IID assumption holds while GNN-based agents generalize better in outof-distribution scenarios. We also showed that search can boost the generalization ability of agents. We stress that proving theorems in INT is not an end in itself. A hard-coded expert system might perform well on INT but not generalize to real-world mathematical theorems. Therefore, INT should be treated as instrumental when diagnosing generalization of agents. The best practice is to use INT in conjunction with real mathematical datasets.
+
+We believe our benchmark can also be of interest to the learning community, facilitating research in studying generalization beyond the IID assumption. The agents’ abilities to reason and to go beyond the IID assumption are essential in theorem proving, and studying how to acquire these abilities is at the frontier of learning research. In other domains requiring out-of-distribution generalization, such as making novel dialogs (Chen et al., 2017) or confronting unseen opponents in Starcraft (Vinyals et al., 2019), the requirements for data and computation forbid a generally affordable research environment. The INT benchmark provides practical means of studying out-of-distribution generalization.
+
+# ACKNOWLEDGEMENTS
+
+We thank Jay McClelland, Han Huang and Yuanhao Wang for helpful comments and discussions. We also thank anonymous reviewers for valuable and constructive feedbacks. We are grateful to the Vector Institute for providing computing resources. YW was supported by the Google PhD fellowship. AQJ was supported by a Vector Institute research grant.
+
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+
+APPENDIX A AXIOM SPECIFICATIONS   
+
+<table><tr><td>Field axioms</td><td>Definition</td></tr><tr><td>AdditionCommutativity (AC)</td><td>→a+b=b+a</td></tr><tr><td>AdditionAssociativity (AA)</td><td>→a+(b+c)=(a+b)+c</td></tr><tr><td>AdditionSimplification (AS)</td><td>a=b→a+(-b)=0</td></tr><tr><td>MultiplicatoinCommutativity (MC)</td><td>→a·b=b·a</td></tr><tr><td>MultiplicationAssociativity (MA)</td><td>→a·(b·c)=(a·b)·c</td></tr><tr><td>MultiplicationSimplification (MS)</td><td></td></tr><tr><td>AdditionMultiplicationLeftDistribution (AMLD)</td><td>→(a+b)·c=a·c+b·c</td></tr><tr><td>AdditionMultiplicationRightDistribution (AMRD)</td><td>→a.(b+c)=a·b+a·c</td></tr><tr><td>SquareDefinition (SD)</td><td>→a² =a·a</td></tr><tr><td>MultiplicationOne (MO)</td><td>→a·1=a</td></tr><tr><td>AdditionZero (A Z)</td><td>→a+0=a</td></tr><tr><td>PrincipleOfEquality (POE)</td><td>(a=b)^(c=d)→a+c=b+d</td></tr><tr><td>EquMoveTerm(Helper axiom) (EMT)</td><td>a+b=c→a=c+(-b)</td></tr><tr><td>Ordered field axioms</td><td>Definition</td></tr><tr><td>All feld axioms</td><td></td></tr><tr><td>SquareGEQZero (SGEQZ)</td><td>a=b→a·b≥0</td></tr><tr><td>EquivalenceImpliesDoubleInequality (E ID I)</td><td>a=b→(a≥b)&gt;(a≤b)</td></tr><tr><td>IneqMoveTerm (IMT)</td><td>a+b≥c→a≥c+(-b)</td></tr><tr><td>FirstPrincipleOfInequality (FPOI )</td><td>(a≥b)&gt;(c≥d)→a+c≥b+d</td></tr><tr><td>SecondPrincipleOfInequality (SPOI)</td><td>(a≥b)&gt;(c≥0)→a·c≥b·c</td></tr></table>
+
+# APPENDIX B THE MORPH FUNCTION
+
+We detail the morphing of $C$ at each step as follows. For each theorem $a$ , we define two symbolic patterns: $\mathcal { L } _ { a }$ and $\textstyle { \mathcal { R } } _ { a }$ , each represented by an expression (see Appendix $\textrm { C }$ for full details). For example, if $a$ is AdditionCommutativity, we use $\mathcal { L } _ { a } = x _ { 1 } + x _ { 2 }$ to denote any formula that is a sum of two terms ( $x _ { 1 }$ and $x _ { 2 }$ can be arbitrary terms). We check if one of the nodes in the computation graph of $C$ has the structure defined by ${ \mathcal { L } } _ { a }$ . If so, we then transform that node to a formula specified by $\textstyle { \mathcal { R } } _ { a }$ . For example, if $C$ is $( p + q ) + l = ( p + ( q + l ) )$ , $p + q$ is a node that matches the pattern specified by ${ \mathcal { L } } _ { a }$ , in which $x _ { 1 } = p$ and $x _ { 2 } = q$ . Let $\mathscr { R } _ { a } = x _ { 2 } + x _ { 1 }$ . We hence transform the node $p + q$ to $q + p$ as specified by $\textstyle { \mathcal { R } } _ { a }$ . As a result, $C ^ { \prime }$ becomes $( q + p ) + l = ( p + ( q + l ) )$ . If there is no node in the computation graph, we morph the core logic statement using the extension function $\mathcal { E }$ , defined in Appendix D . We sample nodes in available computation graphs and combine them with $C$ , coming up with $C ^ { \prime }$ and optionally a non-empty set of new premises $P _ { n e w }$ .
+
+# Algorithm 2 Theorem Generator (complete)
+
+<table><tr><td colspan="2">1: function GENERATE_THEOREM(initial conditions I,axiom order A)</td></tr><tr><td>2:</td><td>Axiom order length L = len(A). Initialize core logic statement Co ~ Uni form(T),and the set of premises P = {Co}.</td></tr><tr><td>3:</td><td>fort←1toLdo</td></tr><tr><td>4:</td><td></td></tr><tr><td>5: 6:</td><td>Get axiom at ← A[t]. Get new logic statement and premises: Ct,Pt ← MORPH(at, Ct-1).</td></tr><tr><td>7:</td><td>Add new premises to the set of all premises: P ← P U Pt.</td></tr><tr><td>8:</td><td>end for</td></tr><tr><td>9:</td><td>return CL, P</td></tr><tr><td>10:</td><td>end function</td></tr><tr><td></td><td>function MORPH(axiom a, core logic statement C)</td></tr><tr><td>2:</td><td>Collect Nt = {n| n is a node in C and matches the pattern specified by La}</td></tr><tr><td>4:</td><td>if Nt ≠ then Sample node n ~ Uniform(Nt).</td></tr><tr><td>6:</td><td>Transform n into new node n&#x27; using the mapping from La to Ra. C&#x27; ← Replace n with n&#x27; in the graph of C. Pnew ← @.</td></tr><tr><td>8:</td><td>else Collect N, the set of all nodes in the graphs.</td></tr><tr><td></td><td>Extend C and get the set of premises: C&#x27;, Pnew ← ε(a,C,N).</td></tr><tr><td>10:</td><td>end if</td></tr><tr><td>12:end function</td><td>return C&#x27;, Pnew.</td></tr></table>
+
+The reasons that we have two sets of rules for morphing are as follow: 1) Transformation rules can only be applied when the axiom will produce an equality, while extension rules can be applied to any axiom. So in order to generate theorems with all the axioms, we need the extension rules. 2) Almost all the extension rules will complicate the core logic statement while none of the transformation rules will. If we only have extension rules, the goal generated can be very complex even the proof is of moderate length. In order to generate compact theorems (goal not too complicated) with long proofs, the transformation rules are preferred. Therefore we only apply extension rules when transformation rules are not applicable.
+
+# APPENDIX C TRANSFORMATION RULES
+
+The implementations of the transformation rules $\mathcal { L }$ and $\mathcal { R }$ .
+
+<table><tr><td>Axiom (a)</td><td>La</td><td>Ra</td></tr><tr><td>AdditionCommutativity</td><td>x1+x2</td><td>x2+x1</td></tr><tr><td>AdditionAssociativity</td><td>x1+(x2+x3)</td><td>(x1+x2)+x3</td></tr><tr><td>AdditionSimplification</td><td>x1+(-x1)</td><td>0</td></tr><tr><td>MultiplicatoinCommutativity</td><td>x1:x2</td><td>X2:X1</td></tr><tr><td>MultiplicationAssociativity</td><td>x1·(x2:x3)</td><td>(x1:x2)·x3</td></tr><tr><td>MultiplicationSimplification</td><td>m1</td><td>1</td></tr><tr><td>AdditionMultiplicationLeftDistribution</td><td>(x1 +x2)·x3</td><td>X1:X3+X2·X3</td></tr><tr><td>AdditionMultiplicationRightDistribution</td><td>x1·(x2+x3)</td><td>X1:X2+x1:X3</td></tr><tr><td>SquareDefinition</td><td>m</td><td>x1:x1</td></tr><tr><td>MultiplicationOne</td><td>x1 ·1or1·x1</td><td>X1</td></tr><tr><td>AdditionZero</td><td>x1+0or0+x1</td><td>双1</td></tr><tr><td>SquareGEQZero</td><td>NA</td><td>NA</td></tr><tr><td>PrincipleOfEquality</td><td>NA</td><td>NA</td></tr><tr><td>EquMoveTerm</td><td>NA</td><td>NA</td></tr><tr><td>EquivalenceImpliesDoubleInequality</td><td>NA</td><td>NA</td></tr><tr><td>IneqMoveTerm</td><td>NA</td><td>NA</td></tr><tr><td>FirstPrincipleOfInequality</td><td>NA</td><td>NA</td></tr><tr><td>SecondPrincipleOfInequality</td><td>NA</td><td>NA</td></tr></table>
+
+# APPENDIX D EXTENSION FUNCTION
+
+For these axioms, the core logic statement $C$ needs to be of the form $\mathrm { L H S } ( C ) = \mathrm { R H S } ( C )$ .   
+Table 5   
+
+<table><tr><td>Axiom (a)</td><td>Extension function ε(C,a,N)</td></tr><tr><td>AdditionCommutativity</td><td>Sample node n ~ Uniform(N) return RHS(C) +n = n+ LHS(C), </td></tr><tr><td>AdditionAssociativity</td><td>Sample nodes n1,n2 ~ Uniform(N) return RHS(C)+(n1 + n2) =LHS(C)+n1 + n2,</td></tr><tr><td>AdditionSimplification</td><td>return 0 =LHS(C)+(-RHS(C)), </td></tr><tr><td>MultiplicatoinCommutativity</td><td>Sample node n ~ Uniform(N) return RHS(C)·n = n·LHS(C), </td></tr><tr><td>MultiplicationAssociativity</td><td>Sample nodes n1,n2 ~ Uni form(N) return RHS(C).(n1 : n2) =LHS(C)·n1 : n2,</td></tr><tr><td>MultiplicationSimplification</td><td>return 1 =LHS(C)· i 1 RHS(C），</td></tr><tr><td>AdditionMultiplicationLeftDistribution</td><td>Sample nodes n1,n2 ~ Uniform(N) return (n1 + n2) :RHS(C) = n1 · LHS(C) + n2 ·LHS(C), δ</td></tr><tr><td>AdditionMultiplicationRightDistribution</td><td>Sample nodes n1,n2 ~ Uni form (N) 1 return RHS(C) ·(n1 + n2) = LHS(C) · n1 + LHS(C) · n2, @</td></tr><tr><td>SquareDefinition</td><td>return LHS(C) : RHS(C) = LHS(C)², δ</td></tr><tr><td>MultiplicationOne</td><td>return Uniform({LHS(C) :1 = RHS(C), 1·LHS(C)=RHS(C) }), </td></tr><tr><td>AdditionZero</td><td>return Uniform({LHS(C) + 0 = RHS(C), 0 +LHS(C)=RHS(C) }), α</td></tr><tr><td>SquareGEQZero</td><td>return LHS(C) · RHS(C) ≥ 0, @</td></tr><tr><td>PrincipleOfEquality</td><td>Sample nodes n1, n2 ~ N, where n1 = n2 return LHS(C) + n1 = RHS(C) + n2,{n1 = n2}</td></tr><tr><td>EquMoveTerm</td><td>Only execute when LHS(C) is of the form x + y return x = RHS(C) +(-y),</td></tr><tr><td>EquivalenceImpliesDoubleInequality</td><td>return LHS(C) ≥RHS(C), </td></tr></table>
+
+For these axioms, the core logic statement $C$ needs to be of the form $\mathrm { L H S } ( C ) \geq \mathrm { R H S } ( C )$ .   
+Table 6   
+
+<table><tr><td>Axiom (a)</td><td>Extension function ε(C,a,N)</td></tr><tr><td>IneqMoveTerm</td><td>Only execute when LHS(C) is of the form x + y return x ≥ RHS(C) +(-y), @</td></tr><tr><td>FirstPrincipleOfInequality</td><td>Sample nodes n1,n2 ~ N,where n1 ≥ n2 return LHS(C) + n1 ≥ RHS(C) + n2, {n1 ≥ n2}</td></tr><tr><td>SecondPrincipleOfInequality</td><td>Sample node n ~ N,where n ≥ 0 return LHS(C) ·n ≥ RHS(C) ·n, {n ≥ 0}</td></tr></table>
+
+# APPENDIX E DATASET STATISTICS
+
+# APPENDIX E.1 THEOREM LENGTH
+
+We compare the length of the theorems generated in characters and plot their distributions in Figure 5. The length of the theorem in characters is a measure for how complicated it is. As is expected, the more complicated the theorem is, the longer the proof(bigger $L$ ). It is also worth noting that as $L$ becomes bigger, the distribution of theorem length becomes less concentrated. This is likely a consequence of a more spread-out theorem length range.
+
+![](images/065d55955681654ec6f89dcf82c0cb2a6d79a42ed2cd1da465be56bf02a3df08.jpg)  
+Figure 5: The distribution of theorem length in characters for field axioms(left) and ordered-field axioms(right) generated with parameters $K 3 L 3$ , $K 3 L 5$ , and $K 3 L 7$ . As the length of the proof is increased, so is the number of characters in the theorem, while the distribution of latter is less concentrated.
+
+# APPENDIX E.2 AXIOM DISTRIBUTIONS
+
+The frequency at which each axiom is applied influences the distribution of theorems our generator is able to produce. In Figure 6, we present the proportions of axioms that are applied in generating 10,000 theorems. Their frequencies are a measure of how easy it is to satisfy the conditions to apply them. For the field axioms, the PrincipleOfEquality axiom is the most frequently used $( 9 . 3 0 \% )$ and the EquMoveTerm axiom is the most rarely used $( 2 . 3 8 \% )$ . EquMoveTerm has a strict condition for application: the left hand side of the core logic statement has to be of the form $x + y$ , therefore not frequently applied. For the ordered-field axioms, the EquivalenceImpliesDoubleInequality axiom is the most frequently used $1 0 . 1 8 \%$ . Since we start with a trivial equality in generation and want to end up with an inequality, a transition from equality to inequality is needed. Among the ways of transitioning, this conditions to apply this axiom is easiest to satisfy. Its popularity is followed by the group of Field axioms, from MultiplicationCommutativity $( 4 . 6 9 \% )$ to AdditionAssociativity $( 5 . 9 8 \% )$ . The rest are ordered-field axioms which define the properties of inequalities, proportions ranging from IneqMoveTerm $( 1 . 1 4 \% )$ to FirstPrincipleOfInequality $( 5 . 7 4 \% )$ .
+
+![](images/fc2e0e6b480ddac4968c778400ecf145cd5369e703638a68a2071b3bd99ca8c2.jpg)  
+Figure 6
+
+# APPENDIX E.3 NUMBER OF NODES
+
+Since an action in the MDP consists of an axiom and a list of nodes as its arguments and the number of axioms is fixed, the number of nodes available determines the size the action space. Therefore it is interesting to investigate how many nodes are available in a proof. In Figure 7 we present the average number of nodes in proofs of different length. It can be told from the figure that the longer the proofs, the more nodes there will be, as expected. Comparing the axiom sets used, we find that the average number of nodes for ordered-field axioms is larger than that of field axioms. This is likely the consequence of ordered-field axioms, in generation, being more capable of producing new premises(e.g. First Principle of Inequality will produce an inequality premise(see Table 6), thus adding more nodes in the graphs).
+
+![](images/1fdd360256aafc4f315ee35230ecfbec25e2fd15638fb80ae89f84e32de3cf17.jpg)  
+Figure 7
+
+# APPENDIX F MORE EXPERIMENTAL DETAILS FOR GENERALIZATION WITH SEARCH
+
+We give more experimental details for the use of MCTS. Following (Silver et al., 2017), in the selection step of the MCTS tree construction, we use the following formula to select the next action,
+
+$$
+a ^ { * } = \mathrm { a r g m a x } _ { a } \left( Q ( s , a ) + c _ { \mathrm { p u c t } } P ( s , a ) \frac { \sqrt { \sum _ { b } N ( s , b ) } } { 1 + N ( s , a ) } \right) ,
+$$
+
+where $Q ( s , a )$ represents the action value function, $N ( s , a )$ denotes the visit counts, $P ( s , a )$ is the prior probability, and $c _ { \mathrm { p u c t } }$ is a constant hyperparameter. In all of our experiments, we used the behavior cloning policy for computing $P ( s , a )$ , and we used $c _ { \mathrm { p u c t } } = 1$ . After the MCTS tree is built, the action is sampled from the policy distribution $\pi ( a | s ) = N ( s , a ) ^ { \frac { 1 } { \tau } }$ , where $\tau$ is a hyperparameter and was chosen as 1 in our experiments.
+
+# APPENDIX G MORE TRAINING AND EVALUATION RESULTS
+
+![](images/08c233066186fb6c45cf54733a90dcbe9f4164c42cbe731dc93cbcbf47166186.jpg)  
+APPENDIX G.1 LEARNING CURVES OF GNN-BASED AGENTS   
+Figure 8: Proof success rates for field axioms(left) and ordered-field axioms(right) of GNN-based agents trained on different $K$ and $L$ parameters. We keep the $K$ the same and vary the $L$ . The agents converge slower and to a lower success rate when the proof length is increased. Also, the agents on field axioms are easier to train than those on ordered-field axioms.
+
+# APPENDIX G.2 PERFORMANCE VARIATION OF TRAINED AGENTS
+
+To verify that the experimental results are statistically significant, we ran the experiments on proof length generalization in subsection 4.3 with 5 random seeds and tabled the results.
+
+Table 7: Success rates of agents trained and tested on problems of different parameters (mean $\pm$ std) in percentage.   
+
+<table><tr><td>Transformers</td><td>Tested on</td><td>K3 L3</td><td>K3 L5</td><td>K3 L7</td></tr><tr><td rowspan="3">Trained on</td><td>K3 L3</td><td>97.6± 0.9</td><td>31.5 ± 1.6</td><td>10.9 ± 1.0</td></tr><tr><td>K3 L5</td><td>97.2 ± 0.7</td><td>88.3 ± 1.2</td><td>59.5 ± 1.6</td></tr><tr><td>K3L7</td><td>96.6 ± 1.2</td><td>87.0 ± 1.6</td><td>75.1 ± 1.2</td></tr><tr><td>GNNs</td><td>Tested on</td><td>K3 L3</td><td>K3 L5</td><td>K3 L7</td></tr><tr><td rowspan="3">Trained on</td><td>K3L3</td><td>91.5 ± 0.5</td><td>45.6 ± 1.7</td><td>16.5 ± 0.8</td></tr><tr><td>K3L5</td><td>86.4 ± 0.9</td><td>77.8 ± 0.9</td><td>58.4 ± 1.5</td></tr><tr><td>K3 L7</td><td>82.0 ± 1.3</td><td>71.4 ± 1.1</td><td>56.5 ± 1.5</td></tr></table>
+
+![](images/1f082c61df141e55512f96c5242d34857244df6a5cb7367113af5071ab84c7b2.jpg)  
+APPENDIX G.3 GNN-BASED AGENTS ON IID GENERALIZATION   
+Figure 9: Proof success rates on problems generated with different $K$ and $L$ parameters ( $K$ denotes the cardinality of the axiom combination of a proof, $L$ denotes the length of the proof). When the IID assumption holds, the success rate decreases as the two generation parameters $K$ and $L$ are increased.
+
+![](images/b3048d29a65cb47b3a1eeceb889d73cd5116dff6b4a6c1eaabc01f1474978123.jpg)  
+APPENDIX G.4 GNN-BASED AGENTS ON INITIAL CONDITION GENERALIZATION   
+Figure 10: Proof success rates on problems generated with different $K$ and $L$ parameters ( $K$ denotes the cardinality of the axiom combination of a proof, $L$ denotes the length of the proof). When generalizing to different initial conditions, there are no obvious trends as to how the proof success rate changes as the degree of the initial entities is varied.
+
+Table 8: Top: Proof success rates (in $\%$ ) of agents trained on different numbers of axiom orders. Bottom: Proof success rates (in $\%$ ) of agents trained on different numbers of axiom combinations. $K$ denotes the cardinality of the axiom combination of a proof, $L$ denotes the length of the proof.   
+
+<table><tr><td rowspan="2">Architecture</td><td rowspan="2">Axiom orders</td><td colspan="2">100</td><td colspan="2">500</td><td colspan="2">2000</td><td colspan="2">5000</td></tr><tr><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td></tr><tr><td rowspan="6">Transformer</td><td>K3 L3</td><td>98.4</td><td>32.6</td><td>99.5</td><td>90.0</td><td>98.8</td><td>98.7</td><td>97.6</td><td>97.6</td></tr><tr><td>K3L5</td><td>95.3</td><td>6.3</td><td>94.0</td><td>56.3</td><td>94.0</td><td>94.9</td><td>96.5</td><td>94.9</td></tr><tr><td>K3 L7</td><td>87.8</td><td>3.8</td><td>88.0</td><td>46.4</td><td>88.3</td><td>77.5</td><td>88.4</td><td>85.5</td></tr><tr><td>K5L5</td><td>94.7</td><td>5.6</td><td>97.0</td><td>72.9</td><td>97.4</td><td>93.1</td><td>97.5</td><td>96.9</td></tr><tr><td>K5L7</td><td>89.7</td><td>1.8</td><td>88.6</td><td>48.6</td><td>89.3</td><td>75.2</td><td>88.6</td><td>84.0</td></tr><tr><td>Average</td><td>93.2</td><td>10.0</td><td>93.4</td><td>62.8</td><td>93.6</td><td>87.9</td><td>93.7</td><td>91.8</td></tr><tr><td rowspan="6">GNN</td><td>K3 L3</td><td>84.3</td><td>38.6</td><td>94.4</td><td>73.9</td><td>93.7</td><td>89.0</td><td>90.5</td><td>92.3</td></tr><tr><td>K3L5</td><td>92.7</td><td>17.1</td><td>86.3</td><td>60.0</td><td>84.4</td><td>72.9</td><td>77.7</td><td>77.1</td></tr><tr><td>K3 L7</td><td>82.4</td><td>14.1</td><td>82.4</td><td>33.8</td><td>68.6</td><td>57.7</td><td>70.2</td><td>63.5</td></tr><tr><td>K5 L5</td><td>91.0</td><td>23.0</td><td>89.7</td><td>61.2</td><td>81.8</td><td>75.0</td><td>78.3</td><td>80.8</td></tr><tr><td>K5L7</td><td>87.5</td><td>12.9</td><td>80.2</td><td>39.0</td><td>66.5</td><td>57.4</td><td>61.6</td><td>60.0</td></tr><tr><td>Average</td><td>87.6</td><td>21.1</td><td>86.6</td><td>53.6</td><td>79.0</td><td>70.4</td><td>75.7</td><td>74.7</td></tr></table>
+
+<table><tr><td rowspan="2">Architecture</td><td rowspan="2">Axiom combos</td><td colspan="2">25</td><td colspan="2">100</td><td colspan="2">200</td><td colspan="2">300</td></tr><tr><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td><td>Train</td><td>Test</td></tr><tr><td rowspan="6">Transformer</td><td>K3L3</td><td>99.2</td><td>34.1</td><td>99.0</td><td>72.8</td><td>99.5</td><td>96.1</td><td>98.6</td><td>98.2</td></tr><tr><td>K3 L5</td><td>97.8</td><td>29.3</td><td>98.6</td><td>66.3</td><td>97.5</td><td>89.5</td><td>94.3</td><td>90.4</td></tr><tr><td>K3L7</td><td>93.6</td><td>25.0</td><td>91.9</td><td>55.9</td><td>91.5</td><td>80.0</td><td>91.9</td><td>85.9</td></tr><tr><td>K5L5</td><td>98.5</td><td>27.4</td><td>98.4</td><td>87.6</td><td>97.0</td><td>93.6</td><td>97.3</td><td>94.9</td></tr><tr><td>K5L7</td><td>91.2</td><td>30.5</td><td>92.2</td><td>76.3</td><td>91.7</td><td>82.9</td><td>90.0</td><td>87.0</td></tr><tr><td>Average</td><td>96.1</td><td>29.3</td><td>96.0</td><td>71.8</td><td>95.4</td><td>88.4</td><td>94.4</td><td>91.3</td></tr><tr><td rowspan="6">GNN</td><td>K3L3</td><td>96.3</td><td>61.6</td><td>96.0</td><td>90.1</td><td>92.7</td><td>91.2</td><td>95.3</td><td>92.0</td></tr><tr><td>K3 L5</td><td>82.1</td><td>43.4</td><td>80.3</td><td>68.9</td><td>78.5</td><td>74.9</td><td>76.5</td><td>76.1</td></tr><tr><td>K3L7</td><td>72.1</td><td>34.3</td><td>68.1</td><td>57.2</td><td>62.3</td><td>63.7</td><td>62.5</td><td>62.0</td></tr><tr><td>K5 L5</td><td>77.8</td><td>61.6</td><td>78.9</td><td>71.0</td><td>74.5</td><td>78.4</td><td>72.8</td><td>74.9</td></tr><tr><td>K5L7</td><td>67.2</td><td>36.8</td><td>59.7</td><td>52.7</td><td>54.9</td><td>54.0</td><td>56.7</td><td>54.5</td></tr><tr><td>Average</td><td>79.1</td><td>47.5</td><td>76.6</td><td>68.0</td><td>72.6</td><td>72.4</td><td>72.8</td><td>71.9</td></tr></table>
+
+![](images/414529975117152b385b6edb06d5dba50322c2d0d71726c6284ee117a417f499.jpg)  
+Figure 11: Proof success rates on problems generated with different parameters ( $\boldsymbol { \mathcal { K } }$ denotes the cardinality of the axiom combination of a proof, $L$ denotes the length of the proof). We keep parameter $L$ the same and vary parameter $K$ . The success rate is likely to decrease when an agent is evaluated on problems that have different $K$ than the problems it is trained on.
+
+![](images/c550cbd50e0ffc707f84de21a8d104842f523b862214c80f364accac6d0159ea.jpg)  
+APPENDIX G.7 GNN-BASED AGENTS ON PROOF LENGTH GENERALIZATION   
+Figure 12: Proof success rates on problems generated with different parameters ( $K$ denotes the cardinality of the axiom combination of a proof, $L$ denotes the length of the proof). We keep parameter $K$ the same and vary parameter $L$ . For all agents, the proof success rate is lower on theorems that require longer proofs. The best-performing agent for problems of a given length is usually the agent trained on problems of the same length.
+
+# APPENDIX H THEOREM PROVING AS A MARKOV DECISION PROCESS (MDP)
+
+We model theorem proving as a Markov Decision Process. A state $s$ in the MDP is the proof state maintained by the assistant, namely, the goal, the premises and the proven facts, represented by computation graphs. An action $a$ is a tuple of an axiom and a sequence of arguments. We denote the axiom space as $\mathcal { X }$ and the argument space, the set of all the nodes in available computation graphs, as $\mathcal { N }$ . The maximum number of arguments for one axiom within our axiomizations is 3, therefore the action space is $\mathcal { A } = \mathcal { X } \times \mathcal { N } ^ { 3 }$ . The assistant ignores redundant arguments if fewer than 3 are needed for the axiom considered. We show in Appendix E.3 the distribution of the number of nodes for proofs of different length. The size of the discrete action space can be as large as $1 8 \times 4 2 ^ { 3 } \approx 1 . 3 3 { \overset { \cdot } { \times } } 1 0 ^ { 6 }$ . The deterministic state transition function $P ( s , a )$ is implicitly determined by the proof assistant. When the proof assistant deems the proof complete and the theorem proven, the episode terminates and a reward of one is given. Otherwise, the reward is zero at each step. When the step limit for a proof is exhausted, the episode terminates with a reward of zero. For experiments in this paper, we used a step limit of 15.
+
+# APPENDIX I EXAMPLE PROBLEMS
+
+# Equality theorems
+
+# Theorem 1
+
+Goal: $( ( 0 \cdot 1 ) \cdot ( ( - ( a ^ { 2 } ) ) \cdot c ) ) = ( ( ( - ( a ^ { 2 } ) ) \cdot ( ( a \cdot a ) + ( - ( a ^ { 2 } ) ) ) ) \cdot c )$
+
+# Theorem 2
+
+Theorem 3
+
+Goal: $\begin{array} { r } { 0 = ( ( ( ( c + 0 ) \cdot ( a + a ) ) \cdot ( \frac { 1 } { ( ( c \cdot a ) + ( c \cdot a ) ) } ) ) + ( - ( 0 + 1 ) ) ) } \end{array}$
+
+# Theorem 4
+
+Premises: $( b + d ) = b$ Goal: $\begin{array} { r } { ( 1 + ( - ( ( b + b ) \cdot ( \frac { 1 } { ( ( b + ( b + d ) ) \cdot 1 ) } ) ) ) ) = ( 0 + 0 ) } \end{array}$
+
+# Theorem 5
+
+Premises: $( a + d ) = b$ Goal: $\begin{array} { r } { 1 = ( ( ( d \cdot ( ( a + d ) + ( ( c + ( a + d ) ) + 0 ) ) ) \cdot ( ( d \cdot ( a + d ) ) + ( d \cdot ( c + b ) ) ) ) \cdot ( \frac { 1 } { ( ( d \cdot ( ( a + d ) + ( ( c + ( a + d ) ) + 0 ) ) ) ^ { 2 } ) } ) } \end{array}$
+
+# Theorem 6
+
+# Theorem 7
+
+Goal: $( ( a \cdot ( a + 0 ) ) + ( ( - ( 0 + a ) ) \cdot ( a + 0 ) ) ) = ( ( a \cdot 0 ) + ( 0 \cdot 0 ) )$
+
+# Theorem 8
+
+# Theorem 9
+
+# Theorem 10
+
+# Theorem 11
+
+Goal: $( 1 \cdot ( b + a ) ) = ( ( 0 + ( a + b ) ) + 0 )$
+
+# Theorem 12
+
+Goal: $( ( ( - c ) \cdot ( - c ) ) + ( ( ( - c ) \cdot c ) + ( ( - c ) \cdot ( - c ) ) ) = ( ( ( - c ) \cdot ( - c ) ) + ( 0 \cdot ( - c ) ) )$
+
+# Theorem 13
+
+Goal: $( ( ( a ^ { 2 } ) \cdot ( a \cdot ( a + 0 ) ) ) + ( a \cdot ( a \cdot ( a + 0 ) ) ) ) = ( ( ( ( a ^ { 2 } ) \cdot ( a ^ { 2 } ) ) + ( a \cdot ( a ^ { 2 } ) ) ) + 0 )$
+
+# Theorem 14
+
+Goal: $\begin{array} { r } { \small \{ ( ( ( b \cdot 1 ) \cdot ( a \cdot c ) ) \cdot ( b \cdot a ) ) + ( ( ( b \cdot 1 ) \cdot ( a \cdot c ) ) \cdot ( b \cdot a ) ) ) = ( ( ( ( ( b \cdot a ) \cdot c ) \cdot ( b \cdot a ) ) + ( ( ( ( b \cdot a ) \cdot c ) \cdot ( b \cdot a ) ) ) \cdot 1 ) } \end{array}$
+
+Theorem 15
+
+Theorem 16
+
+Goal: $0 = ( ( 0 + ( - ( ( a \cdot b ) + ( - ( b \cdot a ) ) ) ) ) + ( - ( 0 \cdot 1 ) ) )$
+
+# Theorem 17
+
+$$
+\begin{array} { r } { ( ( ( b \cdot d ) + ( b \cdot ( b + ( a + d ) ) ) ) + ( ( b + c ) + e ) ) = ( ( ( ( b \cdot d ) + ( b \cdot ( b + c ) ) ) \cdot 1 ) + ( a + d ) ) } \end{array}
+$$
+
+# Theorem 18
+
+Goal: $( ( ( \left( { \frac { 1 } { b } } \right) \cdot b ) \cdot b ) \cdot 1 ) = ( ( b \cdot 1 ) \cdot 1 )$
+
+# Theorem 19
+
+Goal: $\because ( ( 1 \cdot ( b \cdot ( c + a ) ) ) + ( b \cdot a ) ) + 1 ) = ( 1 \cdot ( ( 1 \cdot ( ( b \cdot c ) + ( b \cdot a ) ) ) + ( ( b \cdot a ) + 1 ) ) )$
+
+# Theorem 20
+
+Premises: $( b + d ) = c$ ; $( ( 1 \cdot a ) + e ) = a$ Goal: $\begin{array} { r } { ( ( ( a + ( b + d ) ) \cdot ( \frac { 1 } { ( ( 1 \cdot a ) + c ) } ) ) + ( ( 1 \cdot a ) + e ) ) = ( ( 1 \cdot 1 ) + a ) } \end{array}$
+
+# Theorem 21
+
+Goal: $\begin{array} { r } { ( ( ( ( c ^ { 2 } ) \cdot ( ( c ^ { 2 } ) \cdot c ) ) + ( - ( ( ( c \cdot c ) \cdot ( c ^ { 2 } ) ) \cdot c ) ) ) + ( b + b ) ) = ( ( 1 \cdot ( ( 0 + b ) + b ) ) + ( - 0 ) ) } \end{array}$
+
+# Theorem 22
+
+Premises: $( b + d ) = ( a \cdot b )$ Goal: ( $\cdot ( ( ( ( c + c ) \cdot ( ( ( a \cdot b ) \cdot c ) + ( c + c ) ) ) + ( ( c + c ) \cdot ( c + c ) ) ) + ( a \cdot b ) ) ) = ( ( ( ( ( c + c ) \cdot ( ( ( a \cdot ( b \cdot c ) ) + c ) + ( c + c ) ) ) + ( b + d ) ) \cdot 1 ) + 0 )$
+
+# Theorem 23
+
+Premises: $( ( 0 \cdot 1 ) + d ) = ( 1 \cdot 0 )$ Goal: $( ( ( ( ( a + ( 0 \cdot 1 ) ) \cdot ( 1 \cdot 0 ) ) ) + ( - b ) ) + ( 1 \cdot 0 ) ) ) + ( 1 \cdot 0 ) ) = ( ( ( ( ( a \cdot ( 1 \cdot 0 ) ) + ( ( b + ( - b ) ) \cdot ( 1 \cdot 0 ) ) ) + ( ( - b ) + ( 1 \cdot 0 ) ) ) + ( ( 0 \cdot 1 ) + d ) ) + 0 )$
+
+# Theorem 24
+
+Premises: $( a + d ) = ( 1 + c )$ Goal: (((((1·b)+(c·b))+(1+c))2 )·((1+c)·b)) = ((((((1·b)+(c·b))+(1+c))·(((b·(1+(1·c)))+(a+d))·1))·(1+c))·b)
+
+# Theorem 25
+
+Premises: $( a + d ) = ( b \cdot 1 )$ Goal: $0 = ( ( b + ( a + d ) ) + ( - ( ( b \cdot 1 ) + ( b \cdot 1 ) ) )$ )
+
+# Theorem 26
+
+# Theorem 27
+
+Premises: $( c + d ) = ( b + c )$ Goal: $( 1 \cdot ( ( ( ( b + c ) \cdot c ) + ( b \cdot ( b + c ) ) ) + ( c + d ) ) ) = ( ( ( ( b + c ) ^ { 2 } ) + ( b + c ) ) \cdot 1 )$
+
+# Theorem 28
+
+Premises: $( ( 1 \cdot b ) + d ) = b$ Goal: $ ' ( ( ( ( 1 \cdot b ) + b ) \cdot ( a \cdot 1 ) ) \cdot ( ( ( b + ( ( 1 \cdot b ) + d ) ) \cdot a ) \cdot 1 ) ) + 0 ) = ( ( ( ( b + ( 1 \cdot b ) ) \cdot ( a \cdot 1 ) ) ^ { 2 } ) \cdot 1 )$
+
+# Theorem 29
+
+Goal: $( ( ( b \cdot 1 ) + 0 ) \cdot ( 1 \cdot 0 ) ) = ( ( ( b \cdot 1 ) \cdot ( ( - ( 0 + b ) ) + ( 1 \cdot b ) ) ) + ( 0 \cdot ( ( - ( 0 + b ) ) + ( 1 \cdot b ) ) ) )$
+
+# Theorem 30
+
+# Theorem 31
+
+Goal: $( ( 1 \cdot ( b \cdot b ) ) \cdot b ) = ( 1 \cdot ( 0 + ( ( ( 0 + b ) \cdot b ) \cdot b ) ) )$ )
+
+# Theorem 32
+
+Goal: $( ( ( c \cdot ( c \cdot 1 ) ) + 0 ) \cdot 1 ) = ( ( ( c \cdot c ) + 0 ) \cdot 1 )$
+
+# Theorem 33
+
+Goal: $\begin{array} { r } { 1 = \big ( 1 \cdot \big ( \frac { 1 } { ( ( 1 + 0 ) \cdot ( \frac { 1 } { ( ( b \cdot ( \frac { 1 } { b } ) ) + 0 ) } ) ) } \big ) \big ) } \end{array}$
+
+# Theorem 34
+
+Goal: $( ( ( ( ( ( ( ( c + a ) \cdot a ) \cdot ( c + a ) ) \cdot c ) \cdot ( a + c ) ) \cdot ( c + a ) ) \cdot ( c + a ) ) = ( ( ( ( ( ( ( ( a + c ) \cdot ( c + a ) ) \cdot a ) \cdot c ) \cdot ( a + c ) ) \cdot ( c + a ) ) \cdot ( c + a ) )$
+
+# Theorem 35
+
+Goal: $0 = ( ( - ( 1 \cdot 0 ) ) + ( ( - ( c + c ) ) + ( ( 1 \cdot c ) + c ) ) )$ )
+
+# Theorem 36
+
+$\begin{array} { r } { 1 = \big ( 1 \cdot \big ( \frac { 1 } { ( a \cdot ( \frac { 1 } { ( ( ( a + c ) + a ) + ( - ( c + a ) ) ) } ) } \big ) \big ) } \end{array}$
+
+# Theorem 37
+
+Premises: $( a + d ) = a$ ; $\begin{array} { r } { \left( \left( { \frac { 1 } { c } } \right) + e \right) = b } \end{array}$ Goal: ( $\begin{array} { r } { ( ( 1 \cdot ( 1 \cdot ( \frac { 1 } { \left( c \cdot ( \frac { 1 } { c } ) \right) } ) ) ) + a ) + b ) = ( 1 \cdot ( ( ( 1 \cdot 1 ) + ( a + d ) ) + ( ( \frac { 1 } { c } ) + e ) ) ) } \end{array}$
+
+# Theorem 38
+
+Goal: $0 = ( ( b \cdot ( b + ( - b ) ) ) + ( - ( ( ( 0 + 0 ) \cdot b ) + 0 ) ) )$ )
+
+# Theorem 39
+
+Goal: $( ( 1 \cdot c ) + ( - ( 1 \cdot ( c \cdot 1 ) ) ) ) \cdot 1 ) = ( ( 0 \cdot 1 ) \cdot 1 )$
+
+# Theorem 40
+
+# Theorem 41
+
+Goal: (0 + ( $0 + ( ( c + c ) \cdot c ) ) \cdot ( a \cdot b ) ) ) = ( 0 + ( ( ( c \cdot c ) \cdot a ) + ( ( c \cdot c ) \cdot a ) ) \cdot b$ ))
+
+# Theorem 42
+
+# Theorem 43
+
+# Theorem 44
+
+# Theorem 45
+
+$$
+0 = ( 0 + ( - ( ( ( ( 0 \cdot 0 ) + ( a \cdot 0 ) ) + ( - ( ( ( ( ( a \cdot 5 ) + ( a \cdot b ) ) + ( ( b + b ) + b ) ) + ( - ( ( ( a \cdot ( b + b ) ) + ( b + b ) ) + b ) ) ) + a ) \cdot 0 ) \cdot 1 ) ) ) )
+$$
+
+Theorem 46
+
+Premises: $( ( a + b ) + d ) = ( a + b )$ ; $( b + e ) = a$ Goal: $( a \cdot a ) = ( 1 \cdot ( a \cdot a ) )$
+
+# Theorem 47
+
+# Theorem 48
+
+Goal: $( b + ( ( ( ( a + a ) \cdot 1 ) \cdot a ) + 0 ) ) \cdot a ) = ( ( b \cdot a ) + ( ( ( a \cdot a ) \cdot 1 ) + ( a \cdot ( a \cdot 1 ) ) ) \cdot a ) )$
+
+# Theorem 49
+
+$$
+\begin{array} { r l } & { ( ( ( 1 + b ) \cdot ( ( a \cdot ( ( c \cdot ( ) ) + ( c ^ { 2 } ) ) ) + 1 ) + b ) ) + ( ( 1 + b ) \cdot ( ( ( a \cdot ( ( c \cdot 1 ) + ( c ^ { 2 } ) ) ) + 1 ) + b ) ) ) = ( ( ( ( 1 + b ) + ( 1 + b ) ) \cdot ( ( a \cdot ( ( c \cdot 1 ) + ( c ^ { 2 } ) ) ) + 1 ) + b ) ) } \\ & { ( c \cdot 1 ) ) + ( a \cdot ( c \cdot ( c \cdot ( 1 ) ) ) ) + ( 1 + b ) \cdot 1 ) ) \cdot 1 ) } \end{array}
+$$
+
+# Theorem 50
+
+$$
+0 = { \big ( } ( ( ( ( 0 + ( ( ( ( c \cdot c ) + ( c \cdot c ) ) + ( ( c \cdot c ) + ( c \cdot c ) ) ) ) ) + 0 ) + c ) \cdot a ) + ( - ( ( ( 0 + ( ( ( ( c + c ) \cdot c ) + ( c + c ) ) + ( c \cdot c ) ) ) \cdot a ) + ( c \cdot a ) ) ) { \big ) }
+$$
+
+# Inequality theorems
+
+# Theorem 1
+
+Premises: $( 1 + d ) \geq 0$ ; $( b + e ) \geq 0$ Goal: $\begin{array} { r } { ( ( ( ( 1 + 1 ) \cdot ( a \cdot ( \frac { 1 } { a } ) ) ) \cdot ( 1 + d ) ) + ( b + e ) ) \geq ( ( ( ( 1 \cdot 1 ) + ( 1 \cdot 1 ) ) \cdot ( 1 + d ) ) + 0 ) } \end{array}$
+
+# Theorem 2
+
+Goal: $( b ^ { 2 } ) \geq ( 0 + ( b \cdot ( 1 \cdot b ) ) )$
+
+# Theorem 3
+
+Premises: $( ( c + 0 ) + d ) \geq 0 ; ( d + e ) \geq b$ Goal: $( ( c \cdot ( ( c + 0 ) + d ) ) + ( d + e ) ) \geq ( ( ( 0 + c ) \cdot ( ( c + 0 ) + d ) ) + b )$
+
+# Theorem 4
+
+Goal: $( b + 0 ) \geq ( ( ( 0 + b ) + c ) + c ) + ( - ( c + c ) ) )$ )
+
+# Theorem 5
+
+# Theorem 6
+
+# Theorem 7
+
+Premises: $( ( 0 + a ) + d ) = 0$ Goal: $\left( \left( ( 0 + a ) \cdot a \right) + \left( \left( 0 + a \right) + d \right) \right) \geq \left( \left( a ^ { 2 } \right) + 0 \right)$
+
+# Theorem 8
+
+Premises: $( b + d ) = a$ Goal: $( ( c \cdot b ) + ( b \cdot b ) ) \geq ( 1 \cdot ( ( ( c + a ) + ( - ( b + d ) ) ) + b ) \cdot b ) )$
+
+# Theorem 9
+
+# Theorem 10
+
+Premises: $( c + d ) \geq 0$ Goal: $( b \cdot ( c + d ) ) \geq ( ( ( ( b + b ) + 0 ) + ( - b ) ) \cdot ( c + d ) )$
+
+# Theorem 11
+
+Goal: $( ( ( b + 0 ) + ( b + c ) ) + 0 ) \geq ( ( ( b + b ) + c ) + 0 )$
+
+# Theorem 12
+
+Goal: $( ( c \cdot ( c + 0 ) ) + 0 ) \geq ( ( c ^ { 2 } ) + 0 )$
+
+# Theorem 13
+
+Goal: $( 1 \cdot ( b \cdot 1 ) ) \geq ( ( 1 \cdot b ) \cdot 1 )$
+
+# Theorem 14
+
+Goal: $\begin{array} { r } { 1 \geq ( ( ( ( b \cdot ( \frac { 1 } { b } ) ) + ( \frac { 1 } { b } ) ) + 1 ) \cdot ( \frac { 1 } { ( ( 1 + ( \frac { 1 } { b } ) ) + 1 ) } ) ) } \end{array}$ )
+
+# Theorem 15
+
+# Theorem 16
+
+# Theorem 17
+
+Goal: $( ( ( c \cdot b ) + a ) \cdot ( ( c \cdot b ) + ( c \cdot b ) ) ) \geq ( ( a \cdot ( ( c \cdot b ) + ( c \cdot b ) ) ) + ( ( c \cdot b ) + ( c \cdot b ) ) )$ ) · ((c · b) + (c · b))))
+
+# Theorem 18
+
+Goal: $( ( a \cdot b ) \cdot 1 ) \geq ( ( ( ( a \cdot 1 ) \cdot b ) \cdot 1 ) \cdot 1 )$
+
+# Theorem 19
+
+Goal: $a \geq ( ( a + c ) + ( - c ) )$
+
+# Theorem 20
+
+Goal: $( ( c \cdot b ) \cdot b ) \geq ( b \cdot ( b \cdot c ) )$
+
+# Theorem 21
+
+Premises: $( a + d ) = a$ ; $( ( a + d ) + e ) \geq 0$ ; $( b + f ) \ge ( 0 \cdot 0 )$ Goal: $\tau ( ( ( ( ( c \cdot 0 ) + ( 0 \cdot 0 ) ) + ( a + d ) ) \cdot ( ( 0 + ( ( c + 0 ) \cdot ( a + ( - a ) ) ) ) + a ) ) \cdot ( ( a + d ) + e ) ) + ( b + f ) ) \geq ( ( 0 \cdot ( ( a + d ) + e ) ) + ( 0 \cdot 0 ) )$
+
+# Theorem 22
+
+Premises: $( c + d ) \geq 0$ ; $( ( 0 + 0 ) + e ) \geq ( 0 + 0 )$ Goal: $\begin{array} { r } { \big ( ( ( ( ( 0 + ( c + ( - c ) ) ) \cdot ( - c ) ) \cdot ( - c ) ) ^ { \prime } \cdot ( \frac { 1 } { ( ( 0 , ( - c ) ) + ( 0 \cdot ( - c ) ) ) } ) ) \cdot ( 0 + 1 ) ) \cdot ( c + d ) ) + ( ( 0 + 0 ) + e ) ) \geq ( ( 0 \cdot ( c + d ) ) + ( 0 + 0 ) ) } \end{array}$
+
+# Theorem 23
+
+Premises: $( ( a ^ { 2 } ) + d ) \geq 0$ Goal: (( $( ( a \cdot a ) + c ) \cdot ( 0 + ( 1 \cdot ( a \cdot a ) ) ) \cdot ( ( a ^ { 2 } ) + d ) ) \geq ( ( ( a \cdot a ) \cdot ( ( a ^ { 2 } ) - ( a \cdot a ) ) ) \cdot ( ( a \cdot a ) ) ^ { 2 } )$ + 0)) + (c · ((a 2 ) + 0))) · ((a 2 ) + d))
+
+# Theorem 24
+
+Premises: $( c + d ) = c ;$ $( ( 0 + a ) + e ) \geq a$
+
+Goal: $\tau ( ( ( a + b ) \cdot ( ( ( a + ( - a ) ) + ( a + b ) ) + ( c + d ) ) ) \cdot ( ( ( ( ( 0 + a ) + b ) + c ) \cdot ( a + b ) ) \cdot 1 ) ) + ( ( 0 + a ) + e ) ) \geq ( 0 + a )$
+
+# Theorem 25
+
+Goal: $\begin{array} { r } { 1 \geq \left( \left( a \cdot \left( c + b \right) \right) \cdot \left( \frac { 1 } { \left( \left( a \cdot c \right) + \left( a \cdot b \right) \right) } \right) \right. } \end{array}$
+
+# Theorem 26
+
+Premises: $( a + d ) \geq b$ Goal: $\begin{array} { r } { ( ( 0 \cdot ( ( ( ( ( ( ( a + c ) + a ) \cdot ( a \cdot c ) ) \cdot ( a \cdot c ) ) \cdot ( a \cdot c ) ) + ( ( a \cdot c ) \cdot ( a \cdot c ) ) ) + ( - ( ( ( ( ( ( a + ( c + a ) ) \cdot a ) \cdot c ) + ( a \cdot c ) ) \cdot ( a \cdot c ) ) + 0 ) ) ) ) + ( a + d ) ) \geq ( 0 + b ) } \end{array}$
+
+# Theorem 27
+
+$$
+\begin{array} { r l } & { \mathfrak { s e s : } ( ( ( c \cdot b ) + d ) = ( b \cdot b ) ; ( ( b \cdot b ) + e ) \ge a } \\ & { ( ( ( ( b + b ) + ( b + b ) ) \cdot ( ( ( ( c \cdot ( b \cdot b ) ) + b ) + ( b ^ { 2 } ) ) ) + ( ( b \cdot b ) + e ) ) \ge ( ( ( ( b + b ) \cdot ( ( ( ( c \cdot b ) \cdot b ) + ( b + b ) ) + ( ( c \cdot b ) + c ) ) } \\ & { - ( ( b + b ) \cdot ( ( ( c \cdot b ) \cdot b ) + ( b + b ) ) + ( ( c \cdot b ) + d ) ) ) ) + a ) } \end{array}
+$$
+
+# Theorem 28
+
+Premises: $( ( b \cdot 0 ) + d ) \geq c$ Goal: (( $( ( b + ( ( ( 0 + c ) + ( 0 + c ) ) + 0 ) ) \cdot 0 ) \cdot ( ( b \cdot 0 ) + ( ( ( 0 + c ) + ( 0 + c ) ) \cdot 0 ) \cdot$ )) · 0))) + ((b · 0) + d)) ≥ (0 + c)
+
+# Theorem 29
+
+Premises: $( a + d ) \geq 0$ Goal: (( $) \cdot ( ( ( ( c \cdot c ) + ( c \cdot 0 ) ) \cdot a ) + ( - ( ( ( c + 0 ) \cdot ( ( c + 0 ) \cdot a ) ) \cdot 1 ) ) ) + ($ (a + d)) ≥ (0 + 0)
+
+# Theorem 30
+
+# Theorem 31
+
+Goal: $( 0 + ( 0 + ( c + b ) ) ) \geq ( 0 + ( ( b + c ) + 0 ) )$
+
+# Theorem 32
+
+Goal: $( a + ( a + 0 ) ) \geq ( ( ( ( 0 + a ) + 0 ) + a ) + 0 )$
+
+# Theorem 33
+
+Premises: g) ≥ 0 Goal: ( $\begin{array} { r l } & { \mathfrak { S } : ( ( c + c ) + a ) \ge a ; ( a + e ) \ge \mathfrak { U } ; ( ( c + c ) + J ) \ge ( 0 + a ) ; ( 0 + g ) \ge 0 } \\ & { \big ( ( ( ( ( c + c ) + ( c + c ) ) \cdot ( ( c + c ) + ( c + c ) ) ) + ( ( c + c ) + d ) ) + ( ( d + e ) ) + ( ( c + c ) + f ) ) + ( b + g ) \big ) \ge \big ( ( ( ( 0 + a ) + 0 ) + ( 0 + a ) ) + 0 \big ) } \end{array}$
+
+# Theorem 34
+
+Goal: $( ( ( 0 + b ) + c ) + a ) \geq ( 0 + ( 0 + ( b + ( c + a ) ) ) )$ )
+
+# Theorem 35
+
+Premi Goal: $\begin{array} { r l } & { \mathrm { s e s : } \ ( a + d ) \geq 0 ; \ ( a + e ) \geq ( c \cdot c ) ; \ ( e + f ) \geq 0 ; \ ( c + g ) \geq 0 ; \ ( c + h ) \geq ( c + g ) ; \ ( c + i ) \geq 0 } \\ & { \mathrm { ( } ( ( ( ( ( c \cdot c ) \cdot ( a + d ) ) + ( a + e ) ) \cdot ( e + f ) ) \cdot ( c + g ) ) + ( c + h ) ) \cdot ( c + i ) ) \geq ( ( ( ( ( 0 \cdot ( a + d ) ) + ( c \cdot c ) ) \cdot ( e + f ) ) \cdot ( c + f ) + ( c + h ) ) \cdot ( c + f ) } \end{array}$ $g ) ) + ( c + g ) ) \cdot ( c + i ) )$
+
+# Theorem 36
+
+Goal: $( 1 \cdot ( 1 \cdot ( 1 \cdot a ) ) ) \geq ( 1 \cdot ( ( a + 0 ) + 0 ) )$ )
+
+# Theorem 37
+
+Premises: $( b + d ) \geq b$ ; $( ( c + b ) + e ) \geq c$ ; $( b + f ) \geq a$ ; $( e + g ) \geq ( b + f )$ Goal: $\begin{array} { r } { ( ( ( c + ( b + d ) ) + ( b + f ) ) + ( e + g ) ) \ge ( ( ( ( ( c + b ) + c ) + ( - ( ( c + b ) + e ) ) ) + a ) + ( b + f ) ) } \end{array}$
+
+# Theorem 38
+
+Goal: $( ( a + ( ( ( b + c ) \cdot ( b + c ) ) + ( ( c + b ) \cdot b ) ) ) \cdot ( ( c + b ) + ( c + b ) ) ) \geq ( ( ( ( ( ( b + c ) \cdot ( c + b ) ) + ( ( b + c ) \cdot b ) ) + a ) \cdot ( c + b ) ) + a$ ( $( ( ( ( b + c ) \cdot ( c + b ) ) + ( ( b + c ) \cdot b ) ) + a ) \cdot ( c + b ) )$ )
+
+# Theorem 39
+
+$$
+\begin{array} { r l } & { \mathfrak { c s } ; ( c + d ) = b ; ( ( c + b ) + e ) = ( c + d ) ; ( a + f ) \ge 0 ; ( 0 + g ) \ge 0 ; ( g + h ) \ge 0 ; ( d + i ) \ge 0 } \\ & { ( ( ( ( ( c + ( c + d ) ) + ( ( c + b ) + e ) ) \cdot ( a + f ) ) \cdot ( 0 + g ) ) \cdot ( g + h ) ) \cdot ( d + i ) ) \ge ( ( ( ( ( c + b ) + ( c + d ) ) \cdot ( a + f ) ) \cdot ( g + h ) ) \cdot ( d + i ) ) } \end{array}
+$$
+
+# Theorem 40
+
+Goal: $( ( ( ( c + a ) \cdot b ) \cdot b ) + ( a + c ) ) \geq ( ( a + c ) + ( ( ( a + c ) \cdot b ) \cdot b ) )$
+
+# Theorem 41
+
+Goal: $\begin{array} { r } { ( ( ( c + b ) + ( a + ( c + b ) ) ) \cdot ( \frac { 1 } { ( ( ( ( 1 \cdot c ) + b ) + a ) + ( c + b ) ) } ) ) \geq ( 1 \cdot 1 ) } \end{array}$
+
+# Theorem 42
+
+Premises: $( c + d ) = b$ Goal: (((((c·b)+(c 2 ))·((b+c)·(c·b)))+(c+d))·(((((c·(b+c))·(b+c))·c)·b)+b)) ≥ (((((c·b)+(c 2 ))·((b+c)·(c·b)))+(c+d))2 )
+
+# Theorem 43
+
+$$
+\begin{array} { r l } & { \mathrm { ~ e s : ~ } ( a + d ) = b ; ( d + e ) = a ; ( c + f ) \ge 0 ; ( ( b + b ) + g ) \ge 0 } \\ & { ( ( 1 \cdot ( c + f ) ) \cdot ( ( b + b ) + g ) ) \ge ( ( ( ( ( b + b ) + a ) \cdot ( \frac { 1 } { ( 0 + ( ( b + ( a + d ) ) + ( d + e ) ) ) } ) ) \cdot ( c + f ) ) \cdot ( ( b + b ) + g ) ) } \end{array}
+$$
+
+# Theorem 44
+
+Goal: (( $\left( \left( \left( a \cdot 1 \right) \cdot a \right) \cdot 1 \right) \cdot b \right) + \left( \left( \left( a \cdot 1 \right) \cdot \left( a \cdot 1 \right) \right) \cdot \left( a \cdot a \right) \right) ) \geq \left( 1 \cdot \left( \left( \left( a \cdot a \right) \right. \right.$ $a \cdot 1 ) ) \cdot ( a \cdot a ) ) ) \geq ( 1 \cdot ( ( ( a \cdot a ) \cdot 1 ) \cdot b ) + ( ( ( a \cdot a ) \cdot 1 ) \cdot ( a \cdot a ) ) ) \}$
+
+# Theorem 45
+
+Premises: $( ( c + 0 ) + d ) \geq b ;$ $( 1 + e ) \geq a$ Goal: ( $( 0 + ( ( c + 0 ) + d ) ) + ( 1 + e ) ) \geq ( ( ( 0 + ( - ( ( c \cdot 1 ) + ( - ( c + 0 ) ) ) ) ) + b ) + a )$
+
+# Theorem 46
+
+# Theorem 47
+
+# Theorem 48
+
+Premises: $( a + d ) = b$ Goal: $( 1 \cdot ( ( b + b ) + ( - ( 1 \cdot ( b + ( a + d ) ) ) ) ) ) \geq ( 1 \cdot ( 0 \cdot 1 ) )$
+
+# Theorem 49
+
+Premises: $( ( c \cdot b ) + d ) = a$ ; $( ( c \cdot b ) + e ) \geq b$ Goal: $( ( ( b \cdot b ) \cdot ( a \cdot ( c \cdot b ) ) ) + ( ( c \cdot b ) + e ) ) \geq ( ( ( ( b \cdot b ) \cdot a ) \cdot ( c \cdot b ) ) + b )$
+
+# Theorem 50
+
+Goal: $( ( ( a + c ) \cdot ( c + a ) ) + ( ( a \cdot ( c + a ) ) + ( ( c \cdot c ) + ( c \cdot a ) ) ) \geq ( ( ( a + c ) \cdot ( ( c + a ) + ( c + a ) ) ) \cdot 1 )$

parse/train/Ogga20D2HO-/Ogga20D2HO-.md

@@ -0,0 +1,438 @@
+# FEDMIX: APPROXIMATION OF MIXUP UNDER MEAN AUGMENTED FEDERATED LEARNING
+
+Tehrim Yoon & Sumin Shin & Sung Ju Hwang & Eunho Yang
+
+Korea Advanced Institute of Science and Technology (KAIST) Daejeon, South Korea {tryoon93,sym807,sjhwang82,eunhoy}@kaist.ac.kr
+
+# ABSTRACT
+
+Federated learning (FL) allows edge devices to collectively learn a model without directly sharing data within each device, thus preserving privacy and eliminating the need to store data globally. While there are promising results under the assumption of independent and identically distributed (iid) local data, current state-of-the-art algorithms suffer from performance degradation as the heterogeneity of local data across clients increases. To resolve this issue, we propose a simple framework, Mean Augmented Federated Learning (MAFL), where clients send and receive averaged local data, subject to the privacy requirements of target applications. Under our framework, we propose a new augmentation algorithm, named FedMix, which is inspired by a phenomenal yet simple data augmentation method, Mixup, but does not require local raw data to be directly shared among devices. Our method shows greatly improved performance in the standard benchmark datasets of FL, under highly non-iid federated settings, compared to conventional algorithms.
