Add files using upload-large-folder tool

parse/train/8xoN9ZdSW8/8xoN9ZdSW8.md

@@ -0,0 +1,240 @@
+# Multimodal Virtual Point 3D Detection
+
+Tianwei Yin UT Austin yintianwei@utexas.edu
+
+Xingyi Zhou UT Austin zhouxy@cs.utexas.edu
+
+Philipp Krähenbühl UT Austin philkr@cs.utexas.edu
+
+# Abstract
+
+Lidar-based sensing drives current autonomous vehicles. Despite rapid progress, current Lidar sensors still lag two decades behind traditional color cameras in terms of resolution and cost. For autonomous driving, this means that large objects close to the sensors are easily visible, but far-away or small objects comprise only one measurement or two. This is an issue, especially when these objects turn out to be driving hazards. On the other hand, these same objects are clearly visible in onboard RGB sensors. In this work, we present an approach to seamlessly fuse RGB sensors into Lidar-based 3D recognition. Our approach takes a set of 2D detections to generate dense 3D virtual points to augment an otherwise sparse 3D point cloud. These virtual points naturally integrate into any standard Lidar-based 3D detectors along with regular Lidar measurements. The resulting multi-modal detector is simple and effective. Experimental results on the large-scale nuScenes dataset show that our framework improves a strong CenterPoint baseline by a significant $6 . 6 \ \mathrm { m A P }$ , and outperforms competing fusion approaches. Code and more visualizations are available at https://tianweiy.github.io/mvp/.
+
+# 1 Introduction
+
+3D perception is a core component in safe autonomous driving [1, 55]. A 3D Lidar sensor provides accurate depth measurements of the surrounding environment [23, 49, 75], but is costly and has low resolution at long range. A top-of-the-line 64-lane Lidar sensor can easily cost more than a small car with an input resolution that is at least two orders of magnitude lower than a $\$ 50\mathrm { \text R G B }$ sensor. This Lidar sensor receives one or two measurements for small or far away objects, whereas a corresponding RGB sensor sees hundreds of pixels. However, the RGB sensor does not perceive the depth and cannot directly place its measurements into a scene.
+
+In this paper, we present a simple and effective framework to fuse 3D Lidar and high-resolution color measurements. We lift RGB measurements into 3D virtual points by mapping them into the scene using close-by depth measurements of a Lidar sensor (See Figure 1 for an example). Our Multi-modal Virtual Point detector, MVP, generates high-resolution 3D point-cloud near target objects. A center-based 3D detector [66] then identifies all objects in the scene. Specifically, MVP uses 2D object detections to crop the original point cloud into instance frustums. MVP then generates dense 3D virtual points near these foreground points by lifting 2D pixels into 3D space. We use depth completion in image space to infer the depth of each virtual point. Finally, MVP combines virtual points with the original Lidar measurements as input to a standard center-based 3D detector [66].
+
+Our multi-modal virtual point method has several key advantages: First, 2D object detections are well optimized [17, 74] and highly accurate even for small objects. See Figure 2 for a comparison of two state-of-the-art 2D and 3D detectors on the same scene. The 2D detector has a significantly higher 2D detection accuracy but lacks the necessary 3D information used in the downstream driving task. Secondly, virtual points reduce the density imbalance between close and faraway objects. MVP augments objects at different distances with the same number of virtual points, making the point cloud measurement of these objects more consistent. Finally, our framework is a plug-andplay module to any existing or new 2D or 3D detectors. We test our model on the large-scale nuScenes dataset [2]. Adding multi-modal virtual points brings ${ \bf 6 . 6 \ m A P }$ improvements over a strong CenterPoint baseline [66]. Without any ensembles or test-time augmentation, our best model achieves ${ \bf 6 6 . 4 \ m A P }$ and ${ \bf 7 0 . 5  N D S }$ on nuScenes, outperforming all competing non-ensembled methods on the nuScenes leaderboard at the time of submission.
+
+![](images/802ca41f707f79ce428e72b18cbf58e49acac072296129e4f9b9be5cda82590f.jpg)  
+Figure 1: We augment sparse Lidar point cloud with dense semantic virtual points generated from 2D detections. Left: the augmented point-cloud in the scene. We show the original points in gray and augmented points in red. Right: three cutouts with the origial points on top and virtual points below. The virtual points are up to two orders of magnitude denser.
+
+# 2 Related work
+
+2D Object Detection has great progress in recent years. Standard approaches include the RCNN family [13, 17, 43] which first predict class-agnostic bounding boxes based on predefined anchor boxes and then classify and refine them in a two-stage fashion with deep neural networks. YOLO [42], SSD [33], and RetinaNet [30] predicts the class specific bounded boxes in one shot. Recent anchorfree detectors like CornerNet [24] and CenterNet [74] directly localize objects through keypoints without the need of predefined anchors. In our approach, we use CenterNet [74] as our 2D detector for its simplicity and superior performance for detecting small objects. See Figure 2 for an example of a 2D detectors output.
+
+Lidar-based 3D Object Detection estimates rotated 3D bounding boxes from 3D point clouds [7, 12, 23, 37, 60–63, 65, 76]. 3D detectors share a common output representation and network structure with 2D detectors but encode the input differently. VoxelNet [75] uses a PointNe-based feature extractor to generate a voxel-wise feature representation from which a backbone consisted of sparse 3D convolutions and bird-eye view 2D convolution produces detection outputs. SECOND [60] introduces more efficient sparse convolution operations. PIXOR [61] and PointPillars [23] directly process point clouds in bird-eye view, further improving efficiency. Two-stage 3D detectors [8, 45– 47, 63] use a PointNet-based set abstraction layer [39] to aggregate RoI-specifc features inside first stage proposals to refine outputs. Anchor-free approaches [5, 36, 57, 59, 61, 66] remove the need for axis-aligned bird-eye view anchor boxes. VoteNet [36] detects 3D objects through Hough voting and clustering. CenterPoint [66] proposes a center-based representation for 3D object detection and tracking and achieved state-of-the-art performance on nuScenes and Waymo benchmarks. However, as Figure 2 shows a Lidar-only detector still misses small or far-away objects due to the sparsity of depth measurements. In this work, we build upon the CenterPoint detector and demonstrate significant ${ \bf 6 . 6 \ m A P }$ improvements by adding our multi-modal virtual point approach.
+
+Camera-based 3D Object Detection Camera-based 3D object detection predicts 3D bounding boxes from camera images. Mono3D [6] uses the ground-plane assumption to generate 3D candidate boxes and scores the proposals using 2D semantic cues. CenterNet [74] first detects 2D objects in images and predicts the corresponding 3D depth and bounding box attributes using center features. Despite rapid progress, monocular 3D object detectors still perform far behind the Lidar-based methods. On state-of-the-art 3D detection benchmarks [2, 12], state-of-the-art monocular methods [26, 41] achieve about half the mAP detection accuracy, of standard Lidar based baselines [60]. PseudoLidar [56] based methods produce a virtual point cloud from RGB images, similar to our approach. However, they rely on noisy stereo depth estimates [25, 40, 56] while we use more accurate Lidar measurements. Again, the performance of purely color-based approaches lags slightly behind Lidar or fusion-based methods [2, 12].
+
+![](images/a55c340fde8c8833b29c6154a86673938c751a9c42cb9d55d9d976e5de2cbfcd.jpg)  
+Figure 2: Comparison between state-of-the-art image-based 2D detector [73] and point cloud based 3D detector [66]. We show detection from the 2D detector in blue and detection from 3D detector in green. For the 3D detector, we project the predicted 3D bounding boxes into images to get the 2D detections. For the 2D detector, we train the model using projected 2D boxes from 3D annotations. Compared to 2D detector, 3D detector often misses faraway or small objects.A quantitative comparison between 2D and 3D detectors is included in Section 5.2.
+
+Multi-modal 3D Object Detection fuses information of Lidar and color cameras [20, 28, 28, 29, 35, 37, 53, 67, 71, 72]. Frustum PointNet [38] and Frustum ConvNet [58] first detect objects in image space to identify regions of interest in the point cloud for further processing. It improves the efficiency and precision of 3D detection but is fundamentally limited by the quality of 2D detections. In contrast, we adopt a standard 3D backbone [75] to process the augmented Lidar point cloud, combining the benefit of both sensor modalities. MV3D [7] and AVOD [22] performs object-centric fusion in a two-stage framework. Objects are first detected in each sensor and fused at the proposal stage using RoIPooling [43]. Continuous fusion [20, 29] shares image and Lidar features between their backbones. Closest to our approach are MVX-Net [48], PointAugmenting [54], and PointPainting [52], which utilize point-wise correspondence to annotate each lidar point with image-based segmentation or CNN features. We instead augment the 3D lidar point cloud with additional points surrounding 3D measurements. These additional points make full use of the higher dimensional RGB measurements.
+
+Point Cloud Augmentation generates denser point clouds from sparse Lidar measurements. Lidarbased methods like PUNet [69], PUGAN [27], and Wang et al. [64] learn high level point-wise features from raw Lidar scans. They then reconstruct multiple upsampled point clouds from each high dimensional feature vector. Image-based methods [18, 21, 51] perform depth completion from sparse measurements. We build upon these depth completion methods and demonstrate state-of-the-art 3D detection results through point upsampling.
+
+# 3 Preliminary
+
+Our framework relies on both 2D detection, existing 3D detectors, and a mapping between 2D and 3D. We introduce the necessary concepts and notations below.
+
+2D Detection. Let $I$ be a camera image. A 2D object detector aims to localize and classify all objects in $I$ . A bounding-box $b _ { i } \in \mathbb { R } ^ { 4 }$ describes the objects location. A class score $s _ { i } ( c )$ predicts the likelihood of detection $b _ { i }$ to be of class $c$ . An optional instance mask $m _ { i } \in [ 0 , 1 ] ^ { W \times H }$ predicts a pixel-level segmentation of each object. In this paper, we use the popular CenterNet [74] detector. CenterNet detects objects through keypoint estimation. It takes the input image $I$ and predicts a heatmap for each class $c$ . Peaks (local maxima) of the heatmap corresponds to an object. The model regresses to other bounding box attributes using peak features with an L1 [74] or box IoU [44] objective. For instance segmentation, we use CenterNet2 [73] which adds a cascade RoI heads [3] on top of the first stage proposal network. The overall network runs at 40 FPS and achieves 43.3 instance segmentation mAP on the nuScenes image dataset [2].
+
+3D Detection. Let $P = \{ ( x , y , z , r ) _ { i } \}$ be a point cloud with 3D location $( x , y , z )$ and reflectance $r$ . The goal of a 3D detector is to predict a set of 3D bounding boxes $\{ b _ { i } \}$ from the point cloud $P$ . The bounding box $b = ( u , v , o , w , \bar { l } , h , \theta )$ includes the 3D center location $( u , v , o )$ , object size $( w , l , h )$ and the yaw rotation along $\mathbf { Z }$ axis $\theta$ . In this paper, we build upon the state-of-the-art CenterPoint [66] detector. We experiment with two popular 3D backbones: VoxelNet [75] and PointPillars [23]. VoxelNet quantizes the irregular point clouds into regular bins followed by a simple average pooling to extract features from all points inside a bin [60]. After that, a backbone consisted of sparse 3D convolutions [14] processes the quantized 3D feature volumes and the output is a map view feature map $M \in \mathbb { R } ^ { \bar { W } \times \bar { H } \times F }$ . PointPillar directly processes point clouds as bird-eye view pillar, a single elongated voxel per map location, and extracts features with fast 2D convolution to get the map view feature map $M$ .
+
+With the map view features, a detection head inspired by CenterNet [74] localizes objects in bird-eye view and regress to other box parameters using center features.