+
+# 1 INTRODUCTION
+
+As we enter the era of edge computing, more data is being collected directly from edge devices such as mobile phones, vehicles, facilities, and so on. By decoupling the ability to learn from the delicate process of merging sensitive personal data, Federated learning (FL) proposes a paradigm that allows a global neural network to learn to be trained collaboratively from individual clients without directly accessing the local data of other clients, thus preserving the privacy of each client (Konecnˇ y et al. ´ , 2016; McMahan et al., 2017). Federated learning lets clients do most of the computation using its local data, with the global server only aggregating and updating the model parameters based on those sent by clients.
+
+One of the standard and most widely used algorithm for federated learning is FedAvg (McMahan et al., 2017), which simply averages model parameters trained by each client in an element-wise manner, weighted proportionately by the size of data used by clients. FedProx (Li et al., 2020b) is a variant of FedAvg that adds a proximal term to the objective function of clients, improving statistical stability of the training process. While several other methods have been proposed until recently (Mohri et al., 2019; Yurochkin et al., 2019; Wang et al., 2020), they all build on the idea that updated model parameters from clients are averaged in certain manners.
+
+Although conceptually it provides an ideal learning environment for edge devices, the federated learning still has some practical challenges that prevent the widespread application of it (Li et al., 2020a; Kairouz et al., 2019). Among such challenges, the one that we are interested in this paper is the heterogeneity of the data, as data is distributed non-iid across clients in many real-world settings; in other words, each local client data is not fairly drawn from identical underlying distribution. Since each client will learn from different data distributions, it becomes harder for the model to be trained efficiently, as reported in (McMahan et al., 2017). While theoretical evidence on the convergence of FedAvg with non-iid case has recently been shown in (Li et al., 2020c), efficient algorithms suitable for this setting have not yet been developed or systematically examined despite some efforts (Zhao et al., 2018; Hsieh et al., 2020).
+
+![](images/33e2864a5e6f382503714917d3564db22011d598f6834f61bf385c580a967973.jpg)  
+Figure 1: Brief comparisons of Mixup strategies in FL and MAFL. (a) Global Mixup: Raw data is exchanged and directly used for Mixup between local and received data, which violates privacy. (b) Local Mixup: Mixup is only applied within client’s local data. (c) NaiveMix: Under MAFL, Mixup is performed between local data and received averaged data. (d) FedMix: Under MAFL, our novel algorithm approximates Global Mixup using input derivatives and averaged data.
+
+In addition to non-iid problem, another important issue is that updating model parameters individually trained by each client is very costly and becomes even heavier as the model complexity increases. Some existing works Smith et al. (2016); Sattler et al. (2019) target this issue to decrease the amount of communications while maintaining the performance of FedAvg. A more practical approach to reduce communication cost is to selectively update individual models at each round, rather than having all clients participate in parameter updates. This partial participation of clients per round hardly affects test performance in ideal iid settings but it can exacerbate the heterogeneity of weight updates across clients and as a result, the issue of non-iid (McMahan et al., 2017).
+
+In order to mitigate the heterogeneity across clients while protecting privacy, we provide a novel yet simple framework, mean augmented federated learning (MAFL), in which each client exchanges the updated model parameters as well as its mashed (or averaged) data. MAFL framework allows the trade-off between the amount of meaningful information exchanged and the privacy across clients, depending on several factors such as the number of data instances used in computing the average. We first introduce a naive approach in our framework that simply applies Mixup (Zhang et al., 2018) between local data and averaged external data from other clients to reduce a myopic bias.
+
+Here, we go further in our framework and ask the following seemingly impossible question: can only averaged data in our framework that has lost most of the discriminative information, bring the similar effect as a global Mixup in which clients directly access others’ private data without considering privacy issues? Toward this, we introduce our second and more important approach in our framework, termed Federated Mixup (FedMix), that simply approximates the loss function of global Mixup via Taylor expansion (it turns out that such approximation only involves the averaged data from other clients!). Figure 1 briefly describes the concept of our methods.
+
+We validate our method on standard benchmark datasets for federated learning, and show its effectiveness against the standard federated learning methods especially for non-iid settings. In particular, we claim that FedMix shows better performance and smaller drop in accuracy with more heterogeneity or fewer clients update per communication round, further increasing difficulty of federated learning.
+
+Our contribution is threefold:
+
+• We propose a simple framework for federated learning that averages and exchanges each local data. Even naive approach in this framework performing Mixup with other clients’ mashed data shows performance improvement over existing baselines on several settings. We further develop a novel approximation for insecure global Mixup accessing other clients’ local data, and find out that Taylor expansion of global Mixup only involves the averaged data from other clients. Based on this observation, we propose FedMix in our framework approximating global Mixup without accessing others’ raw data. We validate FedMix on several FL benchmark datasets especially focusing on non-iid data settings where our method significantly outperforms existing baselines while still preserving privacy with minimal increases in communication cost.
+
+# 2 RELATED WORK
+
+Federated learning Federated learning was first proposed in Konecnˇ y et al. ´ (2016) where the prevalent asynchronous SGD (Dean et al., 2012) is used to update a global model in a distributed fashion. A pioneering work in this field proposed the currently most widely used algorithm, FedAvg (McMahan et al., 2017), which is also the first synchronous algorithm dedicated to federated setting. Shortly after, Li et al. (2020b) proposed a variant of FedAvg, named FedProx, where the authors claimed to overcome statistical heterogeneity and increase stability in federated learning. Recent studies attempt to expand federated learning with the aim of providing learning in more diverse and practical environments such as multi-task learning (Smith et al., 2017), generative models (Augenstein et al., 2020), continual learning (Yoon et al., 2020), semi-supervised learning (Jeong et al., 2020), and data with noisy labels (Tuor et al., 2020). Our paper focuses on general federated settings, but it could be considered in such various situations.
+
+However, these algorithms may obtain suboptimal performance when clients participating in FL have non-iid (Zhao et al., 2018; Hsieh et al., 2020) distributions. While the convergence of FedAvg on such settings was initially shown by experiments in McMahan et al. (2017) and later proved in Li et al. (2020c), it does not guarantee performance as good as it would have been for iid setting. Existing algorithms that pointed out this issue have major limitations, such as privacy violation by partial global sharing of local data (Zhao et al., 2018) or no indication of improvement over baseline algorithms such as FedAvg (Hsieh et al., 2020). Our method aims to improve performance particularly on these non-iid situations, without compromising privacy.
+
+Mixup Mixup (Zhang et al., 2018) is a popular data augmentation technique that generates additional data by linear interpolation between actual data instances. Mixup has been usually applied to image classification tasks and shown to improve test accuracy on various datasets such as CIFAR10 and ImageNet-2012 (Russakovsky et al., 2015), and, on popular architectures such as ResNet (He et al., 2016) and ResNeXt (Xie et al., 2017), for various model complexity. It is also reported in Zhang et al. (2018) that Mixup helps with stability, adversarial robustness (Zhang et al., 2018), calibration, and predictive certainty (Thulasidasan et al., 2019). Mixup is expanding from various angles due to its simplicity and popularity. First, beyond image classification tasks, its effectiveness has been proven in various domains such as image segmentation (Eaton-Rosen et al., 2020), speech recognition (Warden, 2018), and natural language processing (Guo et al., 2019). Also, several extensions such as Manifold Mixup (Verma et al., 2018), which performs Mixup in latent space, or CutMix (Yun et al., 2019), which replaces specific regions with others patches, have been proposed.
+
+In most of the previous studies on federated learning, Mixup was partially (or locally) used as a general data augmentation technique. Some recent studies (Oh et al., 2020; Shin et al., 2020) proposed to send blended data to server using Mixup, but they require sending locally- and linearly-mixed (mostly from two instances) data to server at every round, therefore being susceptible to privacy issues with huge communication costs. Our work properly modifies Mixup under the restrictions of federated learning and mitigates the major challenges of federated learning such as non-iid clients.
+
+# 3 MEAN AUGMENTED FEDERATED LEARNING (MAFL) AND FEDMIX
+
+We now provide our framework exchanging averaged data for federated learning and main method approximating insecure global Mixup under our framework, after briefly introducing the setup.
+
+# 3.1 SETUP AND BACKGROUND
+
+Federated learning and FedAvg Federated Averaging (FedAvg) (McMahan et al., 2017) has been the most popular algorithmic framework for federated learning. For every communication round $t = 0 , \ldots , T - 1$ , a client $k \in { 1 , \ldots , N }$ selected for local training sends back its model $\mathbf { \Delta } w _ { t } ^ { k }$ (or only difference to reduce communication cost) to a global server. For every round, $K$ number of clients are selected to locally update and send model parameters. The server simply averages parameters received, so that the global model ${ \pmb w } _ { t }$ after $t$ rounds of communications becomes $\begin{array} { r } { \mathbf { w } _ { t } = \sum _ { k = 1 } ^ { n } p _ { k } \mathbf { w } _ { t } ^ { k } } \end{array}$ where $p _ { k }$ is the importance of client $k$ based on the relative number of data in $k$ among all selected clients at $t$ . The updated global model is sent back to clients for the next round, which undergoes the following $E$ local updates via stochastic gradient descent (SGD):
+
+$\pmb { w } _ { t + 1 , i + 1 } ^ { k }  \pmb { w } _ { t + 1 , i } ^ { k } - \eta _ { t + 1 } \nabla \ell ( f ( \pmb { x } _ { i } ^ { k } ; \pmb { w } _ { t + 1 , i } ) , y _ { i } ^ { k } )$ for $i = 0 , 1 , \ldots , E - 1$ , batch size $B$ , and local learning rate $\eta$ . Here, $\ell$ is the loss function for learning and $f ( \pmb { x } ; \pmb { w } _ { t } )$ is the model output for input $_ { \textbf { \em x } }$ given model weight ${ \pmb w } _ { t }$ .
+
+Mixup Mixup (Zhang et al., 2018) is a simple data augmentation technique using a linear interpolation between two input-label pairs $( { \pmb x } _ { i } , y _ { i } )$ and $( \boldsymbol { x } _ { j } , \boldsymbol { y } _ { j } )$ to augment $\tilde { \mathbfit { x } } = \lambda \mathbfit { x } _ { i } + ( 1 - \lambda ) \mathbfit { x } _ { j }$ and $\tilde { y } = \lambda y _ { i } + ( 1 - \lambda ) y _ { j }$ . The variable $\lambda \in [ 0 , 1 ]$ is a hyperparameter that is chosen from the beta distribution for each training step.
+
+# 3.2 MAFL: MEAN AUGMENTED FEDERATED LEARNING
+
+The most obvious and powerful way for local models to receive information about data from other clients is to simply receive raw individual data. However, under typical federated setting, each client does not have direct access to individual external data due to privacy constraints, leading to overall performance degradation. We propose a federated learning framework that relaxes the limitation of accessing others’ raw data and allows a more granular level of privacy depending on applications. In our new framework, termed mean augmented federated learning (MAFL), clients not only exchange model parameters but also its mashed (or averaged) data.
+
+In MAFL, only the part that exchanges averaged data of each client has been added to the standard FL paradigm (Algorithm 1). Here, the number of data instances used in computing the average, $M _ { k }$ , controls the key features of MAFL such as privacy and communication costs. Lower $M _ { k }$ value results in more relevant information passed over, but only in cost of less secure privacy and larger communication cost. In one extreme of $M _ { k } = 1$ , raw data is thoroughly exchanged and privacy is not protected at all, which is clearly inappropriate for FL. But, in the other extreme, all data of each client is averaged to ensure a considerable degree of privacy. In addition, it also has an advantage on communication cost; each client sends a set of $n _ { k } / { \bar { M } } _ { k }$ averaged data where $n _ { k }$ is local data size of client $k$ . The remaining question is whether it is possible to improve performance even when exchanging information that is averaged from all local data and loses discriminative characteristics.
+
+The most naive way we can consider in our MAFL framework is to directly use the mashed data from other clients, just like regular local data. However, since mashed data has a lot less usable
+
+<table><tr><td>Algorithm 1: Mean Augmented Federated Learning (MAFL) Input: Dk = {Xk,Yk} for k =1,...,N</td></tr><tr><td>Mk:number of data instances used for computing average x, y</td></tr><tr><td>Initialize wo for global server fort=0,...,T-1do forclient k with updated local data do Split local data into M sized batches Compute x,y for each batch</td></tr><tr><td>Send allx,y to server end St ←Kclients selected at random Send wt to clients k ∈ St if updated then Aggregate all x, y to Xg,Yg</td></tr></table>
+
+# Algorithm 2: FedMix
+
+LocalUpdate $\left( k , \pmb { w } _ { t } ; \pmb { X } _ { g } , \pmb { Y } _ { g } \right)$ under   
+MAFL (Algorithm 1):   
+${ \pmb w } \gets { \pmb w } _ { t }$   
+for $e = 0 , \ldots , E - 1$ do Split $\mathbb { D } _ { k }$ into batches of size $B$ for batch $( X , Y )$ do Select an entry $\mathbfit { \Delta } x _ { g } , \mathbfit { y } _ { g }$ from $\begin{array} { r l } & { X _ { g } , Y _ { g } } \\ & { \ell _ { 1 } = } \\ & { ( 1 - \lambda ) \ell \big ( f ( ( 1 - \lambda ) X ; w ) , Y \big ) } \\ & { \ell _ { 2 } = \lambda \ell \big ( f ( ( 1 - \lambda ) X ; w ) , y _ { g } \big ) } \\ & { \ell _ { 3 } = \lambda \frac { \partial \ell _ { 1 } } { \partial x } \cdot x _ { g } } \end{array}$ (derivative calculated at ${ \pmb x } = ( 1 - \lambda ) { \pmb x } _ { i }$ and $y = y _ { i }$ for each of $\mathbf { { x } } _ { i } , y _ { i }$ in $X , Y )$ $\begin{array} { r l } & { \ell = \ell _ { 1 } + \ell _ { 2 } + \ell _ { 3 } } \\ & { \pmb { w }  \pmb { w } - \eta _ { t + 1 } \nabla \ell } \end{array}$ end   
+end   
+return $\pmb { w }$
+
+information than local data, we can think of a method of mixing it with local data:
+
+$$
+\ell _ { \mathtt { N a i v e M i x } } = ( 1 - \lambda ) \ell \Big ( f \big ( ( 1 - \lambda ) \pmb { x } _ { i } + \lambda \bar { \pmb { x } } _ { j } \big ) , y _ { i } \Big ) + \lambda \ell \Big ( f \big ( ( 1 - \lambda ) \pmb { x } _ { i } + \lambda \bar { \pmb { x } } _ { j } \big ) , \bar { y } _ { j } \Big )
+$$
+
+where $( { \pmb x } _ { i } , y _ { i } )$ is an entry from local data and $( \bar { \pmb { x } } _ { j } , \bar { y _ { j } } )$ corresponds to means of (inputs,labels) from other client $j$ . Note that Eq. (1) can be understood as the generalization of the loss of directly using the mashed data mentioned above in the sense that such loss can be achieved if $\lambda$ in Eq. (1) is set deterministically to 0 and 1.
+
+In the experimental section, we confirm the effectiveness of MAFL using $\ell _ { \tt N a i v e M i x }$ . However, in the next subsection, we will show how to achieve better performance by approximating the global Mixup in a more systematical way in our MAFL framework.
+
+# 3.3 FEDMIX: APPROXIMATING GLOBAL MIXUP VIA INPUT DERIVATIVE
+
+We now provide our main approach in the MAFL framework that aims to approximate the effect of global Mixup only using averaged data from other clients. Consider some client $i$ with its local data $( { \pmb x } _ { i } , y _ { i } )$ . It is not allowed in federated learning, but let us assume that client $i$ has access to client $j$ ’s local data $( \boldsymbol { x } _ { j } , \boldsymbol { y } _ { j } )$ . Then, client $i$ would leverage $( \boldsymbol { x } _ { j } , \boldsymbol { y } _ { j } )$ to improve the performance of its local model especially in non-iid settings by augmenting additional data via Mixup:
+
+$$
+\begin{array} { r } { \tilde { \mathbfit { x } } = ( 1 - \lambda ) \mathbfit { x } _ { i } + \lambda \mathbfit { x } _ { j } \quad \mathrm { a n d } \quad \tilde { y } = ( 1 - \lambda ) y _ { i } + \lambda y _ { j } . } \end{array}
+$$
+
+If Mixup rate $\lambda$ is 1, $( \boldsymbol { x } _ { j } , \boldsymbol { y } _ { j } )$ from client $j$ is again directly used like a regular local data, and it would be much more efficient than indirect update of local models through the server.
+
+The essence of our method is to approximate the loss function $\ell ( f ( \tilde { \pmb { x } } ) , \tilde { y } )$ for the augmented data from Eq. (2), with Taylor expansion for the first argument $_ { \textbf { \em x } }$ . Specifically, we derive the following proposition:
+
+Proposition 1 Consider the loss function of the global Mixup modulo the privacy issues,
+
+$$
+\ell _ { \tt G l o b a l M i x u p } \big ( f ( \tilde { x } ) , \tilde { y } \big ) = \ell \Big ( f \big ( ( 1 - \lambda ) x _ { i } + \lambda x _ { j } \big ) , ( 1 - \lambda ) y _ { i } + \lambda y _ { j } \Big )
+$$
+
+for cross-entropy loss $\ell$ . Suppose that Eq. (3) is approximated by applying Taylor series around the place where $\lambda \ll 1$ . Then, $i f$ we ignore the second order term (i.e., $\bar { \mathcal { O } } ( \lambda ^ { 2 } )$ ), we obtain the following approximated loss:
+
+$$
+( 1 - \lambda ) \ell { \Big ( } f { \big ( } ( 1 - \lambda ) \mathbf { { \big ) } } x _ { i } { \Big ) } , y _ { i } { \Big ) } + \lambda \ell { \Big ( } f { \big ( } ( 1 - \lambda ) \mathbf { { \big ) } } x _ { i } { \big ) } , y _ { j } { \Big ) } + \lambda { \frac { \partial \ell } { \partial x } } \cdot \mathbf { { x } } _ { j }
+$$
+
+where the derivative $\textstyle { \frac { \partial \ell } { \partial \mathbf { x } } }$ is evaluated at ${ \pmb x } = ( 1 - \lambda ) { \pmb x } _ { i }$ and $y = y _ { i }$ .
+
+While Eq. (4) still involves $\boldsymbol { \mathscr { x } } _ { j }$ and $y _ { j }$ , invading the privacy of client $j$ , the core value of Proposition 1 gets clearer when mixing up multiple data instances from other clients. Note that the vanilla Mixup is not mixing one specific instance with other data, but performing augmentations among several random selected data. In a non-iid $\mathrm { F L }$ environment, we can also expect that the effect will be greater as we create Mixup data by accessing as much private data as possible from other clients. From this point of view, let us assume that client $i$ has received a set of $M$ private instances, $J$ , from client $j$ . Then, the global Mixup loss in Eq. (3) is
+
+$$
+\frac { 1 } { | J | } \sum _ { j \in J } \ell \Big ( f \big ( ( 1 - \lambda ) \pmb { x } _ { i } + \lambda \pmb { x } _ { j } \big ) , ( 1 - \lambda ) y _ { i } + \lambda y _ { j } \Big ) ,
+$$
+
+and the approximated FedMix loss in Proposition 1 becomes
+
+$$
+\begin{array} { l } { \ell _ { \mathrm { F e d M i x } } = \displaystyle \frac { 1 } { | J | } \sum _ { j \in J } ( 1 - \lambda ) \ell \Big ( f \big ( ( 1 - \lambda ) \mathbf { x } _ { i } \big ) , y _ { i } \Big ) + \lambda \ell \Big ( f \big ( ( 1 - \lambda ) \mathbf { x } _ { i } \big ) , y _ { j } \Big ) + \lambda \frac { \partial \ell } { \partial x } \cdot \mathbf { x } _ { j } } \\ { = ( 1 - \lambda ) \ell \Big ( f \big ( ( 1 - \lambda ) \mathbf { x } _ { i } \big ) , y _ { i } \Big ) + \lambda \ell \Big ( f \big ( ( 1 - \lambda ) \mathbf { x } _ { i } \big ) , \bar { y } _ { j } \Big ) + \lambda \frac { \partial \ell } { \partial x } \cdot \bar { \mathbf { x } } _ { j } } \end{array}
+$$
+
+where we utilize the linearity of Equation 4 in terms of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { j } }$ and $y _ { j }$ , and $\bar { \mathbf { x } } _ { j }$ and $\bar { y } _ { j }$ correspond to mean of $M$ inputs and labels in $J$ , respectively. The algorithmic details are provided in the appendix due to the space constraint (see Algorithm 2 in Appendix A).
+
+# 3.4 PRIVACY ISSUES AND ADDITIONAL COSTS OF MAFL
+
+Privacy issues of MAFL MAFL requires exchanging averaged data by construction. Even though MAFL exchanges only the limited information allowed by the application, it may causes new types of privacy issues. The potential privacy risk of FL or MAFL is beyond the main scope of our study, but in this section, we briefly discuss some basic privacy issues of MAFL and potential solutions.
+
+• There is possibility that local data distribution can be inferred relatively easily from averaged data. This issue simply arises as $M _ { k }$ is not large enough, so that individual data could be inferred from the averaged data easily. On the other hand, if $n _ { k }$ is not big enough, each entry in $X _ { g } , Y _ { g }$ could reveal too much about the whole local distribution of the client it came from.   
+• It could be easy to infer ownership of each entry in $X _ { g } , Y _ { g }$ , if it contains client-id specific information. If clients could identify what other client each entry came from, information about local data of that client could be inferred.   
+• Additional concerns involve identification of data by detecting change in exchanged averaged data, in case of continual learning, which involves local data change across time. This issue is exacerbated as there is update of averaged data for every minute change on local data, which makes the client receiving $X _ { g } , Y _ { g }$ easier to infer the changed portion. One simple suggestion to alleviate this issue would be to only update $X _ { g } , Y _ { g }$ when there is enough change in local data across enough number of clients, so that such changes are not easily exploitable. As a way to strengthen privacy protection under MAFL (and possibly to help with issues mentioned above), we in the server can average within entries of $X _ { g } , Y _ { g }$ . If this additional average is done across every random $m$ entries at the server, it would effectively provide averaged data across all local data of $m$ clients, but would result in an $m$ -fold decrease in the number of averaged data. This variant is considered in Appendix J.   
+• In case where the global server is not credential, the averaged data itself should ensure privacy as it is sent to the server. A most obvious concern comes from when $M _ { k }$ is not large enough, so that each entry of $M _ { k }$ reveals more of information of each individual input. Simply using a sufficiently large value of $M _ { k }$ can alleviate this issue, although this might result in worse performance. However, for clients whose $n _ { k }$ is quite small, there is a limit for $M _ { k }$ to be large enough. One way to alleviate this issue is to introduce a cut-off threshold for allowing clients to send averaged data to server. We report the results in Appendix H.
+
+Communication cost Since MAFL requires sending averaged input data between server and clients, additional communication costs are incurred. However, it turns out that this additional cost is very small compared to communication cost required for exchanging model parameters. This is mainly due to the fact that input dimension is typically much smaller than number of model parameters. Specifically, for input dimension $d _ { i }$ , exchange of averaged data among $N$ clients incurs $2 N d _ { i }$ cost (factor of 2 for server receiving and sending the values). Meanwhile, the cost for exchange of model parameters is $2 N p _ { m }$ where $p _ { m }$ is number of model parameters. Under typical circumstances, averaged data is only exchanged at the beginning of the first communication round, while model parameters have to be exchanged every round. Thus the ratio between the two costs after $T$ communication rounds is $d _ { i } / ( T p _ { m } )$ . Since $d _ { i } \ll p _ { m }$ in general, we consider extra communication burden to be negligible (even in the worst case where we update averaged data every round, the ratio is still $d _ { i } / ( p _ { m } )$ .
+
+FedMix also requires calculation of input derivative term in its loss function, so potentially extra memory is required. We further provide additional computation costs of MAFL in Appendix G.
+
+# 4 EXPERIMENTS
+
+We test our result on various benchmark datasets with NaiveMix (direct mixup between local data and averaged data) and FedMix, then compare the results with FedAvg (McMahan et al., 2017) and FedProx (Li et al., 2020b), as well as other baseline Mixup scenarios. We create a highly non-iid environment to show our methods excel in such situations.
+
+![](images/3062c1ff064458cfa00a93a7dc8bdfbd7b69488cdb89bb83d6aec71a33449b03.jpg)  
+Figure 2: Learning curves for various algorithms on benchmark datasets. Learning curves correspond to results in Table 1. (For simplicity, we only show key algorithms to compare.)
+
+Table 1: Test accuracy after (target rounds) and number of rounds to reach (target test accuracy) on various datasets. Algorithms in conjunction with FedProx are compared separately (bottom). MAFL-based algorithms are marked in bold.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">FEMNIST</td><td colspan="2">CIFAR10</td><td colspan="2">CIFAR100</td></tr><tr><td>test acc.(200)rounds (80%） test acc.(500) rounds (70%） test acc.(500) rounds (40%)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GlobalMixup</td><td>88.2</td><td>8</td><td>88.2</td><td>85</td><td>61.4</td><td>54</td></tr><tr><td>FedAvg</td><td>85.3</td><td>26</td><td>73.8</td><td>283</td><td>50.4</td><td>101</td></tr><tr><td>LocalMix</td><td>82.8</td><td>28</td><td>73.0</td><td>267</td><td>54.8</td><td>91</td></tr><tr><td>NaiveMix</td><td>85.9</td><td>23</td><td>77.4</td><td>198</td><td>53.8</td><td>85</td></tr><tr><td>FedMix</td><td>86.5</td><td>18</td><td>81.2</td><td>162</td><td>56.7</td><td>34</td></tr><tr><td>FedProx</td><td>84.6</td><td>29</td><td>77.3</td><td>266</td><td>51.2</td><td>79</td></tr><tr><td>FedProx+LocalMix</td><td>84.1</td><td>39</td><td>74.1</td><td>314</td><td>54.0</td><td>90</td></tr><tr><td>FedProx+NaiveMix</td><td>85.7</td><td>37</td><td>76.7</td><td>230</td><td>53.1</td><td>74</td></tr><tr><td>FedProx+FedMix</td><td>86.0</td><td>32</td><td>78.9</td><td>223</td><td>54.5</td><td>63</td></tr></table>
+
+# 4.1 EXPERIMENTAL SETUP
+
+Dataset We implement the typical federated setting where clients have their own local data and one centralized server that receives and sends information from/to the clients. We utilize a large number of clients and only utilize partial set of clients chosen each round to locally update. We used three popular image classification benchmark datasets: FEMNIST (Caldas et al., 2019), CIFAR10, and CIFAR100, as well as a popular natural language processing benchmark dataset, Shakespeare. See Appendix B for more details about dataset, models, and hyperparameters used. We introduce data size heterogeneity for FEMNIST dataset: each client has different size of local data, each from a unique writer. Meanwhile, we introduce label distribution heterogeneity for CIFAR datasets, with clients having data with only a limited number of classes.
+
+Algorithms We study the performance of FedMix and NaiveMix and compare with FedAvg and FedProx. We also compare our method against FedAvg with Mixup within local data (labeled LocalMix; see Figure 1(b)), to show whether Mixup within local data is sufficient to allow the model to perform well on external data. To show the effectiveness of FedMix, we also compare our method to the case where we perform direct Mixup with external data (and thus violating privacy, labeled Global Mixup; see Figure 1(a)).
+
+# 4.2 PERFORMANCE OF FEDMIX AND NAIVEMIX ON NON-IID FEDERATED SETTINGS
+
+We compare the learning curves of each method under the same federated settings, in terms of number of communication rounds conducted. Comparing MAFL-based algorithms, NaiveMix shows slight performance increases than FedAvg, FedProx, and Localmix, while FedMix outperforms and shows faster convergence than all of FedAvg, FedProx, Localmix and NaiveMix for all datasets tested as in Figure 2.
+
+While NaiveMix and FedMix is already superior to FedProx, they are parallel to FedProx modification and can be applied in conjunction with FedProx. We compare performances across FedProx variants of various Mixup algorithms in Table 1. FedMix outperforms vanilla FedProx for various datasets, although they do fall short of default version of FedMix used for the main experiment.
+
+Table 2: Test accuracy after 50 rounds on Shakespeare dataset.   
+
+<table><tr><td>Algorithm</td><td>Global Mixup</td><td>FedAvg</td><td>FedProx</td><td>LocalMix</td><td>NaiveMix</td><td>FedMix</td></tr><tr><td>Test Acc. (%)</td><td>54.4</td><td>54.7</td><td>54.4</td><td>53.7</td><td>56.9</td><td>56.9</td></tr></table>
+
+Table 3: Test accuracy on CIFAR10, under varying number of clients $( N )$ . Number of samples per client is kept constant.   
+
+<table><tr><td># of Clients(N)</td><td>20</td><td>40</td><td>60</td></tr><tr><td>Global Mixup</td><td>86.3</td><td>89.2</td><td>88.2</td></tr><tr><td>FedAvg</td><td>65.8</td><td>73.4</td><td>73.8</td></tr><tr><td>LocalMix</td><td>46.9</td><td>71.4</td><td>73.0</td></tr><tr><td>NaiveMix</td><td>62.2</td><td>75.1</td><td>77.4</td></tr><tr><td>FedMix</td><td>68.5</td><td>76.4</td><td>81.2</td></tr></table>
+
+Table 4: Test accuracy on CIFAR10, under varying number of local data per client. Number of clients $( N )$ is kept constant. Number of data is indicated in percentage of the case where all 50,000 data are used.   
+
+<table><tr><td>Local data (%)</td><td>20</td><td>50</td><td>100</td></tr><tr><td>Global Mixup</td><td>71.4</td><td>86.1</td><td>88.2</td></tr><tr><td>FedAvg</td><td>61.8</td><td>74.7</td><td>73.8</td></tr><tr><td>LocalMix</td><td>43.7</td><td>60.3</td><td>73.0</td></tr><tr><td>NaiveMix</td><td>51.5</td><td>69.6</td><td>77.4</td></tr><tr><td>FedMix</td><td>65.2</td><td>77.8</td><td>81.2</td></tr></table>
+
+To confirm whether received information is properly incorporated, we compare FedMix with possible Mixup scenarios under MAFL. We show the results in Appendix D.
+
+While Mixup is usually performed for image classification tasks, it could be applied for language models. For language datasets, since Mixup cannot be performed on input, we perform Mixup on embeddings (for a detailed explanation of Mixup between hidden states, see Appendix E). When tested on Shakespeare dataset, FedMix and NaiveMix both show better performance than baseline algorithms (Table 2). Note that for this task, LocalMix has the lowest performance, and global Mixup does not result in the superior performance above federated algorithms as expected. We think Mixup does not provide performance boost for this specific task, but claim that MAFL algorithms still result in better performance compared to FedAvg.
+
+We also claim that FedMix is superior compared to other methods under various settings, in terms of varying number of clients $( N )$ and varying number of local data per clients. We observe superior performance of FedMix compared to other algorithms for all settings (see Tables 3 and 4). We also vary the number of local epochs $( E )$ between global updates, and still observe that FedMix outperforms other methods (see Appendix F).
+
+FedMix compared to global Mixup with fixed mixup ratio Since FedMix approximates loss function of global Mixup for fixed value of $\lambda \ll 1$ , we can evaluate the efficiency of approximation by comparing between FedMix and a global Mixup scenario with fixed $\lambda$ value. Table 5 shows varying performance between global Mixup and FedMix under various values of $\lambda$ . As $\lambda$ increases, Mixup data reflects more of the features of external data, resulting in better performance in case of global Mixup. However, this also results in our approximation being much less accurate, and we indeed observe performance of FedMix decreasing instead. The result shows that the hyperparameter $\lambda$ should be chosen to balance between better Mixup and better approximation. However, it seems that high $\lambda$ results in significant decrease in both methods, probably due to external data (which is out-of-distribution for local distribution) being overrepresented during local update.
+
+Table 5: Test accuracy on CIFAR10, under varying mixup ratio $\lambda$ .   
+
+<table><tr><td>入</td><td>0.05</td><td>0.1</td><td>0.2</td><td>0.5</td></tr><tr><td>GlobalMixup</td><td>79.4</td><td>80.4</td><td>81.1</td><td>63.6</td></tr><tr><td>FedMix</td><td>81.2</td><td>80.5</td><td>77.7</td><td>67.1</td></tr></table>
+
+![](images/2ccfccc1020cb0d12f73c753a723aabddad830d016334fdae0a311d5495a279c.jpg)  
+Figure 3: Performance of MAFL-based algorithms for various $M _ { k }$ values (left), and samples of averaged images from EMNIST/CIFAR10 for various $M _ { k }$ values (right).
+
+Table 6: Test accuracy after 500 rounds on CIFAR10, under varying number of classes per client.
+
+Table 7: Test accuracy after 500 rounds on CIFAR10, under varying number of clients trained per communication round.   
+
+<table><tr><td></td><td colspan="4">-class/client-</td></tr><tr><td>Algorithm</td><td>2</td><td>3</td><td>5</td><td>10 (iid)</td></tr><tr><td>Global Mixup</td><td>88.2</td><td>90.7</td><td>90.9</td><td>91.4</td></tr><tr><td>FedAvg</td><td>73.8</td><td>84.2</td><td>86.8</td><td>89.3</td></tr><tr><td>Localmix</td><td>73</td><td>83.3</td><td>86.4</td><td>89.1</td></tr><tr><td>NaiveMix</td><td>77.4</td><td>84.5</td><td>87.7</td><td>89.4</td></tr><tr><td>FedMix</td><td>81.2</td><td>85.1</td><td>87.9</td><td>89.1</td></tr></table>
+
+<table><tr><td></td><td colspan="4">-K/N-</td><td></td></tr><tr><td>Algorithm</td><td>0.1</td><td>0.15</td><td>0.25</td><td>0.5</td><td>1.0</td></tr><tr><td>Global Mixup</td><td>89.3</td><td>89.7</td><td>88.2</td><td>91.2</td><td>90.7</td></tr><tr><td>FedAvg</td><td>63.3</td><td>73.2</td><td>73.8</td><td>76.3</td><td>83.1</td></tr><tr><td>Localmix</td><td>64.7</td><td>64.5</td><td>73</td><td>77.9</td><td>79.8</td></tr><tr><td>NaiveMix</td><td>73.6</td><td>74.7</td><td>77.4</td><td>81.4</td><td>83.5</td></tr><tr><td>FedMix</td><td>74.7</td><td>76.9</td><td>80.5</td><td>82.1</td><td>84.3</td></tr></table>
+
+Effect of $M _ { k }$ to compute mean In our algorithm, we chose to calculate $X _ { g } , Y _ { g }$ with all local data for each client. To observe a potential effect of $M _ { k }$ , we varied $M _ { k }$ used to compute the averaged data that is sent from other clients. Inevitably, reducing $M _ { k }$ will result in $X _ { g } , Y _ { g }$ having much more rows, imposing additional computation burden and less preservation of privacy. In general, for both FEMNIST and CIFAR10, there is only small performance decline as privacy is enhanced, as can be seen in Figure 3. We show that using all local data to calculate each mean is sufficient to both preserve privacy and still have good performance.
+
+Mixup between hidden states Manifold Mixup (Verma et al., 2018) was proposed to show improvements over input Mixup (Zhang et al., 2018) in various image classification tasks such as CIFAR10, CIFAR100, and SVHN. We discuss the possibilities and implications of applying Mixup between hidden states in Appendix E. In summary, we show that variants of using hidden states do not show meaningful advances over FedMix using input Mixup, suggesting that in general, it is relatively inefficient since it imposes additional communication burden.
+
+Effect of non-iid-ness and client participation We claim that our method is efficient when faced with non-iid federated settings. For example, our setting of CIFAR10 having only data from 2 classes per client is very non-iid, as in average a pair of clients share only roughly $20 \%$ of data distribution. We test settings for CIFAR10 where clients have data from greater number of classes, and while there is little difference for iid (10 class/client) setting, we observe that FedMix outperform other methods and suffer less from increased heterogeneity from highly non-iid settings (Table 6). In addition, we also observe less decline and better performance for MAFL-based algorithms, FedMix in particular, as we train less number of clients per round, reducing communication burden in cost of performance (Table 7).
+
+# 5 CONCLUSION
+
+We proposed MAFL, a novel framework, that exchanges averaged local data, to gain relevant information while still ensuring privacy. Under the new framework, we first suggested NaiveMix, which is a naive implementation of Mixup between local and received data. More interestingly, we proposed FedMix, which provides approximation of global Mixup only using averaged data. MAFL, and FedMix in particular, showed improved performance over existing algorithms in various benchmarks, particularly in non-iid environments where each client has data distributed heterogeneously. While our method is very effective and still preserving privacy, future work needs to be done to deal with various non-iid environments, desirably with better privacy and beyond image classification tasks.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Research Foundation of Korea (NRF) grants (No.2018R1A5A1059921, No.2019R1C1C1009192) and Institute of Information & Communications Technology Planning & Evaluation (IITP) grants (No.2017-0-01779, XAI, No.2019-0-01371, Development of brain-inspired AI with human-like intelligence, and No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST)) funded by the Korea government (MSIT).
+
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+
+# A ALGORITHMS
+
+We present a brief depiction of FedAvg in Algorithm 3.
+
+# Algorithm 3: FedAvg
+
+Input: $\begin{array} { r } { N , T , K , E , B , p _ { k } , \mathbb { D } _ { k } = \{ X _ { k } , Y _ { k } \} , k = } \end{array}$ LocalUpdate(k, wt): $1 , \dots , N , \eta _ { t } , t = 0 , \dots , T - 1$ w ← wt   
+Initialize $w _ { 0 }$ for global server for e = 0, . . . , E − 1 do   
+for $t = 0 , \ldots , T - 1$ do Split $\mathbb { D } _ { k }$ into batches of size $B$ $\mathbb { S } _ { t }  K$ clients selected at random for batch $( X , Y )$ do Send ${ \pmb w } _ { t }$ to clients $k \in \mathbb S _ { t }$ $\pmb { w } \gets \pmb { w } - \eta _ { t + 1 } \nabla \ell ( f ( \pmb { X } ; \pmb { w } ) , \pmb { Y } )$ for $k \in \mathbb S _ { t }$ do end $\begin{array} { r } { | \begin{array} { r l } & { { \pmb w } _ { t + 1 } ^ { k }  L o c a l U p d a t e ( k , { \pmb w } _ { t } ) } \end{array}  } \end{array}$ end end return w $\begin{array} { r } { \pmb { w } _ { t + 1 }  \frac { 1 } { K } \sum _ { k \in \mathbb { S } _ { t } } p _ { k } \pmb { w } _ { t + 1 } ^ { k } } \end{array}$   
+end
+
+# B EXPERIMENTAL DETAILS
+
+FEMNIST FEMNIST is EMNIST (Cohen et al., 2017), a handwritten MNIST dataset, organized into federated setting, as in Caldas et al. (2019). EMNIST is very similar to MNIST, but has several differences. It includes all 26 capital and small letters of alphabet as classes along with numbers, making it 62 classes in total to classify. Also, each image contains information of the writer of the letter. In a realistic non-iid setting, each client has local data consists of only one writer, which is about 200 to 300 samples per client in average, with differing number of samples. We use $N = 1 0 0$ clients and trained only $K = 1 0$ clients per communication round.
+
+We used LeNet-5 (Lecun et al., 1998) architecture for client training. LeNet is consisted of 2 conv layers followed by 2x2 maxpool layer then 3 fc layers. We used 5x5 conv layers with 6 and 16 channels. Following fc layers have exactly the same hidden dimension of original LeNet-5 model.
+
+CIFAR10 and CIFAR100 CIFAR10 and CIFAR100 are very popular and simple image classification datasets for federated setting. Both contain 50,000 training data and 10,000 test data. We split the data into each client, $N = 6 0$ in case of CIFAR10 and $N = 1 0 0$ in case of CIFAR100. To create an artificial non-iid environment, we allocate data such that each client only has data from 2 (20 for CIFAR100) randomly chosen classes. We train only $K = 1 5$ clients per round for CIFAR10 and $K = 1 0$ for CIFAR100. No validation data was split and we used all training data for local training.
+
+We used modified version of VGG architecture (Simonyan & Zisserman, 2015). Modified VGGnet is consisted of 6 convolutional layers with 3 max pooling layers. 3x3 conv layers are stacked and $2 \mathbf { x } 2$ maxpool layer is stacked after every 2 conv layers. Conv layers have channel sizes of 32, 64, 128, 128, 256, 256. Then 3 fc layers are stacked with hidden dimension 512. We use Dropout layer three times with probability 0.1 after the second, third maxpool layers and before the last fc layer. We remove all batch normalization layers since it is reported that they hurt federated learning performance (Hsieh et al., 2020).
+
+Shakespeare We use dataset from The Complete Works of William Shakespeare, which is a popular dataset for next-character prediction task. We partition the dataset so that each client has conversations of one speaking role, as in Caldas et al. (2019), which naturally results in a heterogeneous setting, as in FEMNIST. We use $N = K = 6 0$ , each with different number of data (minimum is 200). Since input-level Mixup cannot be performed for discrete character labels, we performed Mixup on the embedding layer. Additional concerns for this variation is considered at Appendix E.
+
+We used 2-layer LSTM, both with hidden dimension of 256. The recurret network is followed by an embedding layer. The output of LSTM is passed to a fully-connected layer, with softmax output of one node per character. There are 84 characters used in the dataset.
+
+Local clients are trained by SGD optimizer with learning rate 0.01 and learning decay rate per round 0.999. We set local batch size as 10 for training. Specific hyperparameter setting for each dataset is explained in following Table 8. Throughout the experiment, $M _ { k }$ is fixed to local client’s dataset size. Changes in these parameters are indicated, if made, are stated for all experiments. Note that we use a fixed small value of $\lambda$ for MAFL-based algorithms to show superior performance.
+
+Table 8: Hyperparameter settings for each dataset.   
+
+<table><tr><td>dataset</td><td>FEMNIST</td><td>CIFAR10</td><td>CIFAR100</td><td>Shakespeare</td></tr><tr><td>local epochs (E)</td><td>10</td><td>2</td><td>10</td><td>2</td></tr><tr><td>local batch size</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>class per clients</td><td>-</td><td>2</td><td>20</td><td>1</td></tr><tr><td>fraction of Clients (K/N)</td><td>0.1</td><td>0.25</td><td>0.1</td><td>1</td></tr><tr><td>total dataset classes</td><td>62</td><td>10</td><td>100</td><td>84</td></tr><tr><td>入(for NaiveMix)</td><td>0.2</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>入(forFedMix)</td><td>0.2</td><td>0.05</td><td>0.1</td><td>0.1</td></tr><tr><td>μ (for FedProx)</td><td>0.1</td><td>0.1</td><td>0.01</td><td>0.001</td></tr></table>
+
+# C PROOF OF PROPOSITION 1
+
+We demonstrate mathematical proof of Proposition 1 for FedMix.
+
+Starting from Eq. (3), since the loss function is linear in $y$ for cross-entropy loss $\ell$ , we have
+
+$$
+\ell \big ( f ( \tilde { \pmb x } ) , \tilde { \pmb y } \big ) = ( 1 - \lambda ) \ell \Big ( f \big ( ( 1 - \lambda ) \pmb x _ { i } + \lambda \pmb x _ { j } \big ) , \pmb y _ { i } \Big ) + \lambda \ell \Big ( f \big ( ( 1 - \lambda ) \pmb x _ { i } + \lambda \pmb x _ { j } \big ) , \pmb y _ { j } \Big ) .
+$$
+
+Unlike the original paper, if we assume $\lambda \ll 1$ , we can treat this loss as objective loss function for vicinal risk minimization (VRM). Under this assumption, each term in Eq. (6) can be approximated by independent Taylor expansion on the first and second argument of $\ell$ , so that we have
+
+$$
+\begin{array} { l } { { ( 1 - \lambda ) \ell \Big ( f \big ( ( 1 - \lambda ) \pmb { x } _ { i } \big ) , y _ { i } \Big ) + ( 1 - \lambda ) \times \displaystyle \frac { \partial \ell } { \partial \pmb { x } } \bigg | _ { ( 1 - \lambda ) \pmb { x } _ { i } , y _ { i } } \cdot ( \lambda \pmb { x } _ { j } ) } } \\ { { + \lambda \ell \Big ( f \big ( ( 1 - \lambda ) \pmb { x } _ { i } \big ) , y _ { j } \Big ) + \lambda \times \displaystyle \frac { \partial \ell } { \partial \pmb { x } } \bigg | _ { ( 1 - \lambda ) \pmb { x } _ { i } , y _ { j } } \cdot ( \lambda \pmb { x } _ { j } ) . } } \end{array}
+$$
+
+Since $\lambda \ll 1$ , we can ignore the last term in the second row of Eq. (7), which is $\mathcal { O } ( \lambda ^ { 2 } )$ . We simplify this equation and switch the second term in the first row and the first term in the second row, to finally obtain
+
+$$
+\ell \big ( f ( \boldsymbol { \bar { x } } ) , \boldsymbol { \tilde { y } } \big ) \approx ( 1 - \lambda ) \ell \Big ( f \big ( ( 1 - \lambda ) \boldsymbol { x } _ { i } \big ) , \boldsymbol { y } _ { i } \Big ) + \lambda \ell \Big ( f \big ( ( 1 - \lambda ) \boldsymbol { x } _ { i } \big ) , \boldsymbol { y } _ { j } \Big ) + \lambda \frac { \partial \ell } { \partial \boldsymbol { x } } \cdot \boldsymbol { x } _ { j } .
+$$
+
+The derivative $\textstyle { \frac { \partial \ell } { \partial \mathbf { x } } }$ is calculated at ${ \pmb x } = ( 1 - \lambda ) { \pmb x } _ { i }$ and $y = y _ { i }$ . The coefficient of the last term is changed to $\lambda$ from $\lambda ( 1 - \lambda )$ since we are ignoring $\mathcal { O } ( \lambda ^ { 2 } )$ terms.
+
+# D COMPARISON OF FEDMIX WITH BASELINE MIXUP SCENARIOS
+
+Since our MAFL-based algorithms could be considered as VRM, it could be considered as a data augmentation (Chapelle et al., 2000). Thus, it is important to confirm that increased performance from MAFL not only comes from data augmentation but also from relevant information received from other clients. To check whether this is true, we compare FedMix with algorithms where we use either randomly generated noises as averaged data for Mixup (labeled Mixup w/ random noise) and where we use only locally averaged data for Mixup (labeled Mixup w/ local means).
+
+In Table 9, we observe that if averaged data for MAFL is substituted for randomly generated noise or locally generated images, it does not show the level of performance FedMix is able to show. Thus, we claim that FedMix properly incorporates relevant information received.