+
+2D-3D Correspondence. Multi-modal fusion approaches [48, 52, 53, 72] often rely on a pointwise correspondence between 3D point clouds and 2D pixels. In absence of calibration noise, the projection from the 3D Lidar coordinate into a 2D image coordinate involves an SE(3) transformation from the Lidar measurement to the camera frame and a perspective projection from the camera frame into image coordinates. All transformations may be described with homogeneous, time-dependent transformations. Let $t _ { 1 }$ and $t _ { 2 }$ be the capture time of the Lidar measurement and RGB image respectively. Let $T _ { \mathrm { ( c a r  l i d a r ) } }$ be the transformation from the Lidar sensor to the reference frame of the car. Let $T _ { ( t _ { 1 }  t _ { 2 } ) }$ be the transformation of the car between $t _ { 2 }$ and $t _ { 1 }$ . Let $T _ { \mathrm { ( r g b  c a r ) } }$ be the transformation from the cars reference frame to the RGB sensor. Finally, let $P _ { \mathrm { r g b } }$ be the projection matrix of the RGB camera defined by the camera intrinsic. The transformation from the Lidar to RGB sensor is then defined by
+
+$$
+T _ { \mathrm { r g b }  \mathrm { l i d a r } } ^ { t _ { 1 }  t _ { 2 } } = T _ { ( \mathrm { r g b }  c a r ) } T _ { ( t _ { 1 }  t _ { 2 } ) } T _ { ( \mathrm { c a r }  \mathrm { l i d a r } ) } ,
+$$
+
+followed by a perspective projection with camear matrix $P _ { \mathrm { r g b } }$ and a perspective division. The perspective division makes the mapping from Lidar to RGB surjective and non-invertible. In the next section, we show how to recover an inverse mapping by using depth measurements of Lidar when mapping RGB to Lidar.
+
+# 4 Multimodal Virtual Point
+
+Given a set of 2D object detections, we want to generate dense virtual points $\boldsymbol { v } _ { i } = ( x , y , z , \mathbf { e } )$ where $( x , y , z )$ is the 3D location and $\mathbf { e }$ is the semantic feature from the 2D detector. For simplicity, we use the 2D detectors class scores as semantic features. For each detection $b _ { j }$ with associated instance mask $\mathbf { m } _ { j }$ we generate a fixed number $\tau$ multimodal virtual points.
+
+Virtual Point Generation. We start by projecting the 3D Lidar point cloud onto our detection. Specifically, we transform each Lidar point $( x , y , z , r ) _ { i }$ into the reference frame of the RGB camera following Equation (1), then project it into image coordinates $\mathbf { p } _ { i }$ with associated depth $d _ { i }$ using a perspective projection. Let the collection of all projected points and depth values for a single detection $j$ be the objects frustum $\mathbf { F } _ { j } = \{ ( \mathbf { p } _ { i } , d _ { i } ) | \mathbf { p } _ { i } \overset { \cdot } { \in } \bar { \mathbf { m } } _ { j } \forall _ { i } \}$ . The frustum only considers projected 3D points $\mathbf { p } _ { i }$ that fall within a detection mask $\mathbf { m } _ { j }$ . Any Lidar measurement outside detection masks is discarded. Next, we generate virtual points from each frustum $\mathbf { F } _ { j }$ .
+
+We start by randomly sampling 2D points $\mathbf s \in \mathbf m$ from each instance mask m. We sample $\tau$ points uniformly at random without repetition. For each sampled point $\mathbf { s } _ { k }$ , we retrieve a depth estimate $d _ { k }$ from its nearest neighbor in the frustum $F _ { j }$ : $d _ { k } = \arg \operatorname* { m i n } _ { d _ { i } } \left\| \mathbf { p } _ { i } - \mathbf { s } _ { k } \right\|$ . Given the depth estimate, we unproject the point back into 3D and append the object’s semantic feature $e _ { j }$ to the virtual point. We concatenate the one-hot encoding of the detected class and the detections objectness score in the semantic feature.
+
+Algorithm 1: Multi-modal Virtual Point Generation   
+
+<table><tr><td>Input Hyper parameters :Number of virtual points per object T Output</td><td>:Lidar point cloud L = {(x,y, z,r)i}. Instance masks {m1,...,mn} for n objects. Semantic features {e1,..,en} with ej ∈ RD</td></tr><tr><td>Fj←の ∀j∈{1...n}; for(xi,yi,zi,ri) ∈Ldo /* Perspective projection to 2D point p depth d p,d←Project(PrgbTtda(myi,,1)T);</td><td>: Multi-modal 3D virtual points V ∈ Rn Xτ×(3+D) // Point cloud instance frustums */</td></tr><tr><td>for j ∈{1...n} do ifp∈mj then</td><td></td></tr><tr><td>Fj←FjU{(p,d)}; end</td><td>// Add point to frustum</td></tr><tr><td>end end</td><td></td></tr><tr><td>for j ∈{1...n} do</td><td></td></tr><tr><td>S ← Sample,(mj);</td><td></td></tr><tr><td></td><td></td></tr><tr><td>for s ∈Sdo</td><td></td></tr><tr><td></td><td>// Uniformly sample T 2d points in instance mask</td></tr><tr><td>(p,d) ← NN(s,Fj);</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>// Find closest projected 3D point</td></tr><tr><td></td><td></td></tr><tr><td>(PrgbTtiar)</td><td>/* Unproject the 2D point s using the nearest neighbors depth d*/</td></tr><tr><td>q↑(</td><td></td></tr><tr><td></td><td>Unproject(s,d);</td></tr><tr><td>Add (q,ej) to Vj;</td><td></td></tr><tr><td></td><td></td></tr><tr><td>end</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>end</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr></table>
+
+The virtual point generation is summarized in Algorithm 1 and Figure 3. Next, we show how to incorporate virtual points into a point-based 3D detector.
+
+Virtual Point 3D detection. Voxel-based 3D detectors [60, 66] first voxelize 3D points $( x , y , z ) _ { i }$ and average all point features $( x , y , z , t , r ) _ { i }$ within a voxel. Here $r _ { i }$ is a reflectance measure and $t _ { i }$ is the capture time. A standard 3D convolutional network uses these voxelized features in further processing. For virtual points, this creates an issue. The feature dimensions of real points $( x , y , z , t , r )$ and virtual points differ $( x , y , z , t , \mathbf { e } )$ . A simple solution could be to either concatenate virtual and real points into a larger feature $( x , y , z , t , r , \mathbf { e } )$ and set any missing information to zero. However, this is both wasteful, as the dimension of real points grows by $3 \times$ , and it creates an imbalanced ratio between virtual and real points in different parts of the scene. Furthermore, real measurements are often a bit more precise than virtual points and simple averaging of the two blurs out the information contained in real measurements. To solve this, we modify the average pooling approach by separately averaging features of virtual and real points and concatenating the final averaged features together as input to 3D convolution. For the rest of the architecture, we follow CenterPoint [66].
+
+We further use virtual points in a second-stage refinement. Our MVP model generates dense virtual points near target objects which help two-stage refinement [45, 66]. Here, we follow Yin et al. [66] to extract bird-eye view features from all outward surfaces of the predicted 3D box. The main difference to Yin et al. is that our input is much denser around objects, and hence the second stage refinement has access to richer information.
+
+# 5 Experiments
+
+We evaluate our proposed multimodal virtual point method on the challenging nuScenes benchmark.
+
+nuScenes [2] is a popular multimodal autonomous driving datasets for 3D object detection in urban scenes. The dataset contains 1000 driving sequences, with each 20s long and annotated with 3D bounding boxes. The Lidar frequency is $2 0 \mathrm { H z }$ and the dataset provides sensor and vehicle pose information for each Lidar frame but only includes object annotation every ten frames (0.5s). The dataset hides any personally identifiable information, blurs faces and license plates in color images. There are in total 6 RGB cameras at a resolution of $1 6 0 0 \times 9 0 0$ and a capture frequency of $1 2 \mathrm { H z }$ . We follow the official dataset split to use 700, 150, 150 sequences for training, validation, and testing. This in total results in 28130 frames for training, 6019 frames for validation, and 6008 frames for testing. The annotations include a fine-grained label space of ten classes with a long-tail distribution. For 3D object detection, the official evaluation metrics include the mean Average Precision (mAP)[11] and nuScenes detection score (NDS) [2]. mAP measures the localization precision using a threshold based on the birds-eye view center distance $< 0 . 5 \mathrm { m }$ , 1m, 2m, 4m. NDS is a weighted combination of mAP and regression accuracy of other object attributes including box size, orientation, translation, and class-specific attributes [2]. NDS is the main ranking metric for the benchmark.
+
+![](images/cea7a1071c5d0c5bc301c9e9942c92293f3b2a4e760b5ea509b38ed771cc1c6a.jpg)  
+(a) 2D instance segmentation
+
+![](images/5a2df439a19e6fcc13bada1b5f3b60c478049920845b63d54b385013d873945f.jpg)  
+(c) Sampling and nearest neighbor matching
+
+![](images/9f1823b3fa830acf8f5a3696a1391bef1515e7d3a3b10a89fbb499802d865087.jpg)  
+(b) Lidar point cloud projection
+
+![](images/204de22807dcc359aac30aa34f6961fd838fa62c2772317304f2511914c55425.jpg)  
+(d) Reprojected virtual points   
+Figure 3: Overview of our mlutimodal virtual point generation framework. We start by extracting 2D instance masks for each object in a color image (a). We then project all Lidar measurements into the reference frame of the RGB camera (b). For visualization purposes, points inside the objects are black, other points are grey. We then sample random points inside each 2D instance mask and retrieve a depth estimate from their nearest neighbor Lidar projection (c). For visualization clarity, (c) only shows a subset of virtual points. Finally, all virtual points are reprojected into the original point-cloud (d).
+
+Implementation Details. Our implementation is based on the opensourced code of CenterPoint 1 [66] for 3D detection and CenterNet2 $[ 7 3 ]$ for 2D Detection.
+
+For 2D detection, we train a CenterNet [74] detector on the nuScenes image dataset [2]. We use the DLA-34 [68] backbone with deformable convolutions [10]. We add cascade RoI heads [3] for instance segmentation following Zhou et al. [73]. We train the detector on the nuScenes dataset using the SGD optimizer with a batch size of 16 and a learning rate of 0.02 for 90000 iterations.
+
+For 3D detection, we use the same VoxelNet [75] and PointPillars [23] architectures following [23, 66, 76]. For VoxelNet, the detection range is $[ - 5 4 m , 5 4 m ]$ for the $X$ , $Y$ axis and $[ - 5 m , 3 m ]$ for the $Z$ axis while the range is $[ - 5 1 . 2 m , 5 1 . 2 m ]$ for the $X$ , $Y$ axis for the PointPillar architecture. The voxel size is $( 0 . 0 7 5 m , 0 . 0 7 5 m , 0 . 2 m )$ and $( 0 . 2 m , 0 . 2 m , 8 m )$ for VoxelNet and PointPillar respectively. For data augmentation, we follow CenterPoint and use global random rotations between $[ - \bar { \pi } / 4 , \pi / 4 ]$ , global random scaling between [0.9, 1.1] and global translations between $[ - 0 . 5 m , 0 . 5 m ]$ . To deal with the long-tail class distribution in nuScenes, we use the ground truth sampling in [60] to randomly paste objects into the current frame [60, 76]. We also adopt the class-balanced resampling and class-grouped heads in [76] to improve the average density of rare classes. We train the model for 20 epochs with the AdamW [34] optimizer using the one-cycle policy [16], with a max learning rate of 3e-3 following [66]. The training takes 2.5 days on 4 V100 GPUs with a batch size of 16 (4 frames per GPU).
+
+Table 1: Comparisons with previous methods on nuScenes test set. We show the NDS, mAP, and mAP for each class. Abbreviations are construction vehicle (CV), pedestrian (Ped), motorcycle (Motor), and traffic cone (TC).   
+
+<table><tr><td>Method</td><td>mAP</td><td>NDS</td><td>Car</td><td>Truck</td><td>Bus</td><td>Trailer</td><td>CV</td><td>Ped</td><td>Motor</td><td>Bicycle</td><td>TC</td><td>Barrier</td></tr><tr><td>PointPillars [23]</td><td>30.5</td><td>45.3</td><td>68.4</td><td>23.0</td><td>28.2</td><td>23.4</td><td>4.1</td><td>59.7</td><td>27.4</td><td>1.1</td><td>30.8</td><td>38.9</td></tr><tr><td>WYSIWYG[19]</td><td>35.0</td><td>41.9</td><td>79.1</td><td>30.4</td><td>46.6</td><td>40.1</td><td>7.1</td><td>65.0</td><td>18.2</td><td>0.1</td><td>28.8</td><td>34.7</td></tr><tr><td>3DSSD [62]</td><td>42.6</td><td>56.4</td><td>81.2</td><td>47.2</td><td>61.4</td><td>30.5</td><td>12.6</td><td>70.2</td><td>36.0</td><td>8.6</td><td>31.1</td><td>47.9</td></tr><tr><td>PMPNet [65]</td><td>45.4</td><td>53.1</td><td>79.7</td><td>33.6</td><td>47.1</td><td>43.1</td><td>18.1</td><td>76.5</td><td>40.7</td><td>7.9</td><td>58.8</td><td>48.8</td></tr><tr><td>PointPainting [52]</td><td>46.4</td><td>58.1</td><td>77.9</td><td>35.8</td><td>36.2</td><td>37.3</td><td>15.8</td><td>73.3</td><td>41.5</td><td>24.1</td><td>62.4</td><td>60.2</td></tr><tr><td>CBGS [76]</td><td>52.8</td><td>63.3</td><td>81.1</td><td>48.5</td><td>54.9</td><td>42.9</td><td>10.5</td><td>80.1</td><td>51.5</td><td>22.3</td><td>70.9</td><td>65.7</td></tr><tr><td>CVCNet [4]</td><td>55.3</td><td>64.4</td><td>82.7</td><td>46.1</td><td>46.6</td><td>49.4</td><td>22.6</td><td>79.8</td><td>59.1</td><td>31.4</td><td>65.6</td><td>69.6</td></tr><tr><td>HotSpotNet [5]</td><td>59.3</td><td>66.0</td><td>83.1</td><td>50.9</td><td>56.4</td><td>53.3</td><td>23.0</td><td>81.3</td><td>63.5</td><td>36.6</td><td>73.0</td><td>71.6</td></tr><tr><td>CenterPoint [66]</td><td>58.0</td><td>65.5</td><td>84.6</td><td>51.0</td><td>60.2</td><td>53.2</td><td>17.5 83.4</td><td></td><td>53.7</td><td>28.7</td><td>76.7</td><td>70.9</td></tr><tr><td>MVP (Ours)</td><td>66.4</td><td>70.5</td><td>86.8</td><td>58.5</td><td>67.4</td><td>57.3</td><td>26.1</td><td>89.1</td><td>70.0</td><td>49.3</td><td>85.0</td><td>74.8</td></tr></table>
+
+During the testing, we set the output threshold to be 0.05 for the 2D detector and generate 50 virtual points for each 2D object in the scene. We use an output threshold of 0.1 for the 3D detector after performing non-maxima suppression with an IoU threshold of 0.2 following CenterPoint [66].
+
+# 5.1 State-of-the-art Comparison
+
+We first compare with state-of-the-art approaches on the nuScenes test set. We obtain all results on the public leaderboard by submitting our predictions to an online evaluation server. The submission uses a single MVP model without any ensemble or test-time augmentations. We compare to other methods under the same setting. Table 1 summarizes our results. On the nuScenes dataset, MVP achieves state-of-the-art results of $6 6 . 4 \ : \mathrm { m A P }$ and $7 0 . 5 \ : \mathrm { N D S }$ , outperforming the strong CenterPoint baseline by $8 . 4 \mathrm { m A P }$ and 5.0 NDS. MVP shows consistent improvements across all object categories with significant $1 1 \mathrm { m A P }$ accuracy boosts for small objects $( + 2 0 . 6$ for Bicycle and $+ 1 6 . 3$ for motorcycle). These results clearly verify the effectiveness of our multi-modal virtual point approach.
+
+# 5.2 Ablation Studies
+
+Comparison of 2D and 3D Detector. We first validate the superior detection performance of the camera-based 2D detector compared to the Lidar-based 3D detector. Specifically, we use two state-ofthe-art object detectors: CenterPoint [66] for Lidar-based 3D detection, and CenterNet [74] for imagebased 2D detection. To compare the performance of detectors working in different modalities, we project the predicted 3D bounding boxes into the image space to get the corresponding 2D detections. The 2D CenterNet detector is trained with projected 2D boxes from ground truth 3D annotations. Table 2 summarizes the results over the whole nuScenes validation set. 2D CenterNet [74] significantly outperforms the CenterPoint model by $9 . 8 \mathrm { m A P }$ (using 2D overlap). The improvements are larger for smaller objects with a $1 2 . 6 \mathrm { m A P }$ improvement for objects of medium size and more than $3 \times$ accuracy improvements for small objects $6 . 9 \mathrm { m A P }$ vs. $1 . 6 \mathrm { m A P }$ ). See Figure 2 for a qualitative visualization of these two detectors’ outputs. These results support our motivations for utilizing high-resolution image information to improve 3D detection models with sparse Lidar input.
+
+Table 2: Quantitative comparison between state-of-the-art image-based 2D detector [74] and point cloud based 3D detector [66] on the nuScenes validation set measuring 2D detection accuracy (AP). The comparison use the COCO [31] style mean average precision with 2D IoU threshold between 0.5 and 0.95 in image coordinates. For 3D CenterPoint [66] detector, we project the predicted 3D bounding boxes into images to get the 2D detections. The results show that the 2D detector performs significantly better than Lidar-based 3D detector at localizing small or medium size objects due to high resolution camera input.   
+
+<table><tr><td>Method</td><td>APsmall</td><td>APmedium</td><td>APlarge</td><td>AP</td></tr><tr><td>CenterPoint [66]</td><td>1.6</td><td>11.7</td><td>34.5</td><td>22.7</td></tr><tr><td>CenterNet [74]</td><td>6.9</td><td>24.3</td><td>42.6</td><td>32.5</td></tr></table>
+
+Table 3: Component analysis of our MVP model with VoxelNet [60, 75] and PointPillars [23] backbones on nuScenes validation set.   
+
+<table><tr><td>Encoder</td><td>Baseline</td><td>Virtual Point</td><td>Split Voxelization</td><td>Two-stage</td><td>mAP↑</td><td>NDS↑</td></tr><tr><td rowspan="5">VoxelNet</td><td>?</td><td></td><td></td><td></td><td>59.6</td><td>66.8</td></tr><tr><td></td><td></td><td></td><td>√</td><td>60.5</td><td>67.4</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td>65.9</td><td>69.6</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td>66.0</td><td>70.0</td></tr><tr><td>√</td><td>√</td><td>√</td><td>厂</td><td>67.1</td><td>70.8</td></tr><tr><td rowspan="2">PointPillars</td><td>广</td><td></td><td></td><td></td><td>52.3</td><td>61.3</td></tr><tr><td></td><td>√</td><td></td><td></td><td>62.7</td><td>66.1</td></tr></table>
+
+Component Analysis. Next, we ablate our contributions on the nuScenes validation set. We use the Lidar-only CenterPoint [74] model as our baseline. All hyperparameters and training procedures are the same between all baselines. We change inputs (MVP or regular points), voxelization, or an optional second stage. Table 3 shows the importance of each component of our MVP model. Simply augmenting the Lidar point cloud with multi-modal virtual points gives a $6 . 3 \ \mathrm { m A P }$ and $1 0 . 4 \ \mathrm { m A P }$ improvements for VoxelNet and PointPillars encoder, respectively. For the VoxelNet encoder, split voxelization gives another $0 . 4 ~ \mathrm { N D S }$ improvements due to the better modeling of features inside a voxel. Moreover, two-stage refinement with surface center features brings another $1 . 1 \ \mathrm { m A P }$ and $0 . 8 ~ \mathrm { N D S }$ improvements over our first stage models with small overheads $( 1 - 2 \mathrm { m s } )$ . The improvement of two-stage refinement is slightly larger with virtual points than without. This highlights the effectiveness of our virtual point method to create a finer local structure for better localization and regression using two-stage point-based detection.
+
+Performance Breakdown. To better understand the improvements of our MVP model, we show the performance comparisons on different subsets of the nuScenes validation set based on object distances to the ego-vehicle. We divide all ground truth annotations and predictions into three ranges: $0 { - } 1 5 \mathrm { m }$ , $1 5 { - } 3 0 \mathrm { m }$ , and $3 0 { - } 5 0 \mathrm { m }$ . The baselines include both the Lidar-only two-stage CenterPoint [66] model and the state-of-the-art multi-modal fusion method PointPainting [52]. We reimplement PointPainting using the same 2D detections, backbones, and tricks (including two-stage) as our MVP approach. The main difference to PointPainting [52] is our denser Lidar inputs with multimodal virtual points. Table 4 shows the results. Our MVP model outperforms the Lidar-only baseline by $6 . 6 \mathrm { m A P }$ while achieving a significant $1 0 . 1 \mathrm { m A P }$ improvement for faraway objects. Compared to PointPainting [52], our model achieve a $1 . 1 \mathrm { m A P }$ improvement for faraway objects and performs comparatively for closer objects. This improvement comes from the dense and fine-grained 3D structure generated from our MVP framework. Our method makes better use of the higher dimensional RGB measurements than the simple point-wise semantic feature concatenation as used in prior works [48, 52].
+
+Robustness to 2D Detection We investigate the impact of 2D instance segmentation quality on the final 3D detection performance. With the same image network, we simulate the degradation of 2D segmentation performance with smaller input resolutions. We show the results in Table 5. Our
+
+MVP model is robust to the quality of 2D instance segmentation. The 3D detection performance only decreases by $0 . 8 \mathrm { N D S }$ with a 9 point worse instance segmentation inputs.
+
+Depth Estimation Accuracy We further quantify the depth estimation quality of our nearest neighbor-based depth interpolation algorithm. We choose objects with at least 15 lidar points and randomly mask out $80 \%$ of the points. We then generate virtual points from the projected locations of the masked out lidar points and compute the a bi-directional pointwise chamfer distance between virtual points and masked out real lidar points. Our nearest neighbor approach has bi-directional chamfer distance of 0.33 meter on the nuScenes validation set. We believe more advanced learning based approaches like [18] and [21] may further improve the depth completion and 3D detection performance.
+
+KITTI Results To test the generalization of our method, we add an experiment on the popular KITTI dataset [12]. For 2D detection, we use a pretrained MaskFormer [9] model to generate the instance segmentation masks and create 100 virtual points for each 2D object in the scene. For 3D detection, we use the popular PointPillars [23] detector with augmented point cloud inputs. All other parameters are the same as the default PointPillars model. As shown in Table 6, augmenting the Lidar point cloud with our multimodal virtual points gives a $0 . 5 \mathrm { m A P }$ and $2 . 3 \mathrm { m A P }$ for vehicle and cyclist class, respectively. We didn’t notice an improvement for the pedestrian class, presumable due to inconsistent pedestrian definition between our image model (trained on COCO [32]) and the 3D detector. On COCO, people inside a vehicle or on top of a bike are all considered to be pedestrians while KITTI 3D detectors treat them as vehicle or cyclist.
+
+Table 4: Comparisons between Lidar-only CenterPoint [66] method, fusion-based PointPainting [52] method (denoted as CenterPoint $^ +$ Ours(w/o virtual)), and our multimodal virtual point method for detecting objects of different ranges. All three entries use the VoxelNet backbone. We split the nuScenes validation set into three subsets containing objects at different ranges.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">nuScenes mAP</td></tr><tr><td>0-15m</td><td>15-30m</td><td>30-50m</td><td>Overall</td></tr><tr><td>CenterPoint [66]</td><td>76.2</td><td>60.3</td><td>37.2</td><td>60.5</td></tr><tr><td>CenterPoint + Ours(w/o virtual)</td><td>78.2</td><td>67.4</td><td>46.2</td><td>66.5</td></tr><tr><td>CenterPoint + Ours</td><td>78.1</td><td>67.7</td><td>47.3</td><td>67.1</td></tr></table>
+
+Table 5: Influence of 2D instance segmentation quality for the final 3D detection performance. We show the input resolution, 2D detection mAP, and 3D detection nuScenes detection score (NDS).   