+
+Figure 4: Results for variants of Mixup algorithms for Mixup between hidden states. (a) Learning curves for various algorithms with hidden representation Mixup after $k = 2$ layers. (b) Learning curves for FedMix when Mixup is applied after different numbers of layers.
+
+![](images/d6277fdfdcb4b7a023f929577df3531d37325e1e5299cd44eced148d7beeb1cd.jpg)  
+E VARIANT OF FEDMIX AND NAIVEMIX WITH MIXUP BETWEEN HIDDEN STATES
+
+While input Mixup methods promise significant enhancements, one can expect similar performance from hidden state Mixup, originally proposed by Verma et al. (2018). The authors of this work suggest that Manifold Mixup demonstrates similar, if not greater, advantage in terms of performance and adversarial robustness. We can think of variants of FedMix and NaiveMix that implement hidden state Mixup, and test if this variant outperforms vanilla methods based on input Mixup.
+
+Although the original paper proposed randomizing the layer $k$ just before hidden states that undergo Mixup for each batch, we propose setting this layer $k$ constant. This is to reduce communication cost significantly, since selecting randomized layer for Mixup will require other clients having to send multiple hidden states (which usually have large dimensions), further imposing communication burden.
+
+Another change is that while original Manifold Mixup (Verma et al., 2018) suggests backpropagating the entire computational graph through the whole network, including the encoder (part of model projecting input to designated hidden representation) and the decoder (rest of the model projecting hidden representation to output). Such thing is impossible to do in typical federated setting, since the computational graph to calculate hidden representation of local data and the graph to calculate hidden representation of other clients’ data is separated, and they cannot be updated simultaneously through local update (doing so requires communicating encoder weights across clients every local update, which is highly inefficient in terms of communication). Thus, during Mixup between hidden representations, only the decoder weights can be updated, since only updating encoder of the selected local client will desynchronize encoder weight values for calculating hidden states of local data from those for calculating hidden states of other clients’ data every local update, so that the model does not learn properly.
+
+Table 9: Test accuracy after (target rounds) and number of rounds to reach (target test accuracy) on various datasets. We compare FedMix with baseline Mixup algorithms.   
+
+<table><tr><td rowspan="2">Algorithm</td><td colspan="5"></td></tr><tr><td>FEMNIST</td><td></td><td>CIFAR10</td><td>test acc. (500)</td><td>CIFAR100</td></tr><tr><td>NaiveMix</td><td>test acc.(200)</td><td>rounds (80%)</td><td>test acc.(500)</td><td>rounds (70%)</td><td>rounds (40%)</td></tr><tr><td></td><td>85.9</td><td>23</td><td>77.4</td><td>198 53.8</td><td>85</td></tr><tr><td>Mixup w/ random noise</td><td>86.1</td><td>23</td><td>77.9</td><td>201 51.2</td><td>105</td></tr><tr><td>Mixup w/ local means</td><td>85.5</td><td>21</td><td>73.5</td><td>233 51.0</td><td>87</td></tr><tr><td>FedMix</td><td>86.5</td><td>18</td><td>81.2</td><td>162</td><td>56.7 34</td></tr></table>
+
+Table 10: Test accuracy after 500 rounds on CIFAR10, under varying local epochs $( E )$ .   
+
+<table><tr><td># of Local Epochs (E)</td><td>1</td><td>2</td><td>5</td><td>10</td></tr><tr><td>FedAvg</td><td>74.4</td><td>73.8</td><td>80.7</td><td>78.9</td></tr><tr><td>LocalMix</td><td>63.7</td><td>73.0</td><td>74.7</td><td>80.0</td></tr><tr><td>NaiveMix</td><td>72.0</td><td>77.4</td><td>81.0</td><td>82.6</td></tr><tr><td>FedMix</td><td>75.8</td><td>81.2</td><td>83.0</td><td>82.5</td></tr></table>
+
+To compensate for this downside, we propose performing vanilla SGD updates without Mixup after Manifold Mixup SGD (which updates weights only in decoder). The vanilla updates will have both local encoder and decoder weights to be updated, thus driving the model to have better hidden representations for Mixup. However, this difference not only imposes additional computation cost, but also does not guarantee that it will show better performance compared to input Mixup methods.
+
+While utilizing hidden representations from other clients sound like a safe idea, it does not ensure data privacy, primarily because the updating client has knowledge of the exact encoder weight values used to calculate hidden states received, and based on our modification, the received hidden states are treated as constants during Mixup. Model inversion attacks (Fredrikson et al., 2015) have been suggested to recover input images from hidden states or outputs, with access to weight values. Thus, direct Mixup between hidden states does not guarantee data privacy. Variant of FedMix and NaiveMix can be applied during decoder training phase, so that privacy is ensured while we successfully approximate Mixup.
+
+The performance of proposed algorithms is shown in Figure 4 on CIFAR10 (same settings with main experiment for dataset, model, and training is used; see Appendix B). Comparison between methods in Figure 4(a) shows that while variants of FedMix and NaiveMix show improved performance over existing methods, they still do not outperform our method based on input Mixup (compare with dotted line). Meanwhile, comparison between using different layers for Mixup is shown in Figure 4(b). It is shown that $k = 4$ has fastest learning curve but converges similarly to case of $k = 2$ , both being slightly outperformed by case of input Mixup.
+
+Considering additional computation burden required to communicate hidden states (which often have larger dimensions than raw input) and necessity to communicate the hidden states every communication round (since hidden representations change with encoder weights), we propose that FedMix using input Mixup is superior, and use this method for our main analyses.
+
+# F EFFECT OF LOCAL EPOCHS
+
+Previous works (McMahan et al., 2017; Caldas et al., 2019) show that number of local epochs, $E$ , affects federated learning performance. We tested the effect of $E$ on CIFAR10. In general, we showed that test performance increases as $E$ increases. In addition, we observed that under various values of $E$ , FedMix shows the best performance compared to other algorithms (see Table 10), being a close second after NaiveMix for $E = 1 0$ . MAFL-based algorithms outperform existing algorithms for all values of $E$ tested.
+
+# G ADDITIONAL COMPUTATION COST INCURRED BY MAFL
+
+For FedMix, additional computation and memory are required on edge devices during model training for each communication round, since $\ell _ { \mathrm { F e d M i x } }$ requires additional terms, including gradient by input, $\textstyle { \frac { \partial \ell } { \partial \mathbf { x } } }$ , compared to vanilla FedAvg. We claim that FedMix does not result in an additional computation burden. Specifically, we trained FedMix on CIFAR10 with the same settings as the main experiment to $70 \%$ accuracy in 1.94 hours; FedAvg takes 1.95 hours. FedMix spends a comparable amount of time to reach a similar level of performance of FedAvg. While in the memory aspect, FedMix requires about twice more GPU memory allocation compared to FedAvg, this phenomenon is also observed on LocalMix and NaiveMix. The extra memory burden comes from Mixup by enlarging the input dimension twice. For instance, FedAvg requires 46.00MB to allocate, LocalMix requires 94.00MB and 98.00MB for FedMix. Calculating gradient of the input derivative gives only negligible 2-3MB additional memory usage, which is reasonable concerning the substantial performance increase from LocalMix to FedMix.
+
+# H INTRODUCTION OF CUT-OFF THRESHOLD IN MAFL
+
+To better ensure privacy, a cut-off threshold that prevents clients with fewer data to send averaged data could be introduced. We performed this in FEMNIST, since for such procedure to be effective, heterogeneous size of local client data is necessary. We test with $N = 3 0 0$ clients, and introduce different threshold levels to test its efficiency. In addition, we also test with multiple $\lambda$ values, to see whether threshold level affects optimal value of $\lambda$ for FedMix.
+
+We present the results in Table 11. While threshold does not hugely affect performance, we observe that a moderately small threshold level of 100 results in the best performance. We suggest that as the threshold level is heightened, there is less overfitting to clients with small size local data, but it also results in a decrease in the number of averaged data received by each client. We indeed find an appropriate value of threshold that maximizes performance.
+
+In case where there are a different number of data per client, the sensitivity of $\lambda$ could also be different compared to when all clients have the same number of data. Results in Table 13 show that there is little change in performance by change in $\lambda$ , especially compared to Table 5. In addition, an inspection of the performance of a global model on individual test data of clients does not reveal any noticeable pattern by the size of local data (see Table 13).
+
+Table 11: Test accuracy on FedMix with introduction of cut-off threshold, tested on a number of threshold level. Optimal value of $\lambda$ is also shown.   
+
+<table><tr><td></td><td colspan="4">-threshold-</td></tr><tr><td>入</td><td>1</td><td>100</td><td>150</td><td>200</td></tr><tr><td>Test Acc. (%)</td><td>83.0</td><td>83.3</td><td>83.1</td><td>82.9</td></tr><tr><td>Xoptimal</td><td>0.2</td><td>0.05</td><td>0.1</td><td>0.2</td></tr></table>
+
+Table 12: Test accuracy on FEMNIST, $N =$ 300 under various Mixup ratio $\lambda$ .   
+
+<table><tr><td>入</td><td>0.05</td><td>0.1</td><td>0.2</td></tr><tr><td>Test Acc(%)</td><td>82.8</td><td>82.8</td><td>83.0</td></tr></table>
+
+Table 13: Mean and standard variation of local test accuracy of FedMix on FEMNIST, tested on clients with varying local data size, under varying $\lambda$ .   
+
+<table><tr><td>入</td><td>nk &lt; mean</td><td>100 std</td><td>100 八 mean</td><td>nk ≥199 std</td><td>nk V mean</td><td>199 std</td></tr><tr><td>0.05</td><td>85.8</td><td>15.1</td><td>77.6</td><td>14.8</td><td>85.4</td><td>9.3</td></tr><tr><td>0.1</td><td>81.1</td><td>17.4</td><td>76.4</td><td>14.1</td><td>86.2</td><td>9.5</td></tr><tr><td>0.2</td><td>83.9</td><td>16.2</td><td>77.6</td><td>14.8</td><td>85.8</td><td>10.0</td></tr></table>
+
+# I MAFL IN CONJUNCTION WITH GAUSSIAN NOISE
+
+With results in Figure 3, we expressed concern with small values of $M _ { k }$ causing privacy issues with only a small performance boost, if at all. A common practice of introducing additional privacy is adding Gaussian noise. This is a popular method associated with differential privacy (McMahan et al., 2018), but adding noise alone does not guarantee differential privacy, since the noise level should be explicitly linked to differential privacy levels, $\epsilon$ and $\delta$ . Addition of artificial pixel-wise noise will enhance privacy but will result in a quality drop of averaged data. While privacy added by noise and privacy from averaging data cannot be directly compared, we can select a noise level which in conjunction with small $M _ { k }$ , visually provides data privacy similar to that of maximum $M _ { k }$ .
+
+Results show that the introduction of Gaussian noise does result in a decline in performance (Table 14),although the decline is very small. Interestingly as noise gets larger as $\sigma = 0 . 3$ , random noise provides an effect as data augmentation and results in a performance increase compared to $\sigma = 0$ . This experiment is in line with Appendix D. We conclude that introduction of noise in averaged data could provide us with a reasonable alternative to FedMix with large $M _ { k }$ . While our method does not align directly with differential privacy, we leave as future work how FedMix could be smoothly combined with DP-related methods and how its privacy could be quantified in terms of differential privacy.
+
+Table 14: Performance of FedMix with Gaussian noise. $\sigma$ refers to standard deviation of Gaussian noise.   
+
+<table><tr><td>9</td><td>0</td><td>0.05</td><td>0.075</td><td>0.15</td><td>0.3</td></tr><tr><td>Mk =5</td><td>81.4</td><td>80.1</td><td>81.1</td><td>78.7</td><td>81.5</td></tr><tr><td>Mk =10</td><td>79.9</td><td>80.8</td><td>79.1</td><td>79.4</td><td>81.7</td></tr><tr><td>M = 20</td><td>80.4</td><td>80.5</td><td>79.7</td><td>80.7</td><td>81.0</td></tr></table>
+
+Table 15: Test accuracy on CIFAR10 varying $m$ , number of entries in $X _ { g } , Y _ { g }$ to be further averaged.   
+
+<table><tr><td>m</td><td>1</td><td>4</td><td>10</td></tr><tr><td>Test acc (% )</td><td>81.2</td><td>81.6</td><td>78.4</td></tr></table>
+
+# J ADDITIONAL EXPERIMENTS: VARIATIONS OF FEDMIX
+
+Averaging within $X _ { g } , Y _ { g }$ Further averaging between entries of $X _ { g } , Y _ { g }$ practically provides an extension of the range of viable $M _ { k }$ such that it exceeds $n _ { k }$ , in the sense that each averaged data is from multiple clients’ data. Such a process would also result in fewer data included in $X _ { g } , Y _ { g }$ , so we tested effect of this procedure on model performance. Table 15 shows that for $m$ -fold extra averaging, we even observe increase in performance, but it quickly declines as $m$ gets too large. This method provides an improvement in privacy while even possibly resulting in better performance.
+
+Effect of same-class split for averaging We perform random split of local data for averaging, but an unbalanced split, such as only averaging data with the same class labels, could result in better performance. We compared between random split and same-class split while keeping $M _ { k } = 0 . 5 n _ { k }$ be equal for both methods. Same-class split resulted in a significant decline in performance, and we conclude that there is no advantage of such split over random split that we are using for our main results.
+
+Table 17: Test accuracy of NaiveMix on CIFAR10, under varying Mixup ratio $\lambda$ .
+
+Table 16: Test accuracy on CIFAR10, with class/client $= 2$ , under different split methods. $M _ { k } = 0 . 5 n _ { k }$ for both splits.   
+
+<table><tr><td></td><td>random split</td><td>class split</td></tr><tr><td>FedMix</td><td>81.2</td><td>78.8</td></tr></table>
+
+<table><tr><td>入</td><td>0.05</td><td>0.1</td><td>0.2</td><td>0.5</td></tr><tr><td>NaiveMix</td><td>79.5</td><td>79.9</td><td>80.6</td><td>29.8</td></tr></table>
+
+NaiveMix with varying Mixup ratio $\lambda$ We varied Mixup ratio $\lambda$ for NaiveMix as well. Results in Table 17 shows that NaiveMix also has an intermediate optimal value of $\lambda$ . The drop in performance for $\lambda = 0 . 5$ is much more dramatic than for FedMix (see Table 5 for comparison with Global Mixup and FedMix). We think that NaiveMix loss also suffers as it gives more weight to the averaged data, especially for large $M _ { k }$ .
+
+Heterogeneity from skewed label distribution Recent papers (Yurochkin et al., 2019; Wang et al., 2020) suggested an alternative heterogeneous environment, which does not limit the number of classes per client but skews label distribution in local data. We used a Dirichlet distribution of $\alpha = 0 . 2 , 0 . 5$ as described by Yurochkin et al. (2019) and Wang et al. (2020). Results show that FedMix still outperforms all other algorithms. We think that such label skewing introduces less heterogeneity compared to our practice of limiting the number of classes per client, but nevertheless, FedMix is still the most powerful method in terms of performance.
+
+Table 18: Test accuracy on CIFAR10 under label-skewed heterogeneous environment. We used Dirichlet distribution for uneven label distribution.   
+
+<table><tr><td>α</td><td>FedAvg</td><td>GlobalMix</td><td>LocalMix</td><td>NaiveMix</td><td>FedMix</td></tr><tr><td>0.2</td><td>83.9</td><td>91.1</td><td>84.0</td><td>85.0</td><td>86.4</td></tr><tr><td>0.5</td><td>87.6</td><td>91.1</td><td>88.0</td><td>88.2</td><td>88.4</td></tr></table>

parse/train/S1lhkdKkeV/S1lhkdKkeV.md

@@ -0,0 +1,240 @@
+# Scalable Neural Architecture Search for 3D Medical Image Segmentation
+
+Sungwoong Kim∗1   
+Ildoo Kim∗1   
+Sungbin Lim $^ { * 1 }$   
+Chiheon Kim1   
+Woonhyuk Baek1   
+Hyungjoo Cho2   
+Boogeon Yoon1   
+Taesup Kim1,3
+
+swkim@kakaobrain.com ildoo.kim@kakaobrain.com   
+sungbin.lim@kakaobrain.com   
+chiheon.kim@kakaobrain.com wbaek@kakaobrain.com joysquare@snu.ac.kr eric.yoon@kakaobrain.com taesup.kim@umontreal.ca
+
+1 Kakao Brain, Pangyo, Seongnam, Gyeonggi, Republic of Korea   
+2 Department of Transdisciplinary Studies, Seoul National University, Republic of Korea   
+3 MILA, Universit´e de Montr´eal, Canada
+
+Editors: Under Review for MIDL 2019
+
+# Abstract
+
+In this paper, a neural architecture search (NAS) framework is formulated for 3D medical image segmentation, to automatically optimize a neural architecture from a large design space. For this, a novel NAS framework is proposed to produce the structure of each layer including neural connectivities and operation types in both of the encoder and decoder of a target 3D U-Net. In the proposed NAS framework, having a sufficiently large search space is important in generating an improved network architecture, however optimizing over such a large space is difficult due to the extremely large memory usage and the long run-time originated from high-resolution 3D medical images. Therefore, a novel stochastic sampling algorithm based on the continuous relaxation on the discrete architecture parameters is also proposed for scalable joint optimization of both of the architecture parameters and the neural operation parameters. This makes it possible to maintain a large search space with small computational cost as well as to obtain an unbiased architecture by reducing the discrepancy between the training-time and test-time architectures. On the 3D medical image segmentation tasks with a benchmark dataset, an automatically designed 3D U-Net by the proposed NAS framework outperforms the previous human-designed 3D U-Net as well as the randomly designed 3D U-Net, and moreover this optimized architecture is more compact and also well suited to be transferred for similar but different tasks.
+
+Keywords: AutoML, Neural Architecture Search, Medical Image Segmentation.
+
+# 1. Introduction
+
+Recently, deep neural networks have been extensively used for medical image segmentation tasks (Ronneberger et al., 2015; C¸ i¸cek et al., 2016; Mortazi et al., 2017; Ciresan et al., 2012; Milletari et al., 2016a; Kamnitsas et al., 2016; Havaei et al., 2015; Yu et al., 2017; Kayalibay et al., 2017; Oktay et al., 2018; Isensee et al., 2018). However, such a method in general relies on manual trial-and-error processes for making decisions on the network architecture, hyperparameters for training, and pre-/post-procedures. Due to being restricted to manual tuning, they would have limitations in performance improvement as well as fast transfer to related tasks. Currently, the same problem in the field of general deep learning has promoted the rapid development of automated machine learning (AutoML). Yet, in contrast to the recent intensive studies on the use of advanced AutoML algorithms such as neural architecture search (NAS) (Zoph et al., 2018; Liu et al., 2018a; Bender et al., 2018; Zoph and Le, 2017; Liu et al., 2018b; Pham et al., 2018; Zhang et al., 2018; Cai et al., 2018; Brock et al., 2018) and neural optimizer search (Bello et al., 2017; Alber et al., 2018; Wichrowska et al., 2017; Li and Malik, 2017; Andrychowicz et al., 2016) for general computer vision tasks, only a few naive AutoML approaches using simple hyperparameter optimization have been proposed for medical imaging tasks (Mortazi and Bagci, 2018; Naceur et al., 2018). Therefore, in this paper, we propose a novel NAS framework for AutoML in designing neural networks especially for 3D medical image segmentation.
+
+Since both semantic as well as spatial information can be efficiently exploited through skip connections between an encoder and a decoder, a 3D U-Net has been popularly used in most state-of-the-art deep learning based algorithms for segmenting high-resolution 3D medical images (C¸ i¸cek et al., 2016; Milletari et al., 2016a; Yu et al., 2017; Kayalibay et al., 2017; Oktay et al., 2018; Isensee et al., 2018). However, a convolutional block for each layer in the 3D U-Net has been manually designed with various convolutional filter types, pooling types, skip-connections, and non-linear activation functions. Instead of using the suboptimally designed block, we propose to use a NAS framework to obtain an automatically optimized structure of the block, which is called a cell, for each layer in the 3D U-Net where all cell structures and the corresponding neural operation parameters (e.g. kernel weights) are simultaneously learned in an end-to-end manner. For this, four types of cells - encodernormal cell, reduction cell, decoder-normal cell, expansion cell - are defined to compose the encoder as well as the decoder for the learned U-Net architecture, which is different from the use of two types of cells (normal cell and reduction cell) in previous NAS approaches for encoder-only networks (Zoph et al., 2018; Liu et al., 2018b; Pham et al., 2018). Here, it is noted that in NAS having a sufficiently large search space is important in generating an improved network architecture on a target task. However, optimizing over such a large space for this segmentation task is difficult due to the extreme memory usage and the long runtime when dealing with high-resolution 3D images. Moreover, NAS basically needs to jointly optimize not only the discrete architecture parameters but also the continuous operation parameters, which is so-called bi-level optimization (Liu et al., 2018b; Franceschi et al., 2018), and an exact bi-level optimization over this mixed domain(discrete and continuous) is also difficult, especially with this large search space associated with the 3D U-Net.
+
+Therefore, in this work, a novel stochastic sampling algorithm is applied for bi-level optimization of the mixed parameters in the proposed NAS framework. This can not only search over a large design space but also lead to provide a consistent and unbiased architecture that avoids the retraining of suboptimal operation parameters from the obtained architecture. More specifically, the discrete architecture parameters corresponding to neural connections and operation types in each cell are defined as a set of one-hot discrete variables, and a continuous approximation using Gumbel-softmax (Jang et al., 2017; Maddison et al., 2016) is imposed on these discrete variables. This makes it possible to compute the gradients with respect to both of the approximated architecture variables and the neural operation parameters through a back-propagation and thereby allows to use a stochastic gradient descent (SGD) in bi-level optimization. Furthermore, during the SGD-based bi-level optimization, we utilize an iterative sampling of the candidate architecture, which simulates the test-time final architecture, based on the approximated continuous architecture variables by treating those as logits to provide a categorical distribution. This sampling procedure enables to reduce the computational burden of taking the entire connectivities and operations into account within an outrageously large network originated from the continuous relaxation. Moreover, it also reduces the discrepancy between the training-time and test-time architectures. Namely, the proposed differentiable NAS with stochastic sampling supports great scalability in terms of solvable large search space with small computational cost.
+
+Experimental results on the benchmark 3D medical image segmentation dataset show that in comparison to the previous human-designed 3D U-Net, the network obtained by the proposed scalable NAS leads to better performances even with the less numbers of parameters and FLOPs (multiply-adds). It is furthermore shown that the found architecture from a task having large amounts of labeled data can be transferred to build a network for different segmentation tasks that have small amounts of labeled data and achieves better generalization performances.
+
+To our best knowledge, this is the first work to exploit a complete NAS framework for automatically designing an architecture for the task of 3D medical image segmentation.
+
+# 2. Related Works
+
+NAS can be considered as one of meta-learning processes (Lemke et al., 2015; Vanschoren, 2018) in which a meta-controller performs a guided exploration on a given architecture space via evaluation of each candidate architecture in the inner loop (Zoph et al., 2018; Pham et al., 2018). Several recent works have focused on reducing the computational cost of this architecture evaluation by reusing the trained weights on different architectures (Bender et al., 2018; Liu et al., 2018b; Pham et al., 2018). Especially, they have sampled every candidate network from a single over-parametrized network, called an one-shot model, which allows to train only the one-shot model and directly evaluate any candidate network by inheriting this one-shot model’s trained weights. Among them, DARTS (Liu et al., 2018b) have removed a meta-controller by continuous relaxation of the search space, which leads to simultaneously learn the structure parameters as well as the kernel weights by SGD-based bi-level optimization. Even though DARTS enables efficient SGD-based optimization, it still suffers from the large computational cost to handle all possible neural connectivities and operations in the whole large one-shot model. ProxylessNAS (Cai et al., 2018) have resolved this cost issue by sampling two operation types for each neural connection according to the multinomial distribution during the architecture training. However, it has still used a biased architecture during the training in that there is no guidance for realvalued operation gates (logits) representing the multinomial distribution to be converged to discrete one-hot variables standing for the final architecture at test-time. Hence, we use a stochastic architecture sampling based on the Gumbel-softmax (Jang et al., 2017; Maddison et al., 2016), that is a continuous and differentiable approximation of these one-hot variables, which makes the sampled architecture converged to be the final architecture during the training by gradually reducing the softmax temperature to 0. While the previous NAS approaches have been applied mostly to the tasks of image recognition and language modeling, Nekrasov et al. (2018) has recently adopted NAS for 2D image segmentation. However, they have optimized only the decoder architecture in an encoder-decoder framework with an RNN-based meta-controller trained by reinforcement learning.
+
+Since Ronneberger et al. (2015) first introduced the U-Net for biomedical image segmentation, several modifications have been proposed. For example, C¸ i¸cek et al. (2016) has extended it with 3D convolutional kernels, and then Milletari et al. (2016a) has incorporated the residual blocks into the 3D U-Net. Moreover, Kayalibay et al. (2017) and Yu et al. (2017) have utilized multiple segmentation maps at different scales while Oktay et al. (2018) has adopted attention gates between an encoder and a decoder to simulate multistage cascaded convolutional neural networks (CNNs). Recently, Isensee et al. (2018) has introduced the nnU-Net that is able to dynamically adapt itself to any given segmentation task on the medical domain via non-architectural self-modifications based on the original U-Net. In (Mortazi and Bagci, 2018) the policy gradient algorithm automatically searches for the hyperparameters such as the number of filters, the filter size, and the pooling type for each layer for the 2D cine cardiac MR image segmentation while Naceur et al. (2018) incrementally optimized those hyperparameters as well as the number of layers for the 2D brain tumor segmentation. It is noted that unlike these architecture hyperparameter optimizations, we use the complete NAS to obtain the entire topology of the network architecture in this work.
+
+# 3. Method
+
+In this section, we first describe an architecture search space based on the U-Net-like network for 3D medical image segmentation, and then present a SGD-based bi-level optimization with the proposed stochastic sampling to simultaneously learn both of the architecture and the corresponding neural operation parameters.
+
+# 3.1. Search Space for 3D Medical Image Segmentation
+
+Following the idea of micro search space popularly used in the state-of-the-art NAS approaches (Liu et al., 2018b; Zoph et al., 2018; Pham et al., 2018), U-Net-like networks, which is composed of encoder and decoder layers, are designed as repeated encoder and decoder cells. The neural structure in each cell $C$ is represented as a directed acyclic graph (DAG) (see Figure 1). Let $\mathcal { G } = ( \mathcal { V } ( C ) , \mathcal { E } ( C ) )$ be the DAG where each node $i \in \mathcal V$ corresponds to an intermediate feature vector $\mathbf { x } ^ { i }$ , and each directed edge $( i , j ) \in \mathcal { E }$ stands for a connection between nodes $i$ and $j$ with a certain operation $o ^ { ( i , j ) }$ such that $\begin{array} { r } { \mathbf { x } ^ { j } = \sum _ { ( i , j ) \in \mathcal { E } } o ^ { ( i , j ) } ( \mathbf { x } ^ { i } ) } \end{array}$ . The output of a cell is a channel-wise concatenation of all the intermediate nodes. Here, a cell $C$ is one of four cell-types - encoder-normal $( C _ { \mathsf { e n c } } )$ , reduction $( C _ { \sf r e d } )$ , decoder-normal ( $C _ { \mathrm { d e c } }$ ), and expansion $( C _ { \mathsf { e x p } , }$ ) - such that $C \in \mathcal { C } = \{ C _ { \mathsf { e n c } } , C _ { \mathsf { r e d } } , C _ { \mathsf { d e c } } , C _ { \mathsf { e x p } } \}$ , and the normal cells and resizing cells are stacked alternately with skip connections between the cells in the encoder and the cells in the decoder, layer-by-layer. Note that every cell takes two outputs of the last previous two cells as inputs1 except the first reduction cell which takes an output of the predefined first convolutional block, called a stem cell, and then duplicates it as two inputs. The segmentation output is obtained from the predefined last convolutional block, referred to as an out cell.
+
+![](images/0c021b4e4194dabfd34c1809c978feaae43116fe9c00486f6d6c617f596b64bc.jpg)  
+Figure 1: Architecture search space for 3D medical image segmentation tasks. Both encoder and decoder alternately stack normal cells and resizing cells. The directed arrows between cells indicate the forward paths. Each cell is represented as a DAG which receives two inputs and produces an output.
+
+Since $o ^ { ( i , j ) } \in \mathcal { O }$ where $\mathcal { O }$ denotes the set of all candidate operations, the architecture search problem now amounts to find the best combination of all edge operations in the four cell-types. Basically, even the same type of cells can have different structures according to their layer levels. However, in this work, for simplicity, all cells that have a common type share a common structure regardless of layer levels. It is noted that a special zero operation is also one of the candidate operations to optimize the neural connectivities as well; zero means a lack of connection between two nodes.
+
+# 3.2. Stochastic Bi-Level Optimization
+
+We first represent the selected edge operation using the one-hot indicator vector, $\mathbf { z } ^ { ( i , j ) }$ , as follows:
+
+$$
+o ^ { ( i , j ) } ( { \bf x } ^ { i } ) = \sum _ { o \in \mathcal { O } } z _ { o } ^ { ( i , j ) } o ( { \bf x } ^ { i } ; \boldsymbol { \theta } _ { o } ^ { ( i , j ) } ) ,
+$$
+
+where $\mathbf { z } ^ { ( i , j ) } = \{ z _ { o } ^ { ( i , j ) } \mid o \in \mathcal { O } \}$ , $\theta ^ { ( i , j ) } = \{ \theta _ { o } ^ { ( i , j ) } \mid o \in \mathcal { O } \}$ , and $\theta _ { o } ^ { ( i , j ) }$ denotes the parameter set of the operation $o$ on edge $( i , j )$ , which means that the operation on each edge is differently learned even though the cells from different layers have the same structure, i.e. the same combination of operation types.
+
+Then, finding the best cell architecture corresponds to solving the following bi-level optimization problem:
+
+$$
+\begin{array} { r l } { \underset { Z } { \operatorname* { m i n } } } & { \quad \mathcal { L } _ { \mathsf { v a l } } ( \Theta ^ { * } ( Z ) , Z ) } \\ { \mathrm { s . t . } } & { \quad \Theta ^ { * } ( Z ) = \underset { \Theta } { \operatorname { a r g m i n } } \mathcal { L } _ { \mathsf { t r a i n } } ( \Theta , Z ) , } \end{array}
+$$
+
+where $Z = \{ \mathbf { z } ^ { ( i , j ) } \mid ( i , j ) \in \mathcal { E } ( C ) , C \in \mathcal { C } \}$ , $\Theta = \{ \theta ^ { ( i , j ) } \mid ( i , j ) \in \mathcal { E } ( C ) , C \in \mathcal { C } \}$ , and $\mathcal { L } _ { \mathsf { v a l } }$ and $\scriptstyle { \dot { L } } _ { \mathrm { t r a i n } }$ are validation loss and training loss, respectively. Note that this loss splitting is typically used in meta-learning processes including NAS for better generalization. This is a bi-level program in the mixed domain of continuous variables ( $\Theta$ ) and discrete variables $( Z$ ), which is hard to solve. DARTS (Liu et al., 2018b) and proxylessNAS (Cai et al., 2018) try to circumvent this difficulty by relaxing $Z$ to a continuous operation-weight variables $Z$ such that $\bar { z } _ { o } ^ { ( i , j ) } \in [ 0 , 1 ]$ and $\begin{array} { r } { \sum _ { o \in \mathcal { O } } \bar { z } _ { o } ^ { ( i , j ) } = 1 } \end{array}$ z¯(i,j)o = 1 and making Lval(Θ, Z¯) be differentiable with respect to both of $\Theta$ and $Z$ . This allows to use a SGD-based optimization to obtain an approximate solution $( \Theta ^ { * } , \bar { Z } ^ { * } )$ and derive the final architecture from the relaxed variables $\bar { Z } ^ { * }$ by taking the operation with the highest weight on each edge.
+
+One problem with this method is that the performance of the final architecture is inconsistent with the performance of the relaxed architecture since the relaxed architecture is not guaranteed to be converged to the final architecture. Hence, they necessarily have to retrain $\Theta$ from the scratch after obtaining the final network architecture. Moreover, applying this method directly to a large-scale task such as high-resolution 3D medical image segmentation is infeasible due to the extremely large memory usage and the long run-time during the training to compute the loss functions $\mathcal { L } _ { \mathsf { v a l } }$ and $\mathcal { L } _ { \mathrm { t r a i n } }$ as well as their gradients from the fact that the required resources are proportional to the number of nonzero entries in $Z$ , which scales with the number of candidate operations.
+
+To overcome the aforementioned problems, we propose a modified optimization, called stochastic bi-level optimization, by first treating $Z$ as random discrete variables and then replacing (2) as
+
+$$
+\begin{array} { r l } { \underset { \alpha } { \mathrm { m i n } } \quad } & { \mathbb { E } _ { Z \sim P _ { \alpha } } [ \mathcal { L } _ { \sf v a l } ( \Theta ^ { * } ( Z ) , Z ) ] } \\ { \mathrm { s . t . } \quad } & { \Theta ^ { * } ( Z ) = \underset { \Theta } { \mathrm { a r g m i n } } \ \mathcal { L } _ { \sf t r a i n } ( \Theta , Z ) , } \end{array}
+$$
+
+where $P _ { \alpha }$ is the discrete distribution on $Z$ , parameterized by $\alpha$ . Since it is intractable to exactly compute $\nabla _ { \alpha } \mathbb { E } _ { Z \sim P _ { \alpha } } [ \mathcal { L } _ { \mathsf { v a l } } ( \Theta ^ { * } ( Z ) , Z ) ]$ , this gradient with respect to $\alpha$ is estimated by a continuous relaxation with sampling on $Z$ in order to use the gradient-based bi-level optimization method in this work.
+
+Algorithm 1: Gradient-based stochastic bi-level optimization
+
+Initialize $\alpha$ and $\Theta$ ;   
+while not done do $\hat { Z } \gets$ GumbelSoftmaxSample $( \alpha , \tau )$ ; Update $\Theta$ by a gradient descent using $\nabla _ { \Theta } \mathcal { L } _ { \mathrm { t r a i n } } ( \Theta , \hat { Z } )$ ; Update $\alpha$ by a gradient descent using $\nabla _ { \alpha } \mathcal { L } _ { \mathsf { v a l } } ( \Theta , \hat { Z } )$ ; Anneal $\tau$ ;   
+end
+
+# Algorithm 2: GumbelSoftmaxSample(α, τ )
+
+for $( i , j ) \in \mathcal { E }$ do (i,j)o ∼ Gumbel(0, 1), o ∈ O; $\bar { \mathbf { z } } ^ { ( i , j ) } \longleftarrow 5 \mathsf { o f t m a x } ( ( \boldsymbol { \alpha } ^ { ( i , j ) } + \epsilon ^ { ( i , j ) } ) / \tau )$ ; foreach pair $\{ o _ { 1 } , o _ { 2 } \}$ in $\mathcal { O }$ do $q ^ { \{ o _ { 1 } , o _ { 2 } \} } \longleftarrow \frac { \bar { z } _ { o _ { 1 } } ^ { ( i , j ) } + \bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } { | \mathcal { O } | - 1 }$ ; end Sample $\{ o _ { 1 } , o _ { 2 } \}$ with probability $q ^ { \{ o _ { 1 } , o _ { 2 } \} }$ ; $\begin{array} { r } { ( \hat { z } _ { o _ { 1 } } ^ { ( i , j ) } , \hat { z } _ { o _ { 2 } } ^ { ( i , j ) } ) \longleftarrow ( \frac { \bar { z } _ { o _ { 1 } } ^ { ( i , j ) } } { \bar { z } _ { o _ { 1 } } ^ { ( i , j ) } + \bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } , \frac { \bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } { \bar { z } _ { o _ { 1 } } ^ { ( i , j ) } + \bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } ) , \quad \hat { z } _ { o } ^ { ( i , j ) }  0 , o \notin \{ o _ { 1 } , o _ { 2 } \} ; } \end{array}$   
+end   
+return $\hat { Z }$ ;
+
+# 3.3. Gumbel-Softmax Relaxation with Operation Sampling
+
+The Gumbel-softmax reparametrization technique (Jang et al., 2017; Maddison et al., 2016) can approximate the above gradient by continuous relaxation as
+
+$$
+\nabla _ { \alpha } \mathbb { E } _ { Z \sim P _ { \alpha } } [ \mathcal { L } _ { \mathsf { v a l } } ( \Theta ^ { * } ( Z ) , Z ) ] \approx \mathbb { E } _ { \epsilon \sim \mathsf { G u m b e l } ( 0 , 1 ) } [ \nabla _ { \alpha } \mathcal { L } _ { \mathsf { v a l } } ( \Theta ^ { * } ( \bar { Z } ( \alpha , \epsilon ; \tau ) ) , \bar { Z } ( \alpha , \epsilon ; \tau ) ] ,
+$$
+
+where continuously relaxed variables $\bar { Z } ( \alpha , \epsilon ; \tau ) = 5 \mathsf { o f t m a x } ( ( \alpha + \epsilon ) / \tau )$ , $\tau$ denotes the temperature, and $\epsilon$ is $\alpha$ -independent random variables drawn from the Gumbel distribution. Here, the expectation in (4) is approximated with $\epsilon$ -sampling. It is noted that as $\tau  0$ , the distribution of $Z$ is identical to $P _ { \alpha }$ , which means that by annealing $\tau$ we can enforce $Z$ to be one-hot discrete variables $Z$ during the training; the relaxed architecture is forced to be converged to the final architecture.
+
+Algorithm $1$ summarizes our stochastic bi-level optimization algorithm which alternately updates $\Theta$ and $\alpha$ by respective gradient descents. Here, note that in order to reduce the number of nonzero operation weights in $Z$ and hence to reduce the computational cost, in each iteration during the training we again replace $Z$ with $\hat { Z }$ by sampling two operations from the Gumbel-softmax and then rescaling the corresponding two operation weights to be summed to one with zero weights of the other operations on each edge, as shown in Algorithm 2.
+
+Owing to our continuous relaxation based on the Gumbel-softmax with $\tau$ -annealing, the number of sampled operations on each edge is naturally reduced from two to one during the training. As a result, the proposed differentiable NAS with stochastic operation sampling is able to support improved scalability in terms of solvable large search space with small computational cost.
+
+# 4. Experiments
+
+Dataset The proposed scalable NAS (SCNAS) was evaluated on the three segmentation tasks of 3D MRI data, (1) brain tumor (484 labeled images, 3 classes), (2) heart (20 labeled images, 1 class), and (3) prostate (32 labeled images, 2 classes), from the Medical Segmentation Decathlon challenge (MSD, http://medicaldecathlon.com) where each task has different MRI sequences as well as different foreground classes, which is therefore suitable for evaluating the generalizability and transferability of the SCNAS.
+
+Implementation Details We compared the SCNAS to the state-of-the-art architecture, 3D U-ResNet with the use of multiple segmentation maps (Kayalibay et al., 2017) and attention gates (Oktay et al., 2018), and the random architecture by random selection of edge-operations in each cell from the same architecture search space in the SCNAS. The set of operations $\mathcal { O }$ on each edge in the SCNAS consists of the following eight operations: $3 \times 3 \times 3$ convolutions, depthwise separable dilated $3 \times 3 \times 3$ convolutions with rate 2, 3 and 4, $3 \times 3 \times 3$ max and average 3D pooling, identity (skip connection), and zero. Here, we used the LeakyReLU-Conv-InstanceNorm for convolutional operations.
+
+As shown in Figure 1, the whole network in the SCNAS is composed of 12 automatically designed cells, each of which has 4 nodes. This number of stacked cells is consistent with that of the 3D U-ResNet in terms of respective three times of downsampling and upsampling by a factor of 2. Here, all operations in the reduction cell in the SCNAS are of stride two while the expansion cells perform pre-upsampling for the inputs of the cell. Since the 3D U-ResNet in this evaluation was set to have 32 output channels in the first convolutional block, the number of output channels in the stem cell of the SCNAS was set to 32, and also similar to the 3D U-ResNet, the reduction and expansion cells in the SCNAS respectively double and halve the number of output channels of given inputs.
+
+In both of the 3D U-ResNet and the SCNAS, patch-based training and inference were carried out such that each image was randomly cropped to the region of nonzero values with the predefined resolution during the training, while in testing, the prediction results were obtained by combining patch-based inference results with 50 percent overlap. Similar to Isensee et al. (2018), the predefined resolution for the input patch was set to $1 2 8 \times 1 2 8 \times 1 2 8$ for the tasks of brain tumors and heart while for the prostate task, the length of the $z$ - axis was reduced to 24. Since even the same task provides 3D images with heterogeneous voxel spacings, the input images were first resized for all voxel spacings to be physically equal using the given meta-data, and then $z$ -normalization was separately applied to each input channel. Note that unlike Isensee et al. (2018), any heuristic pre-/post-processing techniques including data augmentation, network-cascade, and prediction-ensemble were not adopted in this evaluation to solely examine the effects by the use of NAS in designing the network architecture.
+
+Since the ground-truth labels for test images are not provided in the MSD dataset, the evaluation is conducted by 5-fold cross-validation (CV) on the training images with the average dice similarity coefficient (DSC) as the metric, and accordingly, we applied the well-known multi-class dice loss function (Milletari et al., 2016b; Isensee et al., 2018). With the ADAM optimizer, the 3D U-ResNet and the SCNAS models were trained for 300 epochs and 400 epochs, respectively, taking their convergences into consideration. The 3D U-ResNet was set with the batch size as 8, initial learning rate as 0.0001, and beta parameters for ADAM optimizer as (0.9, 0.999) while in the SCNAS, with the batch size 1, the initial learning rates / beta parameters were as set to be $0 . 0 2 5 \mathrm { ~ / ~ } ( 0 . 1 , 0 . 0 0 1 )$ for training operation parameters $\Theta$ and $0 . 0 0 3 / \ ( 0 . 5 , 0 . 9 9 9 )$ for training architecture parameters $\alpha$ . If a plateau for 20 epochs on the training loss was detected, the learning rate was reduced by a factor of 10. All experiments were conducted on V100 GPUs, and the implementation was done using PyTorch (Paszke et al., 2017).
+
+Architecture Transfer Since the heart and prostate tasks only have 20 and 32 labeled MRI images, respectively, the 3D U-ResNet as well as the SCNAS can be prone to overfitting on the training set and hence to resulting in performance degradation on the validation set. Therefore, we transferred the optimized architecture obtained from the brain tumor task, which has 484 labeled MRI images, by the SCNAS into these two tasks having scarce data and retrained only the operation parameters on each task, in order to demonstrate that the SCNAS produces a more generalizable neural architecture for the similar tasks of 3D MRI image segmentation. Here, the transferred architecture came from the first CV fold in the brain tumor task.
+
+Results Table 1 shows that the SCNAS produced better architectures than the (humandesigned) 3D U-ResNet as well as the randomly designed 3D U-Net in terms of the overall performances on all three tasks. Especially, on the heart and prostate segmentation tasks, the transferred architecture from the brain tumor task achieved significantly better generalization performances. Note that the obtained architectures by the SCNAS have been also shown that the number of neural operation parameters and the computational complexity for output prediction (in terms of FLOPs) were significantly reduced compared to the 3D UResNet. We observed the performance degradation of the 3D U-ResNet when the number of initial output channels was halved. It is also noted that most previous NAS approaches retrained the neural operation parameters after completing architecture optimization because of the utilization of a biased architecture during the training, while the SCNAS simultaneously optimized both of the architecture parameters and neural operation parameters with an unbiased architecture and thereby removed the requirement of retraining. We conjecture that Isensee et al. (2018) might be benefit from complicated pre-/post-procedures and thus obtained slightly better performances than the SCNAS. Some example images and the corresponding segmentation outputs are included in Appendix A, and the details of the optimized cell architectures by the SCNAS are presented in Appendix B.
+
+# 5. Conclusion
+
+In this work, a complete NAS framework for automatically designing an architecture is proposed and demonstrated on the benchmark dataset of 3D medical image segmentation tasks. In the proposed framework, NAS is formulated as finding the optimal structure of four types of cells composing an encoder as well as a decoder, and both the architecture parameters and the neural operation parameters are learned by gradient descent in an end-to-end manner. We introduce a novel stochastic sampling algorithm which results in significant improvement in terms of the scalability suitable for handling high-resolution 3D medical images and also reduces the inconsistency of the train-time architecture against the final architecture, which leads to avoid the retraining of the operation parameters. Empirical evaluation demonstrates that the automatically optimized network via the proposed NAS outperforms the manually designed 3D U-Net. Moreover, the architecture learned from a task with the large number of training data is successfully transferred to different MRI segmentation tasks with the small number of data.
+
+Table 1: Mean DSC in Brain Tumor, Heart, and Prostate.   
+
+<table><tr><td></td><td></td><td colspan="4">Brain Tumor</td></tr><tr><td>Model</td><td>GFLOPs, Params</td><td>Edema</td><td>Non-Enhancing</td><td>Enhancing</td><td>Average</td></tr><tr><td>3D U-ResNet</td><td>881, 7.6M</td><td>79.10 ± 1.80</td><td>58.38 ± 1.29</td><td>77.37 ± 2.76</td><td>71.61</td></tr><tr><td>Random Search</td><td>152, 2.7M</td><td>79.59 ± 1.28</td><td>57.97 ± 1.36</td><td>77.85 ± 1.35</td><td>71.80</td></tr><tr><td> SCNAS</td><td>129, 2.2M</td><td>79.42 ± 1.45</td><td>58.01 ± 1.46</td><td>78.68 ± 1.80</td><td>72.04</td></tr><tr><td></td><td></td><td>Heart</td><td colspan="3">Prostate</td></tr><tr><td>Model</td><td>GFLOPs, Params</td><td>Left Atrium</td><td>Peripheral</td><td>Transitional</td><td>Average</td></tr><tr><td>3D U-ResNet</td><td>870-163, 7.6M</td><td>89.60 ± 2.35</td><td>48.37 ± 1.44</td><td>79.17 ± 4.30</td><td>63.77</td></tr><tr><td>Random Search</td><td>104-18,1.5M</td><td>89.14 ± 2.74</td><td>50.78 ± 1.22</td><td>79.58 ± 5.01</td><td>65.18</td></tr><tr><td> SCNAS</td><td>136-32, 3.0M</td><td>89.99 ± 1.32</td><td>49.70 ± 1.23</td><td>80.89 ± 3.19</td><td>65.30</td></tr><tr><td> SCNAS(transfer)</td><td>193-37, 4.2M</td><td>90.47 ± 1.70</td><td>53.81 ± 1.30</td><td>82.02 ± 4.52</td><td>67.92</td></tr></table>
+
+The analysis using more candidate operations and different cell structures of the same type at different layer levels are left for future research. In addition, the effects of including non-architectural procedures such as data augmentation, network-cascade, and predictionensemble in the proposed NAS framework need to be analyzed in future works. Another interesting research direction would be applying the NAS framework, either directly or by architecture transfer, to other medical modalities including CT, mammography, and X-ray.
+
+# Acknowledgments
+
+We thank the Kakao Brain Cloud team for supporting to efficiently use GPU clusters for large-scale experiments.
+
+# References
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+SCNAS for 3D Medical Image Segmentation
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+
+![](images/3b00d8f9b516bee7d381b7ec514a68b9c7f7d0a02b6fc0263387a18d54b3c546.jpg)  
+Figure 2: (a) Raw data. (b) Ground Truth. Segmentation results of (c) 3D U-ResNet (d) SCNAS in Brain Tumor (top), Heart (middle), and Prostate (bottom) from the MSD dataset. The prediction results on the heart and prostate data are obtained from transferred networks which are trained on brain tumor data.
+
+In this appendix, we provide examples of segmentation predictions by the 3D U-ResNet and proposed method. For a qualitative assessment, we compare the two results with ground truth. Additionally, samples of cell architectures found by the SCNAS are illustrated in the sequent section.
+
+# Appendix A. Segmentation Samples
+
+Figure 2 shows samples of 3D segmentation results of 3D U-ResNet and proposed method for each MSD dataset. The above samples demonstrate that the architectures found by SCNAS predict more accurately than 3D U-ResNet. Especially, the architecture found on brain tumor images can be transferred well to the dataset of different MRI segmentation tasks.
+
+![](images/329bb26a81a71c724e5aae63eb86f409757a5aa0fbb5375b3898f613965e1ef9.jpg)  
+Figure 3: Reduction Cell
+
+![](images/835db0a8704c4c9150510ff949d67e08ad0ecc12843b1b9338e4cde3799f78f9.jpg)  
+Figure 4: Encoder Normal Cell
+
+# Appendix B. Cell Architectures
+
+Figure 3-6 show the samples of cell architectures which SCNAS found in the first CV fold experiment on the Brain Tumor dataset at the convergence.