+
+<table><tr><td>Resolution</td><td>2D mAP</td><td>NDS</td></tr><tr><td>900</td><td>43.3</td><td>70.0</td></tr><tr><td>640</td><td>39.5</td><td>69.6</td></tr><tr><td>480</td><td>34.2</td><td>69.2</td></tr></table>
+
+Table 6: Comparison between Lidar-only PointPillars detector and our multimodal virtual point method for 3D detection on KITTI dataset. We show the 3D detection mean average precision for each class under the moderate difficulty level.   
+
+<table><tr><td>Method</td><td>Car</td><td>Cyclist</td><td>Pedestrian</td></tr><tr><td>PointPillars [23]</td><td>77.3</td><td>62.7</td><td>52.3</td></tr><tr><td>PointsPillars+Ours</td><td>77.8</td><td>65.0</td><td>50.5</td></tr></table>
+
+# 6 Discussion and conclusions
+
+We proposed a simple multi-modal virtual point approach for outdoor 3D object detection. The main innovation is a multi-modal virtual point generation algorithm that lifts RGB measurements into
+
+![](images/5a7f2232cbdf4080c6316765137c68f82bd0e04a9c51e683a2d6ef8aacb10ad9.jpg)  
+Figure 4: Example qualitative results of MVP on the nuScenes validation set. We show the raw point-cloud in blue, our detected objects in green bounding boxes, and Lidar points inside bounding boxes in red. Best viewed on screen.
+
+3D virtual points using close-by measurements of a Lidar sensor. Our MVP framework generates high-resolution 3D point clouds near target objects and enables more accurate localization and regression, especially for small and faraway objects. The model significantly improves the strong Lidar-only CenterPoint detector and sets a new state-of-the-art on the nuScenes benchmark. Our framework seamlessly integrates into any current or future 3D detection algorithms.
+
+There are still certain limitations with the current approach. Firstly, we assume that virtual points have the same depth as close-by Lidar measurements. This may not hold in the real world. Objects like cars don’t have a planar shape vertical to the ground plane. In the future, we plan to apply learning-based methods [50, 70] to infer the detailed 3D shape and pose from both Lidar measurements and image features. Secondly, our current two-stage refinement modules only use features from the bird-eye view which may not take full advantage of the high-resolution virtual points generated from our algorithm. We believe point or voxel-based two-stage 3D detectors like PVRCNN [45] and M3Detr [15] may give more significant improvements. Finally, the point-based abstraction connecting 2D and 3D detection may introduce too large of a bottleneck to transmit information from 2D to 3D. For example, no pose information is contained in our current position $^ +$ class based MVP features.
+
+Overall, we believe future methods for scalable 3D perception can benefit from the interplay of camera and Lidar sensor inputs via dense semantic virtual points.
+
+Societal Impacts. First and foremost better 3D detection and tracking will lead to safer autonomous vehicles. However, in the short term, it may lead to earlier adoption of potentially not-yet safe autonomous vehicles, and misleading error rates in 3D detection may lead to real-world accidents. Fusing multiple modalities may also increase the iteration cycle and safety testing requirements of autonomous vehicles, as different modalities clearly adapt differently to changes in weather, geographic locations, or even day-night cycles. A low sun may uniquely distract an RGB sensor, and hence unnecessarily distract a 3D detector through MVPs.
+
+Furthermore, increasing the reliance of autonomous vehicles on color sensors introduces privacy issues. While most human beings look indistinguishable in 3D Lidar measurements, they are clearly identifiable in color images. In the wrong hands, this additional data may be used for mass surveillance.
+
+Acknowledgement We thank the anonymous reviewers for the constructive comments. This material is based upon work supported by the National Science Foundation under Grant No. IIS-1845485 and IIS-2006820.
+
+References   
+[1] Mayank Bansal, Alex Krizhevsky, and Abhijit Ogale. Chauffeurnet: Learning to drive by imitating the best and synthesizing the worst. RSS, 2019.   
+[2] Holger Caesar, Varun Bankiti, Alex H. Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. CVPR, 2020.   
+[3] Zhaowei Cai and Nuno Vasconcelos. Cascade r-cnn: Delving into high quality object detection. CVPR, 2018. [4] Qi Chen, Lin Sun, Ernest Cheung, Kui Jia, and Alan Yuille. Every view counts: Cross-view consistency in 3d object detection with hybrid-cylindrical-spherical voxelization. NeurIPS, 2020.   
+[5] Qi Chen, Lin Sun, Zhixin Wang, Kui Jia, and Alan Yuille. Object as hotspots: An anchor-free 3d object detection approach via firing of hotspots. ECCV, 2020.   
+[6] Xiaozhi Chen, Kaustav Kundu, Ziyu Zhang, Huimin Ma, Sanja Fidler, and Raquel Urtasun. Monocular 3d object detection for autonomous driving. CVPR, 2016.   
+[7] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. Multi-view 3d object detection network for autonomous driving. CVPR, 2017.   
+[8] Yilun Chen, Shu Liu, Xiaoyong Shen, and Jiaya Jia. Fast point r-cnn. ICCV, 2019.   
+[9] Bowen Cheng, Alexander G Schwing, and Alexander Kirillov. Per-pixel classification is not all you need for semantic segmentation. arXiv preprint arXiv:2107.06278, 2021.   
+[10] Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong Zhang, Han Hu, and Yichen Wei. Deformable convolutional networks. ICCV, 2017.   
+[11] Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. IJCV, 2010.   
+[12] Andreas Geiger, Philip Lenz, and Raquel Urtasun. Are we ready for autonomous driving? the kitti vision benchmark suite. CVPR, 2012.   
+[13] Ross Girshick. Fast r-cnn. ICCV, 2015.   
+[14] Benjamin Graham, Martin Engelcke, and Laurens van der Maaten. 3d semantic segmentation with submanifold sparse convolutional networks. CVPR, 2018.   
+[15] Tianrui Guan, Jun Wang, Shiyi Lan, Rohan Chandra, Zuxuan Wu, Larry Davis, and Dinesh Manocha. M3detr: Multi-representation, multi-scale, mutual-relation 3d object detection with transformers. arXiv preprint arXiv:2104.11896, 2021.   
+[16] Sylvain Gugger. The 1cycle policy. https://sgugger.github.io/the-1cycle-policy. html, 2018.   
+[17] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick. Mask r-cnn. ICCV, 2017.   
+[18] Mu Hu, Shuling Wang, Bin Li, Shiyu Ning, Li Fan, and Xiaojin Gong. Penet: Towards precise and efficient image guided depth completion. 2021.   
+[19] Peiyun Hu, Jason Ziglar, David Held, and Deva Ramanan. What you see is what you get: Exploiting visibility for 3d object detection. CVPR, 2020.   
+[20] Tengteng Huang, Zhe Liu, Xiwu Chen, and Xiang Bai. Epnet: Enhancing point features with image semantics for 3d object detection. ECCV, 2020.   
+[21] Saif Imran, Xiaoming Liu, and Daniel Morris. Depth completion with twin surface extrapolation at occlusion boundaries. pages 2583–2592, 2021.   
+[22] Jason Ku, Melissa Mozifian, Jungwook Lee, Ali Harakeh, and Steven L Waslander. Joint 3d proposal generation and object detection from view aggregation. IROS, 2018.   
+[23] Alex H. Lang, Sourabh Vora, Holger Caesar, Lubing Zhou, Jiong Yang, and Oscar Beijbom. Pointpillars: Fast encoders for object detection from point clouds. CVPR, 2019.   
+[24] Hei Law and Jia Deng. Cornernet: Detecting objects as paired keypoints. ECCV, 2018.   
+[25] Peiliang Li, Xiaozhi Chen, and Shaojie Shen. Stereo r-cnn based 3d object detection for autonomous driving. CVPR, 2019.   
+[26] Peixuan Li, Huaici Zhao, Pengfei Liu, and Feidao Cao. Rtm3d: Real-time monocular 3d detection from object keypoints for autonomous driving. arXiv preprint arXiv:2001.03343, 2, 2020.   
+[27] Ruihui Li, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng. Pu-gan: a point cloud upsampling adversarial network. pages 7203–7212, 2019.   
+[28] Ming Liang, Bin Yang, Yun Chen, Rui Hu, and Raquel Urtasun. Multi-task multi-sensor fusion for 3d object detection. CVPR, 2019.   
+[29] Ming Liang, Bin Yang, Shenlong Wang, and Raquel Urtasun. Deep continuous fusion for multi-sensor 3d object detection. ECCV, 2018.   
+[30] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollar. Focal loss for dense object detection. ICCV, 2017.   
+[31] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. ECCV, 2014.   
+[32] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. ECCV, 2014.   
+[33] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg. Ssd: Single shot multibox detector. ECCV, 2016.   
+[34] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. ICLR, 2019.   
+[35] Su Pang, Daniel Morris, and Hayder Radha. Clocs: Camera-lidar object candidates fusion for 3d object detection. arXiv preprint arXiv:2009.00784, 2020.   
+[36] Charles R. Qi, Or Litany, Kaiming He, and Leonidas Guibas. Deep hough voting for 3d object detection in point clouds. ICCV, 2019.   
+[37] Charles R. Qi, Wei Liu, Chenxia Wu, Hao Su, and Leonidas J. Guibas. Frustum pointnets for 3d object detection from rgb-d data. CVPR, 2018.   
+[38] Charles R Qi, Wei Liu, Chenxia Wu, Hao Su, and Leonidas J Guibas. Frustum pointnets for 3d object detection from rgb-d data. CVPR, 2018.   
+[39] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. NeurIPS, 2017.   
+[40] Rui Qian, Divyansh Garg, Yan Wang, Yurong You, Serge Belongie, Bharath Hariharan, Mark Campbell, Kilian Q Weinberger, and Wei-Lun Chao. End-to-end pseudo-lidar for image-based 3d object detection. CVPR, 2020.   
+[41] Cody Reading, Ali Harakeh, Julia Chae, and Steven L Waslander. Categorical depth distribution network for monocular 3d object detection. arXiv preprint arXiv:2103.01100, 2021.   
+[42] Joseph Redmon and Ali Farhadi. Yolo9000: better, faster, stronger. CVPR, 2017.   
+[43] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. NIPS, 2015.   
+[44] Hamid Rezatofighi, Nathan Tsoi, JunYoung Gwak, Amir Sadeghian, Ian Reid, and Silvio Savarese. Generalized intersection over union: A metric and a loss for bounding box regression. CVPR, 2019.   
+[45] Shaoshuai Shi, Chaoxu Guo, Li Jiang, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. Pv-rcnn: Point-voxel feature set abstraction for 3d object detection. CVPR, 2020.   
+[46] Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. Pointrcnn: 3d object proposal generation and detection from point cloud. CVPR, 2019.   
+[47] Shaoshuai Shi, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. From points to parts: 3d object detection from point cloud with part-aware and part-aggregation network. TPAMI, 2020.   
+[48] Vishwanath A Sindagi, Yin Zhou, and Oncel Tuzel. Mvx-net: Multimodal voxelnet for 3d object detection. ICRA, 2019.   
+[49] Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, et al. Scalability in perception for autonomous driving: An open dataset benchmark. CVPR, 2020.   
+[50] Shubham Tulsiani, Abhishek Kar, João Carreira, and Jitendra Malik. Learning category-specific deformable 3d models for object reconstruction. TPAMI, 2017.   
+[51] Jonas Uhrig, Nick Schneider, Lukas Schneider, Uwe Franke, Thomas Brox, and Andreas Geiger. Sparsity invariant cnns. pages 11–20, 2017.   
+[52] Sourabh Vora, Alex H Lang, Bassam Helou, and Oscar Beijbom. Pointpainting: Sequential fusion for 3d object detection. CVPR, 2020.   
+[53] Chunwei Wang, Chao Ma, Ming Zhu, and Xiaokang Yang. Pointaugmenting: Cross-modal augmentation for 3d object detection. CVPR, 2021.   
+[54] Chunwei Wang, Chao Ma, Ming Zhu, and Xiaokang Yang. Pointaugmenting: Cross-modal augmentation for 3d object detection. pages 11794–11803, June 2021.   
+[55] Dequan Wang, Coline Devin, Qi-Zhi Cai, Philipp Krähenbühl, and Trevor Darrell. Monocular plan view networks for autonomous driving. IROS, 2019.   
+[56] Yan Wang, Wei-Lun Chao, Divyansh Garg, Bharath Hariharan, Mark Campbell, and Kilian Q Weinberger. Pseudo-lidar from visual depth estimation: Bridging the gap in 3d object detection for autonomous driving. CVPR, 2019.   
+[57] Yue Wang, Alireza Fathi, Abhijit Kundu, David Ross, Caroline Pantofaru, Tom Funkhouser, and Justin Solomon. Pillar-based object detection for autonomous driving. ECCV, 2020.   
+[58] Zhixin Wang and Kui Jia. Frustum convnet: Sliding frustums to aggregate local point-wise features for amodal 3d object detection. arXiv preprint arXiv:1903.01864, 2019.   
+[59] Kelvin Wong, Shenlong Wang, Mengye Ren, Ming Liang, and Raquel Urtasun. Identifying unknown instances for autonomous driving. CORL, 2019.   
+[60] Yan Yan, Yuxing Mao, and Bo Li. Second: Sparsely embedded convolutional detection. Sensors, 2018.   
+[61] Bin Yang, Wenjie Luo, and Raquel Urtasun. Pixor: Real-time 3d object detection from point clouds. CVPR, 2018.   
+[62] Zetong Yang, Yanan Sun, Shu Liu, and Jiaya Jia. 3dssd: Point-based 3d single stage object detector. CVPR, 2020.   
+[63] Zetong Yang, Yanan Sun, Shu Liu, Xiaoyong Shen, and Jiaya Jia. Std: Sparse-to-dense 3d object detector for point cloud. ICCV, 2019.   
+[64] Wang Yifan, Shihao Wu, Hui Huang, Daniel Cohen-Or, and Olga Sorkine-Hornung. Patch-based progressive 3d point set upsampling. pages 5958–5967, 2019.   
+[65] Junbo Yin, Jianbing Shen, Chenye Guan, Dingfu Zhou, and Ruigang Yang. Lidar-based online 3d video object detection with graph-based message passing and spatiotemporal transformer attention. CVPR, 2020.   
+[66] Tianwei Yin, Xingyi Zhou, and Philipp Krähenbühl. Center-based 3d object detection and tracking. CVPR, 2021.   
+[67] Jin Hyeok Yoo, Yecheol Kim, Ji Song Kim, and Jun Won Choi. 3d-cvf: Generating joint camera and lidar features using cross-view spatial feature fusion for 3d object detection. arXiv preprint arXiv:2004.12636, 3, 2020.   
+[68] Fisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep layer aggregation. CVPR, 2018.   
+[69] Lequan Yu, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng. Pu-net: Point cloud upsampling network. pages 2790–2799, 2018.   
+[70] Wentao Yuan, Tejas Khot, David Held, Christoph Mertz, and Martial Hebert. Pcn: Point completion network. 3DV, 2018.   
+[71] Haolin Zhang, Dongfang Yang, Ekim Yurtsever, Keith A Redmill, and Ümit Özgüner. Farawayfrustum: Dealing with lidar sparsity for 3d object detection using fusion. arXiv preprint arXiv:2011.01404, 2020.   
+[72] Wenwei Zhang, Zhe Wang, and Chen Change Loy. Multi-modality cut and paste for 3d object detection. arXiv preprint arXiv:2012.12741, 2020.   
+[73] Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl. Probabilistic two-stage detection. arXiv preprint arXiv:2103.07461, 2021.   
+[74] Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Objects as points. arXiv:1904.07850, 2019.   
+[75] Yin Zhou and Oncel Tuzel. Voxelnet: End-to-end learning for point cloud based 3d object detection. CVPR, 2018.   
+[76] Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and Gang Yu. Class-balanced grouping and sampling for point cloud 3d object detection. arXiv:1908.09492, 2019.

parse/train/8xoN9ZdSW8/8xoN9ZdSW8_content_list.json

@@ -0,0 +1,947 @@
+[
+    {
+        "type": "text",
+        "text": "Multimodal Virtual Point 3D Detection ",
+        "text_level": 1,
+        "bbox": [
+            261,
+            122,
+            736,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tianwei Yin UT Austin yintianwei@utexas.edu ",
+        "bbox": [
+            200,
+            196,
+            382,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Xingyi Zhou UT Austin zhouxy@cs.utexas.edu ",
+        "bbox": [
+            416,
+            196,
+            589,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Philipp Krähenbühl UT Austin philkr@cs.utexas.edu ",
+        "bbox": [
+            622,
+            196,
+            797,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            275,
+            535,
+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lidar-based sensing drives current autonomous vehicles. Despite rapid progress, current Lidar sensors still lag two decades behind traditional color cameras in terms of resolution and cost. For autonomous driving, this means that large objects close to the sensors are easily visible, but far-away or small objects comprise only one measurement or two. This is an issue, especially when these objects turn out to be driving hazards. On the other hand, these same objects are clearly visible in onboard RGB sensors. In this work, we present an approach to seamlessly fuse RGB sensors into Lidar-based 3D recognition. Our approach takes a set of 2D detections to generate dense 3D virtual points to augment an otherwise sparse 3D point cloud. These virtual points naturally integrate into any standard Lidar-based 3D detectors along with regular Lidar measurements. The resulting multi-modal detector is simple and effective. Experimental results on the large-scale nuScenes dataset show that our framework improves a strong CenterPoint baseline by a significant $6 . 6 \\ \\mathrm { m A P }$ , and outperforms competing fusion approaches. Code and more visualizations are available at https://tianweiy.github.io/mvp/. ",
+        "bbox": [
+            233,
+            306,
+            766,
+            512
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            537,
+            310,
+            555
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "3D perception is a core component in safe autonomous driving [1, 55]. A 3D Lidar sensor provides accurate depth measurements of the surrounding environment [23, 49, 75], but is costly and has low resolution at long range. A top-of-the-line 64-lane Lidar sensor can easily cost more than a small car with an input resolution that is at least two orders of magnitude lower than a $\\$ 50\\mathrm { \\text R G B }$ sensor. This Lidar sensor receives one or two measurements for small or far away objects, whereas a corresponding RGB sensor sees hundreds of pixels. However, the RGB sensor does not perceive the depth and cannot directly place its measurements into a scene. ",
+        "bbox": [
+            174,
+            569,
+            825,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a simple and effective framework to fuse 3D Lidar and high-resolution color measurements. We lift RGB measurements into 3D virtual points by mapping them into the scene using close-by depth measurements of a Lidar sensor (See Figure 1 for an example). Our Multi-modal Virtual Point detector, MVP, generates high-resolution 3D point-cloud near target objects. A center-based 3D detector [66] then identifies all objects in the scene. Specifically, MVP uses 2D object detections to crop the original point cloud into instance frustums. MVP then generates dense 3D virtual points near these foreground points by lifting 2D pixels into 3D space. We use depth completion in image space to infer the depth of each virtual point. Finally, MVP combines virtual points with the original Lidar measurements as input to a standard center-based 3D detector [66]. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            797
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our multi-modal virtual point method has several key advantages: First, 2D object detections are well optimized [17, 74] and highly accurate even for small objects. See Figure 2 for a comparison of two state-of-the-art 2D and 3D detectors on the same scene. The 2D detector has a significantly higher 2D detection accuracy but lacks the necessary 3D information used in the downstream driving task. Secondly, virtual points reduce the density imbalance between close and faraway objects. MVP augments objects at different distances with the same number of virtual points, making the point cloud measurement of these objects more consistent. Finally, our framework is a plug-andplay module to any existing or new 2D or 3D detectors. We test our model on the large-scale nuScenes dataset [2]. Adding multi-modal virtual points brings ${ \\bf 6 . 6 \\ m A P }$ improvements over a strong CenterPoint baseline [66]. Without any ensembles or test-time augmentation, our best model achieves ${ \\bf 6 6 . 4 \\ m A P }$ and ${ \\bf 7 0 . 5  N D S }$ on nuScenes, outperforming all competing non-ensembled methods on the nuScenes leaderboard at the time of submission. ",
+        "bbox": [
+            174,
+            804,
+            825,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/802ca41f707f79ce428e72b18cbf58e49acac072296129e4f9b9be5cda82590f.jpg",
+        "image_caption": [
+            "Figure 1: We augment sparse Lidar point cloud with dense semantic virtual points generated from 2D detections. Left: the augmented point-cloud in the scene. We show the original points in gray and augmented points in red. Right: three cutouts with the origial points on top and virtual points below. The virtual points are up to two orders of magnitude denser. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            85,
+            802,
+            276
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            368,
+            825,
+            438
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            458,
+            316,
+            474
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2D Object Detection has great progress in recent years. Standard approaches include the RCNN family [13, 17, 43] which first predict class-agnostic bounding boxes based on predefined anchor boxes and then classify and refine them in a two-stage fashion with deep neural networks. YOLO [42], SSD [33], and RetinaNet [30] predicts the class specific bounded boxes in one shot. Recent anchorfree detectors like CornerNet [24] and CenterNet [74] directly localize objects through keypoints without the need of predefined anchors. In our approach, we use CenterNet [74] as our 2D detector for its simplicity and superior performance for detecting small objects. See Figure 2 for an example of a 2D detectors output. ",
+        "bbox": [
+            174,
+            491,
+            825,
+            602
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Lidar-based 3D Object Detection estimates rotated 3D bounding boxes from 3D point clouds [7, 12, 23, 37, 60–63, 65, 76]. 3D detectors share a common output representation and network structure with 2D detectors but encode the input differently. VoxelNet [75] uses a PointNe-based feature extractor to generate a voxel-wise feature representation from which a backbone consisted of sparse 3D convolutions and bird-eye view 2D convolution produces detection outputs. SECOND [60] introduces more efficient sparse convolution operations. PIXOR [61] and PointPillars [23] directly process point clouds in bird-eye view, further improving efficiency. Two-stage 3D detectors [8, 45– 47, 63] use a PointNet-based set abstraction layer [39] to aggregate RoI-specifc features inside first stage proposals to refine outputs. Anchor-free approaches [5, 36, 57, 59, 61, 66] remove the need for axis-aligned bird-eye view anchor boxes. VoteNet [36] detects 3D objects through Hough voting and clustering. CenterPoint [66] proposes a center-based representation for 3D object detection and tracking and achieved state-of-the-art performance on nuScenes and Waymo benchmarks. However, as Figure 2 shows a Lidar-only detector still misses small or far-away objects due to the sparsity of depth measurements. In this work, we build upon the CenterPoint detector and demonstrate significant ${ \\bf 6 . 6 \\ m A P }$ improvements by adding our multi-modal virtual point approach. ",
+        "bbox": [
+            174,
+            618,
+            825,
+            825
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Camera-based 3D Object Detection Camera-based 3D object detection predicts 3D bounding boxes from camera images. Mono3D [6] uses the ground-plane assumption to generate 3D candidate boxes and scores the proposals using 2D semantic cues. CenterNet [74] first detects 2D objects in images and predicts the corresponding 3D depth and bounding box attributes using center features. Despite rapid progress, monocular 3D object detectors still perform far behind the Lidar-based methods. On state-of-the-art 3D detection benchmarks [2, 12], state-of-the-art monocular methods [26, 41] achieve about half the mAP detection accuracy, of standard Lidar based baselines [60]. PseudoLidar [56] based methods produce a virtual point cloud from RGB images, similar to our approach. However, they rely on noisy stereo depth estimates [25, 40, 56] while we use more accurate Lidar measurements. Again, the performance of purely color-based approaches lags slightly behind Lidar or fusion-based methods [2, 12]. ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/a55c340fde8c8833b29c6154a86673938c751a9c42cb9d55d9d976e5de2cbfcd.jpg",
+        "image_caption": [
+            "Figure 2: Comparison between state-of-the-art image-based 2D detector [73] and point cloud based 3D detector [66]. We show detection from the 2D detector in blue and detection from 3D detector in green. For the 3D detector, we project the predicted 3D bounding boxes into images to get the 2D detections. For the 2D detector, we train the model using projected 2D boxes from 3D annotations. Compared to 2D detector, 3D detector often misses faraway or small objects.A quantitative comparison between 2D and 3D detectors is included in Section 5.2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            88,