+
+![](images/3c71ee83dfd54e78802e62d46ea7d731012a111e0370ab298f981b2664d5d601.jpg)  
+Figure 5: Expansion Cell
+
+![](images/b42a9d4338c8e4b5e0f899fb2e590c3f9c4456fa2de160717b4c947b03cc1ab7.jpg)  
+Figure 6: Decoder Normal Cell

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+        "text": "Sungwoong Kim∗1   \nIldoo Kim∗1   \nSungbin Lim $^ { * 1 }$   \nChiheon Kim1   \nWoonhyuk Baek1   \nHyungjoo Cho2   \nBoogeon Yoon1   \nTaesup Kim1,3 ",
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+        "text": "swkim@kakaobrain.com ildoo.kim@kakaobrain.com   \nsungbin.lim@kakaobrain.com   \nchiheon.kim@kakaobrain.com wbaek@kakaobrain.com joysquare@snu.ac.kr eric.yoon@kakaobrain.com taesup.kim@umontreal.ca ",
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+        "text": "1 Kakao Brain, Pangyo, Seongnam, Gyeonggi, Republic of Korea   \n2 Department of Transdisciplinary Studies, Seoul National University, Republic of Korea   \n3 MILA, Universit´e de Montr´eal, Canada ",
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+        "text": "Editors: Under Review for MIDL 2019 ",
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+        "text": "Abstract ",
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+        "text": "In this paper, a neural architecture search (NAS) framework is formulated for 3D medical image segmentation, to automatically optimize a neural architecture from a large design space. For this, a novel NAS framework is proposed to produce the structure of each layer including neural connectivities and operation types in both of the encoder and decoder of a target 3D U-Net. In the proposed NAS framework, having a sufficiently large search space is important in generating an improved network architecture, however optimizing over such a large space is difficult due to the extremely large memory usage and the long run-time originated from high-resolution 3D medical images. Therefore, a novel stochastic sampling algorithm based on the continuous relaxation on the discrete architecture parameters is also proposed for scalable joint optimization of both of the architecture parameters and the neural operation parameters. This makes it possible to maintain a large search space with small computational cost as well as to obtain an unbiased architecture by reducing the discrepancy between the training-time and test-time architectures. On the 3D medical image segmentation tasks with a benchmark dataset, an automatically designed 3D U-Net by the proposed NAS framework outperforms the previous human-designed 3D U-Net as well as the randomly designed 3D U-Net, and moreover this optimized architecture is more compact and also well suited to be transferred for similar but different tasks. ",
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+        "text": "Keywords: AutoML, Neural Architecture Search, Medical Image Segmentation. ",
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+        "text": "1. Introduction ",
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+        "text": "Recently, deep neural networks have been extensively used for medical image segmentation tasks (Ronneberger et al., 2015; C¸ i¸cek et al., 2016; Mortazi et al., 2017; Ciresan et al., 2012; Milletari et al., 2016a; Kamnitsas et al., 2016; Havaei et al., 2015; Yu et al., 2017; Kayalibay et al., 2017; Oktay et al., 2018; Isensee et al., 2018). However, such a method in general relies on manual trial-and-error processes for making decisions on the network architecture, hyperparameters for training, and pre-/post-procedures. Due to being restricted to manual tuning, they would have limitations in performance improvement as well as fast transfer to related tasks. Currently, the same problem in the field of general deep learning has promoted the rapid development of automated machine learning (AutoML). Yet, in contrast to the recent intensive studies on the use of advanced AutoML algorithms such as neural architecture search (NAS) (Zoph et al., 2018; Liu et al., 2018a; Bender et al., 2018; Zoph and Le, 2017; Liu et al., 2018b; Pham et al., 2018; Zhang et al., 2018; Cai et al., 2018; Brock et al., 2018) and neural optimizer search (Bello et al., 2017; Alber et al., 2018; Wichrowska et al., 2017; Li and Malik, 2017; Andrychowicz et al., 2016) for general computer vision tasks, only a few naive AutoML approaches using simple hyperparameter optimization have been proposed for medical imaging tasks (Mortazi and Bagci, 2018; Naceur et al., 2018). Therefore, in this paper, we propose a novel NAS framework for AutoML in designing neural networks especially for 3D medical image segmentation. ",
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+        "text": "To our best knowledge, this is the first work to exploit a complete NAS framework for automatically designing an architecture for the task of 3D medical image segmentation. ",
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+    {
+        "type": "text",
+        "text": "2. Related Works ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "NAS can be considered as one of meta-learning processes (Lemke et al., 2015; Vanschoren, 2018) in which a meta-controller performs a guided exploration on a given architecture space via evaluation of each candidate architecture in the inner loop (Zoph et al., 2018; Pham et al., 2018). Several recent works have focused on reducing the computational cost of this architecture evaluation by reusing the trained weights on different architectures (Bender et al., 2018; Liu et al., 2018b; Pham et al., 2018). Especially, they have sampled every candidate network from a single over-parametrized network, called an one-shot model, which allows to train only the one-shot model and directly evaluate any candidate network by inheriting this one-shot model’s trained weights. Among them, DARTS (Liu et al., 2018b) have removed a meta-controller by continuous relaxation of the search space, which leads to simultaneously learn the structure parameters as well as the kernel weights by SGD-based bi-level optimization. Even though DARTS enables efficient SGD-based optimization, it still suffers from the large computational cost to handle all possible neural connectivities and operations in the whole large one-shot model. ProxylessNAS (Cai et al., 2018) have resolved this cost issue by sampling two operation types for each neural connection according to the multinomial distribution during the architecture training. However, it has still used a biased architecture during the training in that there is no guidance for realvalued operation gates (logits) representing the multinomial distribution to be converged to discrete one-hot variables standing for the final architecture at test-time. Hence, we use a stochastic architecture sampling based on the Gumbel-softmax (Jang et al., 2017; Maddison et al., 2016), that is a continuous and differentiable approximation of these one-hot variables, which makes the sampled architecture converged to be the final architecture during the training by gradually reducing the softmax temperature to 0. While the previous NAS approaches have been applied mostly to the tasks of image recognition and language modeling, Nekrasov et al. (2018) has recently adopted NAS for 2D image segmentation. However, they have optimized only the decoder architecture in an encoder-decoder framework with an RNN-based meta-controller trained by reinforcement learning. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Since Ronneberger et al. (2015) first introduced the U-Net for biomedical image segmentation, several modifications have been proposed. For example, C¸ i¸cek et al. (2016) has extended it with 3D convolutional kernels, and then Milletari et al. (2016a) has incorporated the residual blocks into the 3D U-Net. Moreover, Kayalibay et al. (2017) and Yu et al. (2017) have utilized multiple segmentation maps at different scales while Oktay et al. (2018) has adopted attention gates between an encoder and a decoder to simulate multistage cascaded convolutional neural networks (CNNs). Recently, Isensee et al. (2018) has introduced the nnU-Net that is able to dynamically adapt itself to any given segmentation task on the medical domain via non-architectural self-modifications based on the original U-Net. In (Mortazi and Bagci, 2018) the policy gradient algorithm automatically searches for the hyperparameters such as the number of filters, the filter size, and the pooling type for each layer for the 2D cine cardiac MR image segmentation while Naceur et al. (2018) incrementally optimized those hyperparameters as well as the number of layers for the 2D brain tumor segmentation. It is noted that unlike these architecture hyperparameter optimizations, we use the complete NAS to obtain the entire topology of the network architecture in this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3. Method ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we first describe an architecture search space based on the U-Net-like network for 3D medical image segmentation, and then present a SGD-based bi-level optimization with the proposed stochastic sampling to simultaneously learn both of the architecture and the corresponding neural operation parameters. ",
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+    {
+        "type": "text",
+        "text": "3.1. Search Space for 3D Medical Image Segmentation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following the idea of micro search space popularly used in the state-of-the-art NAS approaches (Liu et al., 2018b; Zoph et al., 2018; Pham et al., 2018), U-Net-like networks, which is composed of encoder and decoder layers, are designed as repeated encoder and decoder cells. The neural structure in each cell $C$ is represented as a directed acyclic graph (DAG) (see Figure 1). Let $\\mathcal { G } = ( \\mathcal { V } ( C ) , \\mathcal { E } ( C ) )$ be the DAG where each node $i \\in \\mathcal V$ corresponds to an intermediate feature vector $\\mathbf { x } ^ { i }$ , and each directed edge $( i , j ) \\in \\mathcal { E }$ stands for a connection between nodes $i$ and $j$ with a certain operation $o ^ { ( i , j ) }$ such that $\\begin{array} { r } { \\mathbf { x } ^ { j } = \\sum _ { ( i , j ) \\in \\mathcal { E } } o ^ { ( i , j ) } ( \\mathbf { x } ^ { i } ) } \\end{array}$ . The output of a cell is a channel-wise concatenation of all the intermediate nodes. Here, a cell $C$ is one of four cell-types - encoder-normal $( C _ { \\mathsf { e n c } } )$ , reduction $( C _ { \\sf r e d } )$ , decoder-normal ( $C _ { \\mathrm { d e c } }$ ), and expansion $( C _ { \\mathsf { e x p } , }$ ) - such that $C \\in \\mathcal { C } = \\{ C _ { \\mathsf { e n c } } , C _ { \\mathsf { r e d } } , C _ { \\mathsf { d e c } } , C _ { \\mathsf { e x p } } \\}$ , and the normal cells and resizing cells are stacked alternately with skip connections between the cells in the encoder and the cells in the decoder, layer-by-layer. Note that every cell takes two outputs of the last previous two cells as inputs1 except the first reduction cell which takes an output of the predefined first convolutional block, called a stem cell, and then duplicates it as two inputs. The segmentation output is obtained from the predefined last convolutional block, referred to as an out cell. ",
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+        "img_path": "images/0c021b4e4194dabfd34c1809c978feaae43116fe9c00486f6d6c617f596b64bc.jpg",
+        "image_caption": [
+            "Figure 1: Architecture search space for 3D medical image segmentation tasks. Both encoder and decoder alternately stack normal cells and resizing cells. The directed arrows between cells indicate the forward paths. Each cell is represented as a DAG which receives two inputs and produces an output. "
+        ],
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+        "type": "text",
+        "text": "Since $o ^ { ( i , j ) } \\in \\mathcal { O }$ where $\\mathcal { O }$ denotes the set of all candidate operations, the architecture search problem now amounts to find the best combination of all edge operations in the four cell-types. Basically, even the same type of cells can have different structures according to their layer levels. However, in this work, for simplicity, all cells that have a common type share a common structure regardless of layer levels. It is noted that a special zero operation is also one of the candidate operations to optimize the neural connectivities as well; zero means a lack of connection between two nodes. ",
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+        "type": "text",
+        "text": "3.2. Stochastic Bi-Level Optimization ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first represent the selected edge operation using the one-hot indicator vector, $\\mathbf { z } ^ { ( i , j ) }$ , as follows: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/0526dc1103a2a7e18d9060a1da9d57430b303c226efd313000ab80eb544662fe.jpg",
+        "text": "$$\no ^ { ( i , j ) } ( { \\bf x } ^ { i } ) = \\sum _ { o \\in \\mathcal { O } } z _ { o } ^ { ( i , j ) } o ( { \\bf x } ^ { i } ; \\boldsymbol { \\theta } _ { o } ^ { ( i , j ) } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { z } ^ { ( i , j ) } = \\{ z _ { o } ^ { ( i , j ) } \\mid o \\in \\mathcal { O } \\}$ , $\\theta ^ { ( i , j ) } = \\{ \\theta _ { o } ^ { ( i , j ) } \\mid o \\in \\mathcal { O } \\}$ , and $\\theta _ { o } ^ { ( i , j ) }$ denotes the parameter set of the operation $o$ on edge $( i , j )$ , which means that the operation on each edge is differently learned even though the cells from different layers have the same structure, i.e. the same combination of operation types. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Then, finding the best cell architecture corresponds to solving the following bi-level optimization problem: ",
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+    {
+        "type": "equation",
+        "img_path": "images/f0e0c9a574b5145e6e6912714cb18e088740f6da28fd080e0d7331800e9e15a7.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { Z } { \\operatorname* { m i n } } } & { \\quad \\mathcal { L } _ { \\mathsf { v a l } } ( \\Theta ^ { * } ( Z ) , Z ) } \\\\ { \\mathrm { s . t . } } & { \\quad \\Theta ^ { * } ( Z ) = \\underset { \\Theta } { \\operatorname { a r g m i n } } \\mathcal { L } _ { \\mathsf { t r a i n } } ( \\Theta , Z ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $Z = \\{ \\mathbf { z } ^ { ( i , j ) } \\mid ( i , j ) \\in \\mathcal { E } ( C ) , C \\in \\mathcal { C } \\}$ , $\\Theta = \\{ \\theta ^ { ( i , j ) } \\mid ( i , j ) \\in \\mathcal { E } ( C ) , C \\in \\mathcal { C } \\}$ , and $\\mathcal { L } _ { \\mathsf { v a l } }$ and $\\scriptstyle { \\dot { L } } _ { \\mathrm { t r a i n } }$ are validation loss and training loss, respectively. Note that this loss splitting is typically used in meta-learning processes including NAS for better generalization. This is a bi-level program in the mixed domain of continuous variables ( $\\Theta$ ) and discrete variables $( Z$ ), which is hard to solve. DARTS (Liu et al., 2018b) and proxylessNAS (Cai et al., 2018) try to circumvent this difficulty by relaxing $Z$ to a continuous operation-weight variables $Z$ such that $\\bar { z } _ { o } ^ { ( i , j ) } \\in [ 0 , 1 ]$ and $\\begin{array} { r } { \\sum _ { o \\in \\mathcal { O } } \\bar { z } _ { o } ^ { ( i , j ) } = 1 } \\end{array}$ z¯(i,j)o = 1 and making Lval(Θ, Z¯) be differentiable with respect to both of $\\Theta$ and $Z$ . This allows to use a SGD-based optimization to obtain an approximate solution $( \\Theta ^ { * } , \\bar { Z } ^ { * } )$ and derive the final architecture from the relaxed variables $\\bar { Z } ^ { * }$ by taking the operation with the highest weight on each edge. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One problem with this method is that the performance of the final architecture is inconsistent with the performance of the relaxed architecture since the relaxed architecture is not guaranteed to be converged to the final architecture. Hence, they necessarily have to retrain $\\Theta$ from the scratch after obtaining the final network architecture. Moreover, applying this method directly to a large-scale task such as high-resolution 3D medical image segmentation is infeasible due to the extremely large memory usage and the long run-time during the training to compute the loss functions $\\mathcal { L } _ { \\mathsf { v a l } }$ and $\\mathcal { L } _ { \\mathrm { t r a i n } }$ as well as their gradients from the fact that the required resources are proportional to the number of nonzero entries in $Z$ , which scales with the number of candidate operations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To overcome the aforementioned problems, we propose a modified optimization, called stochastic bi-level optimization, by first treating $Z$ as random discrete variables and then replacing (2) as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f657b4e026d8fb963c20f7a793cd4f7f6ce3e520efaad977ddd0fdd9470e5fcc.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { \\alpha } { \\mathrm { m i n } } \\quad } & { \\mathbb { E } _ { Z \\sim P _ { \\alpha } } [ \\mathcal { L } _ { \\sf v a l } ( \\Theta ^ { * } ( Z ) , Z ) ] } \\\\ { \\mathrm { s . t . } \\quad } & { \\Theta ^ { * } ( Z ) = \\underset { \\Theta } { \\mathrm { a r g m i n } } \\ \\mathcal { L } _ { \\sf t r a i n } ( \\Theta , Z ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $P _ { \\alpha }$ is the discrete distribution on $Z$ , parameterized by $\\alpha$ . Since it is intractable to exactly compute $\\nabla _ { \\alpha } \\mathbb { E } _ { Z \\sim P _ { \\alpha } } [ \\mathcal { L } _ { \\mathsf { v a l } } ( \\Theta ^ { * } ( Z ) , Z ) ]$ , this gradient with respect to $\\alpha$ is estimated by a continuous relaxation with sampling on $Z$ in order to use the gradient-based bi-level optimization method in this work. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Gradient-based stochastic bi-level optimization ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Initialize $\\alpha$ and $\\Theta$ ;   \nwhile not done do $\\hat { Z } \\gets$ GumbelSoftmaxSample $( \\alpha , \\tau )$ ; Update $\\Theta$ by a gradient descent using $\\nabla _ { \\Theta } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\Theta , \\hat { Z } )$ ; Update $\\alpha$ by a gradient descent using $\\nabla _ { \\alpha } \\mathcal { L } _ { \\mathsf { v a l } } ( \\Theta , \\hat { Z } )$ ; Anneal $\\tau$ ;   \nend ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2: GumbelSoftmaxSample(α, τ ) ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "for $( i , j ) \\in \\mathcal { E }$ do \u000f(i,j)o ∼ Gumbel(0, 1), o ∈ O; $\\bar { \\mathbf { z } } ^ { ( i , j ) } \\longleftarrow 5 \\mathsf { o f t m a x } ( ( \\boldsymbol { \\alpha } ^ { ( i , j ) } + \\epsilon ^ { ( i , j ) } ) / \\tau )$ ; foreach pair $\\{ o _ { 1 } , o _ { 2 } \\}$ in $\\mathcal { O }$ do $q ^ { \\{ o _ { 1 } , o _ { 2 } \\} } \\longleftarrow \\frac { \\bar { z } _ { o _ { 1 } } ^ { ( i , j ) } + \\bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } { | \\mathcal { O } | - 1 }$ ; end Sample $\\{ o _ { 1 } , o _ { 2 } \\}$ with probability $q ^ { \\{ o _ { 1 } , o _ { 2 } \\} }$ ; $\\begin{array} { r } { ( \\hat { z } _ { o _ { 1 } } ^ { ( i , j ) } , \\hat { z } _ { o _ { 2 } } ^ { ( i , j ) } ) \\longleftarrow ( \\frac { \\bar { z } _ { o _ { 1 } } ^ { ( i , j ) } } { \\bar { z } _ { o _ { 1 } } ^ { ( i , j ) } + \\bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } , \\frac { \\bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } { \\bar { z } _ { o _ { 1 } } ^ { ( i , j ) } + \\bar { z } _ { o _ { 2 } } ^ { ( i , j ) } } ) , \\quad \\hat { z } _ { o } ^ { ( i , j ) }  0 , o \\notin \\{ o _ { 1 } , o _ { 2 } \\} ; } \\end{array}$   \nend   \nreturn $\\hat { Z }$ ; ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3. Gumbel-Softmax Relaxation with Operation Sampling ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Gumbel-softmax reparametrization technique (Jang et al., 2017; Maddison et al., 2016) can approximate the above gradient by continuous relaxation as ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9819aa32753856b738491cbf15cb95cf9d35efcfa78067facce92f865d8325f7.jpg",
+        "text": "$$\n\\nabla _ { \\alpha } \\mathbb { E } _ { Z \\sim P _ { \\alpha } } [ \\mathcal { L } _ { \\mathsf { v a l } } ( \\Theta ^ { * } ( Z ) , Z ) ] \\approx \\mathbb { E } _ { \\epsilon \\sim \\mathsf { G u m b e l } ( 0 , 1 ) } [ \\nabla _ { \\alpha } \\mathcal { L } _ { \\mathsf { v a l } } ( \\Theta ^ { * } ( \\bar { Z } ( \\alpha , \\epsilon ; \\tau ) ) , \\bar { Z } ( \\alpha , \\epsilon ; \\tau ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where continuously relaxed variables $\\bar { Z } ( \\alpha , \\epsilon ; \\tau ) = 5 \\mathsf { o f t m a x } ( ( \\alpha + \\epsilon ) / \\tau )$ , $\\tau$ denotes the temperature, and $\\epsilon$ is $\\alpha$ -independent random variables drawn from the Gumbel distribution. Here, the expectation in (4) is approximated with $\\epsilon$ -sampling. It is noted that as $\\tau  0$ , the distribution of $Z$ is identical to $P _ { \\alpha }$ , which means that by annealing $\\tau$ we can enforce $Z$ to be one-hot discrete variables $Z$ during the training; the relaxed architecture is forced to be converged to the final architecture. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm $1$ summarizes our stochastic bi-level optimization algorithm which alternately updates $\\Theta$ and $\\alpha$ by respective gradient descents. Here, note that in order to reduce the number of nonzero operation weights in $Z$ and hence to reduce the computational cost, in each iteration during the training we again replace $Z$ with $\\hat { Z }$ by sampling two operations from the Gumbel-softmax and then rescaling the corresponding two operation weights to be summed to one with zero weights of the other operations on each edge, as shown in Algorithm 2. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Owing to our continuous relaxation based on the Gumbel-softmax with $\\tau$ -annealing, the number of sampled operations on each edge is naturally reduced from two to one during the training. As a result, the proposed differentiable NAS with stochastic operation sampling is able to support improved scalability in terms of solvable large search space with small computational cost. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Dataset The proposed scalable NAS (SCNAS) was evaluated on the three segmentation tasks of 3D MRI data, (1) brain tumor (484 labeled images, 3 classes), (2) heart (20 labeled images, 1 class), and (3) prostate (32 labeled images, 2 classes), from the Medical Segmentation Decathlon challenge (MSD, http://medicaldecathlon.com) where each task has different MRI sequences as well as different foreground classes, which is therefore suitable for evaluating the generalizability and transferability of the SCNAS. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Implementation Details We compared the SCNAS to the state-of-the-art architecture, 3D U-ResNet with the use of multiple segmentation maps (Kayalibay et al., 2017) and attention gates (Oktay et al., 2018), and the random architecture by random selection of edge-operations in each cell from the same architecture search space in the SCNAS. The set of operations $\\mathcal { O }$ on each edge in the SCNAS consists of the following eight operations: $3 \\times 3 \\times 3$ convolutions, depthwise separable dilated $3 \\times 3 \\times 3$ convolutions with rate 2, 3 and 4, $3 \\times 3 \\times 3$ max and average 3D pooling, identity (skip connection), and zero. Here, we used the LeakyReLU-Conv-InstanceNorm for convolutional operations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1, the whole network in the SCNAS is composed of 12 automatically designed cells, each of which has 4 nodes. This number of stacked cells is consistent with that of the 3D U-ResNet in terms of respective three times of downsampling and upsampling by a factor of 2. Here, all operations in the reduction cell in the SCNAS are of stride two while the expansion cells perform pre-upsampling for the inputs of the cell. Since the 3D U-ResNet in this evaluation was set to have 32 output channels in the first convolutional block, the number of output channels in the stem cell of the SCNAS was set to 32, and also similar to the 3D U-ResNet, the reduction and expansion cells in the SCNAS respectively double and halve the number of output channels of given inputs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In both of the 3D U-ResNet and the SCNAS, patch-based training and inference were carried out such that each image was randomly cropped to the region of nonzero values with the predefined resolution during the training, while in testing, the prediction results were obtained by combining patch-based inference results with 50 percent overlap. Similar to Isensee et al. (2018), the predefined resolution for the input patch was set to $1 2 8 \\times 1 2 8 \\times 1 2 8$ for the tasks of brain tumors and heart while for the prostate task, the length of the $z$ - axis was reduced to 24. Since even the same task provides 3D images with heterogeneous voxel spacings, the input images were first resized for all voxel spacings to be physically equal using the given meta-data, and then $z$ -normalization was separately applied to each input channel. Note that unlike Isensee et al. (2018), any heuristic pre-/post-processing techniques including data augmentation, network-cascade, and prediction-ensemble were not adopted in this evaluation to solely examine the effects by the use of NAS in designing the network architecture. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since the ground-truth labels for test images are not provided in the MSD dataset, the evaluation is conducted by 5-fold cross-validation (CV) on the training images with the average dice similarity coefficient (DSC) as the metric, and accordingly, we applied the well-known multi-class dice loss function (Milletari et al., 2016b; Isensee et al., 2018). With the ADAM optimizer, the 3D U-ResNet and the SCNAS models were trained for 300 epochs and 400 epochs, respectively, taking their convergences into consideration. The 3D U-ResNet was set with the batch size as 8, initial learning rate as 0.0001, and beta parameters for ADAM optimizer as (0.9, 0.999) while in the SCNAS, with the batch size 1, the initial learning rates / beta parameters were as set to be $0 . 0 2 5 \\mathrm { ~ / ~ } ( 0 . 1 , 0 . 0 0 1 )$ for training operation parameters $\\Theta$ and $0 . 0 0 3 / \\ ( 0 . 5 , 0 . 9 9 9 )$ for training architecture parameters $\\alpha$ . If a plateau for 20 epochs on the training loss was detected, the learning rate was reduced by a factor of 10. All experiments were conducted on V100 GPUs, and the implementation was done using PyTorch (Paszke et al., 2017). ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Architecture Transfer Since the heart and prostate tasks only have 20 and 32 labeled MRI images, respectively, the 3D U-ResNet as well as the SCNAS can be prone to overfitting on the training set and hence to resulting in performance degradation on the validation set. Therefore, we transferred the optimized architecture obtained from the brain tumor task, which has 484 labeled MRI images, by the SCNAS into these two tasks having scarce data and retrained only the operation parameters on each task, in order to demonstrate that the SCNAS produces a more generalizable neural architecture for the similar tasks of 3D MRI image segmentation. Here, the transferred architecture came from the first CV fold in the brain tumor task. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results Table 1 shows that the SCNAS produced better architectures than the (humandesigned) 3D U-ResNet as well as the randomly designed 3D U-Net in terms of the overall performances on all three tasks. Especially, on the heart and prostate segmentation tasks, the transferred architecture from the brain tumor task achieved significantly better generalization performances. Note that the obtained architectures by the SCNAS have been also shown that the number of neural operation parameters and the computational complexity for output prediction (in terms of FLOPs) were significantly reduced compared to the 3D UResNet. We observed the performance degradation of the 3D U-ResNet when the number of initial output channels was halved. It is also noted that most previous NAS approaches retrained the neural operation parameters after completing architecture optimization because of the utilization of a biased architecture during the training, while the SCNAS simultaneously optimized both of the architecture parameters and neural operation parameters with an unbiased architecture and thereby removed the requirement of retraining. We conjecture that Isensee et al. (2018) might be benefit from complicated pre-/post-procedures and thus obtained slightly better performances than the SCNAS. Some example images and the corresponding segmentation outputs are included in Appendix A, and the details of the optimized cell architectures by the SCNAS are presented in Appendix B. ",
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+    },
+    {
+        "type": "text",
+        "text": "5. Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, a complete NAS framework for automatically designing an architecture is proposed and demonstrated on the benchmark dataset of 3D medical image segmentation tasks. In the proposed framework, NAS is formulated as finding the optimal structure of four types of cells composing an encoder as well as a decoder, and both the architecture parameters and the neural operation parameters are learned by gradient descent in an end-to-end manner. We introduce a novel stochastic sampling algorithm which results in significant improvement in terms of the scalability suitable for handling high-resolution 3D medical images and also reduces the inconsistency of the train-time architecture against the final architecture, which leads to avoid the retraining of the operation parameters. Empirical evaluation demonstrates that the automatically optimized network via the proposed NAS outperforms the manually designed 3D U-Net. Moreover, the architecture learned from a task with the large number of training data is successfully transferred to different MRI segmentation tasks with the small number of data. ",
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+    {
+        "type": "table",
+        "img_path": "images/6068dfbe934b3304633a358cf5408d5b3767fce76ed10b1e8a98f23b3aba83df.jpg",
+        "table_caption": [
+            "Table 1: Mean DSC in Brain Tumor, Heart, and Prostate. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"4\">Brain Tumor</td></tr><tr><td>Model</td><td>GFLOPs, Params</td><td>Edema</td><td>Non-Enhancing</td><td>Enhancing</td><td>Average</td></tr><tr><td>3D U-ResNet</td><td>881, 7.6M</td><td>79.10 ± 1.80</td><td>58.38 ± 1.29</td><td>77.37 ± 2.76</td><td>71.61</td></tr><tr><td>Random Search</td><td>152, 2.7M</td><td>79.59 ± 1.28</td><td>57.97 ± 1.36</td><td>77.85 ± 1.35</td><td>71.80</td></tr><tr><td> SCNAS</td><td>129, 2.2M</td><td>79.42 ± 1.45</td><td>58.01 ± 1.46</td><td>78.68 ± 1.80</td><td>72.04</td></tr><tr><td></td><td></td><td>Heart</td><td colspan=\"3\">Prostate</td></tr><tr><td>Model</td><td>GFLOPs, Params</td><td>Left Atrium</td><td>Peripheral</td><td>Transitional</td><td>Average</td></tr><tr><td>3D U-ResNet</td><td>870-163, 7.6M</td><td>89.60 ± 2.35</td><td>48.37 ± 1.44</td><td>79.17 ± 4.30</td><td>63.77</td></tr><tr><td>Random Search</td><td>104-18,1.5M</td><td>89.14 ± 2.74</td><td>50.78 ± 1.22</td><td>79.58 ± 5.01</td><td>65.18</td></tr><tr><td> SCNAS</td><td>136-32, 3.0M</td><td>89.99 ± 1.32</td><td>49.70 ± 1.23</td><td>80.89 ± 3.19</td><td>65.30</td></tr><tr><td> SCNAS(transfer)</td><td>193-37, 4.2M</td><td>90.47 ± 1.70</td><td>53.81 ± 1.30</td><td>82.02 ± 4.52</td><td>67.92</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "The analysis using more candidate operations and different cell structures of the same type at different layer levels are left for future research. In addition, the effects of including non-architectural procedures such as data augmentation, network-cascade, and predictionensemble in the proposed NAS framework need to be analyzed in future works. Another interesting research direction would be applying the NAS framework, either directly or by architecture transfer, to other medical modalities including CT, mammography, and X-ray. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+        "text": "We thank the Kakao Brain Cloud team for supporting to efficiently use GPU clusters for large-scale experiments. ",
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+        "image_caption": [
+            "Figure 2: (a) Raw data. (b) Ground Truth. Segmentation results of (c) 3D U-ResNet (d) SCNAS in Brain Tumor (top), Heart (middle), and Prostate (bottom) from the MSD dataset. The prediction results on the heart and prostate data are obtained from transferred networks which are trained on brain tumor data. "
+        ],
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+        "bbox": [
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+        "type": "text",
+        "text": "In this appendix, we provide examples of segmentation predictions by the 3D U-ResNet and proposed method. For a qualitative assessment, we compare the two results with ground truth. Additionally, samples of cell architectures found by the SCNAS are illustrated in the sequent section. ",
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+        "text": "Appendix A. Segmentation Samples ",
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+        "type": "text",
+        "text": "Figure 2 shows samples of 3D segmentation results of 3D U-ResNet and proposed method for each MSD dataset. The above samples demonstrate that the architectures found by SCNAS predict more accurately than 3D U-ResNet. Especially, the architecture found on brain tumor images can be transferred well to the dataset of different MRI segmentation tasks. ",
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+    {
+        "type": "image",
+        "img_path": "images/329bb26a81a71c724e5aae63eb86f409757a5aa0fbb5375b3898f613965e1ef9.jpg",
+        "image_caption": [
+            "Figure 3: Reduction Cell "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            126,
+            826,
+            186
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/835db0a8704c4c9150510ff949d67e08ad0ecc12843b1b9338e4cde3799f78f9.jpg",
+        "image_caption": [
+            "Figure 4: Encoder Normal Cell "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            217,
+            270,
+            779,
+            426
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Appendix B. Cell Architectures ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            503,
+            455,
+            522
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 3-6 show the samples of cell architectures which SCNAS found in the first CV fold experiment on the Brain Tumor dataset at the convergence. ",
+        "bbox": [
+            143,
+            531,
+            854,
+            565
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/3c71ee83dfd54e78802e62d46ea7d731012a111e0370ab298f981b2664d5d601.jpg",
+        "image_caption": [
+            "Figure 5: Expansion Cell "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            227,
+            784,
+            342
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b42a9d4338c8e4b5e0f899fb2e590c3f9c4456fa2de160717b4c947b03cc1ab7.jpg",
+        "image_caption": [
+            "Figure 6: Decoder Normal Cell "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            267,
+            614,
+            730,
+            742
+        ],
+        "page_idx": 15
+    }
+]

parse/train/SkxwY3NtPr/SkxwY3NtPr.md

@@ -0,0 +1,430 @@
+# TOWARDS A UNIFIED EVALUATION OF EXPLANATION METHODS WITHOUT GROUND TRUTH
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper proposes a set of criteria to evaluate the objectiveness of explanation methods of neural networks, which is crucial for the development of explainable AI, but it also presents significant challenges. The core challenge is that people usually cannot obtain ground-truth explanations of the neural network. To this end, we design four metrics to evaluate the explanation result without ground-truth explanations. Our metrics can be broadly applied to nine benchmark methods of interpreting neural networks, which provides new insights of explanation methods.
+
+# 1 INTRODUCTION
+
+Nowadays, many methods are proposed to explain the logic of a deep neural network (DNN) in a post-hoc manner. In this research, we limit our attention to existing methods of estimating the importance/attribution/saliency of input pixels or intermediate-layer neural units w.r.t. the network output (Shrikumar et al., 2016; Lundberg & Lee, 2017; Ribeiro et al., 2016; Binder et al., 2016), which present the mainstream of explaining neural networks. To avoid ambiguity, the estimated importance/saliency/attribution maps are all termed “attribution maps” in this paper.
+
+However, some methods usually pursue attribution maps which look reasonable from the perspective of human users, instead of objectively reflecting the true logic of information processing in the DNN. A trustworthy evaluation of the objectiveness of attribution maps is crucial for the development of deep learning and proposes significant challenges to state-of-the-art algorithms.
+
+Existing metrics (Cui et al., 2019; Arras et al., 2019; Vu et al., 2019; Yang & Kim, 2019; Kim et al., 2017; Adebayo et al., 2018; Ghorbani et al., 2019; Alvarez-Melis & Jaakkola, 2018) of evaluating explanation methods have certain shortcomings.
+
+Issue 1, evaluation of the accuracy of a $\mathbf { D N N } \neq$ evaluation of the objectiveness of attribution maps: Some methods only evaluate whether a DNN encodes a correct logic, instead of examining whether an attribution map objectively reflects the true logic of a DNN. (Cui et al., 2019) used human cognition to evaluate the explanation result. (Yang & Kim, 2019; Kim et al., 2017) aimed to construct a specific dataset with ground-truth explanations for evaluation. For example, they added an irrelevant object into the image. Pixels from the irrelevant object are expected to be assigned with zero attributions.
+
+However, strictly speaking, it is impossible to religiously annotate ground-truth explanations for a DNN. Currently, the ground-truth explanation is constructed under the assumption that a DNN cannot learn irrelevant objects for classification with the purpose of evaluating the logic of the DNN, instead of examining whether an explanation method mistakenly generates attribution maps with seemingly correct logic for an incorrectly learned DNN.
+
+Issue 2, broad applicability: We aim to design an evaluation metric that can be broadly applied to various tasks. In aforementioned methods (Yang & Kim, 2019; Kim et al., 2017), the requirement for constructing a new testing dataset limits the applicability of the evaluation.
+
+Issue 3, quantification of the objectiveness: Some methods quantitatively evaluate the accuracy and robustness of attribution maps. However, there is no strict mechanism to ensure the objectiveness of each numerical value in the attribution map. I.e., if the attribution value of a pixel is twice of that of another pixel, then the first pixel is supposed to contribute twice numerical values to the prediction w.r.t. the second pixel.
+
+Table 1: Review of explanation methods   
+
+<table><tr><td>Method</td><td>What to explain</td><td>Quantitative evaluation of limitations in ap- plication</td></tr><tr><td>CAM (Zhou et al., 2016)</td><td>Attribution 1distribu- tion at intermediate layer</td><td>1.Requirement for global average pooling. 2.Usually explain features at high layers</td></tr><tr><td>Grad_CAM (Selvaraju et al., 2017)</td><td>Attribution 1distribu- tion at intermediate layer</td><td>Usually explain features at high layers</td></tr><tr><td>Grad (Simonyan et al., 2013)</td><td>Pixel-wise attribution</td><td></td></tr><tr><td>GI (Shrikumar et al., 2016)</td><td>Pixel-wise attribution</td><td></td></tr><tr><td>GB (Springenberg et al., 2014)</td><td>Pixel-wise attribution</td><td>Requirement for using ReLU as non-linear layers</td></tr><tr><td>Shapley Value (Shapley,1953) DeepSHAP</td><td>Pixel-wise attribution</td><td>NP-complete problem</td></tr><tr><td>(Lundberg &amp; Lee,2017)</td><td>Pixel-wise attribution</td><td>Similar to LRP,DeepLIFT(Shrikumar et al., 2016) with a designed backward rule</td></tr><tr><td>LIME (Ribeiro et al.,2016)</td><td>Pixel-wise attribution</td><td>Attribution maps at the super-pixel level, rather than at the pixel level</td></tr><tr><td>LRP (Binder et al.,2016)</td><td>Pixel-wise attribution</td><td>Relevance propagation rules of every layer should be defined</td></tr><tr><td>Pert (Fong&amp; Vedaldi,2017)</td><td>Pixel-wise attribution</td><td></td></tr></table>
+
+Except for the objectiveness, previous studies mainly conducted the evaluation from other perspectives. (Arras et al., 2019; Vu et al., 2019) evaluated attribution maps from the perspective of adversarial attacks by adding random noise. (Adebayo et al., 2018; Ghorbani et al., 2019; Alvarez-Melis & Jaakkola, 2018) proposed methods to evaluate the robustness of explanation methods w.r.t. the perturbation. (Adebayo et al., 2018) randomized the layer of DNN from the top to the bottom and visualized the change of attribution maps.
+
+Note that in most applications, people cannot faithfully obtain the ground-truth logic of a DNN. Therefore, considering the above three issues, in this study, we aim to fairly evaluate the objectiveness and robustness of attribution maps from the following four perspectives without ground-truth explanations.
+
+Perspective 1, bias of the attribution map at the pixel level: In order to evaluate the bias of the attribution map, we first need to propose a standard metric to evaluate the accuracy of explanation methods. The Shapley value is the unique solution to model the attribution value of each pixel that satisfies desirable properties including efficiency, symmetry and monotonicity (Lundberg & Lee, 2017). However, the computation of the Shapley value is an NP-complete problem, and previous studies (Lundberg & Lee, 2017) showed that the accurate estimation of the Shapley value is still a significant challenge. To this end, we extend the theory of Shapley sampling (Castro et al., 2009) and design a new evaluation metric, which achieves high accuracy without significantly boosting the computational cost.
+
+We use the new evaluation metric to quantify the bias of the attribution map. Note that this evaluation has no partiality to the Shapley-value-based explanation methods. For example, experimental results showed that LRP (Binder et al., 2016) exhibited significant lower bias than DeepSHAP (Lundberg & Lee, 2017).
+
+Perspective 2, quantification of unexplainable feature components: Given an input image and its attribution map, we revise the input image to generate a new image that reflects the logic of the attribution map. We then compare the intermediate-layer feature of the original image with that of the generated image, so as to disentangle feature components that can and cannot be explained by the attribution map.
+
+Perspective 3, robustness of the explanation: Robustness of the explanation means whether the attribution map is robust to spatial masking of the input image. When we randomly mask a certain region of the input image, we admit that spatial masking destroys global contexts and affects pixelwise attribution value to some extent. The quantification of the robustness of the explanation is an important perspective of evaluating an explanation method.
+
+Perspective 4, mutual verification: The mutual verification means whether different explanation methods can verify each other. Methods generating similar attribution maps are usually believed more reliable.
+
+In this paper, we used our metrics to evaluate nine widely used explanation methods listed in Table 1. We conducted experiments using the LeNet, VGG and ResNet on different benchmark datasets including the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset and the Pascal VOC 2012 (Everingham et al., 2010) dataset. Our experimental results proved the effectiveness of the proposed evaluation methods and provided an insightful understanding of various explanation methods.
+
+The contribution of our work can be concluded as follows.   
+1. In this study, we invent a set of standard metrics to evaluate the objectiveness and the robustness of the attribution map without knowing ground-truth explanations.   
+2. The metric of evaluating the pixel-wise bias of the attribution map can be estimated with a relatively low computational cost, which avoids falling into the computational bottleneck of estimating accurate pixel-wise attributions.   
+3. Since our metrics do not need any annotations of ground-truth explanations, our metrics can be applied to different neural networks trained on different datasets.
+
+# 2 RELATED WORK
+
+Explainable AI is an emerging direction in artificial intelligence, and different explanation methods have been proposed. In this section, we briefly review the Shapley value and limit our discussions to existing methods of evaluating methods of extracting attribution/importance/saliency maps to simplify the story. Appendix C discusses other research directions of explainable AI.
+
+The Shapley value: The Shapley value (Shapley, 1953) was proposed to compute the attribution distribution over all players in a particular cooperative game. However, it is an NP-complete problem to compute the accurate Shapley value. The Shapley value approximated by sampling strategy could be very inaccurate due to the high variance. We extend the theory of the Shapley value to obtain an evaluation metric with a high accuracy but a low computational cost.
+
+Qualitative evaluation: Some studies used a qualitative criterion for evaluation. (Cui et al., 2019) qualitatively defined basic concepts in the evaluation of explanation results, including the complexity of the explanation, the correlation, and the completeness. In contrast, this paper aims to evaluate the methods quantitatively, which makes our metrics more objective and reliable.
+
+Accuracy evaluation: To evaluate the accuracy of attribution maps, (Arras et al., 2019; Vu et al., 2019) used the noise/occlusion to perturb the original image according to the attribution value. There was no mechanism to ensure the prediction result objectively reflected the logic of a DNN. (Yang & Kim, 2019; Kim et al., 2017) built a dataset to help them generate ground-truth explanations. Essentially, these methods tried to obtain the “correct” logic for an input image. However, a rigorous study should not assume that the DNN encodes the correct logic. As a result, this paper proposes to evaluate the objectiveness of explanation results without knowing ground-truth explanations.
+
+Stability evaluation: (Adebayo et al., 2018; Ghorbani et al., 2019; Alvarez-Melis & Jaakkola, 2018) mainly paid attention to the attribution map change when the model input was perturbed. (Adebayo et al., 2018) visualized the change in the attribution map when the weights of the model were destroyed from the top to the bottom. (Ghorbani et al., 2019; Alvarez-Melis & Jaakkola, 2018) used the adversarial image to alter the attribution map. In comparison, we propose a metric to evaluate the robustness to spatial masking.
+
+# 3 ALGORITHM
+
+# 3.1 PRELIMINARIES: THE SHAPLEY VALUE
+
+The Shapley value measures the instancewise feature importance ranking problem. Let $\Omega$ be the set of all pixels of an image $I$ . $I _ { P }$ denotes an image that replaces all pixels in set $\Omega \setminus P$ with average pixel value over images. $F ( I _ { P } )$ denotes the scalar output of a DNN based on a subset of pixels ${ \hat { P } } \subset { \Omega }$ . To compute the Shapley value of the $i$ -th feature, (Shapley, 1953) considered all subsets of $\Omega$ not containing the $i$ -th feature and defined the Shapley value $\mathbf { \bar { A } } _ { i } ^ { * }$ as follows:
+
+$$
+A _ { i } ^ { * } = \sum _ { P \subset \Omega \backslash \{ i \} } \frac { | P | ! ( | \Omega | - | P | - 1 ) ! } { | P | ! } \left[ F ( I _ { P \cup \{ i \} } ) - F ( I _ { P } ) \right]
+$$
+
+It is the unique solution that satisfies desirable properties to assign attribution value to each feature dimension in the input (Chen et al., 2018). Appendix A shows properties of the Shapley value.
+
+# 3.2 EVALUATING THE BIAS OF THE ATTRIBUTION MAP AT THE PIXEL LEVEL
+
+In this section, we design a metric to accurately evaluate the objectiveness of the attribution map. Given an image $I \in \mathbf { I }$ , let us consider the DNN $F$ with a single scalar output $y = F ( I )$ . For DNNs with multiple outputs, existing methods usually explain each individual output dimension independently. Let $\left\{ { a } _ { i } \right\}$ denote the pixel-wise attribution map estimated by a specific explanation method. We aim to evaluate the bias of $\left\{ { a } _ { i } \right\}$ . People usually formulate the network output as the sum of pixel-wise attribution value, i.e. the output $y$ can be decomposed as follows.
+
+$$
+y = b + \sum _ { i \in \Omega } A _ { i } , \qquad \mathrm { s . t . } \quad A _ { i } = \lambda a _ { i }
+$$
+
+$b$ denotes the bias; $i$ denotes the index of each pixel in the input image; $\Omega$ denotes the set of all pixels in the image. Aforementioned $\{ A _ { i } ^ { * } \}$ can be considered as the ground-truth of $\left\{ A _ { i } \right\}$ (Lundberg & Lee, 2017). Since many explanation methods (Selvaraju et al., 2017; Simonyan et al., 2013) mainly compute relative values of attributions $\left\{ { a } _ { i } \right\}$ , instead of a strict attribution map $\left\{ A _ { i } \right\}$ . We use $\lambda$ to bridge $\left\{ A _ { i } \right\}$ and $\left\{ { a } _ { i } \right\}$ . $\lambda$ is a constant for normalization, which can be eliminated during the implementation of the evaluation.
+
+The estimated attribution of each pixel can be assumed to follow a Gaussian distribution $A _ { i } \sim$ $\mathcal { N } ( \mu _ { i } , \sigma _ { i } ^ { 2 } )$ (Castro et al., 2009). Attribution distributions of different pixels can be further assumed to share a unified variance, i.e. $\sigma _ { 1 } ^ { 2 } \approx \sigma _ { 2 } ^ { 2 } \approx . . . \approx \sigma _ { n } ^ { 2 }$ . The evaluation of the attribution distribution $\left\{ { a } _ { i } \right\}$ has two aspects, i.e.
+
+1. the sampling of pixels whose attributions that are more likely to have large deviations;   
+2. the evaluation of the bias of the sampled attributions.
+
+First, for the sampling of attributions of interest, we sample the set of pixels $S$ with top-ranked high (or low) attributions. Attribution values of pixels in $S$ are sampled as those with the highest (or the lowest) values, and these pixels are supposed to be more likely to be significantly biased towards high (or low) attribution values. Meanwhile, from another perspective, the distribution of the sampled attribution values is close to the Gumbel distribution.
+
+Second, although the Shapley value can be considered as a standard formulation of the pixel-wise attribution, it usually cannot be accurately computed because of its high computational cost. In order to accurately evaluate the sampled attribution values without significantly increasing the computational cost, we applied the Shapley value approximated by the sampling method. Just like the target attribution distribution $A _ { i }$ , the approximated Shapley value $A _ { i } ^ { s h a p }$ is assumed to follow a normal distribution $\mathcal { N } ( A _ { i } ^ { * } , ( \sigma ^ { s h a p } ) ^ { 2 } )$ . $A _ { i } ^ { s h a p }$ is an unbiased approximation of the true Shapley value $A _ { i } ^ { * }$ , Thus, the average value over different pixels in $S$ satisfies $\begin{array} { r } { \frac { \sum _ { i \in S } A _ { i } ^ { s h a p } } { | S | } \sim \mathcal { N } ( \frac { \sum _ { i \in S } A _ { i } ^ { * } } { | S | } , \frac { ( \sigma ^ { s h a p } ) ^ { 2 } } { | S | } ) } \end{array}$ .
+
+We can prove that the measurement of the average attribution among all sampled pixel s Pi∈S Ashapi is of much higher accuracy than the raw Shapley value with the same computational cost. The difference between the highest (or lowest) values and its true values $\begin{array} { r } { \left| \frac { \sum _ { i \in S } A _ { i } ^ { s h a p } } { | S | \| A ^ { s h a p } \| } - \frac { \sum _ { i \in S } A _ { i } } { | S | \| A \| } \right| } \end{array}$ can reflect the system bias, as follows.
+
+$$
+\begin{array} { r } { M _ { \mathrm { p i x e l } } = \mathbb { E } _ { I } \left[ \left| \displaystyle \frac { \sum _ { i \in S } A _ { i } ^ { s h a p } } { \left| S \right| \left| \left| A ^ { s h a p } \right| \right| } - \displaystyle \frac { \sum _ { i \in S } A _ { i } } { \left| S \right| \left| A \right| } \right| \right] = \mathbb { E } _ { I } \left[ \displaystyle \frac { 1 } { \left| S \right| } \displaystyle \lvert \frac { \sum _ { i \in S } A _ { i } ^ { s h a p } } { \left| \left| A ^ { s h a p } \right| \right| } - \displaystyle \frac { \sum _ { i \in S } a _ { i } } { \left| \left| a \right| \right]}  \right] } \\ { \mathrm { s . t . } \qquad \forall i \in S , j \in \Omega \setminus S , a _ { i } \geq a _ { j } \quad \mathrm { o r } \quad \forall i \in S , j \in \Omega \setminus S , a _ { i } \leq a _ { j } } \end{array}
+$$
+
+where $\Omega$ is the set of all pixels in an image. $\| A ^ { s h a p } \|$ and $\| a \|$ are used for normalization. A small value of $M _ { \mathrm { p i x e l } }$ indicates the low bias of the attribution map.
+
+Analysis of the high computational efficiency: Suppose that the computational complexity of processing one sample is $\mathcal O ( N )$ , then the computational complexity of sampling $m$ times is $\mathcal { O } ( m N )$ .
+
+If the raw Shapley value needs to obtain the same accuracy, it needs significantly more samples, and the computational complexity is $\mathcal { O } ( \vert S \vert m N )$ . Please see Appendix B for the theoretical analysis of the save of computational cost.
+
+In addition, the proposed metric can also be used to evaluate the attribution of neural activations in the intermediate layer, such as those generated by Grad-CAM. In this case, we can regard the target intermediate-layer feature as the input image to compute attributions, so as to implement the evaluation. For each image, we need to sample multiple times to increase the accuracy. We compute the average performance over different images for evaluation. We need to sample multiple times with different images to increase the accuracy of the evaluation. Note that although the metric is designed based on the Shapley value, experimental results showed that LRP outperforms DeepSHAP.
+
+# 3.3 QUANTIFICATION OF UNEXPLAINABLE FEATURE COMPONENTS
+
+We propose another metric to quantify unexplainable feature components. Given an image $I$ and its attribution map $\{ a _ { i } \}$ , we generate a new image, which reflects the logic of the attribution map. In this way, we can consider the feature of the newly generated image $\widetilde { f }$ as feature components that can be explained. Let $f$ denote the feature of the original image $I$ . Then, $f - \tilde { f }$ corresponds to the unexplainable feature components.
+
+To generate the new image, we mask specific pixels in the original image $I$ , which have the lowest attributions. We select and mask a set of pixels $S$ with the lowest absolute attributions, and the number of the selected points is determined subjects to $\begin{array} { r } { \sum _ { i \in S } \left| a _ { i } \right| = 0 . 1 \sum _ { i \in \Omega } \left| a _ { i } \right| } \end{array}$ to generate the new image $\widetilde { I }$ . The metric is formulated as
+
+$$
+M _ { \mathrm { f e a t u r e } } = \alpha \mathbb { E } _ { I } \left[ \lVert \widetilde { f } _ { I } - f _ { I } \rVert \right]
+$$
+
+where $\begin{array} { r } { \alpha = \frac { 1 } { \mathbb { E } _ { I ^ { \prime } } [ \parallel f _ { I ^ { \prime } } - \mathbb { E } _ { I ^ { \prime \prime } } [ f _ { I ^ { \prime \prime } } ] \parallel ] } } \end{array}$ is used for normalization. A small value of $M _ { \mathrm { f e a t u r e } }$ indicates most feature components in $f$ are explainable.
+
+# 3.4 EVALUATING THE ROBUSTNESS OF EXPLANATION
+
+This metric is used to measure the robustness of explanation methods to the spatial masking. We believe that the method, which is robust to spatial masking, can be considered more convincing. The robustness is an important perspective of evaluating explanation methods.
+
+Given an input image $I \in \mathbf { I }$ and its attribution map $\{ a _ { i } \} ~ w . r . t$ a DNN, we use a mask $M$ to cover specific parts of the image to get a masked image $\hat { I }$ . For each input image $I$ , we can generate four masked images by masking the right, left, top, and bottom half of the image, respectively. For each masked image $\hat { I }$ , the explanation method estimates the attribution $\hat { a } _ { i }$ for each pixel. We compare pixel-level attributions of the unmask pixels between original images and masked images, as follows.
+
+$$
+M _ { \mathrm { n o n - r o b u s t } } = \mathbb { E } _ { I } \left[ \frac { 1 } { \lVert a \rVert } \sqrt { \sum _ { i \in I \backslash I _ { m a s k } } ( a _ { i } - \hat { a } _ { i } ) ^ { 2 } } \right]
+$$
+
+We used $\| a \|$ for normalization, and a large value of $M _ { \mathrm { n o n - r o b u s t } }$ indicates a high non-robustness.
+
+# 3.5 EVALUATING THE MUTUAL VERIFICATION
+
+This metric aims to quantitatively measure the mutual verification between different explanation methods. It is usually believed that two explanation methods are more reliable if they can verify each other. Given a DNN $F$ and an image $I \in \textbf { I }$ , two different explanation methods $\alpha$ and $\beta$ produce attribution maps $a _ { \alpha }$ and $a _ { \beta }$ , respectively. We measure their difference as follows.
+
+$$
+M _ { \mathrm { m u t u a l } } = \mathbb { E } _ { I } \left[ \Vert \frac { a _ { \alpha } } { \Vert a _ { \alpha } \Vert } - \frac { a _ { \beta } } { \Vert a _ { \beta } \Vert } \Vert \right]
+$$
+
+Attribution maps from different explanation methods are normalized by their L2-norm. A lower value of $M _ { \mathrm { m u t u a l } }$ indicates a more convincing mutual verification between explanation methods $\alpha$ and $\beta$ .
+
+![](images/939b1586227e75090f34f1aafa4ce66bffd382787e4fe5f83a6b344c1410b84b.jpg)  
+Figure 1: Example of attribution maps of different methods.
+
+# 4 EXPERIMENT
+
+To evaluate explanation methods, we conducted experiments on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset and the Pascal VOC 2012 (Everingham et al., 2010) dataset. For images in the Pascal VOC 2012 dataset, we cropped objects using their bounding boxes and used the cropped objects as inputs to train DNNs for object classification. We trained and explained LeNet (LeCun et al., 1998), ResNet-20/32/44/56 (He et al., 2016) using the CIFAR-10 dataset. AlexNet (Krizhevsky et al., 2012), VGG-16/19 (Simonyan & Zisserman, 2015), ResNet-50/101 (He et al., 2016) were trained using the Pascal VOC 2012 dataset.
+
+# 4.1 BASELINE
+
+In our experiments, we mainly evaluated the following explanation methods. Figure 1 shows attribution maps yielded by these explanation methods. Appendix D provides more attribution maps.
+
+Grad: Given an input, (Simonyan et al., 2013) quantified the attribution value with the gradient of the input. We termed this algorithm as Grad. For RGB images with multiple channels, Grad selected the maximum magnitude across all channels for each pixel.
+
+GI: (Shrikumar et al., 2016) proposed a method, namely GI, which used the pixel-wise product of the input and its gradient as attribution value. Attribution values for RGB channels were summed to get the final attribution value.
+
+GB: Guided Back-propagation, namely GB, corresponded to Grad where the back-propagation rule at ReLU units was redefined (Springenberg et al., 2014).
+
+LRP: Layer-wise relevance propagation (LRP) (Binder et al., 2016) redefined back-propagation rules for each layer to decompose the output of a DNN over the input. We used LRP- $\cdot \epsilon$ and set the parameter $\epsilon = 1$ in experiments.
+
+DeepSHAP: DeepSHAP adapted DeepLIFT (Shrikumar et al., 2016) to approximate pixel-wise Shapley values for the input image (Lundberg & Lee, 2017). We applied the code released by (Lundberg & Lee, 2017).
+
+LIME: LIME (Ribeiro et al., 2016) trained an interpretable model to compute the attribute value for each super-pixel. We used the code released by (Ribeiro et al., 2016).
+
+Pert: (Fong & Vedaldi, 2017) explained a prediction by training a mask to perturb the input image. Mask values ranging between 0 and 1 indicated the saliency of each pixel. We termed this method Pert.
+
+CAM: CAM computed attribution map over the feature from the last convolutional layer (Zhou et al., 2016). It required the special structure with a global average pooling layer and a fully connected layer at the end of the DNN.
+
+Grad CAM: Grad CAM was similar to CAM (Selvaraju et al., 2017). Grad CAM used gradients over the feature map, instead of the parameters of the fully connected layer.
+
+# 4.2 IMPLEMENTATION DETAILS
+
+Bias of the attribution map at the pixel level: To approximate the Shapley value for each image, we sampled 1000 times for each image in the CIFAR-10 dataset and sampled 100 times for each image in the Pascal VOC 2012 dataset. We sampled the top- $10 \%$ , $30 \%$ , $50 \%$ , $70 \%$ , $90 \%$ pixels with the highest/lowest values.
+
+![](images/6b88e11eef2fa57cd8d3b5235c119022a63d9a1e445f9ba7e0924225eab2a94e.jpg)  
+Figure 2: Results of the bias of the attribution map at the pixel level. Row (I) and row (II) used trained LeNet, ResNet20, ResNet32, ResNet44, ResNet56 on the CIFAR-10 dataset from left to right; row (III) and row (IV) used trained ResNet50, ResNet101, VGG16, VGG19, AlexNet on the Pascal VOC 2012 dataset from left to right. Row (I) and row (III) sampled pixels with the highest attribution values; row (II) and row (IV) sampled the pixels with the lowest attribution values.