+            813,
+            270
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            387,
+            826,
+            470
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-modal 3D Object Detection fuses information of Lidar and color cameras [20, 28, 28, 29, 35, 37, 53, 67, 71, 72]. Frustum PointNet [38] and Frustum ConvNet [58] first detect objects in image space to identify regions of interest in the point cloud for further processing. It improves the efficiency and precision of 3D detection but is fundamentally limited by the quality of 2D detections. In contrast, we adopt a standard 3D backbone [75] to process the augmented Lidar point cloud, combining the benefit of both sensor modalities. MV3D [7] and AVOD [22] performs object-centric fusion in a two-stage framework. Objects are first detected in each sensor and fused at the proposal stage using RoIPooling [43]. Continuous fusion [20, 29] shares image and Lidar features between their backbones. Closest to our approach are MVX-Net [48], PointAugmenting [54], and PointPainting [52], which utilize point-wise correspondence to annotate each lidar point with image-based segmentation or CNN features. We instead augment the 3D lidar point cloud with additional points surrounding 3D measurements. These additional points make full use of the higher dimensional RGB measurements. ",
+        "bbox": [
+            173,
+            484,
+            825,
+            651
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Point Cloud Augmentation generates denser point clouds from sparse Lidar measurements. Lidarbased methods like PUNet [69], PUGAN [27], and Wang et al. [64] learn high level point-wise features from raw Lidar scans. They then reconstruct multiple upsampled point clouds from each high dimensional feature vector. Image-based methods [18, 21, 51] perform depth completion from sparse measurements. We build upon these depth completion methods and demonstrate state-of-the-art 3D detection results through point upsampling. ",
+        "bbox": [
+            174,
+            666,
+            825,
+            750
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Preliminary ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
+            305,
+            785
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our framework relies on both 2D detection, existing 3D detectors, and a mapping between 2D and 3D. We introduce the necessary concepts and notations below. ",
+        "bbox": [
+            174,
+            799,
+            825,
+            827
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2D Detection. Let $I$ be a camera image. A 2D object detector aims to localize and classify all objects in $I$ . A bounding-box $b _ { i } \\in \\mathbb { R } ^ { 4 }$ describes the objects location. A class score $s _ { i } ( c )$ predicts the likelihood of detection $b _ { i }$ to be of class $c$ . An optional instance mask $m _ { i } \\in [ 0 , 1 ] ^ { W \\times H }$ predicts a pixel-level segmentation of each object. In this paper, we use the popular CenterNet [74] detector. CenterNet detects objects through keypoint estimation. It takes the input image $I$ and predicts a heatmap for each class $c$ . Peaks (local maxima) of the heatmap corresponds to an object. The model regresses to other bounding box attributes using peak features with an L1 [74] or box IoU [44] objective. For instance segmentation, we use CenterNet2 [73] which adds a cascade RoI heads [3] on top of the first stage proposal network. The overall network runs at 40 FPS and achieves 43.3 instance segmentation mAP on the nuScenes image dataset [2]. ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3D Detection. Let $P = \\{ ( x , y , z , r ) _ { i } \\}$ be a point cloud with 3D location $( x , y , z )$ and reflectance $r$ . The goal of a 3D detector is to predict a set of 3D bounding boxes $\\{ b _ { i } \\}$ from the point cloud $P$ . The bounding box $b = ( u , v , o , w , \\bar { l } , h , \\theta )$ includes the 3D center location $( u , v , o )$ , object size $( w , l , h )$ and the yaw rotation along $\\mathbf { Z }$ axis $\\theta$ . In this paper, we build upon the state-of-the-art CenterPoint [66] detector. We experiment with two popular 3D backbones: VoxelNet [75] and PointPillars [23]. VoxelNet quantizes the irregular point clouds into regular bins followed by a simple average pooling to extract features from all points inside a bin [60]. After that, a backbone consisted of sparse 3D convolutions [14] processes the quantized 3D feature volumes and the output is a map view feature map $M \\in \\mathbb { R } ^ { \\bar { W } \\times \\bar { H } \\times F }$ . PointPillar directly processes point clouds as bird-eye view pillar, a single elongated voxel per map location, and extracts features with fast 2D convolution to get the map view feature map $M$ . ",
+        "bbox": [
+            173,
+            172,
+            826,
+            327
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the map view features, a detection head inspired by CenterNet [74] localizes objects in bird-eye view and regress to other box parameters using center features. ",
+        "bbox": [
+            176,
+            333,
+            820,
+            361
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2D-3D Correspondence. Multi-modal fusion approaches [48, 52, 53, 72] often rely on a pointwise correspondence between 3D point clouds and 2D pixels. In absence of calibration noise, the projection from the 3D Lidar coordinate into a 2D image coordinate involves an SE(3) transformation from the Lidar measurement to the camera frame and a perspective projection from the camera frame into image coordinates. All transformations may be described with homogeneous, time-dependent transformations. Let $t _ { 1 }$ and $t _ { 2 }$ be the capture time of the Lidar measurement and RGB image respectively. Let $T _ { \\mathrm { ( c a r  l i d a r ) } }$ be the transformation from the Lidar sensor to the reference frame of the car. Let $T _ { ( t _ { 1 }  t _ { 2 } ) }$ be the transformation of the car between $t _ { 2 }$ and $t _ { 1 }$ . Let $T _ { \\mathrm { ( r g b  c a r ) } }$ be the transformation from the cars reference frame to the RGB sensor. Finally, let $P _ { \\mathrm { r g b } }$ be the projection matrix of the RGB camera defined by the camera intrinsic. The transformation from the Lidar to RGB sensor is then defined by ",
+        "bbox": [
+            173,
+            375,
+            825,
+            527
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7a236cbacf886781d09c17da3b28fd2577925d4c8dcead32a360c4bd7c298b3.jpg",
+        "text": "$$\nT _ { \\mathrm { r g b }  \\mathrm { l i d a r } } ^ { t _ { 1 }  t _ { 2 } } = T _ { ( \\mathrm { r g b }  c a r ) } T _ { ( t _ { 1 }  t _ { 2 } ) } T _ { ( \\mathrm { c a r }  \\mathrm { l i d a r } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            529,
+            647,
+            549
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "followed by a perspective projection with camear matrix $P _ { \\mathrm { r g b } }$ and a perspective division. The perspective division makes the mapping from Lidar to RGB surjective and non-invertible. In the next section, we show how to recover an inverse mapping by using depth measurements of Lidar when mapping RGB to Lidar. ",
+        "bbox": [
+            174,
+            550,
+            825,
+            606
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Multimodal Virtual Point ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            625,
+            418,
+            642
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a set of 2D object detections, we want to generate dense virtual points $\\boldsymbol { v } _ { i } = ( x , y , z , \\mathbf { e } )$ where $( x , y , z )$ is the 3D location and $\\mathbf { e }$ is the semantic feature from the 2D detector. For simplicity, we use the 2D detectors class scores as semantic features. For each detection $b _ { j }$ with associated instance mask $\\mathbf { m } _ { j }$ we generate a fixed number $\\tau$ multimodal virtual points. ",
+        "bbox": [
+            173,
+            655,
+            825,
+            710
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Virtual Point Generation. We start by projecting the 3D Lidar point cloud onto our detection. Specifically, we transform each Lidar point $( x , y , z , r ) _ { i }$ into the reference frame of the RGB camera following Equation (1), then project it into image coordinates $\\mathbf { p } _ { i }$ with associated depth $d _ { i }$ using a perspective projection. Let the collection of all projected points and depth values for a single detection $j$ be the objects frustum $\\mathbf { F } _ { j } = \\{ ( \\mathbf { p } _ { i } , d _ { i } ) | \\mathbf { p } _ { i } \\overset { \\cdot } { \\in } \\bar { \\mathbf { m } } _ { j } \\forall _ { i } \\}$ . The frustum only considers projected 3D points $\\mathbf { p } _ { i }$ that fall within a detection mask $\\mathbf { m } _ { j }$ . Any Lidar measurement outside detection masks is discarded. Next, we generate virtual points from each frustum $\\mathbf { F } _ { j }$ . ",
+        "bbox": [
+            173,
+            723,
+            825,
+            821
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We start by randomly sampling 2D points $\\mathbf s \\in \\mathbf m$ from each instance mask m. We sample $\\tau$ points uniformly at random without repetition. For each sampled point $\\mathbf { s } _ { k }$ , we retrieve a depth estimate $d _ { k }$ from its nearest neighbor in the frustum $F _ { j }$ : $d _ { k } = \\arg \\operatorname* { m i n } _ { d _ { i } } \\left\\| \\mathbf { p } _ { i } - \\mathbf { s } _ { k } \\right\\|$ . Given the depth estimate, we unproject the point back into 3D and append the object’s semantic feature $e _ { j }$ to the virtual point. We concatenate the one-hot encoding of the detected class and the detections objectness score in the semantic feature. ",
+        "bbox": [
+            174,
+            828,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b2e5fe5fedd61ddc46af5b0876944c38bdf336f7c783b29a1b6504c82489147.jpg",
+        "table_caption": [
+            "Algorithm 1: Multi-modal Virtual Point Generation "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input Hyper parameters :Number of virtual points per object T Output</td><td>:Lidar point cloud L = {(x,y, z,r)i}. Instance masks {m1,...,mn} for n objects. Semantic features {e1,..,en} with ej ∈ RD</td></tr><tr><td>Fj←の ∀j∈{1...n}; for(xi,yi,zi,ri) ∈Ldo /* Perspective projection to 2D point p depth d p,d←Project(PrgbTtda(myi,,1)T);</td><td>: Multi-modal 3D virtual points V ∈ Rn Xτ×(3+D) // Point cloud instance frustums */</td></tr><tr><td>for j ∈{1...n} do ifp∈mj then</td><td></td></tr><tr><td>Fj←FjU{(p,d)}; end</td><td>// Add point to frustum</td></tr><tr><td>end end</td><td></td></tr><tr><td>for j ∈{1...n} do</td><td></td></tr><tr><td>S ← Sample,(mj);</td><td></td></tr><tr><td></td><td></td></tr><tr><td>for s ∈Sdo</td><td></td></tr><tr><td></td><td>// Uniformly sample T 2d points in instance mask</td></tr><tr><td>(p,d) ← NN(s,Fj);</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>// Find closest projected 3D point</td></tr><tr><td></td><td></td></tr><tr><td>(PrgbTtiar)</td><td>/* Unproject the 2D point s using the nearest neighbors depth d*/</td></tr><tr><td>q↑(</td><td></td></tr><tr><td></td><td>Unproject(s,d);</td></tr><tr><td>Add (q,ej) to Vj;</td><td></td></tr><tr><td></td><td></td></tr><tr><td>end</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>end</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr></table>",
+        "bbox": [
+            169,
+            109,
+            808,
+            500
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The virtual point generation is summarized in Algorithm 1 and Figure 3. Next, we show how to incorporate virtual points into a point-based 3D detector. ",
+        "bbox": [
+            176,
+            527,
+            823,
+            556
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Virtual Point 3D detection. Voxel-based 3D detectors [60, 66] first voxelize 3D points $( x , y , z ) _ { i }$ and average all point features $( x , y , z , t , r ) _ { i }$ within a voxel. Here $r _ { i }$ is a reflectance measure and $t _ { i }$ is the capture time. A standard 3D convolutional network uses these voxelized features in further processing. For virtual points, this creates an issue. The feature dimensions of real points $( x , y , z , t , r )$ and virtual points differ $( x , y , z , t , \\mathbf { e } )$ . A simple solution could be to either concatenate virtual and real points into a larger feature $( x , y , z , t , r , \\mathbf { e } )$ and set any missing information to zero. However, this is both wasteful, as the dimension of real points grows by $3 \\times$ , and it creates an imbalanced ratio between virtual and real points in different parts of the scene. Furthermore, real measurements are often a bit more precise than virtual points and simple averaging of the two blurs out the information contained in real measurements. To solve this, we modify the average pooling approach by separately averaging features of virtual and real points and concatenating the final averaged features together as input to 3D convolution. For the rest of the architecture, we follow CenterPoint [66]. ",
+        "bbox": [
+            173,
+            570,
+            825,
+            736
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We further use virtual points in a second-stage refinement. Our MVP model generates dense virtual points near target objects which help two-stage refinement [45, 66]. Here, we follow Yin et al. [66] to extract bird-eye view features from all outward surfaces of the predicted 3D box. The main difference to Yin et al. is that our input is much denser around objects, and hence the second stage refinement has access to richer information. ",
+        "bbox": [
+            174,
+            742,
+            823,
+            813
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            830,
+            312,
+            848
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We evaluate our proposed multimodal virtual point method on the challenging nuScenes benchmark. ",
+        "bbox": [
+            173,
+            862,
+            823,
+            877
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "nuScenes [2] is a popular multimodal autonomous driving datasets for 3D object detection in urban scenes. The dataset contains 1000 driving sequences, with each 20s long and annotated with 3D bounding boxes. The Lidar frequency is $2 0 \\mathrm { H z }$ and the dataset provides sensor and vehicle pose information for each Lidar frame but only includes object annotation every ten frames (0.5s). The dataset hides any personally identifiable information, blurs faces and license plates in color images. There are in total 6 RGB cameras at a resolution of $1 6 0 0 \\times 9 0 0$ and a capture frequency of $1 2 \\mathrm { H z }$ . We follow the official dataset split to use 700, 150, 150 sequences for training, validation, and testing. This in total results in 28130 frames for training, 6019 frames for validation, and 6008 frames for testing. The annotations include a fine-grained label space of ten classes with a long-tail distribution. For 3D object detection, the official evaluation metrics include the mean Average Precision (mAP)[11] and nuScenes detection score (NDS) [2]. mAP measures the localization precision using a threshold based on the birds-eye view center distance $< 0 . 5 \\mathrm { m }$ , 1m, 2m, 4m. NDS is a weighted combination of mAP and regression accuracy of other object attributes including box size, orientation, translation, and class-specific attributes [2]. NDS is the main ranking metric for the benchmark. ",
+        "bbox": [
+            173,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/cea7a1071c5d0c5bc301c9e9942c92293f3b2a4e760b5ea509b38ed771cc1c6a.jpg",
+        "image_caption": [
+            "(a) 2D instance segmentation "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            89,
+            495,
+            227
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a2df439a19e6fcc13bada1b5f3b60c478049920845b63d54b385013d873945f.jpg",
+        "image_caption": [
+            "(c) Sampling and nearest neighbor matching "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            255,
+            495,
+            392
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9f1823b3fa830acf8f5a3696a1391bef1515e7d3a3b10a89fbb499802d865087.jpg",
+        "image_caption": [
+            "(b) Lidar point cloud projection "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            503,
+            89,
+            816,
+            226
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/204de22807dcc359aac30aa34f6961fd838fa62c2772317304f2511914c55425.jpg",
+        "image_caption": [
+            "(d) Reprojected virtual points ",
+            "Figure 3: Overview of our mlutimodal virtual point generation framework. We start by extracting 2D instance masks for each object in a color image (a). We then project all Lidar measurements into the reference frame of the RGB camera (b). For visualization purposes, points inside the objects are black, other points are grey. We then sample random points inside each 2D instance mask and retrieve a depth estimate from their nearest neighbor Lidar projection (c). For visualization clarity, (c) only shows a subset of virtual points. Finally, all virtual points are reprojected into the original point-cloud (d). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            563,
+            260,
+            761,
+            397
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            551,
+            826,
+            718
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. Our implementation is based on the opensourced code of CenterPoint 1 [66] for 3D detection and CenterNet2 $[ 7 3 ]$ for 2D Detection. ",
+        "bbox": [
+            174,
+            733,
+            821,
+            761
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For 2D detection, we train a CenterNet [74] detector on the nuScenes image dataset [2]. We use the DLA-34 [68] backbone with deformable convolutions [10]. We add cascade RoI heads [3] for instance segmentation following Zhou et al. [73]. We train the detector on the nuScenes dataset using the SGD optimizer with a batch size of 16 and a learning rate of 0.02 for 90000 iterations. ",
+        "bbox": [
+            174,
+            767,
+            825,
+            824
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For 3D detection, we use the same VoxelNet [75] and PointPillars [23] architectures following [23, 66, 76]. For VoxelNet, the detection range is $[ - 5 4 m , 5 4 m ]$ for the $X$ , $Y$ axis and $[ - 5 m , 3 m ]$ for the $Z$ axis while the range is $[ - 5 1 . 2 m , 5 1 . 2 m ]$ for the $X$ , $Y$ axis for the PointPillar architecture. The voxel size is $( 0 . 0 7 5 m , 0 . 0 7 5 m , 0 . 2 m )$ and $( 0 . 2 m , 0 . 2 m , 8 m )$ for VoxelNet and PointPillar respectively. For data augmentation, we follow CenterPoint and use global random rotations between $[ - \\bar { \\pi } / 4 , \\pi / 4 ]$ , global random scaling between [0.9, 1.1] and global translations between $[ - 0 . 5 m , 0 . 5 m ]$ . To deal with the long-tail class distribution in nuScenes, we use the ground truth sampling in [60] to randomly paste objects into the current frame [60, 76]. We also adopt the class-balanced resampling and class-grouped heads in [76] to improve the average density of rare classes. We train the model for 20 epochs with the AdamW [34] optimizer using the one-cycle policy [16], with a max learning rate of 3e-3 following [66]. The training takes 2.5 days on 4 V100 GPUs with a batch size of 16 (4 frames per GPU). ",
+        "bbox": [
+            176,
+            830,
+            825,
+            872
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/78548d34669a26866e7ea69ace84a52b81990fcc429e6e0e749843da3e8dc6b3.jpg",
+        "table_caption": [
+            "Table 1: Comparisons with previous methods on nuScenes test set. We show the NDS, mAP, and mAP for each class. Abbreviations are construction vehicle (CV), pedestrian (Ped), motorcycle (Motor), and traffic cone (TC). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>mAP</td><td>NDS</td><td>Car</td><td>Truck</td><td>Bus</td><td>Trailer</td><td>CV</td><td>Ped</td><td>Motor</td><td>Bicycle</td><td>TC</td><td>Barrier</td></tr><tr><td>PointPillars [23]</td><td>30.5</td><td>45.3</td><td>68.4</td><td>23.0</td><td>28.2</td><td>23.4</td><td>4.1</td><td>59.7</td><td>27.4</td><td>1.1</td><td>30.8</td><td>38.9</td></tr><tr><td>WYSIWYG[19]</td><td>35.0</td><td>41.9</td><td>79.1</td><td>30.4</td><td>46.6</td><td>40.1</td><td>7.1</td><td>65.0</td><td>18.2</td><td>0.1</td><td>28.8</td><td>34.7</td></tr><tr><td>3DSSD [62]</td><td>42.6</td><td>56.4</td><td>81.2</td><td>47.2</td><td>61.4</td><td>30.5</td><td>12.6</td><td>70.2</td><td>36.0</td><td>8.6</td><td>31.1</td><td>47.9</td></tr><tr><td>PMPNet [65]</td><td>45.4</td><td>53.1</td><td>79.7</td><td>33.6</td><td>47.1</td><td>43.1</td><td>18.1</td><td>76.5</td><td>40.7</td><td>7.9</td><td>58.8</td><td>48.8</td></tr><tr><td>PointPainting [52]</td><td>46.4</td><td>58.1</td><td>77.9</td><td>35.8</td><td>36.2</td><td>37.3</td><td>15.8</td><td>73.3</td><td>41.5</td><td>24.1</td><td>62.4</td><td>60.2</td></tr><tr><td>CBGS [76]</td><td>52.8</td><td>63.3</td><td>81.1</td><td>48.5</td><td>54.9</td><td>42.9</td><td>10.5</td><td>80.1</td><td>51.5</td><td>22.3</td><td>70.9</td><td>65.7</td></tr><tr><td>CVCNet [4]</td><td>55.3</td><td>64.4</td><td>82.7</td><td>46.1</td><td>46.6</td><td>49.4</td><td>22.6</td><td>79.8</td><td>59.1</td><td>31.4</td><td>65.6</td><td>69.6</td></tr><tr><td>HotSpotNet [5]</td><td>59.3</td><td>66.0</td><td>83.1</td><td>50.9</td><td>56.4</td><td>53.3</td><td>23.0</td><td>81.3</td><td>63.5</td><td>36.6</td><td>73.0</td><td>71.6</td></tr><tr><td>CenterPoint [66]</td><td>58.0</td><td>65.5</td><td>84.6</td><td>51.0</td><td>60.2</td><td>53.2</td><td>17.5 83.4</td><td></td><td>53.7</td><td>28.7</td><td>76.7</td><td>70.9</td></tr><tr><td>MVP (Ours)</td><td>66.4</td><td>70.5</td><td>86.8</td><td>58.5</td><td>67.4</td><td>57.3</td><td>26.1</td><td>89.1</td><td>70.0</td><td>49.3</td><td>85.0</td><td>74.8</td></tr></table>",
+        "bbox": [
+            173,
+            145,
+            839,
+            314
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            353,
+            825,
+            478
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "During the testing, we set the output threshold to be 0.05 for the 2D detector and generate 50 virtual points for each 2D object in the scene. We use an output threshold of 0.1 for the 3D detector after performing non-maxima suppression with an IoU threshold of 0.2 following CenterPoint [66]. ",
+        "bbox": [
+            176,
+            484,
+            825,
+            526
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 State-of-the-art Comparison ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
+            410,
+            566
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We first compare with state-of-the-art approaches on the nuScenes test set. We obtain all results on the public leaderboard by submitting our predictions to an online evaluation server. The submission uses a single MVP model without any ensemble or test-time augmentations. We compare to other methods under the same setting. Table 1 summarizes our results. On the nuScenes dataset, MVP achieves state-of-the-art results of $6 6 . 4 \\ : \\mathrm { m A P }$ and $7 0 . 5 \\ : \\mathrm { N D S }$ , outperforming the strong CenterPoint baseline by $8 . 4 \\mathrm { m A P }$ and 5.0 NDS. MVP shows consistent improvements across all object categories with significant $1 1 \\mathrm { m A P }$ accuracy boosts for small objects $( + 2 0 . 6$ for Bicycle and $+ 1 6 . 3$ for motorcycle). These results clearly verify the effectiveness of our multi-modal virtual point approach. ",
+        "bbox": [
+            173,
+            580,
+            825,
+            691
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Ablation Studies ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            717,
+            328,
+            731
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparison of 2D and 3D Detector. We first validate the superior detection performance of the camera-based 2D detector compared to the Lidar-based 3D detector. Specifically, we use two state-ofthe-art object detectors: CenterPoint [66] for Lidar-based 3D detection, and CenterNet [74] for imagebased 2D detection. To compare the performance of detectors working in different modalities, we project the predicted 3D bounding boxes into the image space to get the corresponding 2D detections. The 2D CenterNet detector is trained with projected 2D boxes from ground truth 3D annotations. Table 2 summarizes the results over the whole nuScenes validation set. 2D CenterNet [74] significantly outperforms the CenterPoint model by $9 . 8 \\mathrm { m A P }$ (using 2D overlap). The improvements are larger for smaller objects with a $1 2 . 6 \\mathrm { m A P }$ improvement for objects of medium size and more than $3 \\times$ accuracy improvements for small objects $6 . 9 \\mathrm { m A P }$ vs. $1 . 6 \\mathrm { m A P }$ ). See Figure 2 for a qualitative visualization of these two detectors’ outputs. These results support our motivations for utilizing high-resolution image information to improve 3D detection models with sparse Lidar input. ",
+        "bbox": [
+            174,
+            744,
+            826,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1ee9c49f4298ee27ab67866ba09e039f5065bae527f195903579c6ea38c65d36.jpg",
+        "table_caption": [
+            "Table 2: Quantitative comparison between state-of-the-art image-based 2D detector [74] and point cloud based 3D detector [66] on the nuScenes validation set measuring 2D detection accuracy (AP). The comparison use the COCO [31] style mean average precision with 2D IoU threshold between 0.5 and 0.95 in image coordinates. For 3D CenterPoint [66] detector, we project the predicted 3D bounding boxes into images to get the 2D detections. The results show that the 2D detector performs significantly better than Lidar-based 3D detector at localizing small or medium size objects due to high resolution camera input. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>APsmall</td><td>APmedium</td><td>APlarge</td><td>AP</td></tr><tr><td>CenterPoint [66]</td><td>1.6</td><td>11.7</td><td>34.5</td><td>22.7</td></tr><tr><td>CenterNet [74]</td><td>6.9</td><td>24.3</td><td>42.6</td><td>32.5</td></tr></table>",