+
+Table 2: Quantification of unexplainable feature components.   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td><td>Pert</td></tr><tr><td>CIFAR10-LeNet</td><td>0.96821</td><td>1.10399</td><td>0.91546</td><td>1.10371</td><td>0.67032</td><td>1.08469</td><td>0.70177</td></tr><tr><td>CIFAR10-ResNet20</td><td>1.22212</td><td>1.28065</td><td>1.14596</td><td>1.26002</td><td>1.05286</td><td>=</td><td>1.13032</td></tr><tr><td>CIFAR10-ResNet32</td><td>1.22803</td><td>1.29324</td><td>1.16028</td><td>1.27681</td><td>1.04412</td><td>=</td><td>1.14261</td></tr><tr><td>CIFAR10-ResNet44</td><td>1.22382</td><td>1.26434</td><td>1.10411</td><td>1.24961</td><td>1.02039</td><td></td><td>1.10772</td></tr><tr><td>CIFAR10-ResNet56</td><td>1.22487</td><td>1.24123</td><td>1.11306</td><td>1.24146</td><td>1.00916</td><td>=</td><td>1.10122</td></tr><tr><td>VOC2012-AlexNet</td><td>1.02425</td><td>1.09375</td><td>1.04981</td><td>1.0809</td><td>1.01942</td><td>1.11513</td><td>1.04048</td></tr><tr><td>VOC2012-VGG16</td><td>1.18959</td><td>1.22715</td><td>1.22724</td><td>1.32364</td><td>1.12339</td><td>1.32674</td><td>1.12878</td></tr><tr><td>VOC2012-VGG19</td><td>1.08084</td><td>1.1226</td><td>1.10411</td><td>1.17121</td><td>1.08387</td><td>1.23784</td><td>1.13043</td></tr><tr><td>VOC2012-ResNet50</td><td>1.23345</td><td>1.24123</td><td>1.2156</td><td>1.23441</td><td>1.22411</td><td>-</td><td>1.25732</td></tr><tr><td>VOC2012-ResNet101</td><td>1.08592</td><td>1.09363</td><td>1.07853</td><td>1.07856</td><td>1.11289</td><td>=</td><td>1.23279</td></tr></table>
+
+Quantification of unexplainable feature components: Given an image, we masked the pixels with the lowest absolute attribution value. The number of the masked pixels was determined to ensure that the sum of masked absolute attribution value took $10 \%$ of the total absolute attribution value. On average, around $30 \%$ pixels were masked. The masked pixels were assigned with the average pixel value over images. We used features of the last convolutional layer to compute $M _ { f e a t u r e }$ .
+
+# 4.3 EXPERIMENT RESULT AND ANALYSIS
+
+Bias of the attribution map at the pixel level: Figure 2 shows curves of evaluation results on different models learned using different datasets. According to these curves, GI and LIME provided the least biased attribution maps for ResNet at the pixel level. For AlexNet, VGG-16/19 and LeNet, LRP outperformed other methods. Detailed numbers corresponding curves in Figure 2 are listed in the Appendix E.
+
+Some methods could not be evaluated using the bias at the pixel level. For example, Pert computed an importance map without negative values instead of an attribution map for each image. The code of CAM (Zhou et al., 2016) projected attribution values to the range between 0 and 1. Grad CAM and LRP were not used on residual networks, because there was only one fully connected layer behind the last convolutional layer in residual networks. In this case, Grad CAM could not diagnose the logic contained in the cascaded non-liner layers of the DNN. For LRP, the relevance propagation rules of some structures in ResNet were not defined to the best of our knowledge.
+
+Quantification of unexplainable feature components: Table 2 compares the amount of unexplainable feature components between explanation methods. We found that LIME, GB and Pert explained more feature components than other methods. We noticed that the quantification of unexplainable feature components of most explanation methods were considerable larger than expected. It was because the attribution maps from some methods contained relatively larger noise. Thus, the masked pixels were almost uniformly distributed over images, which destroyed the context information and led to worse results.
+
+![](images/cd700db41983ae1d904c06b910fd90afa9ccc9d541f4f2614149abe8e83321d0.jpg)  
+Figure 3: Heat maps of mutual verification. A low value of $M _ { m u t u a l }$ between two methods indicates a more convincing mutual verification between them.
+
+We did not evaluate CAM and Grad CAM, because they calculated attribution maps at the feature level, which were not comparable with attribution values at the pixel level.
+
+Robustness of explanation: Table 3 shows the quantitative results of $M _ { \mathrm { n o n - r o b u s t } }$ on different models trained using different datasets. We found that GB and Grad CAM exhibited a lower non-robustness to spatial masking. For more results, Appendix F shows examples of attribution maps of masked images.
+
+Mutual verification: Figure 3 visualizes the mutual verification $M _ { \mathrm { m u t u a l } }$ between different explanation methods, which indicates a high level mutual verification between LRP, GI and DeepSHAP. Appendix G provides more detailed results. Note that we did not compare CAM and Grad CAM with other methods. It was because they computed attribution maps on intermediate-layer features.
+
+# 5 CONCLUSION
+
+In this paper, we have proposed four metrics to evaluate explanation methods from four different perspectives. The proposed evaluation metrics are computed without requirements for ground-truth explanations. Our metrics can be broadly applied to different methods, w.r.t. DNNs learned using different datasets. These metrics evaluate the bias of the attribution map at the pixel level, quantify the unexplainable feature components, the robustness of the explanation and the mutual verification. In experiments, we used our metrics to evaluate nine widely used explanation methods. Experimental results showed that attribution maps from LRP, GI and LIME exhibited lower bias at the pixel level. LIME and GB explained more feature components than other methods. Regarding the robustness, GB, CAM and Grad CAM were more robust to spatial masking than other explanation methods. DeepSHAP, GI and LRP can better verified each other.
+
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+
+# A PROPERTIES OF THE SHAPLEY VALUE
+
+Let $I$ denote the input image; let $\Omega$ denote the set of all pixels in $I$ . We can use $I _ { \emptyset }$ to denote a baseline image, i.e. all pixels in $I _ { \emptyset }$ equal to the average value over all images. For a subset $S \subset \Omega$ , $I _ { S }$ denotes an image that satisfies
+
+$$
+( I _ { S } ) _ { i } = \left\{ { \left( I \right) _ { i } , \ \mathrm { \Omega } _ { i } \in S } \right.
+$$
+
+where $i$ is the index of the pixel in $I$ and $I _ { \Omega }$ is the same image as $I$ . Let $F$ and $G$ denote two models with scalar output. The Shapley value of the $i$ -th pixel is represented by $A _ { i } ^ { * }$ , and they have the following properties.
+
+Efficiency: The sum of Shapley values $\textstyle \sum _ { i \in \Omega } A _ { i } ^ { * }$ is equal to $F ( I _ { \Omega } ) - F ( I _ { \emptyset } )$ .
+
+Symmetry: The features that are treated equally by the model are treated equally by the Shapley value. If $F ( I _ { S \cup \{ i \} } ) = F ( I _ { S \cup \{ j \} } )$ for all subsets S, then $A _ { i } ^ { * } = A _ { j } ^ { * }$ .
+
+Additivity: For any two models $F$ and $G$ , if they are combined into one model $F + G$ , the Shapley value must be added pixel by pixel: $( A ^ { * } ) _ { i } ^ { F + G } = \stackrel { \cdot } { = } ( A ^ { * } ) _ { i } ^ { F } + ( A ^ { * } ) _ { i } ^ { G }$ .
+
+Monotonicity: For any two models $F$ and $G$ , if for all subsets $S$ we have $F ( I _ { S \cup \{ i \} } ) - F ( I _ { S } ) \geq$ $G ( I _ { S \cup \{ i \} } ) - G ( I _ { S } )$ for all subsets S, then we have $( A ^ { * } ) _ { i } ^ { F } \geq ( A ^ { * } ) _ { i } ^ { G }$ .
+
+# B ANALYSIS OF THE COMPUTATIONAL COST
+
+In this section, we continue using the notation in Section 3.1 and Section 3.2. Suppose that we sample m times to approximate the Shapley value. The variance of Ashapi is σ 2 $\frac { \sigma ^ { 2 } } { m }$ where $\sigma ^ { 2 }$ satisfies (Castro et al., 2009)
+
+$$
+\sigma ^ { 2 } = \sum _ { P \subset \Omega \setminus \{ i \} } \frac { | P | ! ( | \Omega | - | P | - 1 ) ! } { | P | ! } \left[ F ( I _ { P \cup \{ i \} } ) - F ( I _ { P } ) - A _ { i } ^ { * } \right] ^ { 2 }
+$$
+
+So we have $( \sigma ^ { s h a p } ) ^ { 2 } = \sigma ^ { 2 } / m$ . For the set of sampled pixels $S$ , the variance of their average Shapley value is $\begin{array} { r } { \frac { | S | ( \sigma ^ { s h a p } ) ^ { 2 } } { | S | ^ { 2 } } = \frac { ( \sigma ^ { s h a p } ) ^ { 2 } } { | S | } = \frac { \sigma ^ { 2 } } { m | S | } } \end{array}$ . Apparently, if we want to get the same accuracy for a single pixel as the set of pixels, we need to sample $m | S |$ times, which needs much more computational cost than our metric, especially when the number of sampled pixels is large.
+
+# C STUDIES OF EXPLANATION METHODS BESIDES THE ESTIMATION OF ATTRIBUTION MAPS.
+
+Network visualization: The visualization of feature representations inside a neural network is the most direct way of opening the black-box of the neural network. Related techniques include gradient-based visualization (Zeiler & Fergus, 2014; Mahendran & Vedaldi, 2015; Yosinski et al., 2015) and up-convolutional nets (Dosovitskiy & Brox, 2016) to invert feature maps of conv-layers into images.
+
+Network diagnosis: Some studies diagnose feature representations inside a neural network. (Yosinski et al., 2014) measured features transferability in intermediate layers of a neural network. (Aubry & Russell, 2015) visualized feature distributions of different categories in the feature space. (Kindermans et al., 2018) extracted rough pixel-level correlations between network inputs and outputs, i.e. estimating image regions that directly contribute the network output. Network-attack methods (Koh & Liang, 2017; Szegedy et al., 2014) computed adversarial samples to diagnose a CNN. (Lakkaraju et al., 2017) discovered knowledge blind spots of a CNN in a weakly-supervised manner. However, above methods usually analyzed a neural network at the pixel level and did not summarize the network knowledge into clear visual concepts. (Bau et al., 2017) defined six types of semantics for CNN filters, i.e. objects, parts, scenes, textures, materials, and colors. Then, (Zhou et al., 2015) proposed a method to compute the image-resolution receptive field of neural activations in a feature map. Fong and Vedaldi (Fong & Vedaldi, 2018) analyzed how multiple filters jointly represented a certain semantic concept. Other studies retrieved intermediate-layer features from CNNs representing clear concepts. Simon & Rodner (2015) retrieved features to describe objects from feature maps, respectively. (Zhou et al., 2015) selected neural units to describe scenes.
+
+Learning interpretable representations: A new trend in the scope of network interpretability is to learn interpretable feature representations in neural networks Hu et al. (2016); Stone et al. (2017); Liao et al. (2016) in an un-/weakly-supervised manner. Capsule nets Sabour et al. (2017) and interpretable RCNN Wu et al. (2017b) learned interpretable features in intermediate layers. InfoGAN Chen et al. (2016) and $\beta$ -VAE Higgins et al. (2017) learned well-disentangled codes for generative networks.
+
+Explaining neural networks via knowledge distillation: Distilling knowledge from a black-box model into an explainable model is an emerging direction in recent years. (Choi et al., 2017) learned an explainable additive model, and (Vaughan et al., 2018) distilled knowledge of a network into an additive model. In order to disentangle feature representations of object parts from intermediate layers of a CNN, (Zhang et al., 2018) distilled the CNN’s knowledge into an explainer network with interpretable conv-layers, in which each filter represented a specific object part. (Frosst & Hinton, 2017; Tan et al., 2018; Che et al., 2016; Wu et al., 2017a) distilled representations of neural networks into tree structures. These methods did not explain the network knowledge using humaninterpretable semantic concepts.
+
+# D MORE EXAMPLES OF ATTRIBUTION MAPS
+
+![](images/74c7043eae9678f374e1b610ef62c9a903ee1e64e5b1f50dfb82cba0deba7acf.jpg)  
+Figure 4: Example of attribution maps.
+
+E DETAILED RESULTS OF THE BIAS OF THE ATTRIBUTION MAP AT THE PIXEL LEVEL
+
+Table 4: Bias of the attribution map at the pixel level on CIFAR-10-LeNet   
+
+<table><tr><td>Method</td><td>Grad_CAM</td><td>Grad|</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>top-10%</td><td>0.17256</td><td>0.05302</td><td>0.04060</td><td>0.04701</td><td>0.04924</td><td>0.03138</td><td>0.04164</td></tr><tr><td>top-30%</td><td>0.04921</td><td>0.03090</td><td>0.01892</td><td>0.02109</td><td>0.02408</td><td>0.03051</td><td>0.01798</td></tr><tr><td>top-50%</td><td>0.01961</td><td>0.02057</td><td>0.01126</td><td>0.01225</td><td>0.01443</td><td>0.02919</td><td>0.01041</td></tr><tr><td>top-70%</td><td>0.01531</td><td>0.01137</td><td>0.00671</td><td>0.00746</td><td>0.00902</td><td>0.02727</td><td>0.00634</td></tr><tr><td>top-90%</td><td>0.01785</td><td>0.00314</td><td>0.00240</td><td>0.00300</td><td>0.00334</td><td>0.02537</td><td>0.00277</td></tr><tr><td>bottom-10%</td><td>0.05317</td><td>0.06326</td><td>0.04907</td><td>0.06730</td><td>0.05755</td><td>0.02160</td><td>0.04271</td></tr><tr><td>bottom-30%</td><td>0.04510</td><td>0.03948</td><td>0.02626</td><td>0.03224</td><td>0.03100</td><td>0.02150</td><td>0.02198</td></tr><tr><td>bottom-50%</td><td>0.04732</td><td>0.02840</td><td>0.01767</td><td>0.02126</td><td>0.02047</td><td>0.02222</td><td>0.01481</td></tr><tr><td>bottom-70%</td><td>0.04693</td><td>0.01886</td><td>0.01270</td><td>0.01551</td><td>0.01462</td><td>0.02286</td><td>0.01086</td></tr><tr><td>bottom-90%</td><td>0.04022</td><td>0.01026</td><td>0.00808</td><td>0.01025</td><td>0.00882</td><td>0.02385</td><td>0.00708</td></tr></table>
+
+Table 5: Bias of the attribution map at the pixel level on CIFAR-10-ResNet20   
+
+<table><tr><td>Method</td><td>Gradl</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>top-10%</td><td>0.04692</td><td>0.03962</td><td>0.04801</td><td>0.04702</td><td>0.02716</td></tr><tr><td>top-30%</td><td>0.02552</td><td>0.01522</td><td>0.01848</td><td>0.02175</td><td>0.02706</td></tr><tr><td>top-50%</td><td>0.01616</td><td>0.00795</td><td>0.00943</td><td>0.01241</td><td>0.02666</td></tr><tr><td>top-70%</td><td>0.00762</td><td>0.00416</td><td>0.00425</td><td>0.00643</td><td>0.02456</td></tr><tr><td>top-90%</td><td>0.00288</td><td>0.00283</td><td>0.00336</td><td>0.00270</td><td>0.02185</td></tr><tr><td>bottom-10%</td><td>0.06610</td><td>0.06449</td><td>0.06726</td><td>0.06866</td><td>0.01790</td></tr><tr><td>bottom-30%</td><td>0.04272</td><td>0.03437</td><td>0.03833</td><td>0.03962</td><td>0.01765</td></tr><tr><td>bottom-50%</td><td>0.03117</td><td>0.02296</td><td>0.02704</td><td>0.02722</td><td>0.01836</td></tr><tr><td>bottom-70%</td><td>0.02168</td><td>0.01726</td><td>0.02053</td><td>0.01992</td><td>0.01897</td></tr><tr><td>bottom-90%</td><td>0.01357</td><td>0.01275</td><td>0.01513</td><td>0.01346</td><td>0.02005</td></tr></table>
+
+Table 6: Bias of the attribution map at the pixel level on CIFAR-10-ResNet32   
+
+<table><tr><td>Method</td><td>Gradl</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>top-10%</td><td>0.04836</td><td>0.04126</td><td>0.03984</td><td>0.04773</td><td>0.02699</td></tr><tr><td>top-30%</td><td>0.02600</td><td>0.01641</td><td>0.01728</td><td>0.02247</td><td>0.02702</td></tr><tr><td>top-50%</td><td>0.01639</td><td>0.00874</td><td>0.00972</td><td>0.01289</td><td>0.02630</td></tr><tr><td>top-70%</td><td>0.00793</td><td>0.00459</td><td>0.00565</td><td>0.00683</td><td>0.02465</td></tr><tr><td>top-90%</td><td>0.00268</td><td>0.00264</td><td>0.00242</td><td>0.00260</td><td>0.02267</td></tr><tr><td>bottom-10%</td><td>0.06535</td><td>0.06389</td><td>0.07271</td><td>0.06787</td><td>0.02059</td></tr><tr><td>bottom-30%</td><td>0.04217</td><td>0.03433</td><td>0.03612</td><td>0.03917</td><td>0.02053</td></tr><tr><td>bottom-50%</td><td>0.03064</td><td>0.02304</td><td>0.02359</td><td>0.02686</td><td>0.02065</td></tr><tr><td>bottom-70%</td><td>0.02135</td><td>0.01726</td><td>0.01733</td><td>0.01963</td><td>0.02103</td></tr><tr><td>bottom-90%</td><td>0.01331</td><td>0.01254</td><td>0.01214</td><td>0.01307</td><td>0.02145</td></tr></table>
+
+Table 7: Bias of the attribution map at the pixel level on CIFAR-10-ResNet44   
+
+<table><tr><td>Method</td><td>Grad|</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>top-10%</td><td>0.04795</td><td>0.04155</td><td>0.04055</td><td>0.04751</td><td>0.02712</td></tr><tr><td>top-30%</td><td>0.02576</td><td>0.01626</td><td>0.01698</td><td>0.02254</td><td>0.02709</td></tr><tr><td>top-50%</td><td>0.01609</td><td>0.00848</td><td>0.00935</td><td>0.01307</td><td>0.02637</td></tr><tr><td>top-70%</td><td>0.00748</td><td>0.00433</td><td>0.00502</td><td>0.00686</td><td>0.02485</td></tr><tr><td>top-90%</td><td>0.00248</td><td>0.00228</td><td>0.00230</td><td>0.00226</td><td>0.02271</td></tr><tr><td>bottom-10%</td><td>0.06510</td><td>0.06452</td><td>0.07143</td><td>0.06802</td><td>0.01924</td></tr><tr><td>bottom-30%</td><td>0.04196</td><td>0.03449</td><td>0.03663</td><td>0.03977</td><td>0.01879</td></tr><tr><td>bottom-50%</td><td>0.03082</td><td>0.02323</td><td>0.02448</td><td>0.02736</td><td>0.01967</td></tr><tr><td>bottom-70%</td><td>0.02159</td><td>0.01752</td><td>0.01811</td><td>0.01989</td><td>0.02064</td></tr><tr><td>bottom-90%</td><td>0.01353</td><td>0.01282</td><td>0.01293</td><td>0.01323</td><td>0.02137</td></tr></table>
+
+Table 8: Bias of the attribution map at the pixel level on CIFAR-10-ResNet56   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>top-10%</td><td>0.04716</td><td>0.04129</td><td>0.04780</td><td>0.04601</td><td>0.02690</td></tr><tr><td>top-30%</td><td>0.02537</td><td>0.01659</td><td>0.01855</td><td>0.02089</td><td>0.02714</td></tr><tr><td>top-50%</td><td>0.01607</td><td>0.00900</td><td>0.01022</td><td>0.01215</td><td>0.02679</td></tr><tr><td>top-70%</td><td>0.00791</td><td>0.00482</td><td>0.00569</td><td>0.00652</td><td>0.02509</td></tr><tr><td>top-90%</td><td>0.00212</td><td>0.00194</td><td>0.00214</td><td>0.00234</td><td>0.02252</td></tr><tr><td>bottom-10%</td><td>0.06625</td><td>0.06551</td><td>0.06065</td><td>0.06727</td><td>0.01828</td></tr><tr><td>bottom-30%</td><td>0.04164</td><td>0.03449</td><td>0.03158</td><td>0.03779</td><td>0.01812</td></tr><tr><td>bottom-50%</td><td>0.03001</td><td>0.02299</td><td>0.02129</td><td>0.02574</td><td>0.01911</td></tr><tr><td>bottom-70%</td><td>0.02086</td><td>0.01712</td><td>0.01587</td><td>0.01868</td><td>0.02023</td></tr><tr><td>bottom-90%</td><td>0.01300</td><td>0.01237</td><td>0.01146</td><td>0.01267</td><td>0.02088</td></tr></table>
+
+Table 9: Bias of the attribution map at the pixel level on VOC2012-ResNet50   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>top-10%</td><td>0.00766</td><td>0.00728</td><td>0.00633</td><td>0.00738</td><td>0.00876</td></tr><tr><td>top-30%</td><td>0.00437</td><td>0.00345</td><td>0.00276</td><td>0.00359</td><td>0.00606</td></tr><tr><td>top-50%</td><td>0.00297</td><td>0.00213</td><td>0.00170</td><td>0.00223</td><td>0.00451</td></tr><tr><td>top-70%</td><td>0.00173</td><td>0.00138</td><td>0.00113</td><td>0.00144</td><td>0.00335</td></tr><tr><td>top-90%</td><td>0.00060</td><td>0.00060</td><td>0.00053</td><td>0.00062</td><td>0.00235</td></tr><tr><td>bottom-10%</td><td>0.00831</td><td>0.00824</td><td>0.00850</td><td>0.00839</td><td>0.00430</td></tr><tr><td>bottom-30%</td><td>0.00500</td><td>0.00417</td><td>0.00386</td><td>0.00432</td><td>0.00240</td></tr><tr><td>bottom-50%</td><td>0.00354</td><td>0.00269</td><td>0.00244</td><td>0.00280</td><td>0.00151</td></tr><tr><td>bottom-70%</td><td>0.00228</td><td>0.00189</td><td>0.00171</td><td>0.00194</td><td>0.00113</td></tr><tr><td>bottom-90%</td><td>0.00117</td><td>0.00112</td><td>0.00111</td><td>0.00114</td><td>0.00128</td></tr></table>
+
+Table 10: Bias of the attribution map at the pixel level on VOC2012-ResNet101   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>top-10%</td><td>0.00790</td><td>0.00678</td><td>0.00610</td><td>0.00685</td><td>0.00827</td></tr><tr><td>top-30%</td><td>0.00438</td><td>0.00307</td><td>0.00260</td><td>0.00318</td><td>0.00608</td></tr><tr><td>top-50%</td><td>0.00291</td><td>0.00185</td><td>0.00158</td><td>0.00194</td><td>0.00473</td></tr><tr><td>top-70%</td><td>0.00175</td><td>0.00121</td><td>0.00104</td><td>0.00127</td><td>0.00371</td></tr><tr><td>top-90%</td><td>0.00067</td><td>0.00055</td><td>0.00049</td><td>0.00056</td><td>0.00278</td></tr><tr><td>bottom-10%</td><td>0.00861</td><td>0.00754</td><td>0.00839</td><td>0.00767</td><td>0.00387</td></tr><tr><td>bottom-30%</td><td>0.00496</td><td>0.00369</td><td>0.00377</td><td>0.00381</td><td>0.00222</td></tr><tr><td>bottom-50%</td><td>0.00343</td><td>0.00237</td><td>0.00238</td><td>0.00245</td><td>0.00142</td></tr><tr><td>bottom-70%</td><td>0.00225</td><td>0.00169</td><td>0.00169</td><td>0.00173</td><td>0.00124</td></tr><tr><td>bottom-90%</td><td>0.00117</td><td>0.00104</td><td>0.00112</td><td>0.00105</td><td>0.00174</td></tr></table>
+
+Table 11: Bias of the attribution map at the pixel level on VOC2012-AlexNet   
+
+<table><tr><td>Method</td><td>Grad_CAM</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>top-10%</td><td>0.05403</td><td>0.00744</td><td>0.00505</td><td>0.00646</td><td>0.00845</td><td>0.00854</td><td>0.00453</td></tr><tr><td>top-30%</td><td>0.03467</td><td>0.00434</td><td>0.00243</td><td>0.00296</td><td>0.00447</td><td>0.00598</td><td>0.00197</td></tr><tr><td>top-50%</td><td>0.02459</td><td>0.00298</td><td>0.00149</td><td>0.00182</td><td>0.00295</td><td>0.00443</td><td>0.00119</td></tr><tr><td>top-70%</td><td>0.01711</td><td>0.00176</td><td>0.00094</td><td>0.00120</td><td>0.00209</td><td>0.00329</td><td>0.00080</td></tr><tr><td>top-90%</td><td>0.01115</td><td>0.00060</td><td>0.00035</td><td>0.00058</td><td>0.00140</td><td>0.00222</td><td>0.00043</td></tr><tr><td>bottom-10%</td><td>0.01026</td><td>0.00906</td><td>0.00638</td><td>0.00876</td><td>0.00438</td><td>0.00466</td><td>0.00430</td></tr><tr><td>bottom-30%</td><td>0.00978</td><td>0.00534</td><td>0.00327</td><td>0.00400</td><td>0.00215</td><td>0.00267</td><td>0.00202</td></tr><tr><td>bottom-50%</td><td>0.00803</td><td>0.00372</td><td>0.00215</td><td>0.00253</td><td>0.00131</td><td>0.00165</td><td>0.00129</td></tr><tr><td>bottom-70%</td><td>0.00610</td><td>0.00239</td><td>0.00151</td><td>0.00178</td><td>0.00075</td><td>0.00103</td><td>0.00091</td></tr><tr><td>bottom-90%</td><td>0.00690</td><td>0.00124</td><td>0.00093</td><td>0.00111</td><td>0.00030</td><td>0.00104</td><td>0.00056</td></tr></table>
+
+Table 12: Bias of the attribution map at the pixel level on VOC2012-VGG16   
+
+<table><tr><td>Method</td><td>Grad_CAM</td><td>Gradl</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>top-10%</td><td>0.02984</td><td>0.00764</td><td>0.00656</td><td>0.00563</td><td>0.00768</td><td>0.00864</td><td>0.00461</td></tr><tr><td>top-30%</td><td>0.01876</td><td>0.00405</td><td>0.00292</td><td>0.00253</td><td>0.00310</td><td>0.00554</td><td>0.00159</td></tr><tr><td>top-50%</td><td>0.01365</td><td>0.00267</td><td>0.00176</td><td>0.00154</td><td>0.00187</td><td>0.00387</td><td>0.00092</td></tr><tr><td>top-70%</td><td>0.00947</td><td>0.00162</td><td>0.00115</td><td>0.00098</td><td>0.00132</td><td>0.00272</td><td>0.00065</td></tr><tr><td>top-90%</td><td>0.00650</td><td>0.00061</td><td>0.00051</td><td>0.00038</td><td>0.00089</td><td>0.00172</td><td>0.00044</td></tr><tr><td>bottom-10%</td><td>0.00767</td><td>0.00869</td><td>0.00768</td><td>0.00865</td><td>0.00396</td><td>0.00580</td><td>0.00288</td></tr><tr><td>bottom-30%</td><td>0.00708</td><td>0.00483</td><td>0.00372</td><td>0.00406</td><td>0.00173</td><td>0.00344</td><td>0.00117</td></tr><tr><td>bottom-50%</td><td>0.00654</td><td>0.00330</td><td>0.00238</td><td>0.00261</td><td>0.00106</td><td>0.00223</td><td>0.00072</td></tr><tr><td>bottom-70%</td><td>0.00540</td><td>0.00219</td><td>0.00170</td><td>0.00185</td><td>0.00075</td><td>0.00153</td><td>0.00054</td></tr><tr><td>bottom-90%</td><td>0.00509</td><td>0.00120</td><td>0.00107</td><td>0.00122</td><td>0.00041</td><td>0.00114</td><td>0.00040</td></tr></table>
+
+Table 13: Bias of the attribution map at the pixel level on VOC2012-VGG19   
+
+<table><tr><td>Method</td><td>Grad_CAM</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>top-10%</td><td>0.02564</td><td>0.00761</td><td>0.00672</td><td>0.00580</td><td>0.00802</td><td>0.00851</td><td>0.00468</td></tr><tr><td>top-30%</td><td>0.01991</td><td>0.00403</td><td>0.00301</td><td>0.00261</td><td>0.00337</td><td>0.00565</td><td>0.00157</td></tr><tr><td>top-50%</td><td>0.01517</td><td>0.00263</td><td>0.00181</td><td>0.00159</td><td>0.00205</td><td>0.00402</td><td>0.00091</td></tr><tr><td>top-70%</td><td>0.01139</td><td>0.00161</td><td>0.00118</td><td>0.00102</td><td>0.00143</td><td>0.00282</td><td>0.00064</td></tr><tr><td>top-90%</td><td>0.00799</td><td>0.00062</td><td>0.00053</td><td>0.00043</td><td>0.00096</td><td>0.00183</td><td>0.00045</td></tr><tr><td>bottom-10%</td><td>0.00932</td><td>0.00867</td><td>0.00782</td><td>0.00871</td><td>0.00442</td><td>0.00587</td><td>0.00236</td></tr><tr><td>bottom-30%</td><td>0.00764</td><td>0.00480</td><td>0.00378</td><td>0.00406</td><td>0.00193</td><td>0.00352</td><td>0.00095</td></tr><tr><td>bottom-50%</td><td>0.00683</td><td>0.00326</td><td>0.00243</td><td>0.00260</td><td>0.00120</td><td>0.00231</td><td>0.00059</td></tr><tr><td>bottom-70%</td><td>0.00630</td><td>0.00217</td><td>0.00173</td><td>0.00184</td><td>0.00084</td><td>0.00140</td><td>0.00046</td></tr><tr><td>bottom-90%</td><td>0.00649</td><td>0.00119</td><td>0.00109</td><td>0.00120</td><td>0.00043</td><td>0.00107</td><td>0.00036</td></tr></table>
+
+# F ATTRIBUTION MAPS OF MASKED IMAGES
+
+G DETAILED RESULTS OF THE MUTUAL VERIFICATION
+
+![](images/84255e01c2a68ee7187460f4e43b594534feb150938c83b1a932b728b5f35483.jpg)  
+Figure 5: Example of attribution maps after spatial masking.
+
+Table 14: Mutual verification on VOC2012-VGG19   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.6661</td><td>1.4054</td><td>1.4177</td><td>1.4142</td><td>1.4375</td></tr><tr><td>GI</td><td>1.6661</td><td>0.0000</td><td>1.4159</td><td>1.4008</td><td>1.4140</td><td>1.3437</td></tr><tr><td>GB</td><td>1.4054</td><td>1.4159</td><td>0.0000</td><td>1.5834</td><td>1.4244</td><td>1.5242</td></tr><tr><td>DeepSHAP</td><td>1.4177</td><td>1.4009</td><td>1.5835</td><td>0.0000</td><td>1.3691</td><td>1.1106</td></tr><tr><td>LIME</td><td>1.4142</td><td>1.4141</td><td>1.4245</td><td>1.3691</td><td>0.0000</td><td>1.3874</td></tr><tr><td>LRP</td><td>1.4375</td><td>1.3437</td><td>1.5243</td><td>1.1106</td><td>1.3874</td><td>0.0000</td></tr></table>
+
+Table 15: Mutual verification on VOC2012-VGG16   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.6647</td><td>1.3992</td><td>1.4302</td><td>1.4142</td><td>1.4621</td></tr><tr><td>GI</td><td>1.6647</td><td>0.0000</td><td>1.4191</td><td>1.3711</td><td>1.4139</td><td>1.2649</td></tr><tr><td>GB</td><td>1.3992</td><td>1.4191</td><td>0.0000</td><td>1.5839</td><td>1.4260</td><td>1.5320</td></tr><tr><td>DeepSHAP</td><td>1.4302</td><td>1.3712</td><td>1.5840</td><td>0.0000</td><td>1.3824</td><td>1.0636</td></tr><tr><td>LIME</td><td>1.4142</td><td>1.4140</td><td>1.4261</td><td>1.3825</td><td>0.0000</td><td>1.3960</td></tr><tr><td>LRP</td><td>1.4621</td><td>1.2649</td><td>1.5321</td><td>1.0636</td><td>1.3960</td><td>0.0000</td></tr></table>
+
+Table 16: Mutual verification on VOC2012-AlexNet   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.5540</td><td>1.3477</td><td>1.4348</td><td>1.4148</td><td>1.4851</td></tr><tr><td>GI</td><td>1.5541</td><td>0.0000</td><td>1.4344</td><td>1.3224</td><td>1.4113</td><td>0.8341</td></tr><tr><td>GB</td><td>1.3477</td><td>1.4344</td><td>0.0000</td><td>1.4477</td><td>1.4146</td><td>1.4573</td></tr><tr><td>DeepSHAP</td><td>1.4349</td><td>1.3225</td><td>1.4478</td><td>0.0000</td><td>1.3221</td><td>1.2251</td></tr><tr><td>LIME</td><td>1.4149</td><td>1.4115</td><td>1.4147</td><td>1.3222</td><td>0.0000</td><td>1.3916</td></tr><tr><td>LRP</td><td>1.4851</td><td>0.8341</td><td>1.4573</td><td>1.2250</td><td>1.3915</td><td>0.0000</td></tr></table>
+
+Table 17: Mutual verification on VOC2012-ResNet50   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.6285</td><td>1.4143</td><td>1.6242</td><td>1.4143</td></tr><tr><td>GI</td><td>1.6285</td><td>0.0000</td><td>1.4137</td><td>0.6537</td><td>1.4139</td></tr><tr><td>GB</td><td>1.4143</td><td>1.4136</td><td>0.0000</td><td>1.4137</td><td>1.4238</td></tr><tr><td>DeepSHAP</td><td>1.6242</td><td>0.6537</td><td>1.4137</td><td>0.0000</td><td>1.4139</td></tr><tr><td>LIME</td><td>1.4144</td><td>1.4141</td><td>1.4240</td><td>1.4140</td><td>0.0000</td></tr></table>
+
+Table 18: Mutual verification on VOC2012-ResNet101   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.6877</td><td>1.4136</td><td>1.6428</td><td>1.4145</td></tr><tr><td>GI</td><td>1.6877</td><td>0.0000</td><td>1.4140</td><td>0.6778</td><td>1.4139</td></tr><tr><td>GB</td><td>1.4136</td><td>1.4140</td><td>0.0000</td><td>1.4140</td><td>1.4275</td></tr><tr><td>DeepSHAP</td><td>1.6428</td><td>0.6778</td><td>1.4140</td><td>0.0000</td><td>1.4138</td></tr><tr><td>LIME</td><td>1.4146</td><td>1.4140</td><td>1.4277</td><td>1.4139</td><td>0.0000</td></tr></table>
+
+Table 19: Mutual verification on CIFAR10-ResNet56   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.4183</td><td>1.4107</td><td>1.4135</td><td>1.4142</td></tr><tr><td>GI</td><td>1.4183</td><td>0.0000</td><td>1.4140</td><td>1.4104</td><td>1.4140</td></tr><tr><td>GB</td><td>1.4107</td><td>1.4140</td><td>0.0000</td><td>1.4128</td><td>1.3879</td></tr><tr><td>DeepSHAP</td><td>1.4135</td><td>1.4104</td><td>1.4128</td><td>0.0000</td><td>1.4097</td></tr><tr><td>LIME</td><td>1.4142</td><td>1.4140</td><td>1.3879</td><td>1.4098</td><td>0.0000</td></tr></table>
+
+Table 20: Mutual verification on CIFAR10-ResNet44   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.3999</td><td>1.4128</td><td>1.4147</td><td>1.4141</td></tr><tr><td>GI</td><td>1.3999</td><td>0.0000</td><td>1.4125</td><td>1.4114</td><td>1.4143</td></tr><tr><td>GB</td><td>1.4128</td><td>1.4125</td><td>0.0000</td><td>1.4152</td><td>1.4175</td></tr><tr><td>DeepSHAP</td><td>1.4147</td><td>1.4114</td><td>1.4152</td><td>0.0000</td><td>1.4097</td></tr><tr><td>LIME</td><td>1.4142</td><td>1.4144</td><td>1.4152</td><td>1.4175</td><td>0.0000</td></tr></table>
+
+Table 21: Mutual verification on CIFAR10-ResNet32   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.4259</td><td>1.4129</td><td>1.4117</td><td>1.4143</td></tr><tr><td>GI</td><td>1.4259</td><td>0.0000</td><td>1.4164</td><td>1.4113</td><td>1.4145</td></tr><tr><td>GB</td><td>1.4129</td><td>1.4164</td><td>0.0000</td><td>1.4144</td><td>1.4111</td></tr><tr><td>DeepSHAP</td><td>1.4117</td><td>1.4113</td><td>1.4144</td><td>0.0000</td><td>1.4117</td></tr><tr><td>LIME</td><td>1.4144</td><td>1.4145</td><td>1.4112</td><td>1.4117</td><td>0.0000</td></tr></table>
+
+Table 22: Mutual verification on CIFAR10-ResNet20   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.3895</td><td>1.4010</td><td>1.4134</td><td>1.4143</td></tr><tr><td>GI</td><td>1.3895</td><td>0.0000</td><td>1.4140</td><td>1.4133</td><td>1.4145</td></tr><tr><td>GB</td><td>1.4010</td><td>1.4140</td><td>0.0000</td><td>1.4164</td><td>1.4382</td></tr><tr><td>DeepSHAP</td><td>1.4134</td><td>1.4133</td><td>1.4164</td><td>0.0000</td><td>1.4118</td></tr><tr><td>LIME</td><td>1.4144</td><td>1.4145</td><td>1.4382</td><td>1.4118</td><td>0.0000</td></tr></table>
+
+Table 23: Mutual verification on CIFAR10-LeNet   
+
+<table><tr><td>Method</td><td>Grad</td><td>GI</td><td>GB</td><td>DeepSHAP</td><td>LIME</td><td>LRP</td></tr><tr><td>Grad</td><td>0.0000</td><td>1.5079</td><td>1.1955</td><td>1.4352</td><td>1.4130</td><td>1.4904</td></tr><tr><td>GI</td><td>1.5079</td><td>0.0000</td><td>1.4714</td><td>1.3025</td><td>1.4003</td><td>0.7179</td></tr><tr><td>GB</td><td>1.1955</td><td>1.4714</td><td>0.0000</td><td>1.4329</td><td>1.4249</td><td>1.4881</td></tr><tr><td>DeepSHAP</td><td>1.4352</td><td>1.3025</td><td>1.4329</td><td>0.0000</td><td>1.3958</td><td>1.3063</td></tr><tr><td>LIME</td><td>1.4131</td><td>1.4004</td><td>1.4249</td><td>1.3959</td><td>0.0000</td><td>1.3805</td></tr><tr><td>LRP</td><td>1.4904</td><td>0.7179</td><td>1.4881</td><td>1.3063</td><td>1.3804</td><td>0.0000</td></tr></table>

parse/train/SylLYsCcFm/SylLYsCcFm.md

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+# LEARNING TO MAKE ANALOGIES BY CONTRASTING ABSTRACT RELATIONAL STRUCTURE
+
+Felix Hill\*, Adam Santoro∗, David G. T. Barrett, Ari Morcos & Tim Lillicrap Deepmind, London {felixhill,adamsantoro,barrettdavid,arimorcos,countzero}@google.com
+
+# ABSTRACT
+
+Analogical reasoning has been a principal focus of various waves of AI research. Analogy is particularly challenging for machines because it requires relational structures to be represented such that they can be flexibly applied across diverse domains of experience. Here, we study how analogical reasoning can be induced in neural networks that learn to perceive and reason about raw visual data. We find that the critical factor for inducing such a capacity is not an elaborate architecture, but rather, careful attention to the choice of data and the manner in which it is presented to the model. The most robust capacity for analogical reasoning is induced when networks learn analogies by contrasting abstract relational structures in their input domains, a training method that uses only the input data to force models to learn about important abstract features. Using this technique we demonstrate capacities for complex, visual and symbolic analogy making and generalisation in even the simplest neural network architectures.
+
+# 1 INTRODUCTION
+
+The ability to make analogies – that is, to flexibly map familiar relations from one domain of experience to another – is a fundamental ingredient of human intelligence and creativity (Gentner, 1983; Hofstadter, 1996; Hummel & Holyoak, 1997; Lovett & Forbus, 2017). As noted, for instance, by Holyoak & Thagard (1995), analogies gave Roman scientists a deeper understanding of sound when they leveraged knowledge of a familiar source domain (water waves in the sea) to better understand the structure of an unfamiliar target domain (acoustics). The Romans ‘aligned’ relational principles about water waves (periodicity, bending round corners, rebounding off solids) to phenomena observed in acoustics, in spite of the numerous perceptual and physical differences between water and sound. This flexible alignment, or mapping, of relational structure between source and target domains, independent of perceptual congruence, is a prototypical example of analogy making.
+
+It has proven particularly challenging to replicate processes of analogical thought in machines. Many classical or symbolic AI models lack the flexibility to apply predicates or operations across diverse domains, particularly those that may have never previously been observed. It is natural to consider, however, whether the strengths of modern neural network-based models can be exploited to solve difficult analogical problems, given their capacity to represent stimuli at different levels of abstraction and to enable flexible, context-dependent computation over noisy and ambiguous inputs.
+
+In this work we demonstrate that well-known neural network architectures can indeed learn to make analogies with remarkable generality and flexibility. This ability, however, is critically dependent on a method of training we call learning analogies by contrasting abstract relational structure (LABC). We show that simple architectures can be trained using this approach to apply abstract relations to never-before-seen source-target domain mappings, and even to entirely unfamiliar target domains.
+
+Our work differs from previous computational models of analogy in two important ways. First, unlike previous neural network models of analogy, we optimize a single model to perform both stimulus representation and cross-domain mapping jointly. This allows us to explore the potential benefit of interactions between perception, representation and inter-domain alignment, a question of some debate in the analogy literature (Forbus et al., 1998). Second, we do not instantiate an explicit cognitive theory or analogy-like computation in our model architecture, but instead use this theoretical insight to inform the way in which the model is trained.
+
+![](images/8b03221091c0f3eef96e3dc0f1d096073b5ff4ded47879e16e92bfcc2ede1bbc.jpg)  
+Figure 1: A visual analogy problem. In this example, the model must (1) identify a relation (Progression ) on a particular domain ( shape quantity ) in the source sequence (top), and (2) apply it to a different domain ( line color ) in order to find the candidate answer panel that correctly completes target sequence (bottom). There are seven possible domains and four possible relations in the dataset.
+
+# 2 ANALOGIES AS HIGH-LEVEL PERCEPTION AND STRUCTURE MAPPING
+
+Perhaps the best-known explanation of human analogical reasoning is Structure Mapping Theory (SMT) (Gentner, 1983). SMT emphasizes the distinction between two means of comparing domains of experience; analogy and similarity. According to SMT, two domains are similar if they share many attributes (i.e. properties that can be expressed with a one-place predicate like BLUE(sea) ), whereas they are analogous if they share few attributes but many relations (i.e. properties expressed by many-place predicates like BENDS-AROUND(sea, solid-objects) ). SMT assumes that our perceptions can be represented as collections of attributes and structured relations, and that these representations do not necessarily depend on the subsequent mappings that use them.
+
+The High-Level Perception (HLP) theory of analogy (Chalmers et al., 1992; Mitchell, 1993) instead construes analogy as a function of tightly-interacting perceptual and reasoning processes, positing that the creation of stimulus representations and the alignment of those representations are mutually dependent. For example, when making an analogy between the sea and acoustics, we might represent certain perceptual features (the fact that the sea appears to be moving) and ignore others (the fact that the sea looks blue), because the particular comparison that we make depends on location and direction, and not on colour.
+
+In this work we aim to induce flexible analogy making in neural networks by drawing inspiration from both SMT and HLP. The perceptual and domain-comparison components of our models are connected and jointly optimised end-to-end, which, as posited by HLP, reflects a high degree of interaction between perception and domain comparison. On the other hand, the key insight of this paper, LABC, is directly motivated by SMT. We find that LABC greatly enhances the ability of our networks to resolve analogies in a generalisable way by encouraging them to compare inputs at the more abstract level of relations rather than the less abstract level of attributes. LABC organizes the training data such that the inference and mapping of relational structure is essential for good performance. This means that problems cannot be resolved by considering mere similarity of attributes, or even less appropriately, via spurious surface-level statistics or memorization.
+
+# 3 VISUAL ANALOGY PROBLEMS
+
+Our first experiments involve greyscale visual scenes similar to those previously applied to test both human reasoning ability (Raven, 1983; Geary et al., 2000) and reasoning in machine learning models (Bongard, 1967; Fleuret et al., 2011; Barrett et al., 2018). Each scene is composed of a source sequence, consisting of three panels (distinct images), a target sequence, consisting of two panels, and four candidate answer panels (Fig. 1). In the source sequence a relation $r$ is instantiated, where $r$ is one of four possible relations from the set $R = \left\{ \begin{array} { r l r l } \end{array} \right.$ XOR , OR , AND Progression $\}$ . Models must then consider the two panels in the target sequence, together with the four candidate answer panels, to determine which answer panel best completes the target sequence – by analogy with the source sequence – so that $r$ is also instantiated (Fig. 2).
+
+![](images/9c82dc2f464a457fc9b84da7f1aeffa91069718361d96431eef47cdbaf473c98.jpg)  
+Figure 2: In LABC, the multiple choice candidates are all semantically plausible, in that they are all consistent completions of the target domain using some relation. Only the correct answer uses the same relation as the source domain, so, the only way to solve the problem is to use analogical reasoning. In contrast, perceptually plausible incorrect candidates are consistent with the target domain attributes but not the relations and all other possible candidates are inconsistent with the target domain relation and attributes.
+
+The notion of domain is critical to analogy. In our visual analogy task, a relation can be instantiated in one of seven different domains: line type , line colour , shape type , shape colour , shape size , shape quantity and shape position (see Fig. 1 and Appendix Fig. 7 for examples). Within a panel of a given domain, the attributes present in the scene (such as the colour of the shapes or the positions of the lines) can take one of 10 possible values. A question in the dataset is therefore defined by a relation $r$ , a domain $d _ { s }$ on which $r$ is instantiated in the source sequence, a set of values for the source-domain $v _ { 1 } ^ { s } \cdot \cdot \cdot v _ { 3 } ^ { s }$ , a target domain $d _ { t }$ , values for the target-domain $v _ { 1 } ^ { t } \cdot \cdot \cdot v _ { 3 } ^ { t }$ , the position $k \in \{ 1 \cdots 4 \}$ of the correct answer among the answer candidate panels and whatever is instantiated in the three incorrect candidate answer panels $c _ { i } , i \neq k$ . Note, however, that the values of certain domain attributes that are not relevant to a given question (such as the colour of shapes in the shape quantity domain) still have to be selected, and can vary freely. Thus, despite the small number of latent factors involved, the space of possible questions is of the order of ten million.
+
+The interplay between relations, domains and values makes it possible to construct questions that require increasing degrees of abstraction and analogymaking. The simplest case involves a relation $r$ , a domain $d _ { s } \ = = \ d _ { t }$ , and values $v _ { i } ^ { s } ~ = = ~ v _ { i } ^ { t }$ that are common to both source and target sequences (Fig 3a). To solve such a question a model must identify a candidate answer panel that results in a copy of the source sequence in the target sequence. This does not seem to require any meaningful understanding of $r$ , nor any particular analogy-making. Somewhat greater abstraction and analogymaking is required for questions involving a single domain $\scriptstyle ( d _ { s } = = d _ { t }$ ), but different values in the source and target sequence $v _ { i } ^ { s } \neq v _ { i } ^ { t }$ (Fig 3b). In this case the model must learn that, in a given domain, the relation $r$ can apply to a range of different values. Finally, in the full analogy questions considered in this study, the relation $r$ can be instantiated on different domains in the source and target sequences (i.e. $d _ { t } \neq d _ { s }$ ; Fig 3c). These questions require a sensitivity to the idea that a single relation $r$ can be applied in different (but related) ways to different domains of experience.1
+
+# 3.1 METHODS
+
+Our model consisted of a simple perceptual front-end – a convolutional neural network (CNN) – which provided input for a recurrent neural network (RNN) by producing embeddings for each image panel independently. The RNN processed the source sequence embeddings, the target sequence embeddings, and a single candidate embedding, to produce a scalar score. Four such passes (one for each source-target-candidate set) produced four scalar scores, quantifying how the model evaluated the suitability of the particular candidate. Finally, a softmax was computed across the scores to select the model’s ‘answer’. Further model details are in appendix 7.1.
+
+![](images/03afe4004d338e78df0096b9270a3cb40e2485857265fe4ad9fe922d7eb4938f.jpg)  
+Figure 3: (a), (b), (c) Three types of visual reasoning questions. Each question requires a different degree of analogy making, with the question on the right demanding the most fluid and abstract application of the underlying relation. (d) Learning analogies by contrasting. When learning by contrasting, each answer choice is consistent with a relational structure in the target sequence. Only the correct answer choice is consistent with relations in both the source and target domains. This forces the network to consider the source sequence to infer the correct structure.
+
+Learning Analogies By Contrasting (LABC) In the default setting of our data generator – the normal training regime – for a question involving source domain $d _ { s }$ , target domain $d _ { t }$ and relation $r$ , the candidate answers can contain any (incorrect) values chosen at random from $d _ { t }$ (Fig. 2). By selecting incorrect candidate answers from the same domain $d _ { t }$ as the correct answer, we ensure that they are perceptually plausible, so that the problem cannot be solved trivially by matching the domain of the question to one of the answers. Even so, the baseline training regime may allow models to find perceptual correlations that allow it to arrive at the correct answer consistently over the training data. We can make this less likely by instead training the model to contrast abstract relational structure – the LABC regime – simply by ensuring that incorrect answers are both perceptually and semantically plausible (Fig. 2). More specifically, incorrect answers are selected from $d _ { t }$ such that each answer $c _ { i }$ completes a decoy relation $\boldsymbol { \hat { r } _ { i } } \neq \boldsymbol { r }$ with the target sequence. LABC ensures that, during training, models have no alternative but to first observe a relation $r$ in the source domain and consider and complete the same relation in the target domain - i.e. to execute a full analogical reasoning step.2
+
+![](images/5e3309230dca0505df3671e356defa265364450c63ddf00a0765bd20568366af.jpg)  
+Figure 4: Results of the three experiments in the visual analogy domain for a network that learns from random candidate answers, by contrasting abstract structures or both types of question interleaved. Bar heights depict the means across eight seeds in each condition; standard errors were $< 0 . 0 1$ for each condition (not shown – see the appendix Table 4 for the values)
+
+Note that in all experiments reported below, we generated 600,000 training questions, 10,000 validation questions and test sets of 100,000 questions. These data will be published with the paper.
+
+# 3.2 EXPERIMENT 1: NOVEL DOMAIN TRANSFER
+
+A key aspect of analogy-making is the process of comparing or aligning two domains. We can measure how well models acquire this ability by testing them on analogies involving unfamiliar source domain target domain transfers. For each of the seven possible target domains $d _ { t }$ we randomly selected a source domain $d _ { s } \ne d _ { t }$ , yielding a test set of seven domain transfer pairs $[ d _ { s }  d _ { t } ]$ . Our models were then trained on questions involving one of the remaining $7 \times 7 - 7 = 4 2$ domain transfer pairs. For a test question involving domains $d _ { s }$ and $d _ { t }$ , each model was therefore familiar with $d _ { s }$ and $d _ { t }$ but had not been trained to make an analogy from $d _ { s }$ to $d _ { t }$ .
+
+We found that a network can indeed learn to apply relations by analogy involving novel domain transfers, but that this ability crucially relies on learning by contrasting. The effect is strong; for the most focused test questions involving semantically plausible (contrasting) candidate answers the model trained by contrasting achieves $83 \%$ accuracy (depending on the held-out domain), versus $58 \%$ for a model trained with randomly-chosen candidate answers.
+
+# 3.3 EXPERIMENT 2: NOVEL TARGET DOMAIN
+
+Humans can use analogies to better understand comparatively unfamiliar domains, as in the Roman explanation of acoustics by analogy with the sea. To capture this scenario, we held out two domains ( line type and shape colour , chosen at random) from the model’s training data, and ensured that each test question involved one of these domains. To make sense of the test questions, a model must therefore (presumably) learn to represent the relations in the dataset in a sufficiently general way that this knowledge can be applied to completely novel domains. Any model that resolves such problems successfully must therefore exploit any (rudimentary) perceptual similarity between the test domain and the domains that were observed during training. For instance, the process of applying a relation in the shape colour domain may recruit similar feature detectors to those required when applying it to the line colour domain.
+
+Table 1: Test performance of a selection of visual reasoning network architectures trained in the normal regime and with LABC on the Novel Domain Transfer experiment. The test questions include those with both semantically-plausible and merely perceptually-plausible incorrect answers. Mean $\left( + \mathbf { S } \mathbf { D } \right)$ for 10 models in each condition with different random initialisations.   
+
+<table><tr><td></td><td>LABC training</td><td>Normal training</td></tr><tr><td>RNN (this paper)</td><td>0.90 (0.01)</td><td>0.79 (0.09)</td></tr><tr><td>ResNet-50</td><td>0.82 (0.02)</td><td>0.77 (0.10)</td></tr><tr><td>Parallel ResNet-50</td><td>0.89 (0.01)</td><td>0.75 (0.14)</td></tr><tr><td>Parallel Relation Net (WReN)</td><td>0.95 (0.017)</td><td>0.72 (0.21)</td></tr></table>
+
+Surprisingly, we found that the network can indeed learn to make sense of the unfamiliar target domains in the test questions, although again this capacity is boosted by LABC (Fig 4b). Accuracy in the LABC condition on the most focused (contrasting) test questions is lower than in the Experiment 1 $( \sim 8 0 \%$ , depending on the held-out domain), but well above the model trained with random answer candidates $( \sim 6 0 \% )$ ). Interestingly, a model trained on semantically-plausible sets of candidate answers (LABC) performs somewhat worse on test questions whose answers are merely perceptuallyplausible than a model trained in that (normal) regime. This deficit can be largely recovered by interleaving random-answer and contrasting candidates during training.
+
+# 3.3.1 EXPERIMENT 3: NOVEL DOMAIN VALUES
+
+Another way in which a domain can be unfamiliar to a network is if it involves attributes whose values have not been observed during training. Since each of the seven (source and target) domains our analogy problems permits 10 values, we can measure interpolation by withholding values 1, 3, 5, 7 and 9 and measure extrapolation by withholding values 6, 7, 8, 9 and 10. To extrapolate effectively, a model must therefore be able to resolve questions at test time involving lines or shapes that are darker, larger, more-sided or simply more numerous than those in its training questions.
+
+In the case of interpolation, we found that a model trained with random candidates performs very poorly on the more challenging contrasting test questions $( \mathrm { F i g ~ 4 c ~ 4 5 \% }$ vs $93 \%$ for LABC), which suggests that models trained in the normal regime overfit to a strategy that bears no resemblance to human-like analogical reasoning. We verified this hypothesis by running an analysis where we presented only the target domain sequence and candidate answers to the model. After a long period of training in the normal regime, this ‘source-blind’ model achieved $9 7 \%$ accuracy, which confirms that it indeed finds short-cut solutions that do not require analogical mapping. In contrast, the accuracy of the source-blind model in the LBAC condition converged at $32 \%$ .
+
+We also found, somewhat surprisingly, that LBAC results in a (modest) improvement in how well models can extrapolate to novel input values (Fig 4c); a model trained on questions with both contrasting and random candidate answers performs significantly better than the normal model on the test questions with contrasting candidate answers $62 \%$ vs. $43 \%$ ), and mantains comparable performance on test questions with random candidate answers ( $45 \%$ vs. $44 \%$ ).
+
+# 3.3.2 EXPERIMENT 4: MODEL COMPARISON
+
+Finally, we explore the extent to which LABC improves generalistion for different various different network architectures, taking as a guide the models considered in Barrett et al. (2018). We run the Novel Domain Transfer experiment with each of these models, trained using both LABC and in the normal training regime. We measure generalisation on a mixed set comprised equally of test questions with semantically-plausible incorrect answer candidates (matching LABC training) and those with merely perceptually-plausible incorrect answers (matching normal training). All models perform better at novel domain transfer when trained with LABC (Table 1). This confirms that our method does not depend on the use of a specific architecture. Further, the fact that LABC yields better performance than normal training on a balanced, mixed test set shows that it is the most effective way to train models in problem instances where the exact details of test questions may be unknown.
+
+# 3.4 CONCLUSION
+
+These results demonstrate that LABC increases the ability of models to generalize beyond the distribution of their training data. This effect is observed for the prototypical analogical processes involving novel domain mappings and unfamiliar target domains (Experiments 1 & 2). Interestingly, it also results in moderate improvements to how well models extrapolate to perceptual input outside the range of their training experience (Experiment 3).
+
+# 4 EMERGENT RELATIONAL REPRESENTATIONS
+
+To understand the mechanisms that support this generalisation we analysed neural activity in models trained with LABC compared to models that were not. First, we took the RNN hidden state activity just prior to the input of the candidate panel. For a model trained via LABC, we found that these activities clustered according to relation type (e.g. progression ) more-so than domain (e.g., shape colour ) (Fig 5a, Table 2). In contrast, for models trained normally the relation
+
+<table><tr><td rowspan=1 colspan=1>Cluster</td><td rowspan=1 colspan=1>LABC</td><td rowspan=1 colspan=1>Normal Training</td></tr><tr><td rowspan=1 colspan=1>inter-relation dist.inter-domain dist.</td><td rowspan=1 colspan=1>5.2(± 2.2)1.8 (± 1.8)</td><td rowspan=1 colspan=1>4.4 (± 2.0)2.0 (± 1.8)</td></tr></table>
+
+Table 2: Mean $\mathbf { ( + S D ) }$ distance between clusters of RNN hidden state representations. LABC seems to promote greater distinctions between different abstract relations in the representation space of models. Since the maximum euclidean distance between two points in a 64 dimensional unit cube is P64i 12 = 8, the distances that we observe between relational representations are close to the maximum possible.
+
+based clusters overlapped to a greater extent (Fig 5b, Table 2). Thus, LABC seems to encourage the model to represent relations more explicitly, which could in turn explain its capacity to generalise by analogy to novel domains.
+
+![](images/184c0d754b73faa611c0580c4fb0c482ee946c8f8af1e9a596e6fa668f875e02.jpg)  
+Figure 5: LABC supports emergent relational representations. Principle component analysis (PCA) (right) and t-SNE analysis (left) of RNN hidden states. Each dot represents a (64-dimensional) state coloured according to the relation type and domain of the corresponding question.
+
+# 5 SYMBOLIC ANALOGY PROBLEMS
+
+Many of the most important studies of analogy in AI involve symbolic or numerical patterns (Hofstadter, 1996), and various neural models of semantic cognition more generally also operate on discrete, feature-based representations of input concepts (Rogers & McClelland, 2004; Devereux et al., 2018). To verify our findings in this setting, we implemented a symbolic analogy task based on feature-based stimuli. This more controlled domain allows to show that the construction of appropriate incorrect answer candidates can be learned in a proposal model that is trained jointly with a model that learns to contrast the candidates, widening the potential applications of LABC to task settings where we lack a clear understanding of the underlying abstract relational structure (Sec 5.3).
+
+In our task, inputs are $D$ -dimensional vectors $v$ of discrete, integer-valued features (analogous to semantic properties such as furry or carnivorous in previous work). Across any set $V : v \in V$ of stimuli, each feature dimension then corresponds to a domain (the domains of skin-type or dietary habits in the present example). To simulate a space of abstract relational structures on these domains, we simply take a set $F$ whose elements $f$ are common mathematical functions operating on sets: MIN , MAX , ARGMIN , ARGMAX , SUM and RANGE . Abstract relations in this context can be stated, for instance, as $\mathrm { { S U M } } \left( 1 , \quad 2 , \quad 3 ; \quad 6 \right)$ . Given a such a function $f$ , a set $V$ of stimulus vectors, and a random choice of domain $d$ , we can compute a $D$ -dimensional) answer vector $\boldsymbol { a } _ { [ V , d , f ] }$ as the result of applying $f$ to $d$ on $V$ (i.e. executing $f$ on the $d$ -th dimension of each $v \in \dot { V } .$ ). It is now simple to randomly construct an analogy question in this setting. We select a function $f$ , source $d _ { s }$ and target $d _ { t }$ domains at random, randomly generate source $V _ { s }$ and target $V _ { t }$ stimuli. We then apply $f$ to $V _ { s }$ on $d _ { s }$ and to $V _ { t }$ on $d _ { t }$ to generate the source and target domain solutions, $a _ { [ V _ { s } , d _ { s } , f ] }$ and $a _ { [ V _ { t } , d _ { t } , f ] }$ , respectively. The inputs $V _ { s }$ , $d _ { s }$ , $a _ { [ V _ { s } , d _ { s } , f ] }$ , $V _ { t }$ and $v _ { t }$ are passed to the model, together with a ${ \dot { d } } \times k$ matrix of alternative answer choices that includes $a _ { [ V _ { t } , d _ { t } , f ] }$ $k$ is a fixed number of choices, four in the visual analogies presented previously). We then require the model to select which of these alternatives is the true completion of the analogy $a _ { [ V _ { t } , d _ { t } , f ] }$ . As in the visual analogy tasks, to resolve such a question the model must detect an abstract relationship in the input domain $d _ { s }$ , that explains the connection between the source stimuli $V _ { s }$ and the answer vector $a _ { [ V _ { s } , d _ { s } , f ] }$ . Once this achieved, the model must evaluate the function $f$ that describes this relationship on the source domain $d _ { t }$ with sufficient accuracy that it can identify the result of that evaluation $a _ { [ V _ { t } , d _ { t } , f ] }$ in the context of $k$ distracting alternative (incorrect) answers.
+
+We study generalization in this task by restricting the particular ordered domain mappings $d _ { s }  d _ { t }$ that are observed in the training and test set; for example, aligning structure from domain 3 to domain 1 may be required in the test set, but may have never been seen before in the training set. While we withhold particular alignments (e.g., $3  1 \AA$ ), we ensure that all dimensions are aligned ‘out-of’ $( 3  )$ and ‘into’ $(  1 )$ ) at least once in the training set. Note that this setup is directly analogous to the ‘Novel Domain Transfer’ experiment in the visual analogy problems (Sec. 3.2).
+
+# 5.1 METHODS
+
+The high level model structure was similar to that of the previous experiments: candidates and their context were processed independently to produce scores, which we put through a softmax and trained with a cross-entropy loss. See appendix 7.2 for further details.Producing appropriate candidate answers $C$ to train by contrasting abstract relational structures (LABC) is straightforward. For a problem involving function $f$ , we sample functions ${ \hat { f } } \sim F \setminus \{ f \}$ at random and populate $C$ with the $c _ { \hat { f } }$ , where $c _ { \hat { f } } = \hat { f } ( V _ { t } )$ . In other words, $c _ { \hat { f } }$ adheres to some relational structure, but just not the structure apparent in the source set. Thus, to determine which candidate is correct, the model necessarily has to first infer the particular relational structure of the source set. Because the implementation of this training regime requires knowledge of the underlying structure of the data (i.e. the space of all functions), we refer to this condition as LABC-explicit SMT.
+
+# 5.2 EXPERIMENT 1: NOVEL DOMAIN TRANSFER
+
+We replicated the main findings of the visual analogy task (Experiment 1 S3.2): models trained without LABC performed at just above chance, whereas models trained with LABC achieved accuracies of just under $9 0 \%$ . Note that we tested models with candidates generated using the functions in $F$ not equal to the function used to define the relation in the source set. If instead we were to generate random vectors as candidates, the models could simply learn and embed all possible $f \in F$ into their weights, and choose the correct answer by process of elimination; any candidate that does not satisfy $f ( \bar { V _ { t } } )$ for any $f$ is necessarily an incorrect candidate. This method does not require any analogical reasoning, since the model can outright ignore the relation instantiated in the source set, which is precisely a type of back-door solution characteristic of neural network approaches. These results are intuitive at first glance – a model that cannot use back-door solutions, and instead is required to be more discerning at training time will perform better at test time. However, this intuition is less obvious when testing demands novel domain transfer. In other words, it is not obvious that a model that has learned to discern the various functions in $F$ would necessarily be able to flexibly apply the functions in ways never-before-seen, as is demanded in the test set.
+
+![](images/b94acb5874c6e39cf563f113ea5c3a87d38dfa5d911a8e6a40555e0844afe0cf.jpg)  
+Figure 6: Structure alignment and mapping on symbolic, numeric data. In this task a particular structure is implemented as a set-function, which in the depicted example is $f = \operatorname* { m i n }$ . For example, the “answer” $a _ { 1 }$ could denote the minimum size from the sizes in the source symbolic vector set. The model must then evaluate the minimum for the “aligned” domain, which in the depicted case is intensity. In this depicted example the candidate does not adhere to this structure, so it would be an incorrect candidate. The correct candidate would look like the answer vector to the right of the image.
+
+# 5.3 EXPERIMENT 2: NOVEL DOMAIN TRANSFER WITH AUTOMATIC LABC METHODS
+
+We explored ways to replicate the results of Experiment 1 without hand-crafting the candidate answers. For our first method, (LABC-topk) we uniformly sampled integer values for the $c \in C$ from within some range $( 0 , 6 4 )$ rather than computing them as $\hat { f } ( V _ { t } )$ for some ${ \hat { f } } \in F$ . As mentioned previously, such a method should encourage back-door memorization based solutions, since for most candidates, $c \neq { \hat { f } } ( V _ { t } )$ for any ${ \hat { f } } \in F$ . To counter this, we randomly generated a set of candidates, performed a forward pass with each, selected the top- $k$ scalar scores produced by the model, and backpropagated gradients only through these $k$ candidates. Intuitively, this forces the model to train on only those candidates that are maximally confusing. Thus, we rely on random generation to chose our contrasting candidates, and rely on the model itself to sub-select them from within the randomly generated pool. This method improves performance from chance $( 2 5 \% )$ to approximately $7 7 \%$ .
+
+It is possible that this top- $k$ method simply exploited random sampling to stumble on the candidates that would have otherwise been hand-crafted. Indeed, for more difficult problems (involving real images, for example) a random generator may not produce anything resembling data from the underlying distribution, making this method less suitable. We thus replicated the top- $k$ experiment, but actively excluded randomly generated candidates that satisfied $c = { \hat { f } } ( V _ { t } )$ for some ${ \hat { f } } \in F$ . Performance fell to $4 3 \%$ , confirming this intuition, but interestingly, still greatling improving baseline performance.
+
+Table 3: Test performance on our symbolic analogy in the four different training regimes.   
+
+<table><tr><td>Training method</td><td>Test accuracy</td></tr><tr><td>LABC; explicit SMT</td><td>0.89</td></tr><tr><td>LABC; top-k</td><td>0.77</td></tr><tr><td>LABC;adversarial</td><td>0.62</td></tr><tr><td>Random candidate answers</td><td>0.25</td></tr></table>
+
+Finally, we considered a method for generating candidates that did not depend on random generation (LABC-adversarial), but instead exploited a generator model. The generator was identical to the model used to solve the task except its input consisted only of the target set $V _ { t }$ , its output was a proposed candidate vector $c$ . This candidate vector was then passed to the original model, which solved the analogy problem as before. The generator model was trained to maximize the score given by the analogy model; i.e. it was trained to produce maximally confusing candidates. The overall objective therefore resembled a two-player minimax game (Goodfellow et al., 2014):
+
+$$
+\operatorname* { m i n } _ { \theta } \operatorname* { m a x } _ { \phi } \mathcal { L } ( f _ { \theta } ( S _ { 1 } , a _ { 1 } , S _ { 2 } , a _ { 2 } , g _ { \phi } ( S _ { 2 } ) ) )
+$$
+
+where $f _ { \theta }$ is the analogy model and $g _ { \phi }$ is the candidate proposal model. Using this method to propose candidates improved the model’s test performance from chance $( 2 5 \% )$ to approximately $6 2 \%$ .
+
+These latter experiments show interesting links between LABC, GANs (Goodfellow et al., 2014), and possibly self-play (Silver et al., 2017). Indeed, the latter two approaches may be seen as automated methods that can approximate LABC. For example, in self-play, agents continuously challenge their opponents by proposing maximally challenging data. In the case of analogy, maximally challenging candidates may be those that are semantically-plausible rather than simply perceptually plausible. To our knowledge no prior work has demonstrated the effects of adversarial training regimes on out-of-distribution generalization in a controlled setting like the present context.
+
+# 6 DISCUSSION
+
+Our experiments show that simple neural networks can learn to make analogies with visual and symbolic inputs, but this is critically contingent on the way in which they are trained; during training, the correct answers should be contrasted with alternative incorrect answers that are plausible at the level of relations rather than simple perceptual attributes. This is consistent with the SMT of human analogy-making, which highlights the importance of inter-domain comparison at the level of abstract relational structures. At the same time, in the visual analogy domain, our model reflects the idea of analogy as closely intertwined with perception itself. We find that models that are trained by LABC to reason better by analogy are, perhaps surprisingly, also better able to extrapolate to a wider range of input values. Thus, making better analogies seems connected to the ability of models to perceive and represent their raw experience.
+
+Recent literature has questioned whether neural networks can generalise in systematic ways to data drawn from outside the training distribution (Lake & Baroni, 2018). Our results show that neural networks are not fundamentally limited in this respect. Rather, the capacity needs to be coaxed out through careful learning. The data with which these networks learn, and the manner in which they learn it, are of paramount importance. Such a lesson is not new; indeed, the task of one-shot learning was thought to be difficult, if not impossible to perform using neural networks, but was nonetheless “solved” using appropriate training objectives, models, and optimization innovations (e.g., (Santoro et al., 2016; Finn et al., 2017)). The insights presented here may guide promising, general purpose approaches to obtain similar successes in flexible, generalisable abstract reasoning.
+
+Earlier work on analogical reasoning in AI and cognitive science employed constructed symbolic stimuli or pre-processed perceptual input (Carbonell (1981); Hummel & Holyoak (1997); Hofstadter (1996); Larkey & Love (2003) inter alia; see Gentner & Forbus (2011) for a full review). More recenty, Reed et al. (2015) learn an analogy model on top of pre-trained visual embeddings of geometric figures and rendered graphics, while Mikolov et al. (2013) show how analogies can be made via non-parametric operations on vector-spaces of text-based word representations. While the input to our visual analogy model is less naturalistic than these latter cases, this permits clear control over the semantics of training or test data when designing and evaluating hypotheses. Our study is nonetheless the only that we are aware to demonstrates such flexible, generalisable analogy making in neural networks learning end-to-end from raw perception. It is therefore a proof of principle that even very basic neural networks have the potential for strong analogical reasoning and generalization.
+
+As discussed in Sec. 5.3, in many machine-learning contexts it may not be possible to know exactly what a ‘good quality’ negative example looks like. The experiments there show that, in such cases, we might still achieve notable improvements in generalization via methods that learn to play the role of teacher by presenting alternatives to the main (student) model, as per Shafto et al. (2014). This underlines the fact that, for established learning algorithms involving negative examples such as (noise) contrastive estimation (Smith & Eisner, 2005; Gutmann & Hyvarinen, 2010) or negative ¨ sampling (Mikolov et al., 2013), the way in which negative examples are selected can be critical3. It may also help to explain the power of methods like self-play (Silver et al., 2016), in which a model is encouraged to continually challenge itself by posing increasingly difficult learning challenges.
+
+Analogies as the functions of the mind To check whether a plate is on a table we can look at the space above the table, but to find out whether a picture is on a wall or a person is on a train, the equivalent check would fail. A single on function operating in the same way on all input domains could not explain these entirely divergent outcomes of function evaluation. On the other hand, it seems implausible that our cognitive system encodes the knowledge underpinning these apparently distinct applications of the on relation in entirely independent representations. The findings of this work argue instead for a different perspective; that a single concept of on is indeed exploited in each of the three cases, but that its meaning and representation is sufficiently abstract to permit flexible interaction with, and context-dependent adaptation to, each particular domain of application. If we equate this process with analogy-making, then analogies are something like the functions of the mind. We believe that greater focus on analogy may be critical for replicating human-like cognitive processes, and ultimately human-like intelligent behaviour, in machines. It may now be time to revisit the insights from past waves of AI research on analogy, while bringing to bear the tools, perspectives and computing power of the present day.
+
+# ACKNOWLEDGMENTS
+
+Thanks to Greg Wayne and Jay McClelland for very helpful comments, and to Emilia Santoro, Adam’s most important publication to date.
+
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+
+# 7 MODEL DETAILS
+
+# 7.1 VISUAL ANALOGY PROBLEMS
+
+The CNN was 4-layers deep, with 32 kernels per layer, each of size $3 \times 3$ with a stride of 2. Thus, each layer downsampled the image by half. Each panel in a question was $8 0 \times 8 0$ pixels, and greyscale. The panels were presented one at a time to the CNN to produce 9 total embeddings (3 for the source sequence, 2 for the target sequence, and 4 for each candidate). We then used these embeddings to compile 4 distinct inputs for the RNN. Each input was comprised of the source sequence embeddings, the target sequence embeddings, and a single candidate embedding, for a total of 6 embeddings per RNN-input sequence. We passed these independently to the RNN (with 64 hidden units), whose final output was then passed through a linear layer to produce a single scalar. 4 such passes (one for each source-target-candidate sequence) produced 4 scalar scores, denoting the model’s evaluation of the suitability of the particular candidate for the analogy problem. Finally, a softmax was computed across the scores to select the model’s “answer”. We used a cross entropy loss function and the Adam optimizer with a learning rate of $1 e ^ { - 4 }$ .
+
+# 7.2 SYMBOLIC ANALOGY PROBLEMS
+
+A given input consisted of a set of vectors 16-dimensional vectors. This set included 8 vectors comprising $S _ { 1 }$ , one vector $d _ { 1 }$ , 8 vectors comprising $S _ { 2 }$ , and 8 vectors comprising the set of candidate vectors $C$ . Vectors were given a single digit binary variable tag to denote whether they were members of the source or target set (augmenting their size to 17-dimensions).
+
+We note that the entity vectors have 0’s in their unused dimensions. While this may make the problem easier, this experiment was designed to explicitly test domain-transfer generalization, moreso than an ability to discern the domains that need to be considered, by stripping away any difficulties in perception (i.e., in identifying the relevant domains), and seeing if the effect of LABC persists. Thus, at test time the model should have an easy time identifying the relevant dimensions, but it will never have seen the particular transfer from, say, dimension $i$ to dimension $j$ . So, even though it may have an easy time identifying and processing each dimension $i$ and $j$ , it may be incapable (without LABC) of integrating the information processed from each of these dimensions.
+
+We employed a parallel processing architecture, similar to the visual analogy experiments, with a Relation Network (128 unit, 3 layer MLP with ReLU non-linearities for the $g _ { \theta }$ function and a similar 2-layer MLP for the $f _ { \phi }$ function) replacing the RNN core. Thus, a single model processed $( S _ { 1 } , d _ { 1 } , S _ { 2 } , c _ { n } )$ , with $c _ { n }$ being a different candidate vector from $C$ for each parallel pass. The model’s output was a single scalar denoting the score assigned to the particular candidate $c _ { n }$ – these scores were then passed through a softmax, and training proceeded using a cross entropy loss function. We used batch sizes of 32 and the Adam optimizer with a learning rate of $3 e ^ { - 4 }$ .
+
+# 8 SUPPLEMENTARY RESULTS
+
+Table 4: Results for the visual analogy task (RNN Model).   
+
+<table><tr><td rowspan="2">Extrapolation</td><td rowspan="2">Mean</td><td colspan="2">LABC Normal (Train)</td><td rowspan="2">Mix (Train) 0.94</td><td rowspan="2">LABC (Contrasting) 0.62</td><td rowspan="2">Normal (Contrasting) 0.43</td><td rowspan="2">Mix (Contrasting) 0.56</td><td rowspan="2">LABC (Normal) 0.45</td><td rowspan="2">Normal (Normal) 0.44</td><td rowspan="2">Mix (Normal) 0.39</td></tr><tr><td>0.94</td><td>(Train) 0.95</td></tr><tr><td rowspan="2">Interpolation</td><td>Std</td><td>0.005</td><td>0.007</td><td>0.005</td><td>0.02</td><td>0.009</td><td>0.012</td><td>0.01</td><td>0.01</td><td>0.04</td></tr><tr><td>Mean</td><td>0.94</td><td>0.97</td><td>0.94</td><td>0.93</td><td>0.45</td><td>0.89</td><td>0.65</td><td>0.89</td><td>0.87</td></tr><tr><td rowspan="2">Novel Domain Transfer</td><td>Std</td><td>0.003</td><td>0.003</td><td>0.003</td><td>0.004</td><td>0.004</td><td>0.008</td><td>0.01</td><td>0.006</td><td>0.01</td></tr><tr><td>Mean</td><td>0.88</td><td>0.83</td><td>0.85</td><td>0.87</td><td>0.48</td><td>0.88</td><td>0.7</td><td>0.82</td><td>0.79</td></tr><tr><td rowspan="2">Novel Domain (shape colour)</td><td>Std</td><td>0.015</td><td>0.01</td><td>0.015</td><td>0.005</td><td>0.02</td><td>0.009</td><td>0.01</td><td>0.01</td><td>0.02</td></tr><tr><td>Mean</td><td>0.87</td><td>0.84</td><td>0.85</td><td>0.78</td><td>0.50</td><td>0.80</td><td>0.51</td><td>0.61</td><td>0.58</td></tr><tr><td rowspan="2"></td><td>Std</td><td>0.007</td><td>0.008</td><td>0.007</td><td>0.004</td><td>0.02</td><td>0.006</td><td>0.02</td><td>0.01</td><td>0.02</td></tr><tr><td>Mean</td><td>0.87</td><td>0.85</td><td>0.86</td><td>0.76</td><td>0.45</td><td>0.75</td><td>0.5</td><td>0.57</td><td>0.54</td></tr><tr><td>Novel Domain (line type)</td><td>Std</td><td>0.006</td><td>0.004</td><td>0.006</td><td>0.02</td><td>0.01</td><td>0.02</td><td>0.02</td><td>0.02</td><td>0.01</td></tr></table>
+
+# 9 MODEL COMPARISON DETAILS
+
+Our application of a ResNet-50 processes all nine panels simultaneously (five analogy question panels along with the four multiple choice candidates) as a set of input channels. The Parallel ResNet-50 processes six panels simultaneously as input channels (five analogy question panels along with one multiple choice candidate) to produce a score. Then, similar to the RNN model described above, the candidate with the highest score is chosen by the model. The parallel relation network model also processes six panels simultaneously, using a convnet to obtain panel embeddings and using a relation network (Santoro et al., 2017) for computing a score. For full model architecture details, see the appendix.
+
+Interestingly, the model with strongest generalisation is the parallel relation network, with a particularly high accuracy of $9 5 \%$ on the held out domain-transfer test set. This model was tested on a mixture of multiple choice candidates (that included semantically plausible and perceptually plausible candidates), indicating that models trained with LABC do not over-specialize to problem settings where only semantically plausible candidates are available. We also observe that during normal training, test set performance can oscillate between good solutions and poor solutions, indicated by the high standard deviation in the test set accuracy. These results imply that there are multiple model configurations that have good performance on the training set, but that only some of these configurations have the desired generalisation behaviour on the test set. LABC encourages a model to learn the configurations that generalise at the most abstract semantically-meaningful level, as desired.
+
+We also note that the fact that a model trained in the normal regime performs marinally better than one trained using (normal+)LABC data on test questions involving perceptually-plausible candidates. We believe this may be understood as a symptom of the strong ability of deep learning models to specialize to the exact nature of the problems on which they are trained. The model comparison experiments demonstrate that this negligible but undesirable specialization effect is outweighed by the greater benefits of training with LABC on test questions with semantically-plausible candidates (i.e. those that require a higher-level semantic interpretation of the problem). Training with LABC will therefore yield a much higher expected performance, for instance, in cases where the exact details of the test questions is not known.
+
+# 10 FURTHER DISCUSSION AND RELATED WORK
+
+It is interesting to consider to what extent the effects reported in this work can transfer to a wider class of learning and reasoning problems beyond classical analogies. The importance of teaching concepts (to humans or models) by contrasting with negative examples is relatively established in both cognitive science (Shafto et al., 2014; Smith & Gentner, 2014) and educational research (Silver, 2010; Ali, 1981). Our results underline the importance of this principle when training modern neural networks to replicate human-like cognitive processes and reasoning from raw perceptual input. In cases where expert understanding of potential data exists, for instance in the case of active learning with human interaction, it provides a recipe for achieving more robust representations leading to far greater powers of generalization. We should aspire to select as negative examples those examples that are plausible considering the most abstract principles that describe the data.
+
+A further notable property of our trained networks is the fact they can resolve analogies (even those involving with unfamiliar input domains) in a single rollout (forward pass) of a recurrent network. This propensity for fast reasoning has an interesting parallel with the fast and instinctive way in which humans can execute visual analogical reasoning (Morrison et al., 2001; Qiu et al., 2008).
+
+# 10.1 DISTANCE METRIC APPROACHES
+
+LBC shares similarities with distance metric approaches such as the large-margin nearest neighbor classifier (LMNN) (Weinberger & Saul, 2009), the triplet loss (Schroff et al., 2015), and others. In these approaches the goal is to transform inputs such that the distance between input embeddings from the same class is small, while the distance between input embeddings from different classes is large. Given these improved embeddings, classification can proceed using off-the-shelf classification algorithms, such as k-nearest neighbors. We note that these approaches emphasize the form of the loss function and the quality of the resultant input embeddings on subsequent classification. However, the goal of LBC is not to induce better classification per se, as it is in these methods. Instead, the goal is to induce out-of-distribution generalisation by virtue of improved abstract understanding of the underlying problem. It is unclear, for example, whether the embeddings produced by LMNN or the triplet loss are naturally amenable to this kind of generalisation, and as far as we are aware, it has not beed tested. Nonetheless, LBC places a critical focus on the nature, or quality of the data comprising the incorrect classes, and is agnostic to the exact nature of the loss function. Thus, it is possible to use previous approaches (e.g., LMNN or triplet loss, etc.) in conjunction with LBC, which we do not explore. LBC also shares similarities to recent generative adversarial active learning approaches (Zhu & Bento, 2017). However, these approaches do not explicitly point to the effects of the quality of incorrect samples to out-of-distribution generalisation, nor are we aware of any experiments that test abstract generalisation using networks trained with generative samples.
+
+11 VISUAL ANALOGY EXAMPLES
+
+![](images/6c887c97b9cf374c54da3f73b5c9688d5f16322940c730e406713c7b7ee7ac4f.jpg)  
+relation: AND source domain: shape position target domain: line type
+
+![](images/5f2940c8bccbf6a281a70dafdf5e30f94a29ecd20934768e3c8ec313c7058320.jpg)
+
+relation: XOR source domain: line type target domain: line type relation: progression source domain: shape size target domain: shape size relation: OR source domain: shape type target domain: line colour relation: OR source domain: shape position target domain: shape size
+
+![](images/3f0bb90de47faf36c0b053a33f9795d651ffb76f012301c4dc10de841a3dd3f3.jpg)
+
+![](images/0931a3e0635c36f830dfaa37c1283d307845ad74c3871ca8e8b7880fac8ed6b4.jpg)
+
+![](images/d46e9282ad85f8519139ca5e3c7f2ff6bbb70c7c263d5fcbd2a60651efdfef53.jpg)
+
+![](images/fa4c6a8b57627aea0749d5b3a8bcdf34a6351aa8b7c94cc5ccc1e9f60a5d0887.jpg)
+
+relation: OR source domain: line colour target domain: shape colour relation: XOR source domain: line colour target domain: shape colour relation: progression source domain: shape type target domain: line colour relation: OR source domain: shape type target domain: line colour relation: XOR source domain: line type target domain: line colour relation: XOR source domain: line type target domain: shape colour relation: XOR source domain: shape colour target domain: line colour
+
+![](images/1c7e8822122f4f39cdbb09af7f047629540224cf89eff12d4feeb9caaecfbf3d.jpg)
+
+![](images/df0def5f43ded006398eabe6729b769fa6081580b7c2d49964fafc6f72dee720.jpg)
+
+![](images/a2a3d9c3146c5fcfaa7281d33689119be5f2b626894ca62e044fa567d2026311.jpg)
+
+![](images/63848025d67f9248268dbae3646eac454147bda19faab36ddbf0549696269385.jpg)  
+Figure 7: Examples of visual analogy problems. These visual analogy examples have been selected from the interpolation test set. The correct multiple choice candidate for each problem is highlighted in green. In the top half, we have randomly chosen examples where our RNN model trained with LABC selects the correct answer. In the bottom half, we have randomly selected examples where our RNN model trained with LABC chooses the incorrect candidates (highlighted in red). The performance of this model on the interpolation test set is $9 3 \%$ .
+
+![](images/31d6452f69f457157c84c164b6f03aa362fe3f77dd16d56511d795e47db27157.jpg)
+
+![](images/1088ca6d84fbd4378d0c8f914943b05dd27d26104b0abecd668bde6cc06f21ca.jpg)

parse/train/SylLYsCcFm/SylLYsCcFm_content_list.json

@@ -0,0 +1,1882 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING TO MAKE ANALOGIES BY CONTRASTING ABSTRACT RELATIONAL STRUCTURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Felix Hill\\*, Adam Santoro∗, David G. T. Barrett, Ari Morcos & Tim Lillicrap Deepmind, London {felixhill,adamsantoro,barrettdavid,arimorcos,countzero}@google.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            250,
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+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Analogical reasoning has been a principal focus of various waves of AI research. Analogy is particularly challenging for machines because it requires relational structures to be represented such that they can be flexibly applied across diverse domains of experience. Here, we study how analogical reasoning can be induced in neural networks that learn to perceive and reason about raw visual data. We find that the critical factor for inducing such a capacity is not an elaborate architecture, but rather, careful attention to the choice of data and the manner in which it is presented to the model. The most robust capacity for analogical reasoning is induced when networks learn analogies by contrasting abstract relational structures in their input domains, a training method that uses only the input data to force models to learn about important abstract features. Using this technique we demonstrate capacities for complex, visual and symbolic analogy making and generalisation in even the simplest neural network architectures. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The ability to make analogies – that is, to flexibly map familiar relations from one domain of experience to another – is a fundamental ingredient of human intelligence and creativity (Gentner, 1983; Hofstadter, 1996; Hummel & Holyoak, 1997; Lovett & Forbus, 2017). As noted, for instance, by Holyoak & Thagard (1995), analogies gave Roman scientists a deeper understanding of sound when they leveraged knowledge of a familiar source domain (water waves in the sea) to better understand the structure of an unfamiliar target domain (acoustics). The Romans ‘aligned’ relational principles about water waves (periodicity, bending round corners, rebounding off solids) to phenomena observed in acoustics, in spite of the numerous perceptual and physical differences between water and sound. This flexible alignment, or mapping, of relational structure between source and target domains, independent of perceptual congruence, is a prototypical example of analogy making. ",
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+    {
+        "type": "text",
+        "text": "It has proven particularly challenging to replicate processes of analogical thought in machines. Many classical or symbolic AI models lack the flexibility to apply predicates or operations across diverse domains, particularly those that may have never previously been observed. It is natural to consider, however, whether the strengths of modern neural network-based models can be exploited to solve difficult analogical problems, given their capacity to represent stimuli at different levels of abstraction and to enable flexible, context-dependent computation over noisy and ambiguous inputs. ",
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+        "text": "In this work we demonstrate that well-known neural network architectures can indeed learn to make analogies with remarkable generality and flexibility. This ability, however, is critically dependent on a method of training we call learning analogies by contrasting abstract relational structure (LABC). We show that simple architectures can be trained using this approach to apply abstract relations to never-before-seen source-target domain mappings, and even to entirely unfamiliar target domains. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our work differs from previous computational models of analogy in two important ways. First, unlike previous neural network models of analogy, we optimize a single model to perform both stimulus representation and cross-domain mapping jointly. This allows us to explore the potential benefit of interactions between perception, representation and inter-domain alignment, a question of some debate in the analogy literature (Forbus et al., 1998). Second, we do not instantiate an explicit cognitive theory or analogy-like computation in our model architecture, but instead use this theoretical insight to inform the way in which the model is trained. ",
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+    {
+        "type": "image",
+        "img_path": "images/8b03221091c0f3eef96e3dc0f1d096073b5ff4ded47879e16e92bfcc2ede1bbc.jpg",
+        "image_caption": [
+            "Figure 1: A visual analogy problem. In this example, the model must (1) identify a relation (Progression ) on a particular domain ( shape quantity ) in the source sequence (top), and (2) apply it to a different domain ( line color ) in order to find the candidate answer panel that correctly completes target sequence (bottom). There are seven possible domains and four possible relations in the dataset. "
+        ],
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+        "page_idx": 1
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "2 ANALOGIES AS HIGH-LEVEL PERCEPTION AND STRUCTURE MAPPING",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Perhaps the best-known explanation of human analogical reasoning is Structure Mapping Theory (SMT) (Gentner, 1983). SMT emphasizes the distinction between two means of comparing domains of experience; analogy and similarity. According to SMT, two domains are similar if they share many attributes (i.e. properties that can be expressed with a one-place predicate like BLUE(sea) ), whereas they are analogous if they share few attributes but many relations (i.e. properties expressed by many-place predicates like BENDS-AROUND(sea, solid-objects) ). SMT assumes that our perceptions can be represented as collections of attributes and structured relations, and that these representations do not necessarily depend on the subsequent mappings that use them. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The High-Level Perception (HLP) theory of analogy (Chalmers et al., 1992; Mitchell, 1993) instead construes analogy as a function of tightly-interacting perceptual and reasoning processes, positing that the creation of stimulus representations and the alignment of those representations are mutually dependent. For example, when making an analogy between the sea and acoustics, we might represent certain perceptual features (the fact that the sea appears to be moving) and ignore others (the fact that the sea looks blue), because the particular comparison that we make depends on location and direction, and not on colour. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work we aim to induce flexible analogy making in neural networks by drawing inspiration from both SMT and HLP. The perceptual and domain-comparison components of our models are connected and jointly optimised end-to-end, which, as posited by HLP, reflects a high degree of interaction between perception and domain comparison. On the other hand, the key insight of this paper, LABC, is directly motivated by SMT. We find that LABC greatly enhances the ability of our networks to resolve analogies in a generalisable way by encouraging them to compare inputs at the more abstract level of relations rather than the less abstract level of attributes. LABC organizes the training data such that the inference and mapping of relational structure is essential for good performance. This means that problems cannot be resolved by considering mere similarity of attributes, or even less appropriately, via spurious surface-level statistics or memorization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 VISUAL ANALOGY PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Our first experiments involve greyscale visual scenes similar to those previously applied to test both human reasoning ability (Raven, 1983; Geary et al., 2000) and reasoning in machine learning models (Bongard, 1967; Fleuret et al., 2011; Barrett et al., 2018). Each scene is composed of a source sequence, consisting of three panels (distinct images), a target sequence, consisting of two panels, and four candidate answer panels (Fig. 1). In the source sequence a relation $r$ is instantiated, where $r$ is one of four possible relations from the set $R = \\left\\{ \\begin{array} { r l r l } \\end{array} \\right.$ XOR , OR , AND Progression $\\}$ . Models must then consider the two panels in the target sequence, together with the four candidate answer panels, to determine which answer panel best completes the target sequence – by analogy with the source sequence – so that $r$ is also instantiated (Fig. 2). ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9c82dc2f464a457fc9b84da7f1aeffa91069718361d96431eef47cdbaf473c98.jpg",
+        "image_caption": [
+            "Figure 2: In LABC, the multiple choice candidates are all semantically plausible, in that they are all consistent completions of the target domain using some relation. Only the correct answer uses the same relation as the source domain, so, the only way to solve the problem is to use analogical reasoning. In contrast, perceptually plausible incorrect candidates are consistent with the target domain attributes but not the relations and all other possible candidates are inconsistent with the target domain relation and attributes. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The notion of domain is critical to analogy. In our visual analogy task, a relation can be instantiated in one of seven different domains: line type , line colour , shape type , shape colour , shape size , shape quantity and shape position (see Fig. 1 and Appendix Fig. 7 for examples). Within a panel of a given domain, the attributes present in the scene (such as the colour of the shapes or the positions of the lines) can take one of 10 possible values. A question in the dataset is therefore defined by a relation $r$ , a domain $d _ { s }$ on which $r$ is instantiated in the source sequence, a set of values for the source-domain $v _ { 1 } ^ { s } \\cdot \\cdot \\cdot v _ { 3 } ^ { s }$ , a target domain $d _ { t }$ , values for the target-domain $v _ { 1 } ^ { t } \\cdot \\cdot \\cdot v _ { 3 } ^ { t }$ , the position $k \\in \\{ 1 \\cdots 4 \\}$ of the correct answer among the answer candidate panels and whatever is instantiated in the three incorrect candidate answer panels $c _ { i } , i \\neq k$ . Note, however, that the values of certain domain attributes that are not relevant to a given question (such as the colour of shapes in the shape quantity domain) still have to be selected, and can vary freely. Thus, despite the small number of latent factors involved, the space of possible questions is of the order of ten million. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The interplay between relations, domains and values makes it possible to construct questions that require increasing degrees of abstraction and analogymaking. The simplest case involves a relation $r$ , a domain $d _ { s } \\ = = \\ d _ { t }$ , and values $v _ { i } ^ { s } ~ = = ~ v _ { i } ^ { t }$ that are common to both source and target sequences (Fig 3a). To solve such a question a model must identify a candidate answer panel that results in a copy of the source sequence in the target sequence. This does not seem to require any meaningful understanding of $r$ , nor any particular analogy-making. Somewhat greater abstraction and analogymaking is required for questions involving a single domain $\\scriptstyle ( d _ { s } = = d _ { t }$ ), but different values in the source and target sequence $v _ { i } ^ { s } \\neq v _ { i } ^ { t }$ (Fig 3b). In this case the model must learn that, in a given domain, the relation $r$ can apply to a range of different values. Finally, in the full analogy questions considered in this study, the relation $r$ can be instantiated on different domains in the source and target sequences (i.e. $d _ { t } \\neq d _ { s }$ ; Fig 3c). These questions require a sensitivity to the idea that a single relation $r$ can be applied in different (but related) ways to different domains of experience.1 ",
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+    {
+        "type": "text",
+        "text": "3.1 METHODS ",
+        "text_level": 1,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our model consisted of a simple perceptual front-end – a convolutional neural network (CNN) – which provided input for a recurrent neural network (RNN) by producing embeddings for each image panel independently. The RNN processed the source sequence embeddings, the target sequence embeddings, and a single candidate embedding, to produce a scalar score. Four such passes (one for each source-target-candidate set) produced four scalar scores, quantifying how the model evaluated the suitability of the particular candidate. Finally, a softmax was computed across the scores to select the model’s ‘answer’. Further model details are in appendix 7.1. ",
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+    },
+    {
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+        "img_path": "images/03afe4004d338e78df0096b9270a3cb40e2485857265fe4ad9fe922d7eb4938f.jpg",
+        "image_caption": [
+            "Figure 3: (a), (b), (c) Three types of visual reasoning questions. Each question requires a different degree of analogy making, with the question on the right demanding the most fluid and abstract application of the underlying relation. (d) Learning analogies by contrasting. When learning by contrasting, each answer choice is consistent with a relational structure in the target sequence. Only the correct answer choice is consistent with relations in both the source and target domains. This forces the network to consider the source sequence to infer the correct structure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Learning Analogies By Contrasting (LABC) In the default setting of our data generator – the normal training regime – for a question involving source domain $d _ { s }$ , target domain $d _ { t }$ and relation $r$ , the candidate answers can contain any (incorrect) values chosen at random from $d _ { t }$ (Fig. 2). By selecting incorrect candidate answers from the same domain $d _ { t }$ as the correct answer, we ensure that they are perceptually plausible, so that the problem cannot be solved trivially by matching the domain of the question to one of the answers. Even so, the baseline training regime may allow models to find perceptual correlations that allow it to arrive at the correct answer consistently over the training data. We can make this less likely by instead training the model to contrast abstract relational structure – the LABC regime – simply by ensuring that incorrect answers are both perceptually and semantically plausible (Fig. 2). More specifically, incorrect answers are selected from $d _ { t }$ such that each answer $c _ { i }$ completes a decoy relation $\\boldsymbol { \\hat { r } _ { i } } \\neq \\boldsymbol { r }$ with the target sequence. LABC ensures that, during training, models have no alternative but to first observe a relation $r$ in the source domain and consider and complete the same relation in the target domain - i.e. to execute a full analogical reasoning step.2 ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/5e3309230dca0505df3671e356defa265364450c63ddf00a0765bd20568366af.jpg",
+        "image_caption": [
+            "Figure 4: Results of the three experiments in the visual analogy domain for a network that learns from random candidate answers, by contrasting abstract structures or both types of question interleaved. Bar heights depict the means across eight seeds in each condition; standard errors were $< 0 . 0 1$ for each condition (not shown – see the appendix Table 4 for the values) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Note that in all experiments reported below, we generated 600,000 training questions, 10,000 validation questions and test sets of 100,000 questions. These data will be published with the paper. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 EXPERIMENT 1: NOVEL DOMAIN TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            594,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A key aspect of analogy-making is the process of comparing or aligning two domains. We can measure how well models acquire this ability by testing them on analogies involving unfamiliar source domain target domain transfers. For each of the seven possible target domains $d _ { t }$ we randomly selected a source domain $d _ { s } \\ne d _ { t }$ , yielding a test set of seven domain transfer pairs $[ d _ { s }  d _ { t } ]$ . Our models were then trained on questions involving one of the remaining $7 \\times 7 - 7 = 4 2$ domain transfer pairs. For a test question involving domains $d _ { s }$ and $d _ { t }$ , each model was therefore familiar with $d _ { s }$ and $d _ { t }$ but had not been trained to make an analogy from $d _ { s }$ to $d _ { t }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We found that a network can indeed learn to apply relations by analogy involving novel domain transfers, but that this ability crucially relies on learning by contrasting. The effect is strong; for the most focused test questions involving semantically plausible (contrasting) candidate answers the model trained by contrasting achieves $83 \\%$ accuracy (depending on the held-out domain), versus $58 \\%$ for a model trained with randomly-chosen candidate answers. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 EXPERIMENT 2: NOVEL TARGET DOMAIN ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Humans can use analogies to better understand comparatively unfamiliar domains, as in the Roman explanation of acoustics by analogy with the sea. To capture this scenario, we held out two domains ( line type and shape colour , chosen at random) from the model’s training data, and ensured that each test question involved one of these domains. To make sense of the test questions, a model must therefore (presumably) learn to represent the relations in the dataset in a sufficiently general way that this knowledge can be applied to completely novel domains. Any model that resolves such problems successfully must therefore exploit any (rudimentary) perceptual similarity between the test domain and the domains that were observed during training. For instance, the process of applying a relation in the shape colour domain may recruit similar feature detectors to those required when applying it to the line colour domain. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ee7cf291ece3eb6d65ac621ae8b3c95affe3e7dfe4967c01cb5ee77d41775f37.jpg",
+        "table_caption": [
+            "Table 1: Test performance of a selection of visual reasoning network architectures trained in the normal regime and with LABC on the Novel Domain Transfer experiment. The test questions include those with both semantically-plausible and merely perceptually-plausible incorrect answers. Mean $\\left( + \\mathbf { S } \\mathbf { D } \\right)$ for 10 models in each condition with different random initialisations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>LABC training</td><td>Normal training</td></tr><tr><td>RNN (this paper)</td><td>0.90 (0.01)</td><td>0.79 (0.09)</td></tr><tr><td>ResNet-50</td><td>0.82 (0.02)</td><td>0.77 (0.10)</td></tr><tr><td>Parallel ResNet-50</td><td>0.89 (0.01)</td><td>0.75 (0.14)</td></tr><tr><td>Parallel Relation Net (WReN)</td><td>0.95 (0.017)</td><td>0.72 (0.21)</td></tr></table>",
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+        "type": "text",
+        "text": "Surprisingly, we found that the network can indeed learn to make sense of the unfamiliar target domains in the test questions, although again this capacity is boosted by LABC (Fig 4b). Accuracy in the LABC condition on the most focused (contrasting) test questions is lower than in the Experiment 1 $( \\sim 8 0 \\%$ , depending on the held-out domain), but well above the model trained with random answer candidates $( \\sim 6 0 \\% )$ ). Interestingly, a model trained on semantically-plausible sets of candidate answers (LABC) performs somewhat worse on test questions whose answers are merely perceptuallyplausible than a model trained in that (normal) regime. This deficit can be largely recovered by interleaving random-answer and contrasting candidates during training. ",
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+    {
+        "type": "text",
+        "text": "3.3.1 EXPERIMENT 3: NOVEL DOMAIN VALUES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another way in which a domain can be unfamiliar to a network is if it involves attributes whose values have not been observed during training. Since each of the seven (source and target) domains our analogy problems permits 10 values, we can measure interpolation by withholding values 1, 3, 5, 7 and 9 and measure extrapolation by withholding values 6, 7, 8, 9 and 10. To extrapolate effectively, a model must therefore be able to resolve questions at test time involving lines or shapes that are darker, larger, more-sided or simply more numerous than those in its training questions. ",
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+    },
+    {
+        "type": "text",
+        "text": "In the case of interpolation, we found that a model trained with random candidates performs very poorly on the more challenging contrasting test questions $( \\mathrm { F i g ~ 4 c ~ 4 5 \\% }$ vs $93 \\%$ for LABC), which suggests that models trained in the normal regime overfit to a strategy that bears no resemblance to human-like analogical reasoning. We verified this hypothesis by running an analysis where we presented only the target domain sequence and candidate answers to the model. After a long period of training in the normal regime, this ‘source-blind’ model achieved $9 7 \\%$ accuracy, which confirms that it indeed finds short-cut solutions that do not require analogical mapping. In contrast, the accuracy of the source-blind model in the LBAC condition converged at $32 \\%$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "We also found, somewhat surprisingly, that LBAC results in a (modest) improvement in how well models can extrapolate to novel input values (Fig 4c); a model trained on questions with both contrasting and random candidate answers performs significantly better than the normal model on the test questions with contrasting candidate answers $62 \\%$ vs. $43 \\%$ ), and mantains comparable performance on test questions with random candidate answers ( $45 \\%$ vs. $44 \\%$ ). ",
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+    {
+        "type": "text",
+        "text": "3.3.2 EXPERIMENT 4: MODEL COMPARISON ",
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+        "type": "text",
+        "text": "Finally, we explore the extent to which LABC improves generalistion for different various different network architectures, taking as a guide the models considered in Barrett et al. (2018). We run the Novel Domain Transfer experiment with each of these models, trained using both LABC and in the normal training regime. We measure generalisation on a mixed set comprised equally of test questions with semantically-plausible incorrect answer candidates (matching LABC training) and those with merely perceptually-plausible incorrect answers (matching normal training). All models perform better at novel domain transfer when trained with LABC (Table 1). This confirms that our method does not depend on the use of a specific architecture. Further, the fact that LABC yields better performance than normal training on a balanced, mixed test set shows that it is the most effective way to train models in problem instances where the exact details of test questions may be unknown. ",
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+        "type": "text",
+        "text": "3.4 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "These results demonstrate that LABC increases the ability of models to generalize beyond the distribution of their training data. This effect is observed for the prototypical analogical processes involving novel domain mappings and unfamiliar target domains (Experiments 1 & 2). Interestingly, it also results in moderate improvements to how well models extrapolate to perceptual input outside the range of their training experience (Experiment 3). ",
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+    {
+        "type": "text",
+        "text": "4 EMERGENT RELATIONAL REPRESENTATIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To understand the mechanisms that support this generalisation we analysed neural activity in models trained with LABC compared to models that were not. First, we took the RNN hidden state activity just prior to the input of the candidate panel. For a model trained via LABC, we found that these activities clustered according to relation type (e.g. progression ) more-so than domain (e.g., shape colour ) (Fig 5a, Table 2). In contrast, for models trained normally the relation",
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+        "type": "table",
+        "img_path": "images/ca51b7ec17f81566b5629eb94b887994659a0e6d187dffb08d47473b62814894.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Cluster</td><td rowspan=1 colspan=1>LABC</td><td rowspan=1 colspan=1>Normal Training</td></tr><tr><td rowspan=1 colspan=1>inter-relation dist.inter-domain dist.</td><td rowspan=1 colspan=1>5.2(± 2.2)1.8 (± 1.8)</td><td rowspan=1 colspan=1>4.4 (± 2.0)2.0 (± 1.8)</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 2: Mean $\\mathbf { ( + S D ) }$ distance between clusters of RNN hidden state representations. LABC seems to promote greater distinctions between different abstract relations in the representation space of models. Since the maximum euclidean distance between two points in a 64 dimensional unit cube is P64i 12 = 8, the distances that we observe between relational representations are close to the maximum possible. ",
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+    {
+        "type": "text",
+        "text": "based clusters overlapped to a greater extent (Fig 5b, Table 2). Thus, LABC seems to encourage the model to represent relations more explicitly, which could in turn explain its capacity to generalise by analogy to novel domains. ",
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+    {
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+        "img_path": "images/184c0d754b73faa611c0580c4fb0c482ee946c8f8af1e9a596e6fa668f875e02.jpg",
+        "image_caption": [
+            "Figure 5: LABC supports emergent relational representations. Principle component analysis (PCA) (right) and t-SNE analysis (left) of RNN hidden states. Each dot represents a (64-dimensional) state coloured according to the relation type and domain of the corresponding question. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5 SYMBOLIC ANALOGY PROBLEMS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Many of the most important studies of analogy in AI involve symbolic or numerical patterns (Hofstadter, 1996), and various neural models of semantic cognition more generally also operate on discrete, feature-based representations of input concepts (Rogers & McClelland, 2004; Devereux et al., 2018). To verify our findings in this setting, we implemented a symbolic analogy task based on feature-based stimuli. This more controlled domain allows to show that the construction of appropriate incorrect answer candidates can be learned in a proposal model that is trained jointly with a model that learns to contrast the candidates, widening the potential applications of LABC to task settings where we lack a clear understanding of the underlying abstract relational structure (Sec 5.3). ",
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+    {
+        "type": "text",
+        "text": "In our task, inputs are $D$ -dimensional vectors $v$ of discrete, integer-valued features (analogous to semantic properties such as furry or carnivorous in previous work). Across any set $V : v \\in V$ of stimuli, each feature dimension then corresponds to a domain (the domains of skin-type or dietary habits in the present example). To simulate a space of abstract relational structures on these domains, we simply take a set $F$ whose elements $f$ are common mathematical functions operating on sets: MIN , MAX , ARGMIN , ARGMAX , SUM and RANGE . Abstract relations in this context can be stated, for instance, as $\\mathrm { { S U M } } \\left( 1 , \\quad 2 , \\quad 3 ; \\quad 6 \\right)$ . Given a such a function $f$ , a set $V$ of stimulus vectors, and a random choice of domain $d$ , we can compute a $D$ -dimensional) answer vector $\\boldsymbol { a } _ { [ V , d , f ] }$ as the result of applying $f$ to $d$ on $V$ (i.e. executing $f$ on the $d$ -th dimension of each $v \\in \\dot { V } .$ ). It is now simple to randomly construct an analogy question in this setting. We select a function $f$ , source $d _ { s }$ and target $d _ { t }$ domains at random, randomly generate source $V _ { s }$ and target $V _ { t }$ stimuli. We then apply $f$ to $V _ { s }$ on $d _ { s }$ and to $V _ { t }$ on $d _ { t }$ to generate the source and target domain solutions, $a _ { [ V _ { s } , d _ { s } , f ] }$ and $a _ { [ V _ { t } , d _ { t } , f ] }$ , respectively. The inputs $V _ { s }$ , $d _ { s }$ , $a _ { [ V _ { s } , d _ { s } , f ] }$ , $V _ { t }$ and $v _ { t }$ are passed to the model, together with a ${ \\dot { d } } \\times k$ matrix of alternative answer choices that includes $a _ { [ V _ { t } , d _ { t } , f ] }$ $k$ is a fixed number of choices, four in the visual analogies presented previously). We then require the model to select which of these alternatives is the true completion of the analogy $a _ { [ V _ { t } , d _ { t } , f ] }$ . As in the visual analogy tasks, to resolve such a question the model must detect an abstract relationship in the input domain $d _ { s }$ , that explains the connection between the source stimuli $V _ { s }$ and the answer vector $a _ { [ V _ { s } , d _ { s } , f ] }$ . Once this achieved, the model must evaluate the function $f$ that describes this relationship on the source domain $d _ { t }$ with sufficient accuracy that it can identify the result of that evaluation $a _ { [ V _ { t } , d _ { t } , f ] }$ in the context of $k$ distracting alternative (incorrect) answers. ",
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+    {
+        "type": "text",
+        "text": "We study generalization in this task by restricting the particular ordered domain mappings $d _ { s }  d _ { t }$ that are observed in the training and test set; for example, aligning structure from domain 3 to domain 1 may be required in the test set, but may have never been seen before in the training set. While we withhold particular alignments (e.g., $3  1 \\AA$ ), we ensure that all dimensions are aligned ‘out-of’ $( 3  )$ and ‘into’ $(  1 )$ ) at least once in the training set. Note that this setup is directly analogous to the ‘Novel Domain Transfer’ experiment in the visual analogy problems (Sec. 3.2). ",
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+    {
+        "type": "text",
+        "text": "5.1 METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The high level model structure was similar to that of the previous experiments: candidates and their context were processed independently to produce scores, which we put through a softmax and trained with a cross-entropy loss. See appendix 7.2 for further details.Producing appropriate candidate answers $C$ to train by contrasting abstract relational structures (LABC) is straightforward. For a problem involving function $f$ , we sample functions ${ \\hat { f } } \\sim F \\setminus \\{ f \\}$ at random and populate $C$ with the $c _ { \\hat { f } }$ , where $c _ { \\hat { f } } = \\hat { f } ( V _ { t } )$ . In other words, $c _ { \\hat { f } }$ adheres to some relational structure, but just not the structure apparent in the source set. Thus, to determine which candidate is correct, the model necessarily has to first infer the particular relational structure of the source set. Because the implementation of this training regime requires knowledge of the underlying structure of the data (i.e. the space of all functions), we refer to this condition as LABC-explicit SMT. ",
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+    {
+        "type": "text",
+        "text": "5.2 EXPERIMENT 1: NOVEL DOMAIN TRANSFER ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We replicated the main findings of the visual analogy task (Experiment 1 S3.2): models trained without LABC performed at just above chance, whereas models trained with LABC achieved accuracies of just under $9 0 \\%$ . Note that we tested models with candidates generated using the functions in $F$ not equal to the function used to define the relation in the source set. If instead we were to generate random vectors as candidates, the models could simply learn and embed all possible $f \\in F$ into their weights, and choose the correct answer by process of elimination; any candidate that does not satisfy $f ( \\bar { V _ { t } } )$ for any $f$ is necessarily an incorrect candidate. This method does not require any analogical reasoning, since the model can outright ignore the relation instantiated in the source set, which is precisely a type of back-door solution characteristic of neural network approaches. These results are intuitive at first glance – a model that cannot use back-door solutions, and instead is required to be more discerning at training time will perform better at test time. However, this intuition is less obvious when testing demands novel domain transfer. In other words, it is not obvious that a model that has learned to discern the various functions in $F$ would necessarily be able to flexibly apply the functions in ways never-before-seen, as is demanded in the test set. ",
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+    {
+        "type": "image",
+        "img_path": "images/b94acb5874c6e39cf563f113ea5c3a87d38dfa5d911a8e6a40555e0844afe0cf.jpg",
+        "image_caption": [
+            "Figure 6: Structure alignment and mapping on symbolic, numeric data. In this task a particular structure is implemented as a set-function, which in the depicted example is $f = \\operatorname* { m i n }$ . For example, the “answer” $a _ { 1 }$ could denote the minimum size from the sizes in the source symbolic vector set. The model must then evaluate the minimum for the “aligned” domain, which in the depicted case is intensity. In this depicted example the candidate does not adhere to this structure, so it would be an incorrect candidate. The correct candidate would look like the answer vector to the right of the image. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.3 EXPERIMENT 2: NOVEL DOMAIN TRANSFER WITH AUTOMATIC LABC METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We explored ways to replicate the results of Experiment 1 without hand-crafting the candidate answers. For our first method, (LABC-topk) we uniformly sampled integer values for the $c \\in C$ from within some range $( 0 , 6 4 )$ rather than computing them as $\\hat { f } ( V _ { t } )$ for some ${ \\hat { f } } \\in F$ . As mentioned previously, such a method should encourage back-door memorization based solutions, since for most candidates, $c \\neq { \\hat { f } } ( V _ { t } )$ for any ${ \\hat { f } } \\in F$ . To counter this, we randomly generated a set of candidates, performed a forward pass with each, selected the top- $k$ scalar scores produced by the model, and backpropagated gradients only through these $k$ candidates. Intuitively, this forces the model to train on only those candidates that are maximally confusing. Thus, we rely on random generation to chose our contrasting candidates, and rely on the model itself to sub-select them from within the randomly generated pool. This method improves performance from chance $( 2 5 \\% )$ to approximately $7 7 \\%$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is possible that this top- $k$ method simply exploited random sampling to stumble on the candidates that would have otherwise been hand-crafted. Indeed, for more difficult problems (involving real images, for example) a random generator may not produce anything resembling data from the underlying distribution, making this method less suitable. We thus replicated the top- $k$ experiment, but actively excluded randomly generated candidates that satisfied $c = { \\hat { f } } ( V _ { t } )$ for some ${ \\hat { f } } \\in F$ . Performance fell to $4 3 \\%$ , confirming this intuition, but interestingly, still greatling improving baseline performance. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/01f180bb5d40c50b56af59bd680d2a2dd753710ab687aac1bff5adfc4f4b2772.jpg",
+        "table_caption": [
+            "Table 3: Test performance on our symbolic analogy in the four different training regimes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Training method</td><td>Test accuracy</td></tr><tr><td>LABC; explicit SMT</td><td>0.89</td></tr><tr><td>LABC; top-k</td><td>0.77</td></tr><tr><td>LABC;adversarial</td><td>0.62</td></tr><tr><td>Random candidate answers</td><td>0.25</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Finally, we considered a method for generating candidates that did not depend on random generation (LABC-adversarial), but instead exploited a generator model. The generator was identical to the model used to solve the task except its input consisted only of the target set $V _ { t }$ , its output was a proposed candidate vector $c$ . This candidate vector was then passed to the original model, which solved the analogy problem as before. The generator model was trained to maximize the score given by the analogy model; i.e. it was trained to produce maximally confusing candidates. The overall objective therefore resembled a two-player minimax game (Goodfellow et al., 2014): ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b01df5a7d430866ee4bc5cf31c2121d2c836ed486169027ab71ba8e4fad56878.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\operatorname* { m a x } _ { \\phi } \\mathcal { L } ( f _ { \\theta } ( S _ { 1 } , a _ { 1 } , S _ { 2 } , a _ { 2 } , g _ { \\phi } ( S _ { 2 } ) ) )\n$$",
+        "text_format": "latex",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { \\theta }$ is the analogy model and $g _ { \\phi }$ is the candidate proposal model. Using this method to propose candidates improved the model’s test performance from chance $( 2 5 \\% )$ to approximately $6 2 \\%$ . ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "These latter experiments show interesting links between LABC, GANs (Goodfellow et al., 2014), and possibly self-play (Silver et al., 2017). Indeed, the latter two approaches may be seen as automated methods that can approximate LABC. For example, in self-play, agents continuously challenge their opponents by proposing maximally challenging data. In the case of analogy, maximally challenging candidates may be those that are semantically-plausible rather than simply perceptually plausible. To our knowledge no prior work has demonstrated the effects of adversarial training regimes on out-of-distribution generalization in a controlled setting like the present context. ",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our experiments show that simple neural networks can learn to make analogies with visual and symbolic inputs, but this is critically contingent on the way in which they are trained; during training, the correct answers should be contrasted with alternative incorrect answers that are plausible at the level of relations rather than simple perceptual attributes. This is consistent with the SMT of human analogy-making, which highlights the importance of inter-domain comparison at the level of abstract relational structures. At the same time, in the visual analogy domain, our model reflects the idea of analogy as closely intertwined with perception itself. We find that models that are trained by LABC to reason better by analogy are, perhaps surprisingly, also better able to extrapolate to a wider range of input values. Thus, making better analogies seems connected to the ability of models to perceive and represent their raw experience. ",
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+    {
+        "type": "text",
+        "text": "Recent literature has questioned whether neural networks can generalise in systematic ways to data drawn from outside the training distribution (Lake & Baroni, 2018). Our results show that neural networks are not fundamentally limited in this respect. Rather, the capacity needs to be coaxed out through careful learning. The data with which these networks learn, and the manner in which they learn it, are of paramount importance. Such a lesson is not new; indeed, the task of one-shot learning was thought to be difficult, if not impossible to perform using neural networks, but was nonetheless “solved” using appropriate training objectives, models, and optimization innovations (e.g., (Santoro et al., 2016; Finn et al., 2017)). The insights presented here may guide promising, general purpose approaches to obtain similar successes in flexible, generalisable abstract reasoning. ",
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+    {
+        "type": "text",
+        "text": "Earlier work on analogical reasoning in AI and cognitive science employed constructed symbolic stimuli or pre-processed perceptual input (Carbonell (1981); Hummel & Holyoak (1997); Hofstadter (1996); Larkey & Love (2003) inter alia; see Gentner & Forbus (2011) for a full review). More recenty, Reed et al. (2015) learn an analogy model on top of pre-trained visual embeddings of geometric figures and rendered graphics, while Mikolov et al. (2013) show how analogies can be made via non-parametric operations on vector-spaces of text-based word representations. While the input to our visual analogy model is less naturalistic than these latter cases, this permits clear control over the semantics of training or test data when designing and evaluating hypotheses. Our study is nonetheless the only that we are aware to demonstrates such flexible, generalisable analogy making in neural networks learning end-to-end from raw perception. It is therefore a proof of principle that even very basic neural networks have the potential for strong analogical reasoning and generalization. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Sec. 5.3, in many machine-learning contexts it may not be possible to know exactly what a ‘good quality’ negative example looks like. The experiments there show that, in such cases, we might still achieve notable improvements in generalization via methods that learn to play the role of teacher by presenting alternatives to the main (student) model, as per Shafto et al. (2014). This underlines the fact that, for established learning algorithms involving negative examples such as (noise) contrastive estimation (Smith & Eisner, 2005; Gutmann & Hyvarinen, 2010) or negative ¨ sampling (Mikolov et al., 2013), the way in which negative examples are selected can be critical3. It may also help to explain the power of methods like self-play (Silver et al., 2016), in which a model is encouraged to continually challenge itself by posing increasingly difficult learning challenges. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Analogies as the functions of the mind To check whether a plate is on a table we can look at the space above the table, but to find out whether a picture is on a wall or a person is on a train, the equivalent check would fail. A single on function operating in the same way on all input domains could not explain these entirely divergent outcomes of function evaluation. On the other hand, it seems implausible that our cognitive system encodes the knowledge underpinning these apparently distinct applications of the on relation in entirely independent representations. The findings of this work argue instead for a different perspective; that a single concept of on is indeed exploited in each of the three cases, but that its meaning and representation is sufficiently abstract to permit flexible interaction with, and context-dependent adaptation to, each particular domain of application. If we equate this process with analogy-making, then analogies are something like the functions of the mind. We believe that greater focus on analogy may be critical for replicating human-like cognitive processes, and ultimately human-like intelligent behaviour, in machines. It may now be time to revisit the insights from past waves of AI research on analogy, while bringing to bear the tools, perspectives and computing power of the present day. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+    {
+        "type": "text",
+        "text": "Thanks to Greg Wayne and Jay McClelland for very helpful comments, and to Emilia Santoro, Adam’s most important publication to date. ",
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+    {
+        "type": "text",
+        "text": "7 MODEL DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7.1 VISUAL ANALOGY PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            421,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The CNN was 4-layers deep, with 32 kernels per layer, each of size $3 \\times 3$ with a stride of 2. Thus, each layer downsampled the image by half. Each panel in a question was $8 0 \\times 8 0$ pixels, and greyscale. The panels were presented one at a time to the CNN to produce 9 total embeddings (3 for the source sequence, 2 for the target sequence, and 4 for each candidate). We then used these embeddings to compile 4 distinct inputs for the RNN. Each input was comprised of the source sequence embeddings, the target sequence embeddings, and a single candidate embedding, for a total of 6 embeddings per RNN-input sequence. We passed these independently to the RNN (with 64 hidden units), whose final output was then passed through a linear layer to produce a single scalar. 4 such passes (one for each source-target-candidate sequence) produced 4 scalar scores, denoting the model’s evaluation of the suitability of the particular candidate for the analogy problem. Finally, a softmax was computed across the scores to select the model’s “answer”. We used a cross entropy loss function and the Adam optimizer with a learning rate of $1 e ^ { - 4 }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.2 SYMBOLIC ANALOGY PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            343,
+            442,
+            357
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A given input consisted of a set of vectors 16-dimensional vectors. This set included 8 vectors comprising $S _ { 1 }$ , one vector $d _ { 1 }$ , 8 vectors comprising $S _ { 2 }$ , and 8 vectors comprising the set of candidate vectors $C$ . Vectors were given a single digit binary variable tag to denote whether they were members of the source or target set (augmenting their size to 17-dimensions). ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We note that the entity vectors have 0’s in their unused dimensions. While this may make the problem easier, this experiment was designed to explicitly test domain-transfer generalization, moreso than an ability to discern the domains that need to be considered, by stripping away any difficulties in perception (i.e., in identifying the relevant domains), and seeing if the effect of LABC persists. Thus, at test time the model should have an easy time identifying the relevant dimensions, but it will never have seen the particular transfer from, say, dimension $i$ to dimension $j$ . So, even though it may have an easy time identifying and processing each dimension $i$ and $j$ , it may be incapable (without LABC) of integrating the information processed from each of these dimensions. ",
+        "bbox": [
+            174,
+            431,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We employed a parallel processing architecture, similar to the visual analogy experiments, with a Relation Network (128 unit, 3 layer MLP with ReLU non-linearities for the $g _ { \\theta }$ function and a similar 2-layer MLP for the $f _ { \\phi }$ function) replacing the RNN core. Thus, a single model processed $( S _ { 1 } , d _ { 1 } , S _ { 2 } , c _ { n } )$ , with $c _ { n }$ being a different candidate vector from $C$ for each parallel pass. The model’s output was a single scalar denoting the score assigned to the particular candidate $c _ { n }$ – these scores were then passed through a softmax, and training proceeded using a cross entropy loss function. We used batch sizes of 32 and the Adam optimizer with a learning rate of $3 e ^ { - 4 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "8 SUPPLEMENTARY RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            667,
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+            684
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/331b00f3d6c97c929a562fc669f0f621134dc8ad5e6d9891a0dc758c01cdda50.jpg",
+        "table_caption": [
+            "Table 4: Results for the visual analogy task (RNN Model). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Extrapolation</td><td rowspan=\"2\">Mean</td><td colspan=\"2\">LABC Normal (Train)</td><td rowspan=\"2\">Mix (Train) 0.94</td><td rowspan=\"2\">LABC (Contrasting) 0.62</td><td rowspan=\"2\">Normal (Contrasting) 0.43</td><td rowspan=\"2\">Mix (Contrasting) 0.56</td><td rowspan=\"2\">LABC (Normal) 0.45</td><td rowspan=\"2\">Normal (Normal) 0.44</td><td rowspan=\"2\">Mix (Normal) 0.39</td></tr><tr><td>0.94</td><td>(Train) 0.95</td></tr><tr><td rowspan=\"2\">Interpolation</td><td>Std</td><td>0.005</td><td>0.007</td><td>0.005</td><td>0.02</td><td>0.009</td><td>0.012</td><td>0.01</td><td>0.01</td><td>0.04</td></tr><tr><td>Mean</td><td>0.94</td><td>0.97</td><td>0.94</td><td>0.93</td><td>0.45</td><td>0.89</td><td>0.65</td><td>0.89</td><td>0.87</td></tr><tr><td rowspan=\"2\">Novel Domain Transfer</td><td>Std</td><td>0.003</td><td>0.003</td><td>0.003</td><td>0.004</td><td>0.004</td><td>0.008</td><td>0.01</td><td>0.006</td><td>0.01</td></tr><tr><td>Mean</td><td>0.88</td><td>0.83</td><td>0.85</td><td>0.87</td><td>0.48</td><td>0.88</td><td>0.7</td><td>0.82</td><td>0.79</td></tr><tr><td rowspan=\"2\">Novel Domain (shape colour)</td><td>Std</td><td>0.015</td><td>0.01</td><td>0.015</td><td>0.005</td><td>0.02</td><td>0.009</td><td>0.01</td><td>0.01</td><td>0.02</td></tr><tr><td>Mean</td><td>0.87</td><td>0.84</td><td>0.85</td><td>0.78</td><td>0.50</td><td>0.80</td><td>0.51</td><td>0.61</td><td>0.58</td></tr><tr><td rowspan=\"2\"></td><td>Std</td><td>0.007</td><td>0.008</td><td>0.007</td><td>0.004</td><td>0.02</td><td>0.006</td><td>0.02</td><td>0.01</td><td>0.02</td></tr><tr><td>Mean</td><td>0.87</td><td>0.85</td><td>0.86</td><td>0.76</td><td>0.45</td><td>0.75</td><td>0.5</td><td>0.57</td><td>0.54</td></tr><tr><td>Novel Domain (line type)</td><td>Std</td><td>0.006</td><td>0.004</td><td>0.006</td><td>0.02</td><td>0.01</td><td>0.02</td><td>0.02</td><td>0.02</td><td>0.01</td></tr></table>",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "9 MODEL COMPARISON DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+    {
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+        "text": "Our application of a ResNet-50 processes all nine panels simultaneously (five analogy question panels along with the four multiple choice candidates) as a set of input channels. The Parallel ResNet-50 processes six panels simultaneously as input channels (five analogy question panels along with one multiple choice candidate) to produce a score. Then, similar to the RNN model described above, the candidate with the highest score is chosen by the model. The parallel relation network model also processes six panels simultaneously, using a convnet to obtain panel embeddings and using a relation network (Santoro et al., 2017) for computing a score. For full model architecture details, see the appendix. ",
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+    {
+        "type": "text",
+        "text": "Interestingly, the model with strongest generalisation is the parallel relation network, with a particularly high accuracy of $9 5 \\%$ on the held out domain-transfer test set. This model was tested on a mixture of multiple choice candidates (that included semantically plausible and perceptually plausible candidates), indicating that models trained with LABC do not over-specialize to problem settings where only semantically plausible candidates are available. We also observe that during normal training, test set performance can oscillate between good solutions and poor solutions, indicated by the high standard deviation in the test set accuracy. These results imply that there are multiple model configurations that have good performance on the training set, but that only some of these configurations have the desired generalisation behaviour on the test set. LABC encourages a model to learn the configurations that generalise at the most abstract semantically-meaningful level, as desired. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also note that the fact that a model trained in the normal regime performs marinally better than one trained using (normal+)LABC data on test questions involving perceptually-plausible candidates. We believe this may be understood as a symptom of the strong ability of deep learning models to specialize to the exact nature of the problems on which they are trained. The model comparison experiments demonstrate that this negligible but undesirable specialization effect is outweighed by the greater benefits of training with LABC on test questions with semantically-plausible candidates (i.e. those that require a higher-level semantic interpretation of the problem). Training with LABC will therefore yield a much higher expected performance, for instance, in cases where the exact details of the test questions is not known. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "10 FURTHER DISCUSSION AND RELATED WORK ",
+        "text_level": 1,
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+        "text": "It is interesting to consider to what extent the effects reported in this work can transfer to a wider class of learning and reasoning problems beyond classical analogies. The importance of teaching concepts (to humans or models) by contrasting with negative examples is relatively established in both cognitive science (Shafto et al., 2014; Smith & Gentner, 2014) and educational research (Silver, 2010; Ali, 1981). Our results underline the importance of this principle when training modern neural networks to replicate human-like cognitive processes and reasoning from raw perceptual input. In cases where expert understanding of potential data exists, for instance in the case of active learning with human interaction, it provides a recipe for achieving more robust representations leading to far greater powers of generalization. We should aspire to select as negative examples those examples that are plausible considering the most abstract principles that describe the data. ",
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+        "text": "A further notable property of our trained networks is the fact they can resolve analogies (even those involving with unfamiliar input domains) in a single rollout (forward pass) of a recurrent network. This propensity for fast reasoning has an interesting parallel with the fast and instinctive way in which humans can execute visual analogical reasoning (Morrison et al., 2001; Qiu et al., 2008). ",
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+    {
+        "type": "text",
+        "text": "10.1 DISTANCE METRIC APPROACHES ",
+        "text_level": 1,
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+        "text": "LBC shares similarities with distance metric approaches such as the large-margin nearest neighbor classifier (LMNN) (Weinberger & Saul, 2009), the triplet loss (Schroff et al., 2015), and others. In these approaches the goal is to transform inputs such that the distance between input embeddings from the same class is small, while the distance between input embeddings from different classes is large. Given these improved embeddings, classification can proceed using off-the-shelf classification algorithms, such as k-nearest neighbors. We note that these approaches emphasize the form of the loss function and the quality of the resultant input embeddings on subsequent classification. However, the goal of LBC is not to induce better classification per se, as it is in these methods. Instead, the goal is to induce out-of-distribution generalisation by virtue of improved abstract understanding of the underlying problem. It is unclear, for example, whether the embeddings produced by LMNN or the triplet loss are naturally amenable to this kind of generalisation, and as far as we are aware, it has not beed tested. Nonetheless, LBC places a critical focus on the nature, or quality of the data comprising the incorrect classes, and is agnostic to the exact nature of the loss function. Thus, it is possible to use previous approaches (e.g., LMNN or triplet loss, etc.) in conjunction with LBC, which we do not explore. LBC also shares similarities to recent generative adversarial active learning approaches (Zhu & Bento, 2017). However, these approaches do not explicitly point to the effects of the quality of incorrect samples to out-of-distribution generalisation, nor are we aware of any experiments that test abstract generalisation using networks trained with generative samples. ",
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+        "text": "11 VISUAL ANALOGY EXAMPLES ",
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+            "Figure 7: Examples of visual analogy problems. These visual analogy examples have been selected from the interpolation test set. The correct multiple choice candidate for each problem is highlighted in green. In the top half, we have randomly chosen examples where our RNN model trained with LABC selects the correct answer. In the bottom half, we have randomly selected examples where our RNN model trained with LABC chooses the incorrect candidates (highlighted in red). The performance of this model on the interpolation test set is $9 3 \\%$ . "
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parse/train/q1eCa1kMfDd/q1eCa1kMfDd.md

@@ -0,0 +1,234 @@
+# Flattening Sharpness for Dynamic Gradient Projection Memory Benefits Continual Learning
+
+Danruo Deng1, Guangyong Chen2∗, Jianye $\mathbf { H a o ^ { 3 , 4 } }$ , Qiong Wang2, Pheng-Ann Heng1,2 1The Chinese University of Hong Kong, 2Guangdong-Hong Kong-Macao Joint Laboratory of Human-Machine Intelligence-Synergy Systems, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, 3College of Intelligence and Computing, Tianjin University, 4Huawei Noah’s Ark Lab {drdeng,pheng}@cse.cuhk.edu.hk, {gy.chen, wangqiong}@siat.ac.cn, jianye.hao@tju.edu.cn
+
+# Abstract
+
+The backpropagation networks are notably susceptible to catastrophic forgetting, where networks tend to forget previously learned skills upon learning new ones. To address such the ’sensitivity-stability’ dilemma, most previous efforts have been contributed to minimizing the empirical risk with different parameter regularization terms and episodic memory, but rarely exploring the usages of the weight loss landscape. In this paper, we investigate the relationship between the weight loss landscape and sensitivity-stability in the continual learning scenario, based on which, we propose a novel method, Flattening Sharpness for Dynamic Gradient Projection Memory (FS-DGPM). In particular, we introduce a soft weight to represent the importance of each basis representing past tasks in GPM, which can be adaptively learned during the learning process, so that less important bases can be dynamically released to improve the sensitivity of new skill learning. We further introduce Flattening Sharpness (FS) to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all seen tasks. As demonstrated empirically, our proposed method consistently outperforms baselines with the superior ability to learn new skills while alleviating forgetting effectively.2.
+
+# 1 Introduction
+
+Humans have the ability to continually learn new knowledge without forgetting their previously learned ones through mediating a rich set of neurocognitive mechanisms [41, 15, 39]. This ability, often known as continual learning or lifelong learning [29], is crucial for computational systems, such as deep neural networks (DNNs), which are required to sequentially learn and deal with multiple tasks when implemented in the dynamically changing environment. Continual learning remains a long-standing challenge for DNNs since these networks are typically trained with stationary training batches by stochastic gradient descent methods [19], which generally leads to an abrupt performance decrease on previously learned tasks as new tasks are learned. To address such catastrophic forgetting, we can brutally retrain an oracle network on the entire dataset containing all tasks to capture dynamic changes in the data distribution, but this methodology is obviously too inefficient to hinder the learning of novel data in real time.
+
+During the last few years, lots of research efforts have been devoted to improving the stability of DNNs on old tasks while keeping sensitive to new information. The first intuitive idea is to introduce an independent branch for each new task while freezing the old task parameters to preserve the old knowledge [33, 45, 43, 25, 35, 22]. However, in this way, the network will inevitably become redundant as the task number continually increases. As presented in the neurocognitive works [41, 15], the reactivation of neuronal activity patterns, representing old memories, plays an important role in the continual learning of humans [39]. Thus, forgetness can be effectively mitigated by training a single network for new tasks by considering diverse information stored in the memory, including the original training samples of old tasks [31, 5, 13], the gradients induced from old tasks [9] and the feature subspace representing old tasks [34]. However, their continual learning performance is still limited because DNNs can easily overfit the limited information stored in the small-size memory.
+
+The overfitting problem of DNNs is often attributed to the complex loss landscape containing multiple local optima, and the sharpness of the loss landscape has been widely used to characterize the generalization gap in standard training scenarios from both theoretical and empirical perspectives [27, 21, 42, 10, 6]. While this characterization has inspired new approaches for model training with better generalization, practical algorithms that especially seek out flatter minima to effectively address the ’sensitivity-stability’ dilemma for continual learning have thus far been elusive. In this paper, our first contribution is to characterize the weight loss landscape for the continual learning scenario and identify that a flatter loss landscape with lower loss value often leads to better continual learning performance, as shown in Figure 1 and Figure 3.
+
+Further, based on our characterization of the weight loss landscape, we find that the recently proposed Gradient Projection Memory (GPM) method [34] maintains the lowest loss value on old tasks among the previously proposed methods by taking gradient steps orthogonal to the subspace representing old tasks. However, its loss landscape on newly learned tasks is the sharpest due to the lack of sufficient subspace left for new task learning. To improve the network’s sensitivity, our second contribution is to predict the importance of bases spanning the subspace for old tasks, so that less important bases can be dynamically released. In particular, we introduce a soft weight to indicate the bases importance, which can be dynamically adjusted by combining the Flattening Sharpness $( F S )$ to minimize the loss value and loss sharpness simultaneously. Intuitively, a basis will be regarded as important for preserving old knowledge if the gradients induced by new tasks and old ones are aligned in the opposite direction on that basis. As demonstrated through extensive experiments, our proposed method can consistently outperform the state-of-the-art methods [17, 30, 24, 31, 5, 13, 34] by a notable margin across a range of widely used benchmark datasets.
+
+# 2 Related Work
+
+In this section, we briefly survey the representative works of continual learning by highlighting their contributions. To simplify our presentation, this section is organized by dividing the representative works into three categories, parameter isolation, regularization-based, memory-based methods.
+
+Parameter isolation methods address forgetting by assigning a different subset of network parameters to each task. Without restrictions on network architecture, new neurons or layers or modules can be added for new tasks, while the previous task parameters can be frozen or copied to preserve old knowledge. For instance, Progressive Neural Network (PGN) [33] freezes the parameters trained with previous knowledge while expands the architecture by allocating new sub-networks with fixed capacity for new tasks. Dynamically Expandable Networks (DEN) [45] selectively retrains or expands network capacity by splitting/duplicating important units on new tasks. Reinforced Continual Learning (RCL) [43] uses reinforcement learning strategy to adaptively expand the network of each layer, while [22] use neural architecture search to find optimal network structures for each sequential task. Alternatively, with the architecture remaining static, a fixed part is allocated to each task. During the training of a new task, previous task parts are masked out to prevent interference. The mask sets are imposed at parameter level [25], or unit level [35]. PackNet [25] uses iterative pruning to fully restrict gradient updates on important weights via a binary mask, whereas HAT [35] limits the update of important units recognized by the hard attention mask through gradient backpropagation.
+
+Regularization-based methods introduce an additional regularization term in the loss function to consolidate previous knowledge without using replay. This involves using knowledge distillation [23, 14] or penalizing changes in weights deemed important for previous tasks [17, 46, 28, 2, 3] to reduce forgetting. There are many ways to measure the importance. Elastic Weight Consolidation (EWC) [17] identifies important weights based on the diagonal values of Fisher information matrix after training, while Synaptic Intelligence (SI) [46] calculates them online and over the entire learning trajectory in parameter space. Memory Aware Synapses (MAS) [2] estimates importance weights in an unsupervised manner, while Variational Continual Learning (VCL) [28] introduces a variational framework that spawned some Bayesian-based works [32, 8, 1, 7]. For example, [32] recursively uses a Gaussian Laplace approximation of the Hessian to approximate the posterior after every task, [8] adjusts the learning rate according to the uncertainty defined by the probability distribution of the network weights. [1] introduces an interpretation of node-wise uncertainty on the Kullback-Leibler (KL) divergence term of the variational lower bound for Gaussian mean-field approximation.
+
+Our method mainly follows memory-based methods, which mitigate forgetting based on information extracted from old tasks or based on a generative model to generate pseudo samples. For example, iCaRL [30] selects and stores samples closest to the feature mean of each class. ER [5, 31] suggests reservoir sampling under the limited and fixed budget for replay buffer. Deep Generative Replay (DGR) [36] trains a deep generative model in the Generative Adversarial Network (GAN) framework [11] to simulate past data. These previous task samples are mainly reused as model inputs for replay in the above methods. However, replay might be prone to overfitting the subset of stored samples. Alternatively, samples stored in memory can also be used to constrain the optimization of the new task loss to prevent previous task interference, thereby leaving more leeway for backward and forward transfer. Gradient Episodic Memory (GEM) [24] projects the estimated gradients in the feasible region, which is outlined by previous task gradients calculated from the episodic memory samples. Averaged-GEM (A-GEM) [4] relaxes the projection to a direction that is estimated from samples randomly selected from memory. [12] proposes a unified view of episodic memory-based continual learning methods, including GEM and A-GEM, and improves performance over these methods by using a loss-balancing update scheme. A few other works have utilized gradient information to protect previous knowledge. [31, 13] adopt optimization-based meta-learning to enforce gradient alignment between samples from different tasks. GPM [34] minimizes interference between sequential tasks by ensuring that gradient updates only occur in directions orthogonal to the input of previous tasks.
+
+# 3 The Weight Loss Landscape of Continual Learning
+
+In this section, we first introduce our formulation of continual learning, and then characterize the weight loss landscape for the continual learning scenario from stability and sensitivity. Finally, some insights combining the weight loss landscape and continual learning are provided.
+
+# 3.1 Problem Formulation
+
+Throughout the paper, we denote scalars as $a$ , vectors as $\textbf { \em a }$ , matrices as $\pmb { A }$ , and sets as $\mathcal { A }$ . We consider a supervised learning setup where $T$ tasks are sequentially learned from their training data. Each task has an identical task descriptor, $\tau \in \{ 1 , 2 , \ldots , T \}$ , with its dataset $\mathscr { D } _ { \tau } = \{ \pmb { x } _ { i , \tau } , y _ { i , \tau } \} _ { i = 1 } ^ { n _ { \tau } }$ containing $n _ { \tau }$ samples randomly generated from a latent distribution $\mathcal { D } _ { \tau }$ . At any time-step during the learning process, we minimize the empirical risk of the model on all $t$ tasks seen so far, with just limited size of memory $\mathcal { M }$ to summarize the training data of previous tasks $\{ \mathcal { D } _ { \tau } \} _ { \tau = 1 } ^ { t - 1 }$ . To simplify the notation, we denote $\begin{array} { r } { L _ { \mathcal { A } } ( \pmb { w } ) = \frac { 1 } { | \mathcal { A } | } \sum _ { ( \pmb { x } , \pmb { y } ) \in \mathcal { A } } \left[ \ell \left( f _ { \pmb { w } } \left( \pmb { x } \right) , \pmb { y } \right) \right] } \end{array}$ as the average empirical loss for the set , where $\ell ( \cdot , \cdot )$ is an arbitrary loss function (e.g. the cross-entropy (CE) loss), $| { \cal A } |$ is the sample size of the set $\mathcal { A }$ , and $f _ { w }$ is the DNN with weight $\textbf { \em w }$ . Our final goal is to find an optimal parameter $\pmb { w }$ , which minimizes the overall risk $\scriptstyle \sum _ { \tau = 1 } ^ { T } L _ { \mathcal { D } _ { \tau } } ( \pmb { w } )$ for all tasks.
+
+# 3.2 Connection of Weight Loss Landscape and Continual Learning
+
+After learning a new task, we visualize the weight loss landscape of each task seen so far by plotting changes in its training loss when moving the weights $\pmb { w }$ in a random direction $^ d$ with magnitude $\alpha$ following [21]:
+
+$$
+g _ { t } ( \alpha ) = L _ { \mathcal { D } _ { t } } ( \pmb { w } + \alpha \pmb { d } ) = \frac { 1 } { | \mathcal { D } _ { t } | } \sum _ { ( \pmb { x } , \pmb { y } ) \in \mathcal { D } _ { t } } \ell \left( f _ { \pmb { w } + \alpha d } \left( \pmb { x } \right) , \pmb { y } \right) ,
+$$
+
+where $\mathcal { D } _ { t }$ is the training set for the $t$ -th task previously learned. To eliminate the scaling invariance of DNNs, $^ d$ is sampled from a Gaussian distribution and further normalized by $\begin{array} { r } { \pmb { d } _ { l , j }  \frac { \mathbf { \bar { d } } _ { l , j } } { \| \pmb { d } _ { l , j } \| _ { F } } \| \pmb { w } _ { l , j } \| _ { F } } \end{array}$ where $d _ { l , j }$ represents the $j$ -th filter at $l$ -th layer of $^ d$ , and $\left\| \cdot \right\| _ { F }$ denotes the Frobenius norm. Compared with our visualization, [26] only consider one task and plot the loss landscape along the directions of its Hessian eigenvectors, which only reflects some of the relationship between forgetting and landscape. Considering $^ d$ is randomly selected, we repeat the visualization 10 times with different $^ d$
+
+![](images/fd37d0fd577ccb1d848a629a229d31c87841d79be1d0b99c0f1ac596de52444b.jpg)  
+Figure 1: The connection between the weight loss landscape and continual learning is investigated on four methods. (a)-( c) shows the stability of the first task. (a) is the test accuracy change curve of the first task; (b) and (c) are the weight loss landscape of the first task after learning the fifth task and all ten tasks. (d)-(e) shows the sensitivity of the fifth task, which are the test accuracy and the weight loss landscape of the fifth task after just learning the fifth task. The shape of the weight loss landscape obtained using ten different random filter-normalized directions. ( $" \mathrm { T } i "$ is abbr. of the $i$ -th task)
+
+We first study the stability of the network by plotting changes of the weight loss landscape for the first task after new task learning. In particular, We use the previously proposed ER [5], La-MAML [13], and GPM [34] to train a MLP network with two hidden layers on the Permuted MNIST (PMNIST) [20] dataset that contains 10 tasks. We also retrain the network on the entire dataset contain all passed tasks as an Oracle network. Early stopping is used to halt the training with up to 10 epochs for each task based on the validation loss as proposed in [35]. As shown in Figure 1(a), all three continual learning methods lose their stability as learning new tasks. It can be observed from Figure 1(b)-(c) that the weight loss landscape becomes sharper and loss value increases simultaneously, when the testing accuracy of the first task continually decreases. We further evaluate the sensitivity of the network by observing the performance of the fifth task just after it has just been learned. As shown in Figure 1(d)-(e), ER shows the best learning capability compared with other methods with the lowest loss value and the flattest loss landscape. Thus, based on these empirical findings, we assume that lower loss value with a flatter neighbor may lead to better continual learning performance.
+
+# 3.3 A Case Study of Flattening Sharpness for Vanilla ER
+
+In this part, we further validate our above assumption by Flattening Sharpness for vanilla ER (FS-ER). Compared with ER that looks for a solution $\pmb { w }$ that jointly minimizes the training loss of current task data and memory data, FS-ER seeks out a solution with both low loss and flat neighbor by minimizing the maximal loss in the neighbor around the parameter value. The schematic of the Flattening Sharpness (FS) is shown in Figure 2. We introduce the adversarial weight perturbation (orange dashed line) to explicitly flatten the weight loss landscape via injecting the worst-case weight perturbation, which is calculated from the current task data and past task data sampled from the replay buffer (Refer to Appendix C.1 and the next section for more details). Figure 3 respectively shows the weight loss landscapes of the fifth task after just learning the fifth task and all tasks. We find that FS-ER successfully gets a solution with a lower loss value and flatter landscape, either after the fifth task has just been learned or all ten tasks have been learned. The average testing accuracy of all tasks using FS-ER is $9 0 . 4 4 \%$ , significantly higher than ER $( 8 6 . 1 6 \% )$ , which means that flattening sharpness does benefit continual learning.
+
+![](images/2e3f3ddf535810155a992fc66e3895a7221d3a75c0786f35ade9210a2420a872.jpg)  
+Figure 2: Schematic of FS-ER update. The dashed line and the solid line indicate the gradient ascent and descent, respectively. Orange denotes the actual update of the parameter $\pmb { w }$
+
+![](images/b58c7460443734cc64e08af56719f4753f34ebec8f27068bdfe89caa7994dc81.jpg)  
+Figure 3: Landscape of the fifth task after just learning (a) the fifth task and (b) all ten tasks.
+
+# 4 Flattening Sharpness for Dynamic Gradient Projection Memory
+
+As shown in Figure 1, GPM achieves the highest testing accuracy on old tasks among all three practical continual learning methods, but shows less sensitivity to new task learning. To address this issue, we propose Flattening Sharpness for Dynamic GPM (FS-DGPM), which dynamically adjusts the gradient subspace representing the past tasks to improve the sensitivity to the new task, while ensuring stability of the previous tasks. In particular, we let $M = [ \pmb { u } _ { 1 } , \pmb { u } _ { 2 } , \cdot \cdot \cdot , \pmb { u } _ { k } ]$ denote the bases matrix that spans the gradient subspace of the previous task, $\pmb { \Lambda } = \mathrm { d i a g } [ \lambda _ { 1 } , \lambda _ { 2 } , . . . , \lambda _ { k } ]$ be the diagonal matrix with its $i$ -th diagonal element $\lambda _ { i } \in [ 0 , 1 ]$ indicating the importance of each basis, and $k$ is the number of bases.
+
+# 4.1 Sharpness Evaluation
+
+Comparing with the classical strategy that perturbs weight in the entire space [40, 16, 42, 10], we focus on characterizing the weight loss landscape on the new task with respect to the important subspace representing old task. The important subspace can be effectively calculated based on the examples sampled from replay buffer $\mathcal { M }$ after each task training. Then, we can find the worst case by maximizing the training loss of the network on the new task in this subspace. Formally, the sharpness of the loss landscape around the solution $\textbf { \em w }$ in the old parameter space can be predicted as,
+
+$$
+\operatorname* { m a x } _ { \pmb { v } \in \mathcal { V } } L _ { \hat { \mathcal { D } } _ { t } } \left( \pmb { w } + \pmb { v } \right) ,
+$$
+
+where $\nu$ denotes the subspace spanned by $M$ and $\pmb { \Lambda }$ , and $\hat { \mathcal { D } } _ { t }$ denotes the batch samples of the current $t$ -th task. As shown in Eq. (1), the high value can be obtained when the network fails to learn the new task (sensitivity) and the new task learning seriously interferes with the past tasks learning (stability). Based on the gradient method, the adversarial weight perturbation $\pmb { v }$ can be solved as,
+
+$$
+\pmb { v }  \pmb { v } + \eta _ { 1 } \pmb { M } \pmb { \Lambda } \pmb { M } ^ { T } ( \nabla _ { ( \pmb { w } + \pmb { v } ) } L _ { \hat { \mathcal { D } } _ { t } } ( \pmb { w } + \pmb { v } ) ) ,
+$$
+
+where $\eta _ { 1 }$ is the update step size. Note that $\textbf {  { v } }$ is initialized as 0 and layer-wise updated. As shown in Appendix $\mathrm { E }$ , two-step for $\textbf {  { v } }$ (default settings) are enough to get good improvements.
+
+# 4.2 Dynamic Gradient Projection Memory
+
+After obtaining the adversarial weight perturbation $\textbf {  { v } }$ , we can further update the bases importance matrix $\pmb { \Lambda } = \mathrm { d i a g } [ \lambda _ { 1 } , \lambda _ { 2 } , . . . , \lambda _ { k } ]$ by jointly considering the current task batch $\hat { \mathcal { D } } _ { t }$ and the batch $\hat { \mathcal { M } }$ sampled from the replay buffer $\mathcal { M }$ as following,
+
+$$
+\lambda _ { i }  \lambda _ { i } - \eta _ { 2 } ( \nabla _ { \lambda _ { i } } L _ { \hat { \mathcal { D } } _ { t } \cup \hat { \mathcal { M } } } ( \pmb { w } + \pmb { v } ) ) ,
+$$
+
+where the sigmoid function is used at the end of gradient update to constrain the importance value $\lambda _ { i }$ between 0 and 1. In addition, the second term on the right side in Eq. (3) can be approximated by the first-order Taylor expansion as,
+
+$$
+\nabla _ { \lambda _ { i } } L _ { \hat { \mathcal { D } } _ { t } \cup \hat { \mathcal { M } } } ( { \boldsymbol w } + { \boldsymbol v } ) \approx \eta _ { 1 } \left( \nabla _ { { \boldsymbol w } } L _ { \hat { \mathcal { D } } _ { t } } \left( { \boldsymbol w } \right) \right) ^ { T } \mathbf { { u } } _ { i } \boldsymbol u _ { i } ^ { T } \left( \nabla _ { { \boldsymbol w } } L _ { \hat { \mathcal { D } } _ { t } \cup \hat { \mathcal { M } } } \left( { \boldsymbol w } \right) \right) .
+$$
+
+The above equation characterizes the relationship between the gradients induced by the current task and the old tasks based on the basis $\mathbf { \delta } \mathbf { u } _ { i }$ . As illustrated in the Figure 4, this equation implies that when the projections of two gradients on the basis $\mathbf { \delta } \mathbf { u } _ { i }$ are aligned in the same direction, the gradient of $\lambda _ { i }$ will be positive, and when there is interference, the gradient will be negative. The positive (negative) gradient will decrease (increase) the importance $\lambda _ { i }$ , thereby releasing (tightening) the update limit of the new task on the corresponding
+
+![](images/0c65063f71562cc6c18285cf0a8ea38b34c5439409e5823be9b371fd66973c10.jpg)  
+Figure 4: A depiction of transfer (a) and interference (b) in the basis $\textbf { \em u }$ of gradient space.
+
+basis $\mathbf { \delta } \mathbf { u } _ { i }$ . We provide the full derivation in the Appendix A.2. Note that the initial value of all importance is set to 1 and dynamically adjusted from the second task.
+
+# 4.3 Weight Updating
+
+Finally, we update the model parameters by minimizing the worst performance of $f _ { w + v }$ with respect to $\textbf { \em w }$ , while adjusting the update magnitude of $\pmb { w }$ on each basis based on its importance to alleviate forgetting. More concretely, the parameter $\textbf { \em w }$ will be updated to:
+
+$$
+\pmb { w }  \pmb { w } - \eta _ { 3 } ( \pmb { I } - \pmb { M } \pmb { \Lambda } \pmb { M } ^ { T } ) \nabla _ { \pmb { w } } L _ { \hat { D } _ { t } \cup \hat { \mathcal { M } } } ( \pmb { w } + \pmb { v } ) .
+$$
+
+Note that the optimization is performed over the model parameters $\pmb { w }$ , whereas the objective is computed using perturbed model $f _ { w + v }$ . In addition, we update the replay buffer $\mathcal { M }$ with reservoir sampling as in [31], and then use Singular Value Decomposition (SVD) to recalculate $M$ based on the sampling data in the replay buffer after learning one task following GPM [34]. Comparing with [34], we calculate the important bases in the entire gradient space and use them to replace the bases calculated last time. Besides that, our method degenerates to GPM when $\eta _ { 1 }$ and $\eta _ { 2 }$ are set to 0. The complete pseudo-code of FS-DGPM is outlined in the Algorithm 1.
+
+Algorithm 1 FS-DGPM (Flattening Sharpness for Dynamic Gradient Projection Memory)   
+
+<table><tr><td>weight step size n2,batch size b. InitializingM←{},M←I,△←I fort=1,2,.,Tdo for ep = 1,2,... , numepochs do</td><td>Input: Network weight w,loss function l, learning rate n3,FS step size 71,FS steps K, Soft</td></tr><tr><td>for batch Dt  Dt do MM</td><td></td></tr><tr><td>for k =1,.·,K do v ←v+niMAMT (V(w+v)Lb(w+ v))</td><td> Sharpness Evaluation</td></tr><tr><td>end for if t ≥ 2 then</td><td></td></tr><tr><td>△ ← △-n2(∀ALD:UM(ω+ ν))</td><td> Dynamic Gradient Projection Memory</td></tr><tr><td>end if w ← w - n3 (1 - MAMT) VwLDtUm(w+v)</td><td> Weight updating</td></tr><tr><td></td><td></td></tr><tr><td>Push Dt to M with reservior sampling</td><td></td></tr><tr><td>end for</td><td></td></tr><tr><td>end for</td><td></td></tr><tr><td></td><td></td></tr><tr><td>M ←UpdateGPM (M)</td><td></td></tr><tr><td></td><td>V see Appendix Alg. 2</td></tr><tr><td></td><td></td></tr><tr><td>end for</td><td></td></tr></table>
+
+# 4.4 Theoretical Understanding
+
+We further provide a theoretical view on why landscape can characterize the continual learning performance and why our proposed FS-DGPM works. To simplify our explanation, we only consider two tasks, which contains the training sets $\mathcal { D } _ { 1 }$ and $\mathcal { D } _ { 2 }$ sampled from the distributions $\mathcal { D } _ { 1 }$ and $\mathcal { D } _ { 2 }$ , respectively. Based on the previous works on PAC-Bayes bound [27, 42, 10], given a "prior" distribution $P$ (a common assumption is zero mean, $\sigma ^ { 2 }$ variance Gaussian distribution) over the weights, the expected error of the classifier for the continual learning scenario can be bounded with probability at least $1 - \delta$ over the draw of $n$ training data:
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { \Delta w } \mathbb { E } _ { v } \left[ L _ { \mathcal { D } _ { 1 } \cup \mathcal { D } _ { 2 } } ( w + \Delta w + v ) \right] \leq \displaystyle \operatorname* { m i n } _ { \Delta w \in \mathcal { V } ^ { \mathcal { C } } } \mathbb { E } _ { v } \left[ L _ { \mathcal { D } _ { 1 } } ( w + \Delta w + v ) \right] + L _ { \mathcal { D } _ { 2 } } ( w + \Delta w ) } \\ { \displaystyle + \operatorname* { m a x } _ { v \in \mathcal { V } } L _ { \mathcal { D } _ { 2 } } ( w + \Delta w + v ) - L _ { \mathcal { D } _ { 2 } } ( w + \Delta w ) + 4 \sqrt { \frac { 1 } { n } \left( K L ( w + \Delta w + v | | P ) + \ln \frac { 2 n } { \delta } \right) } . } \end{array}
+$$
+
+where $\Delta w$ is the update based on the previously optimal solution $\pmb { w }$ learned on the old task $\mathcal { D } _ { 1 }$ when learning the new one $\mathcal { D } _ { 2 }$ , and $\textbf {  { v } }$ is often chosen as a zero mean spherical Gaussian perturbation with variance $\sigma ^ { 2 }$ in every direction. Let $\Delta w \in \mathcal { V } ^ { C }$ , then $\Delta w$ lies in the complementary space of the important space representing the old task $\mathcal { D } _ { 1 }$ , so that $\mathbb { E } _ { \pmb { v } } \left[ L _ { \mathcal { D } _ { 1 } } ( \pmb { w } + \Delta \pmb { w } + \bar { \pmb { v } } ) \right]$ does not increase too much compared with the previously minimized $\mathbb { E } _ { v } \left[ L _ { \mathcal { D } _ { 1 } } ( \pmb { w } + \pmb { v } ) \right]$ . The second term denotes the empirical loss on the second task and the third term represents the sharpness of the weight loss landscape around the $\mathbf { \Delta } w + \Delta w$ . Since we have constrained $\Delta w \in \mathcal { V } ^ { C }$ , then it is natural to assume $\pmb { v } \in \mathcal { V }$ , so that $\Delta w + v$ will cover the full space. Thus, our FS-DGPM exactly optimizes the worstcase of the flatness of weight loss landscape to control the PAC-Bayes bound, which theoretically justifies both lower loss value and flatter landscape lead to better continual learning performance, and why our proposed FS-DGPM works.
+
+# 5 Experiments
+
+In this section, we conduct extensive experiments to compare the performance of our proposed FS-DGPM model with the state-of-the-art methods on widely used continual learning benchmark datasets. Additional results and more details about the datasets, experiment setup, baselines, and model architectures are presented in the Appendix D and E.
+
+# 5.1 Experimental Setup
+
+Datasets: We evaluate our algorithm on four image classification datasets: Permuted MNIST (PMNIST) [20], CIFAR-100 Split [18], CIFAR-100 Superclass [44] and TinyImageNet [37]. The PMNIST dataset is a variant of the MNIST dataset, in which each task applies a fixed random pixel permutation to the original dataset. The PMNIST benchmark dataset consists of 20 tasks, and each contains only 1000 samples from 10 different classes [13]. The CIFAR-100 Split is constructed by randomly dividing 100 classes of CIFAR-100 into 10 tasks with 10 classes per task. The CIFAR-100 Superclass is divided into 20 tasks according to the 20 superclasses of the CIFAR-100 dataset, and each superclass contains 5 different but semantically related classes. Whereas, TinyImageNet is constructed by splitting 200 classes into 40 5-way classification tasks. In our experiments, we do not use any data augmentation. The dataset statistics are given in Appendix D.1.
+
+Network Architecture: For PMNIST, we use a fully connected network with two hidden layers of 100 units each following [24]. For experiments of CIFAR-100 Split and CIFAR-100 Superclass, we use a 5-layer AlexNet and LeNet architecture similar to [34] respectively. For TinyImageNet, we use the same network architecture as [13], which consists of 4 conv layers and 3 fully connected layers. In PMNIST, all tasks share the final classifier layer, while other experiments use a multi-head incremental setting, that is, each task has a separate classifier. Refer to Appendix D.2 for more details.
+
+Baselines: We compare our method against multiple methods described below. (1) EWC [17], a regularization-based method that uses the diagonal of Fisher information to identify important weights; (2) ICARL [30], a memory-based method that uses knowledge-distillation and episodic memory to reduce forgetting; (3) GEM [24], another memory-based method that uses the gradient of episodic memory to constrain optimization to prevent forgetting; (4) ER [5], a simple and competitive method based on reservoir sampling; (5) La-MAML [13] and (6) GPM [34] are memory-based methods inspired by optimization-based meta-learning and based on gradient orthogonal constraints, respectively; (7) Multitask is an oracle baseline that all tasks are learned jointly using the entire dataset at once in a single network. Multitask is not a continual learning strategy but serves as an upper bound on average test accuracy on all tasks.
+
+Training Details: All baselines and our method use stochastic gradient descent (SGD) for training. For each task in PMNIST and TinyImageNet, we train the network in 1 and 10 epochs, respectively, with the batch size as 10. These experimental settings are the same as La-MAML [13], so that we directly compare with their reported results. In the CIFAR-100 Split and CIFAR-100 Superclass experiments, we use the early termination strategy to train up to 50 epochs for each task, which is based on the validation loss as proposed in [35]. For both datasets, the batch size is set to 64. The replay buffer size of PMNIST, CIFAR-100 Split, CIFAR-100 Superclass, and TinyImageNet are 200, 1000, 1000, and 400, respectively. Details about the experimental setting and the hyperparameters considered for each baseline are provided in Appendix D.5 and D.6.
+
+![](images/4e073446d807c347d9eea6966f758b844a885215dc7d4c6ea9417ba0c8bc3f17.jpg)  
+Figure 5: (a) Average accuracy as a function of the number of tasks trained on 20-Split CIFAR-100 Superclass. (b) Training time per task on 20-Split CIFAR-100 Superclass. (c) Memory usage on four datasets. ("Superclass" is abbr. of CIFAR-100 Superclass).
+
+Table 1: Experimental results on 10-Split CIFAR-100, 20-Split CIFAR-100 Superclass and 40-Split TinyImageNet in 50 epochs. Each experiment is run with 5 seeds. † and ∗ denotes results reported by [13] and [44] respectively.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR-100 Split</td><td colspan="2">CIFAR-100 Superclass</td><td colspan="2">TinyImageNet</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>EWC</td><td>72.77 ± 0.45</td><td>-3.59 ± 0.55</td><td>50.26 ± 1.48</td><td>-7.87 ± 1.63</td><td></td><td></td></tr><tr><td>GEM</td><td>70.15 ± 0.34</td><td>-8.61 ± 0.42</td><td>50.35 ± 0.80</td><td>-9.50 ± 0.85</td><td>50.57 ± 0.61†</td><td>-20.50 ± 0.10+</td></tr><tr><td>ICARL</td><td>53.50 ± 0.81</td><td>-20.44 ± 0.82</td><td>49.05 ± 0.51</td><td>-11.24 ± 0.27</td><td>54.77 ± 0.32†</td><td>-3.93 ± 0.55†</td></tr><tr><td>ER</td><td>70.07 ± 0.35</td><td>-7.70 ± 0.59</td><td>51.64 ± 1.09</td><td>-7.86 ± 0.89</td><td>48.32 ± 1.51†</td><td>-19.86 ± 0.70t</td></tr><tr><td>La-MAML</td><td>71.37 ± 0.67</td><td>-5.39 ± 0.53</td><td>54.44 ± 1.36</td><td>-6.65 ± 0.85</td><td>66.90 ± 1.65†</td><td>-9.13 ± 0.90t</td></tr><tr><td>GPM</td><td>73.18 ± 0.52</td><td>-1.17 ± 0.27</td><td>57.33 ± 0.37</td><td>-0.37 ± 0.12</td><td>67.39 ± 0.47</td><td>1.45 ± 0.22</td></tr><tr><td>FS-DGPM</td><td>74.33 ± 0.31</td><td>-2.71 ± 0.17</td><td>58.81 ± 0.34</td><td>-2.97 ± 0.35</td><td>70.41 ± 1.30</td><td>-2.11 ± 0.84</td></tr><tr><td>Multitask</td><td>79.75 ± 0.38</td><td>-</td><td>61.00 ±0.20*</td><td>1</td><td>77.10 ± 1.06†</td><td>-</td></tr></table>
+
+Metrics: We evaluate the continual learning performance by the average accuracy (ACC) and backward transfer (BWT) [24, 4, 5], formulated as following,
+
+$$
+A C C = \frac { 1 } { T } \sum _ { i = 1 } ^ { T } R _ { T , i } , \quad B W T = \frac { 1 } { T - 1 } \sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } ,
+$$
+
+where $T$ is the total number of learned sequential tasks, $R _ { i , j }$ is the test classification accuracy of the model on $j$ -th task after learning the last sample from $i$ -th task. ACC is the average test classification accuracy of all tasks, bigger is better. BWT is the interference of new learning on the past knowledge. More specifically, negative BWT implies (catastrophic) forgetting whereas positive BWT indicates learning new task increases the performance on some preceding tasks.
+
+# 5.2 Results and Discussion
+
+PMNIST: We first evaluate our algorithm for 20 sequential PMNIST tasks with only 1000 samples per task in a single-head setting. From the results, as shown in Table 2, we see that our method (FS-DGPM) achieves the best average accuracy $( 7 6 . 9 6 \% \pm 0 . 7 7 )$ . Moreover, we achieve the least amount of forgetting except GPM, which is essentially a trade-off in accuracy to minimize forgetting. As shown in Figure 5(c), we only use about $31 \%$ of the final memory size of GPM and achieve $\sim 2 . 5 \%$ better accuracy.
+
+Table 2: Experimental results (mean $\pm$ std in 5 runs) on PMNIST in single-epoch.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">PMNIST</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>EWC</td><td>62.25 ± 1.44</td><td>-15.22 ± 1.25</td></tr><tr><td>GEM</td><td>61.82 ± 0.85</td><td>-15.58 ± 1.17</td></tr><tr><td>ER</td><td>68.31 ±0.56</td><td>-13.91 ± 0.67</td></tr><tr><td>La-MAML</td><td>75.98 ±0.60</td><td>-10.21 ± 0.90</td></tr><tr><td>GPM</td><td>74.54 ± 0.36</td><td>-7.17 ± 0.51</td></tr><tr><td>FS-DGPM</td><td>76.96 ± 0.51</td><td>-7.45 ± 0.30</td></tr><tr><td>Multitask</td><td>86.54 ± 1.74</td><td>=</td></tr></table>
+
+![](images/4d317dac45d492cb2d0838a60b80610b725c142e5c467d6df22414edd67b297e.jpg)  
+Figure 6: The ablation study implemented on CIFAR-100 Split with 10 tasks. (a)-(c) shows the stability of the first task. (a) is the test accuracy change curve of the first task; (b) and (c) are the weight loss landscape of the first task after learning the fifth task and all ten tasks. (d)-(e) shows the sensitivity of the fifth task, which are the test accuracy and the weight loss landscape of the fifth task after just learning the fifth task. The shape of the weight loss landscape obtained using ten different random filter-normalized directions. ( $" \mathrm { T } i "$ is abbr. of the $i$ -th task)
+
+CIFAR-100 and TinyImageNet: Next, we use a multi-head setting to evaluate our algorithm under the more challenging visual classification benchmarks. Table 1 reports all results of these experiments. We outperform all baselines on three datasets, with achieving the best average accuracy $7 4 . 3 3 \%$ , $5 8 . 8 1 \%$ and $7 0 . 4 1 \%$ . In these experiments, we observe that GPM is a strong baseline with the least forgetting. At the same time, we highlight that our method achieves the highest accuracy on all datasets and the second-lowest forgetting after GPM. Figure 5(a) shows the process of performance changing with the number of tasks on the CIFAR-100 Superclass. We consistently see the superior performance of our method at any stage of model evolution. It is also worth emphasizing that although our method requires more time for training than GPM, it has lower memory usage and better test accuracy (See Figure 5(b)-(c)). As noted by [4] and [38], EWC performs poorly without multiple passes over the datasets, and GEM is not very effective under the single-head variants. These situations have also been observed in our experiments.
+
+# 5.3 Ablation studies on FS-DGPM
+
+Table 3: The ablation study results on CIFAR-100 Split and Superclass. Each experiment is run with 5 seeds.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR-100 Split</td><td colspan="2">CIFAR-100 Superclass</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>FS-DGPM</td><td>74.33 ± 0.31</td><td>-2.71 ± 0.17</td><td>58.81 ± 0.34</td><td>-2.97± 0.35</td></tr><tr><td>La-DGPM</td><td>73.74 ± 0.61</td><td>-3.05 ± 0.73</td><td>58.18 ± 0.41</td><td>-2.41 ± 0.39</td></tr><tr><td>FS-GPM</td><td>73.96 ± 0.44</td><td>-3.12 ±0.43</td><td>58.61 ± 0.53</td><td>-2.79± 0.30</td></tr><tr><td>DGPM</td><td>73.78 ± 0.32</td><td>-3.67 ± 0.42</td><td>56.78 ± 0.49</td><td>-2.44 ± 0.40</td></tr><tr><td>GPM</td><td>73.18± 0.52</td><td>-1.17 ± 0.27</td><td>57.33 ± 0.37</td><td>-0.37 ± 0.12</td></tr></table>
+
+We further investigate our model performance with an ablation study and summarize it in Table 3. We respectively ablate the effects of flattening sharpness and dynamically adjusting the soft weight for bases. We refer to them as DGPM and FS-GPM. We also construct an ablation referred to as LaDGPM (Look-ahead DGPM), where the adversarial weight perturbation $\pmb { v }$ in Eq. (2) is changed to the direction
+
+of gradient descent. At the same time, we also change the sign in Eq. (3) to ensure that the soft weight of the basis is reduced when the gradients are aligned in the same direction. From the results, shown in Table 3, we observe that flattening sharpness does benefits GPM, with $\sim 1 . 0 \%$ improvement over GPM on both datasets. We can further observe through Figure 6 that all landscapes of FS-DGPM have lower loss values and flatter neighbors than DGPM and La-DGPM on the CIFAR-100 Split experiments. In addition, we see that DGPM performs well in learning new tasks, but it also leads to forgetting previous tasks. This situation can be efficiently alleviated by flattening sharpness. Hence, FS is indeed a powerful method worthy of being widely adopted for continual learning scenarios.
+
+# 6 Conclusion
+
+In this paper, we explore the weight loss landscape to characterize the well-known ’sensitivitystability’ dilemma faced by continual learning algorithms, and find that lower loss value with flatter neighbor often leads to better continual learning performance. Based on this finding, we propose our FS-DGPM algorithm, which introduces a soft weight to represent the importance of each basis representing past tasks in GPM, so that less important bases can be dynamically released to improve the sensitivity of new skill learning. Flattening Sharpness (FS) is also introduced here to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all tasks seen so far. The evaluation of various image classification tasks with different network architectures and the comparison with some state-of-the-art algorithms show the effectiveness of our method in achieving high classification performance while alleviating forgetting. Although our method theoretically and empirically demonstrates the advantages of introducing FS into continual learning, whether there exists a closer upper bound for the continual learning performance remains an unresolved problem and left for our future exploration. Although our method theoretically and empirically demonstrates the advantages of introducing bases soft weight and FS into continual learning, whether there exists a better dynamic adjustment and a closer upper bound for the continual learning performance remains an unresolved problem and left for our future exploration.
+
+# Acknowledgments
+
+We would like to thank Dr. Yongzhe Deng and Zhuo Zhang for their helpful discussions, and anonymous reviewers for their valuable comments to improve this work. This work was supported by a grant from the National Key Research and Development Program of China (Project No. 2020YFB1313900), Hong Kong Research Grants Council under General Research Fund (Project No. 14201620), the National Natural Science Foundation of China (Project No. 62006219, 62072452) and Guangdong Provincial Basic and Applied Basic Research Fund-Regional Joint Fund (Project No. 2020B1515130004).
+
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+        "text": "Flattening Sharpness for Dynamic Gradient Projection Memory Benefits Continual Learning ",
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+        "text": "Danruo Deng1, Guangyong Chen2∗, Jianye $\\mathbf { H a o ^ { 3 , 4 } }$ , Qiong Wang2, Pheng-Ann Heng1,2 1The Chinese University of Hong Kong, 2Guangdong-Hong Kong-Macao Joint Laboratory of Human-Machine Intelligence-Synergy Systems, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, 3College of Intelligence and Computing, Tianjin University, 4Huawei Noah’s Ark Lab {drdeng,pheng}@cse.cuhk.edu.hk, {gy.chen, wangqiong}@siat.ac.cn, jianye.hao@tju.edu.cn ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The backpropagation networks are notably susceptible to catastrophic forgetting, where networks tend to forget previously learned skills upon learning new ones. To address such the ’sensitivity-stability’ dilemma, most previous efforts have been contributed to minimizing the empirical risk with different parameter regularization terms and episodic memory, but rarely exploring the usages of the weight loss landscape. In this paper, we investigate the relationship between the weight loss landscape and sensitivity-stability in the continual learning scenario, based on which, we propose a novel method, Flattening Sharpness for Dynamic Gradient Projection Memory (FS-DGPM). In particular, we introduce a soft weight to represent the importance of each basis representing past tasks in GPM, which can be adaptively learned during the learning process, so that less important bases can be dynamically released to improve the sensitivity of new skill learning. We further introduce Flattening Sharpness (FS) to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all seen tasks. As demonstrated empirically, our proposed method consistently outperforms baselines with the superior ability to learn new skills while alleviating forgetting effectively.2. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Humans have the ability to continually learn new knowledge without forgetting their previously learned ones through mediating a rich set of neurocognitive mechanisms [41, 15, 39]. This ability, often known as continual learning or lifelong learning [29], is crucial for computational systems, such as deep neural networks (DNNs), which are required to sequentially learn and deal with multiple tasks when implemented in the dynamically changing environment. Continual learning remains a long-standing challenge for DNNs since these networks are typically trained with stationary training batches by stochastic gradient descent methods [19], which generally leads to an abrupt performance decrease on previously learned tasks as new tasks are learned. To address such catastrophic forgetting, we can brutally retrain an oracle network on the entire dataset containing all tasks to capture dynamic changes in the data distribution, but this methodology is obviously too inefficient to hinder the learning of novel data in real time. ",
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+        "type": "text",
+        "text": "During the last few years, lots of research efforts have been devoted to improving the stability of DNNs on old tasks while keeping sensitive to new information. The first intuitive idea is to introduce an independent branch for each new task while freezing the old task parameters to preserve the old knowledge [33, 45, 43, 25, 35, 22]. However, in this way, the network will inevitably become redundant as the task number continually increases. As presented in the neurocognitive works [41, 15], the reactivation of neuronal activity patterns, representing old memories, plays an important role in the continual learning of humans [39]. Thus, forgetness can be effectively mitigated by training a single network for new tasks by considering diverse information stored in the memory, including the original training samples of old tasks [31, 5, 13], the gradients induced from old tasks [9] and the feature subspace representing old tasks [34]. However, their continual learning performance is still limited because DNNs can easily overfit the limited information stored in the small-size memory. ",
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+    {
+        "type": "text",
+        "text": "The overfitting problem of DNNs is often attributed to the complex loss landscape containing multiple local optima, and the sharpness of the loss landscape has been widely used to characterize the generalization gap in standard training scenarios from both theoretical and empirical perspectives [27, 21, 42, 10, 6]. While this characterization has inspired new approaches for model training with better generalization, practical algorithms that especially seek out flatter minima to effectively address the ’sensitivity-stability’ dilemma for continual learning have thus far been elusive. In this paper, our first contribution is to characterize the weight loss landscape for the continual learning scenario and identify that a flatter loss landscape with lower loss value often leads to better continual learning performance, as shown in Figure 1 and Figure 3. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Further, based on our characterization of the weight loss landscape, we find that the recently proposed Gradient Projection Memory (GPM) method [34] maintains the lowest loss value on old tasks among the previously proposed methods by taking gradient steps orthogonal to the subspace representing old tasks. However, its loss landscape on newly learned tasks is the sharpest due to the lack of sufficient subspace left for new task learning. To improve the network’s sensitivity, our second contribution is to predict the importance of bases spanning the subspace for old tasks, so that less important bases can be dynamically released. In particular, we introduce a soft weight to indicate the bases importance, which can be dynamically adjusted by combining the Flattening Sharpness $( F S )$ to minimize the loss value and loss sharpness simultaneously. Intuitively, a basis will be regarded as important for preserving old knowledge if the gradients induced by new tasks and old ones are aligned in the opposite direction on that basis. As demonstrated through extensive experiments, our proposed method can consistently outperform the state-of-the-art methods [17, 30, 24, 31, 5, 13, 34] by a notable margin across a range of widely used benchmark datasets. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we briefly survey the representative works of continual learning by highlighting their contributions. To simplify our presentation, this section is organized by dividing the representative works into three categories, parameter isolation, regularization-based, memory-based methods. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Parameter isolation methods address forgetting by assigning a different subset of network parameters to each task. Without restrictions on network architecture, new neurons or layers or modules can be added for new tasks, while the previous task parameters can be frozen or copied to preserve old knowledge. For instance, Progressive Neural Network (PGN) [33] freezes the parameters trained with previous knowledge while expands the architecture by allocating new sub-networks with fixed capacity for new tasks. Dynamically Expandable Networks (DEN) [45] selectively retrains or expands network capacity by splitting/duplicating important units on new tasks. Reinforced Continual Learning (RCL) [43] uses reinforcement learning strategy to adaptively expand the network of each layer, while [22] use neural architecture search to find optimal network structures for each sequential task. Alternatively, with the architecture remaining static, a fixed part is allocated to each task. During the training of a new task, previous task parts are masked out to prevent interference. The mask sets are imposed at parameter level [25], or unit level [35]. PackNet [25] uses iterative pruning to fully restrict gradient updates on important weights via a binary mask, whereas HAT [35] limits the update of important units recognized by the hard attention mask through gradient backpropagation. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Regularization-based methods introduce an additional regularization term in the loss function to consolidate previous knowledge without using replay. This involves using knowledge distillation [23, 14] or penalizing changes in weights deemed important for previous tasks [17, 46, 28, 2, 3] to reduce forgetting. There are many ways to measure the importance. Elastic Weight Consolidation (EWC) [17] identifies important weights based on the diagonal values of Fisher information matrix after training, while Synaptic Intelligence (SI) [46] calculates them online and over the entire learning trajectory in parameter space. Memory Aware Synapses (MAS) [2] estimates importance weights in an unsupervised manner, while Variational Continual Learning (VCL) [28] introduces a variational framework that spawned some Bayesian-based works [32, 8, 1, 7]. For example, [32] recursively uses a Gaussian Laplace approximation of the Hessian to approximate the posterior after every task, [8] adjusts the learning rate according to the uncertainty defined by the probability distribution of the network weights. [1] introduces an interpretation of node-wise uncertainty on the Kullback-Leibler (KL) divergence term of the variational lower bound for Gaussian mean-field approximation. ",
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+    {
+        "type": "text",
+        "text": "Our method mainly follows memory-based methods, which mitigate forgetting based on information extracted from old tasks or based on a generative model to generate pseudo samples. For example, iCaRL [30] selects and stores samples closest to the feature mean of each class. ER [5, 31] suggests reservoir sampling under the limited and fixed budget for replay buffer. Deep Generative Replay (DGR) [36] trains a deep generative model in the Generative Adversarial Network (GAN) framework [11] to simulate past data. These previous task samples are mainly reused as model inputs for replay in the above methods. However, replay might be prone to overfitting the subset of stored samples. Alternatively, samples stored in memory can also be used to constrain the optimization of the new task loss to prevent previous task interference, thereby leaving more leeway for backward and forward transfer. Gradient Episodic Memory (GEM) [24] projects the estimated gradients in the feasible region, which is outlined by previous task gradients calculated from the episodic memory samples. Averaged-GEM (A-GEM) [4] relaxes the projection to a direction that is estimated from samples randomly selected from memory. [12] proposes a unified view of episodic memory-based continual learning methods, including GEM and A-GEM, and improves performance over these methods by using a loss-balancing update scheme. A few other works have utilized gradient information to protect previous knowledge. [31, 13] adopt optimization-based meta-learning to enforce gradient alignment between samples from different tasks. GPM [34] minimizes interference between sequential tasks by ensuring that gradient updates only occur in directions orthogonal to the input of previous tasks. ",
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+    {
+        "type": "text",
+        "text": "3 The Weight Loss Landscape of Continual Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we first introduce our formulation of continual learning, and then characterize the weight loss landscape for the continual learning scenario from stability and sensitivity. Finally, some insights combining the weight loss landscape and continual learning are provided. ",
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+    {
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+        "text": "3.1 Problem Formulation ",
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+    {
+        "type": "text",
+        "text": "Throughout the paper, we denote scalars as $a$ , vectors as $\\textbf { \\em a }$ , matrices as $\\pmb { A }$ , and sets as $\\mathcal { A }$ . We consider a supervised learning setup where $T$ tasks are sequentially learned from their training data. Each task has an identical task descriptor, $\\tau \\in \\{ 1 , 2 , \\ldots , T \\}$ , with its dataset $\\mathscr { D } _ { \\tau } = \\{ \\pmb { x } _ { i , \\tau } , y _ { i , \\tau } \\} _ { i = 1 } ^ { n _ { \\tau } }$ containing $n _ { \\tau }$ samples randomly generated from a latent distribution $\\mathcal { D } _ { \\tau }$ . At any time-step during the learning process, we minimize the empirical risk of the model on all $t$ tasks seen so far, with just limited size of memory $\\mathcal { M }$ to summarize the training data of previous tasks $\\{ \\mathcal { D } _ { \\tau } \\} _ { \\tau = 1 } ^ { t - 1 }$ . To simplify the notation, we denote $\\begin{array} { r } { L _ { \\mathcal { A } } ( \\pmb { w } ) = \\frac { 1 } { | \\mathcal { A } | } \\sum _ { ( \\pmb { x } , \\pmb { y } ) \\in \\mathcal { A } } \\left[ \\ell \\left( f _ { \\pmb { w } } \\left( \\pmb { x } \\right) , \\pmb { y } \\right) \\right] } \\end{array}$ as the average empirical loss for the set , where $\\ell ( \\cdot , \\cdot )$ is an arbitrary loss function (e.g. the cross-entropy (CE) loss), $| { \\cal A } |$ is the sample size of the set $\\mathcal { A }$ , and $f _ { w }$ is the DNN with weight $\\textbf { \\em w }$ . Our final goal is to find an optimal parameter $\\pmb { w }$ , which minimizes the overall risk $\\scriptstyle \\sum _ { \\tau = 1 } ^ { T } L _ { \\mathcal { D } _ { \\tau } } ( \\pmb { w } )$ for all tasks. ",
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+    {
+        "type": "text",
+        "text": "3.2 Connection of Weight Loss Landscape and Continual Learning ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After learning a new task, we visualize the weight loss landscape of each task seen so far by plotting changes in its training loss when moving the weights $\\pmb { w }$ in a random direction $^ d$ with magnitude $\\alpha$ following [21]: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0cf543168bc901a43cda573b70ad79cac9dfa5c3490678689a90c6aa0e768953.jpg",
+        "text": "$$\ng _ { t } ( \\alpha ) = L _ { \\mathcal { D } _ { t } } ( \\pmb { w } + \\alpha \\pmb { d } ) = \\frac { 1 } { | \\mathcal { D } _ { t } | } \\sum _ { ( \\pmb { x } , \\pmb { y } ) \\in \\mathcal { D } _ { t } } \\ell \\left( f _ { \\pmb { w } + \\alpha d } \\left( \\pmb { x } \\right) , \\pmb { y } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            832,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { t }$ is the training set for the $t$ -th task previously learned. To eliminate the scaling invariance of DNNs, $^ d$ is sampled from a Gaussian distribution and further normalized by $\\begin{array} { r } { \\pmb { d } _ { l , j }  \\frac { \\mathbf { \\bar { d } } _ { l , j } } { \\| \\pmb { d } _ { l , j } \\| _ { F } } \\| \\pmb { w } _ { l , j } \\| _ { F } } \\end{array}$ where $d _ { l , j }$ represents the $j$ -th filter at $l$ -th layer of $^ d$ , and $\\left\\| \\cdot \\right\\| _ { F }$ denotes the Frobenius norm. Compared with our visualization, [26] only consider one task and plot the loss landscape along the directions of its Hessian eigenvectors, which only reflects some of the relationship between forgetting and landscape. Considering $^ d$ is randomly selected, we repeat the visualization 10 times with different $^ d$ ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd37d0fd577ccb1d848a629a229d31c87841d79be1d0b99c0f1ac596de52444b.jpg",
+        "image_caption": [
+            "Figure 1: The connection between the weight loss landscape and continual learning is investigated on four methods. (a)-( c) shows the stability of the first task. (a) is the test accuracy change curve of the first task; (b) and (c) are the weight loss landscape of the first task after learning the fifth task and all ten tasks. (d)-(e) shows the sensitivity of the fifth task, which are the test accuracy and the weight loss landscape of the fifth task after just learning the fifth task. The shape of the weight loss landscape obtained using ten different random filter-normalized directions. ( $\" \\mathrm { T } i \"$ is abbr. of the $i$ -th task) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first study the stability of the network by plotting changes of the weight loss landscape for the first task after new task learning. In particular, We use the previously proposed ER [5], La-MAML [13], and GPM [34] to train a MLP network with two hidden layers on the Permuted MNIST (PMNIST) [20] dataset that contains 10 tasks. We also retrain the network on the entire dataset contain all passed tasks as an Oracle network. Early stopping is used to halt the training with up to 10 epochs for each task based on the validation loss as proposed in [35]. As shown in Figure 1(a), all three continual learning methods lose their stability as learning new tasks. It can be observed from Figure 1(b)-(c) that the weight loss landscape becomes sharper and loss value increases simultaneously, when the testing accuracy of the first task continually decreases. We further evaluate the sensitivity of the network by observing the performance of the fifth task just after it has just been learned. As shown in Figure 1(d)-(e), ER shows the best learning capability compared with other methods with the lowest loss value and the flattest loss landscape. Thus, based on these empirical findings, we assume that lower loss value with a flatter neighbor may lead to better continual learning performance. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 A Case Study of Flattening Sharpness for Vanilla ER ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this part, we further validate our above assumption by Flattening Sharpness for vanilla ER (FS-ER). Compared with ER that looks for a solution $\\pmb { w }$ that jointly minimizes the training loss of current task data and memory data, FS-ER seeks out a solution with both low loss and flat neighbor by minimizing the maximal loss in the neighbor around the parameter value. The schematic of the Flattening Sharpness (FS) is shown in Figure 2. We introduce the adversarial weight perturbation (orange dashed line) to explicitly flatten the weight loss landscape via injecting the worst-case weight perturbation, which is calculated from the current task data and past task data sampled from the replay buffer (Refer to Appendix C.1 and the next section for more details). Figure 3 respectively shows the weight loss landscapes of the fifth task after just learning the fifth task and all tasks. We find that FS-ER successfully gets a solution with a lower loss value and flatter landscape, either after the fifth task has just been learned or all ten tasks have been learned. The average testing accuracy of all tasks using FS-ER is $9 0 . 4 4 \\%$ , significantly higher than ER $( 8 6 . 1 6 \\% )$ , which means that flattening sharpness does benefit continual learning. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e3f3ddf535810155a992fc66e3895a7221d3a75c0786f35ade9210a2420a872.jpg",
+        "image_caption": [
+            "Figure 2: Schematic of FS-ER update. The dashed line and the solid line indicate the gradient ascent and descent, respectively. Orange denotes the actual update of the parameter $\\pmb { w }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/b58c7460443734cc64e08af56719f4753f34ebec8f27068bdfe89caa7994dc81.jpg",
+        "image_caption": [
+            "Figure 3: Landscape of the fifth task after just learning (a) the fifth task and (b) all ten tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Flattening Sharpness for Dynamic Gradient Projection Memory ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1, GPM achieves the highest testing accuracy on old tasks among all three practical continual learning methods, but shows less sensitivity to new task learning. To address this issue, we propose Flattening Sharpness for Dynamic GPM (FS-DGPM), which dynamically adjusts the gradient subspace representing the past tasks to improve the sensitivity to the new task, while ensuring stability of the previous tasks. In particular, we let $M = [ \\pmb { u } _ { 1 } , \\pmb { u } _ { 2 } , \\cdot \\cdot \\cdot , \\pmb { u } _ { k } ]$ denote the bases matrix that spans the gradient subspace of the previous task, $\\pmb { \\Lambda } = \\mathrm { d i a g } [ \\lambda _ { 1 } , \\lambda _ { 2 } , . . . , \\lambda _ { k } ]$ be the diagonal matrix with its $i$ -th diagonal element $\\lambda _ { i } \\in [ 0 , 1 ]$ indicating the importance of each basis, and $k$ is the number of bases. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Sharpness Evaluation ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Comparing with the classical strategy that perturbs weight in the entire space [40, 16, 42, 10], we focus on characterizing the weight loss landscape on the new task with respect to the important subspace representing old task. The important subspace can be effectively calculated based on the examples sampled from replay buffer $\\mathcal { M }$ after each task training. Then, we can find the worst case by maximizing the training loss of the network on the new task in this subspace. Formally, the sharpness of the loss landscape around the solution $\\textbf { \\em w }$ in the old parameter space can be predicted as, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a885f3df2ee29aa1ff9d96dd9ac8b0eb91d431a9dc07fb1b433fc0cb30661ec2.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pmb { v } \\in \\mathcal { V } } L _ { \\hat { \\mathcal { D } } _ { t } } \\left( \\pmb { w } + \\pmb { v } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            362,
+            571,
+            386
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\nu$ denotes the subspace spanned by $M$ and $\\pmb { \\Lambda }$ , and $\\hat { \\mathcal { D } } _ { t }$ denotes the batch samples of the current $t$ -th task. As shown in Eq. (1), the high value can be obtained when the network fails to learn the new task (sensitivity) and the new task learning seriously interferes with the past tasks learning (stability). Based on the gradient method, the adversarial weight perturbation $\\pmb { v }$ can be solved as, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee3c67afa7009a5def4c9bf6711ed7fec57055063dd72c06dd0375e6d28424b0.jpg",
+        "text": "$$\n\\pmb { v }  \\pmb { v } + \\eta _ { 1 } \\pmb { M } \\pmb { \\Lambda } \\pmb { M } ^ { T } ( \\nabla _ { ( \\pmb { w } + \\pmb { v } ) } L _ { \\hat { \\mathcal { D } } _ { t } } ( \\pmb { w } + \\pmb { v } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta _ { 1 }$ is the update step size. Note that $\\textbf {  { v } }$ is initialized as 0 and layer-wise updated. As shown in Appendix $\\mathrm { E }$ , two-step for $\\textbf {  { v } }$ (default settings) are enough to get good improvements. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Dynamic Gradient Projection Memory ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After obtaining the adversarial weight perturbation $\\textbf {  { v } }$ , we can further update the bases importance matrix $\\pmb { \\Lambda } = \\mathrm { d i a g } [ \\lambda _ { 1 } , \\lambda _ { 2 } , . . . , \\lambda _ { k } ]$ by jointly considering the current task batch $\\hat { \\mathcal { D } } _ { t }$ and the batch $\\hat { \\mathcal { M } }$ sampled from the replay buffer $\\mathcal { M }$ as following, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b49f3213fc057863159417f5e61106cc8ce8962ec7493248dda3a216c10417b.jpg",
+        "text": "$$\n\\lambda _ { i }  \\lambda _ { i } - \\eta _ { 2 } ( \\nabla _ { \\lambda _ { i } } L _ { \\hat { \\mathcal { D } } _ { t } \\cup \\hat { \\mathcal { M } } } ( \\pmb { w } + \\pmb { v } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            608,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the sigmoid function is used at the end of gradient update to constrain the importance value $\\lambda _ { i }$ between 0 and 1. In addition, the second term on the right side in Eq. (3) can be approximated by the first-order Taylor expansion as, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/920c172093fc49828f5aa429ea4120e251ff7ca56b0dec6d9e5bb9567d1ade9d.jpg",
+        "text": "$$\n\\nabla _ { \\lambda _ { i } } L _ { \\hat { \\mathcal { D } } _ { t } \\cup \\hat { \\mathcal { M } } } ( { \\boldsymbol w } + { \\boldsymbol v } ) \\approx \\eta _ { 1 } \\left( \\nabla _ { { \\boldsymbol w } } L _ { \\hat { \\mathcal { D } } _ { t } } \\left( { \\boldsymbol w } \\right) \\right) ^ { T } \\mathbf { { u } } _ { i } \\boldsymbol u _ { i } ^ { T } \\left( \\nabla _ { { \\boldsymbol w } } L _ { \\hat { \\mathcal { D } } _ { t } \\cup \\hat { \\mathcal { M } } } \\left( { \\boldsymbol w } \\right) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above equation characterizes the relationship between the gradients induced by the current task and the old tasks based on the basis $\\mathbf { \\delta } \\mathbf { u } _ { i }$ . As illustrated in the Figure 4, this equation implies that when the projections of two gradients on the basis $\\mathbf { \\delta } \\mathbf { u } _ { i }$ are aligned in the same direction, the gradient of $\\lambda _ { i }$ will be positive, and when there is interference, the gradient will be negative. The positive (negative) gradient will decrease (increase) the importance $\\lambda _ { i }$ , thereby releasing (tightening) the update limit of the new task on the corresponding ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c65063f71562cc6c18285cf0a8ea38b34c5439409e5823be9b371fd66973c10.jpg",
+        "image_caption": [
+            "Figure 4: A depiction of transfer (a) and interference (b) in the basis $\\textbf { \\em u }$ of gradient space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "basis $\\mathbf { \\delta } \\mathbf { u } _ { i }$ . We provide the full derivation in the Appendix A.2. Note that the initial value of all importance is set to 1 and dynamically adjusted from the second task. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 Weight Updating ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, we update the model parameters by minimizing the worst performance of $f _ { w + v }$ with respect to $\\textbf { \\em w }$ , while adjusting the update magnitude of $\\pmb { w }$ on each basis based on its importance to alleviate forgetting. More concretely, the parameter $\\textbf { \\em w }$ will be updated to: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27590f64dcd27cbf02fda2132f4c1a17a2d674081acbe4608e6c719480b419dd.jpg",
+        "text": "$$\n\\pmb { w }  \\pmb { w } - \\eta _ { 3 } ( \\pmb { I } - \\pmb { M } \\pmb { \\Lambda } \\pmb { M } ^ { T } ) \\nabla _ { \\pmb { w } } L _ { \\hat { D } _ { t } \\cup \\hat { \\mathcal { M } } } ( \\pmb { w } + \\pmb { v } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the optimization is performed over the model parameters $\\pmb { w }$ , whereas the objective is computed using perturbed model $f _ { w + v }$ . In addition, we update the replay buffer $\\mathcal { M }$ with reservoir sampling as in [31], and then use Singular Value Decomposition (SVD) to recalculate $M$ based on the sampling data in the replay buffer after learning one task following GPM [34]. Comparing with [34], we calculate the important bases in the entire gradient space and use them to replace the bases calculated last time. Besides that, our method degenerates to GPM when $\\eta _ { 1 }$ and $\\eta _ { 2 }$ are set to 0. The complete pseudo-code of FS-DGPM is outlined in the Algorithm 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/11f8d88b3416a939e9ec3781d82cb0c44d506c47e6da5d489773b58becaa53bd.jpg",
+        "table_caption": [
+            "Algorithm 1 FS-DGPM (Flattening Sharpness for Dynamic Gradient Projection Memory) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>weight step size n2,batch size b. InitializingM←{},M←I,△←I fort=1,2,.,Tdo for ep = 1,2,... , numepochs do</td><td>Input: Network weight w,loss function l, learning rate n3,FS step size 71,FS steps K, Soft</td></tr><tr><td>for batch Dt  Dt do MM</td><td></td></tr><tr><td>for k =1,.·,K do v ←v+niMAMT (V(w+v)Lb(w+ v))</td><td> Sharpness Evaluation</td></tr><tr><td>end for if t ≥ 2 then</td><td></td></tr><tr><td>△ ← △-n2(∀ALD:UM(ω+ ν))</td><td> Dynamic Gradient Projection Memory</td></tr><tr><td>end if w ← w - n3 (1 - MAMT) VwLDtUm(w+v)</td><td> Weight updating</td></tr><tr><td></td><td></td></tr><tr><td>Push Dt to M with reservior sampling</td><td></td></tr><tr><td>end for</td><td></td></tr><tr><td>end for</td><td></td></tr><tr><td></td><td></td></tr><tr><td>M ←UpdateGPM (M)</td><td></td></tr><tr><td></td><td>V see Appendix Alg. 2</td></tr><tr><td></td><td></td></tr><tr><td>end for</td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4 Theoretical Understanding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We further provide a theoretical view on why landscape can characterize the continual learning performance and why our proposed FS-DGPM works. To simplify our explanation, we only consider two tasks, which contains the training sets $\\mathcal { D } _ { 1 }$ and $\\mathcal { D } _ { 2 }$ sampled from the distributions $\\mathcal { D } _ { 1 }$ and $\\mathcal { D } _ { 2 }$ , respectively. Based on the previous works on PAC-Bayes bound [27, 42, 10], given a \"prior\" distribution $P$ (a common assumption is zero mean, $\\sigma ^ { 2 }$ variance Gaussian distribution) over the weights, the expected error of the classifier for the continual learning scenario can be bounded with probability at least $1 - \\delta$ over the draw of $n$ training data: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e200765fb1e7a4988dbdde787b8d451146a26fcd2339b7e566f7983972487f99.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { \\Delta w } \\mathbb { E } _ { v } \\left[ L _ { \\mathcal { D } _ { 1 } \\cup \\mathcal { D } _ { 2 } } ( w + \\Delta w + v ) \\right] \\leq \\displaystyle \\operatorname* { m i n } _ { \\Delta w \\in \\mathcal { V } ^ { \\mathcal { C } } } \\mathbb { E } _ { v } \\left[ L _ { \\mathcal { D } _ { 1 } } ( w + \\Delta w + v ) \\right] + L _ { \\mathcal { D } _ { 2 } } ( w + \\Delta w ) } \\\\ { \\displaystyle + \\operatorname* { m a x } _ { v \\in \\mathcal { V } } L _ { \\mathcal { D } _ { 2 } } ( w + \\Delta w + v ) - L _ { \\mathcal { D } _ { 2 } } ( w + \\Delta w ) + 4 \\sqrt { \\frac { 1 } { n } \\left( K L ( w + \\Delta w + v | | P ) + \\ln \\frac { 2 n } { \\delta } \\right) } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\Delta w$ is the update based on the previously optimal solution $\\pmb { w }$ learned on the old task $\\mathcal { D } _ { 1 }$ when learning the new one $\\mathcal { D } _ { 2 }$ , and $\\textbf {  { v } }$ is often chosen as a zero mean spherical Gaussian perturbation with variance $\\sigma ^ { 2 }$ in every direction. Let $\\Delta w \\in \\mathcal { V } ^ { C }$ , then $\\Delta w$ lies in the complementary space of the important space representing the old task $\\mathcal { D } _ { 1 }$ , so that $\\mathbb { E } _ { \\pmb { v } } \\left[ L _ { \\mathcal { D } _ { 1 } } ( \\pmb { w } + \\Delta \\pmb { w } + \\bar { \\pmb { v } } ) \\right]$ does not increase too much compared with the previously minimized $\\mathbb { E } _ { v } \\left[ L _ { \\mathcal { D } _ { 1 } } ( \\pmb { w } + \\pmb { v } ) \\right]$ . The second term denotes the empirical loss on the second task and the third term represents the sharpness of the weight loss landscape around the $\\mathbf { \\Delta } w + \\Delta w$ . Since we have constrained $\\Delta w \\in \\mathcal { V } ^ { C }$ , then it is natural to assume $\\pmb { v } \\in \\mathcal { V }$ , so that $\\Delta w + v$ will cover the full space. Thus, our FS-DGPM exactly optimizes the worstcase of the flatness of weight loss landscape to control the PAC-Bayes bound, which theoretically justifies both lower loss value and flatter landscape lead to better continual learning performance, and why our proposed FS-DGPM works. ",
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+        "text": "5 Experiments ",
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+        "text": "In this section, we conduct extensive experiments to compare the performance of our proposed FS-DGPM model with the state-of-the-art methods on widely used continual learning benchmark datasets. Additional results and more details about the datasets, experiment setup, baselines, and model architectures are presented in the Appendix D and E. ",
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+        "text": "5.1 Experimental Setup ",
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+        "text": "Datasets: We evaluate our algorithm on four image classification datasets: Permuted MNIST (PMNIST) [20], CIFAR-100 Split [18], CIFAR-100 Superclass [44] and TinyImageNet [37]. The PMNIST dataset is a variant of the MNIST dataset, in which each task applies a fixed random pixel permutation to the original dataset. The PMNIST benchmark dataset consists of 20 tasks, and each contains only 1000 samples from 10 different classes [13]. The CIFAR-100 Split is constructed by randomly dividing 100 classes of CIFAR-100 into 10 tasks with 10 classes per task. The CIFAR-100 Superclass is divided into 20 tasks according to the 20 superclasses of the CIFAR-100 dataset, and each superclass contains 5 different but semantically related classes. Whereas, TinyImageNet is constructed by splitting 200 classes into 40 5-way classification tasks. In our experiments, we do not use any data augmentation. The dataset statistics are given in Appendix D.1. ",
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+        "type": "text",
+        "text": "Network Architecture: For PMNIST, we use a fully connected network with two hidden layers of 100 units each following [24]. For experiments of CIFAR-100 Split and CIFAR-100 Superclass, we use a 5-layer AlexNet and LeNet architecture similar to [34] respectively. For TinyImageNet, we use the same network architecture as [13], which consists of 4 conv layers and 3 fully connected layers. In PMNIST, all tasks share the final classifier layer, while other experiments use a multi-head incremental setting, that is, each task has a separate classifier. Refer to Appendix D.2 for more details. ",
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+        "text": "Baselines: We compare our method against multiple methods described below. (1) EWC [17], a regularization-based method that uses the diagonal of Fisher information to identify important weights; (2) ICARL [30], a memory-based method that uses knowledge-distillation and episodic memory to reduce forgetting; (3) GEM [24], another memory-based method that uses the gradient of episodic memory to constrain optimization to prevent forgetting; (4) ER [5], a simple and competitive method based on reservoir sampling; (5) La-MAML [13] and (6) GPM [34] are memory-based methods inspired by optimization-based meta-learning and based on gradient orthogonal constraints, respectively; (7) Multitask is an oracle baseline that all tasks are learned jointly using the entire dataset at once in a single network. Multitask is not a continual learning strategy but serves as an upper bound on average test accuracy on all tasks. ",
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+        "type": "text",
+        "text": "Training Details: All baselines and our method use stochastic gradient descent (SGD) for training. For each task in PMNIST and TinyImageNet, we train the network in 1 and 10 epochs, respectively, with the batch size as 10. These experimental settings are the same as La-MAML [13], so that we directly compare with their reported results. In the CIFAR-100 Split and CIFAR-100 Superclass experiments, we use the early termination strategy to train up to 50 epochs for each task, which is based on the validation loss as proposed in [35]. For both datasets, the batch size is set to 64. The replay buffer size of PMNIST, CIFAR-100 Split, CIFAR-100 Superclass, and TinyImageNet are 200, 1000, 1000, and 400, respectively. Details about the experimental setting and the hyperparameters considered for each baseline are provided in Appendix D.5 and D.6. ",
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+        "image_caption": [
+            "Figure 5: (a) Average accuracy as a function of the number of tasks trained on 20-Split CIFAR-100 Superclass. (b) Training time per task on 20-Split CIFAR-100 Superclass. (c) Memory usage on four datasets. (\"Superclass\" is abbr. of CIFAR-100 Superclass). "
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+        "table_caption": [
+            "Table 1: Experimental results on 10-Split CIFAR-100, 20-Split CIFAR-100 Superclass and 40-Split TinyImageNet in 50 epochs. Each experiment is run with 5 seeds. † and ∗ denotes results reported by [13] and [44] respectively. "
+        ],
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+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">CIFAR-100 Split</td><td colspan=\"2\">CIFAR-100 Superclass</td><td colspan=\"2\">TinyImageNet</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>EWC</td><td>72.77 ± 0.45</td><td>-3.59 ± 0.55</td><td>50.26 ± 1.48</td><td>-7.87 ± 1.63</td><td></td><td></td></tr><tr><td>GEM</td><td>70.15 ± 0.34</td><td>-8.61 ± 0.42</td><td>50.35 ± 0.80</td><td>-9.50 ± 0.85</td><td>50.57 ± 0.61†</td><td>-20.50 ± 0.10+</td></tr><tr><td>ICARL</td><td>53.50 ± 0.81</td><td>-20.44 ± 0.82</td><td>49.05 ± 0.51</td><td>-11.24 ± 0.27</td><td>54.77 ± 0.32†</td><td>-3.93 ± 0.55†</td></tr><tr><td>ER</td><td>70.07 ± 0.35</td><td>-7.70 ± 0.59</td><td>51.64 ± 1.09</td><td>-7.86 ± 0.89</td><td>48.32 ± 1.51†</td><td>-19.86 ± 0.70t</td></tr><tr><td>La-MAML</td><td>71.37 ± 0.67</td><td>-5.39 ± 0.53</td><td>54.44 ± 1.36</td><td>-6.65 ± 0.85</td><td>66.90 ± 1.65†</td><td>-9.13 ± 0.90t</td></tr><tr><td>GPM</td><td>73.18 ± 0.52</td><td>-1.17 ± 0.27</td><td>57.33 ± 0.37</td><td>-0.37 ± 0.12</td><td>67.39 ± 0.47</td><td>1.45 ± 0.22</td></tr><tr><td>FS-DGPM</td><td>74.33 ± 0.31</td><td>-2.71 ± 0.17</td><td>58.81 ± 0.34</td><td>-2.97 ± 0.35</td><td>70.41 ± 1.30</td><td>-2.11 ± 0.84</td></tr><tr><td>Multitask</td><td>79.75 ± 0.38</td><td>-</td><td>61.00 ±0.20*</td><td>1</td><td>77.10 ± 1.06†</td><td>-</td></tr></table>",
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+        "text": "Metrics: We evaluate the continual learning performance by the average accuracy (ACC) and backward transfer (BWT) [24, 4, 5], formulated as following, ",
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+        "type": "equation",
+        "img_path": "images/ae4adf11151780cc5ecad63f94cb5997ab9db01a3e577f0e237e474e014a7b62.jpg",
+        "text": "$$\nA C C = \\frac { 1 } { T } \\sum _ { i = 1 } ^ { T } R _ { T , i } , \\quad B W T = \\frac { 1 } { T - 1 } \\sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } ,\n$$",
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+        "type": "text",
+        "text": "where $T$ is the total number of learned sequential tasks, $R _ { i , j }$ is the test classification accuracy of the model on $j$ -th task after learning the last sample from $i$ -th task. ACC is the average test classification accuracy of all tasks, bigger is better. BWT is the interference of new learning on the past knowledge. More specifically, negative BWT implies (catastrophic) forgetting whereas positive BWT indicates learning new task increases the performance on some preceding tasks. ",
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+    {
+        "type": "text",
+        "text": "5.2 Results and Discussion ",
+        "text_level": 1,
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+        "text": "PMNIST: We first evaluate our algorithm for 20 sequential PMNIST tasks with only 1000 samples per task in a single-head setting. From the results, as shown in Table 2, we see that our method (FS-DGPM) achieves the best average accuracy $( 7 6 . 9 6 \\% \\pm 0 . 7 7 )$ . Moreover, we achieve the least amount of forgetting except GPM, which is essentially a trade-off in accuracy to minimize forgetting. As shown in Figure 5(c), we only use about $31 \\%$ of the final memory size of GPM and achieve $\\sim 2 . 5 \\%$ better accuracy. ",
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+        "table_caption": [
+            "Table 2: Experimental results (mean $\\pm$ std in 5 runs) on PMNIST in single-epoch. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">PMNIST</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>EWC</td><td>62.25 ± 1.44</td><td>-15.22 ± 1.25</td></tr><tr><td>GEM</td><td>61.82 ± 0.85</td><td>-15.58 ± 1.17</td></tr><tr><td>ER</td><td>68.31 ±0.56</td><td>-13.91 ± 0.67</td></tr><tr><td>La-MAML</td><td>75.98 ±0.60</td><td>-10.21 ± 0.90</td></tr><tr><td>GPM</td><td>74.54 ± 0.36</td><td>-7.17 ± 0.51</td></tr><tr><td>FS-DGPM</td><td>76.96 ± 0.51</td><td>-7.45 ± 0.30</td></tr><tr><td>Multitask</td><td>86.54 ± 1.74</td><td>=</td></tr></table>",
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+        "image_caption": [
+            "Figure 6: The ablation study implemented on CIFAR-100 Split with 10 tasks. (a)-(c) shows the stability of the first task. (a) is the test accuracy change curve of the first task; (b) and (c) are the weight loss landscape of the first task after learning the fifth task and all ten tasks. (d)-(e) shows the sensitivity of the fifth task, which are the test accuracy and the weight loss landscape of the fifth task after just learning the fifth task. The shape of the weight loss landscape obtained using ten different random filter-normalized directions. ( $\" \\mathrm { T } i \"$ is abbr. of the $i$ -th task) "
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+        "text": "CIFAR-100 and TinyImageNet: Next, we use a multi-head setting to evaluate our algorithm under the more challenging visual classification benchmarks. Table 1 reports all results of these experiments. We outperform all baselines on three datasets, with achieving the best average accuracy $7 4 . 3 3 \\%$ , $5 8 . 8 1 \\%$ and $7 0 . 4 1 \\%$ . In these experiments, we observe that GPM is a strong baseline with the least forgetting. At the same time, we highlight that our method achieves the highest accuracy on all datasets and the second-lowest forgetting after GPM. Figure 5(a) shows the process of performance changing with the number of tasks on the CIFAR-100 Superclass. We consistently see the superior performance of our method at any stage of model evolution. It is also worth emphasizing that although our method requires more time for training than GPM, it has lower memory usage and better test accuracy (See Figure 5(b)-(c)). As noted by [4] and [38], EWC performs poorly without multiple passes over the datasets, and GEM is not very effective under the single-head variants. These situations have also been observed in our experiments. ",
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+    {
+        "type": "text",
+        "text": "5.3 Ablation studies on FS-DGPM ",
+        "text_level": 1,
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+        "table_caption": [
+            "Table 3: The ablation study results on CIFAR-100 Split and Superclass. Each experiment is run with 5 seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">CIFAR-100 Split</td><td colspan=\"2\">CIFAR-100 Superclass</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>FS-DGPM</td><td>74.33 ± 0.31</td><td>-2.71 ± 0.17</td><td>58.81 ± 0.34</td><td>-2.97± 0.35</td></tr><tr><td>La-DGPM</td><td>73.74 ± 0.61</td><td>-3.05 ± 0.73</td><td>58.18 ± 0.41</td><td>-2.41 ± 0.39</td></tr><tr><td>FS-GPM</td><td>73.96 ± 0.44</td><td>-3.12 ±0.43</td><td>58.61 ± 0.53</td><td>-2.79± 0.30</td></tr><tr><td>DGPM</td><td>73.78 ± 0.32</td><td>-3.67 ± 0.42</td><td>56.78 ± 0.49</td><td>-2.44 ± 0.40</td></tr><tr><td>GPM</td><td>73.18± 0.52</td><td>-1.17 ± 0.27</td><td>57.33 ± 0.37</td><td>-0.37 ± 0.12</td></tr></table>",
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+        "type": "text",
+        "text": "We further investigate our model performance with an ablation study and summarize it in Table 3. We respectively ablate the effects of flattening sharpness and dynamically adjusting the soft weight for bases. We refer to them as DGPM and FS-GPM. We also construct an ablation referred to as LaDGPM (Look-ahead DGPM), where the adversarial weight perturbation $\\pmb { v }$ in Eq. (2) is changed to the direction ",
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+        "type": "text",
+        "text": "of gradient descent. At the same time, we also change the sign in Eq. (3) to ensure that the soft weight of the basis is reduced when the gradients are aligned in the same direction. From the results, shown in Table 3, we observe that flattening sharpness does benefits GPM, with $\\sim 1 . 0 \\%$ improvement over GPM on both datasets. We can further observe through Figure 6 that all landscapes of FS-DGPM have lower loss values and flatter neighbors than DGPM and La-DGPM on the CIFAR-100 Split experiments. In addition, we see that DGPM performs well in learning new tasks, but it also leads to forgetting previous tasks. This situation can be efficiently alleviated by flattening sharpness. Hence, FS is indeed a powerful method worthy of being widely adopted for continual learning scenarios. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we explore the weight loss landscape to characterize the well-known ’sensitivitystability’ dilemma faced by continual learning algorithms, and find that lower loss value with flatter neighbor often leads to better continual learning performance. Based on this finding, we propose our FS-DGPM algorithm, which introduces a soft weight to represent the importance of each basis representing past tasks in GPM, so that less important bases can be dynamically released to improve the sensitivity of new skill learning. Flattening Sharpness (FS) is also introduced here to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all tasks seen so far. The evaluation of various image classification tasks with different network architectures and the comparison with some state-of-the-art algorithms show the effectiveness of our method in achieving high classification performance while alleviating forgetting. Although our method theoretically and empirically demonstrates the advantages of introducing FS into continual learning, whether there exists a closer upper bound for the continual learning performance remains an unresolved problem and left for our future exploration. Although our method theoretically and empirically demonstrates the advantages of introducing bases soft weight and FS into continual learning, whether there exists a better dynamic adjustment and a closer upper bound for the continual learning performance remains an unresolved problem and left for our future exploration. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We would like to thank Dr. Yongzhe Deng and Zhuo Zhang for their helpful discussions, and anonymous reviewers for their valuable comments to improve this work. This work was supported by a grant from the National Key Research and Development Program of China (Project No. 2020YFB1313900), Hong Kong Research Grants Council under General Research Fund (Project No. 14201620), the National Natural Science Foundation of China (Project No. 62006219, 62072452) and Guangdong Provincial Basic and Applied Basic Research Fund-Regional Joint Fund (Project No. 2020B1515130004). ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "type": "text",
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