+        "bbox": [
+            302,
+            199,
+            692,
+            258
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d597376ab0a3e43a94b577a75e54c3e33079cd0bb93f177d7bae1f11dfb2f70.jpg",
+        "table_caption": [
+            "Table 3: Component analysis of our MVP model with VoxelNet [60, 75] and PointPillars [23] backbones on nuScenes validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Encoder</td><td>Baseline</td><td>Virtual Point</td><td>Split Voxelization</td><td>Two-stage</td><td>mAP↑</td><td>NDS↑</td></tr><tr><td rowspan=\"5\">VoxelNet</td><td>?</td><td></td><td></td><td></td><td>59.6</td><td>66.8</td></tr><tr><td></td><td></td><td></td><td>√</td><td>60.5</td><td>67.4</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td>65.9</td><td>69.6</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td>66.0</td><td>70.0</td></tr><tr><td>√</td><td>√</td><td>√</td><td>厂</td><td>67.1</td><td>70.8</td></tr><tr><td rowspan=\"2\">PointPillars</td><td>广</td><td></td><td></td><td></td><td>52.3</td><td>61.3</td></tr><tr><td></td><td>√</td><td></td><td></td><td>62.7</td><td>66.1</td></tr></table>",
+        "bbox": [
+            197,
+            321,
+            800,
+            457
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Component Analysis. Next, we ablate our contributions on the nuScenes validation set. We use the Lidar-only CenterPoint [74] model as our baseline. All hyperparameters and training procedures are the same between all baselines. We change inputs (MVP or regular points), voxelization, or an optional second stage. Table 3 shows the importance of each component of our MVP model. Simply augmenting the Lidar point cloud with multi-modal virtual points gives a $6 . 3 \\ \\mathrm { m A P }$ and $1 0 . 4 \\ \\mathrm { m A P }$ improvements for VoxelNet and PointPillars encoder, respectively. For the VoxelNet encoder, split voxelization gives another $0 . 4 ~ \\mathrm { N D S }$ improvements due to the better modeling of features inside a voxel. Moreover, two-stage refinement with surface center features brings another $1 . 1 \\ \\mathrm { m A P }$ and $0 . 8 ~ \\mathrm { N D S }$ improvements over our first stage models with small overheads $( 1 - 2 \\mathrm { m s } )$ . The improvement of two-stage refinement is slightly larger with virtual points than without. This highlights the effectiveness of our virtual point method to create a finer local structure for better localization and regression using two-stage point-based detection. ",
+        "bbox": [
+            174,
+            489,
+            825,
+            655
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Performance Breakdown. To better understand the improvements of our MVP model, we show the performance comparisons on different subsets of the nuScenes validation set based on object distances to the ego-vehicle. We divide all ground truth annotations and predictions into three ranges: $0 { - } 1 5 \\mathrm { m }$ , $1 5 { - } 3 0 \\mathrm { m }$ , and $3 0 { - } 5 0 \\mathrm { m }$ . The baselines include both the Lidar-only two-stage CenterPoint [66] model and the state-of-the-art multi-modal fusion method PointPainting [52]. We reimplement PointPainting using the same 2D detections, backbones, and tricks (including two-stage) as our MVP approach. The main difference to PointPainting [52] is our denser Lidar inputs with multimodal virtual points. Table 4 shows the results. Our MVP model outperforms the Lidar-only baseline by $6 . 6 \\mathrm { m A P }$ while achieving a significant $1 0 . 1 \\mathrm { m A P }$ improvement for faraway objects. Compared to PointPainting [52], our model achieve a $1 . 1 \\mathrm { m A P }$ improvement for faraway objects and performs comparatively for closer objects. This improvement comes from the dense and fine-grained 3D structure generated from our MVP framework. Our method makes better use of the higher dimensional RGB measurements than the simple point-wise semantic feature concatenation as used in prior works [48, 52]. ",
+        "bbox": [
+            174,
+            672,
+            826,
+            852
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Robustness to 2D Detection We investigate the impact of 2D instance segmentation quality on the final 3D detection performance. With the same image network, we simulate the degradation of 2D segmentation performance with smaller input resolutions. We show the results in Table 5. Our ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "MVP model is robust to the quality of 2D instance segmentation. The 3D detection performance only decreases by $0 . 8 \\mathrm { N D S }$ with a 9 point worse instance segmentation inputs. ",
+        "bbox": [
+            173,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Depth Estimation Accuracy We further quantify the depth estimation quality of our nearest neighbor-based depth interpolation algorithm. We choose objects with at least 15 lidar points and randomly mask out $80 \\%$ of the points. We then generate virtual points from the projected locations of the masked out lidar points and compute the a bi-directional pointwise chamfer distance between virtual points and masked out real lidar points. Our nearest neighbor approach has bi-directional chamfer distance of 0.33 meter on the nuScenes validation set. We believe more advanced learning based approaches like [18] and [21] may further improve the depth completion and 3D detection performance. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            244
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "KITTI Results To test the generalization of our method, we add an experiment on the popular KITTI dataset [12]. For 2D detection, we use a pretrained MaskFormer [9] model to generate the instance segmentation masks and create 100 virtual points for each 2D object in the scene. For 3D detection, we use the popular PointPillars [23] detector with augmented point cloud inputs. All other parameters are the same as the default PointPillars model. As shown in Table 6, augmenting the Lidar point cloud with our multimodal virtual points gives a $0 . 5 \\mathrm { m A P }$ and $2 . 3 \\mathrm { m A P }$ for vehicle and cyclist class, respectively. We didn’t notice an improvement for the pedestrian class, presumable due to inconsistent pedestrian definition between our image model (trained on COCO [32]) and the 3D detector. On COCO, people inside a vehicle or on top of a bike are all considered to be pedestrians while KITTI 3D detectors treat them as vehicle or cyclist. ",
+        "bbox": [
+            173,
+            257,
+            825,
+            396
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/70c4f84ecb0a865516f5b71ae7cd255f772c3b660216a12311a8fd1055d3e5ed.jpg",
+        "table_caption": [
+            "Table 4: Comparisons between Lidar-only CenterPoint [66] method, fusion-based PointPainting [52] method (denoted as CenterPoint $^ +$ Ours(w/o virtual)), and our multimodal virtual point method for detecting objects of different ranges. All three entries use the VoxelNet backbone. We split the nuScenes validation set into three subsets containing objects at different ranges. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">nuScenes mAP</td></tr><tr><td>0-15m</td><td>15-30m</td><td>30-50m</td><td>Overall</td></tr><tr><td>CenterPoint [66]</td><td>76.2</td><td>60.3</td><td>37.2</td><td>60.5</td></tr><tr><td>CenterPoint + Ours(w/o virtual)</td><td>78.2</td><td>67.4</td><td>46.2</td><td>66.5</td></tr><tr><td>CenterPoint + Ours</td><td>78.1</td><td>67.7</td><td>47.3</td><td>67.1</td></tr></table>",
+        "bbox": [
+            246,
+            477,
+            750,
+            563
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/dfef2903a7381027637e1bb7917c259fe2df3eb2d2bcd1bd76587a4e337c3981.jpg",
+        "table_caption": [
+            "Table 5: Influence of 2D instance segmentation quality for the final 3D detection performance. We show the input resolution, 2D detection mAP, and 3D detection nuScenes detection score (NDS). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Resolution</td><td>2D mAP</td><td>NDS</td></tr><tr><td>900</td><td>43.3</td><td>70.0</td></tr><tr><td>640</td><td>39.5</td><td>69.6</td></tr><tr><td>480</td><td>34.2</td><td>69.2</td></tr></table>",
+        "bbox": [
+            387,
+            627,
+            611,
+            693
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ce8c11ce2d80e86156912ca8066df1803440ae3fb1008b7b4d79a42d9a9a31f1.jpg",
+        "table_caption": [
+            "Table 6: Comparison between Lidar-only PointPillars detector and our multimodal virtual point method for 3D detection on KITTI dataset. We show the 3D detection mean average precision for each class under the moderate difficulty level. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Car</td><td>Cyclist</td><td>Pedestrian</td></tr><tr><td>PointPillars [23]</td><td>77.3</td><td>62.7</td><td>52.3</td></tr><tr><td>PointsPillars+Ours</td><td>77.8</td><td>65.0</td><td>50.5</td></tr></table>",
+        "bbox": [
+            325,
+            771,
+            669,
+            823
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Discussion and conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            853,
+            429,
+            869
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We proposed a simple multi-modal virtual point approach for outdoor 3D object detection. The main innovation is a multi-modal virtual point generation algorithm that lifts RGB measurements into ",
+        "bbox": [
+            173,
+            882,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a7f2232cbdf4080c6316765137c68f82bd0e04a9c51e683a2d6ef8aacb10ad9.jpg",
+        "image_caption": [
+            "Figure 4: Example qualitative results of MVP on the nuScenes validation set. We show the raw point-cloud in blue, our detected objects in green bounding boxes, and Lidar points inside bounding boxes in red. Best viewed on screen. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            83,
+            818,
+            354
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "3D virtual points using close-by measurements of a Lidar sensor. Our MVP framework generates high-resolution 3D point clouds near target objects and enables more accurate localization and regression, especially for small and faraway objects. The model significantly improves the strong Lidar-only CenterPoint detector and sets a new state-of-the-art on the nuScenes benchmark. Our framework seamlessly integrates into any current or future 3D detection algorithms. ",
+        "bbox": [
+            174,
+            431,
+            823,
+            501
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "There are still certain limitations with the current approach. Firstly, we assume that virtual points have the same depth as close-by Lidar measurements. This may not hold in the real world. Objects like cars don’t have a planar shape vertical to the ground plane. In the future, we plan to apply learning-based methods [50, 70] to infer the detailed 3D shape and pose from both Lidar measurements and image features. Secondly, our current two-stage refinement modules only use features from the bird-eye view which may not take full advantage of the high-resolution virtual points generated from our algorithm. We believe point or voxel-based two-stage 3D detectors like PVRCNN [45] and M3Detr [15] may give more significant improvements. Finally, the point-based abstraction connecting 2D and 3D detection may introduce too large of a bottleneck to transmit information from 2D to 3D. For example, no pose information is contained in our current position $^ +$ class based MVP features. ",
+        "bbox": [
+            174,
+            507,
+            825,
+            646
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Overall, we believe future methods for scalable 3D perception can benefit from the interplay of camera and Lidar sensor inputs via dense semantic virtual points. ",
+        "bbox": [
+            174,
+            652,
+            823,
+            680
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Societal Impacts. First and foremost better 3D detection and tracking will lead to safer autonomous vehicles. However, in the short term, it may lead to earlier adoption of potentially not-yet safe autonomous vehicles, and misleading error rates in 3D detection may lead to real-world accidents. Fusing multiple modalities may also increase the iteration cycle and safety testing requirements of autonomous vehicles, as different modalities clearly adapt differently to changes in weather, geographic locations, or even day-night cycles. A low sun may uniquely distract an RGB sensor, and hence unnecessarily distract a 3D detector through MVPs. ",
+        "bbox": [
+            174,
+            695,
+            825,
+            792
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, increasing the reliance of autonomous vehicles on color sensors introduces privacy issues. While most human beings look indistinguishable in 3D Lidar measurements, they are clearly identifiable in color images. In the wrong hands, this additional data may be used for mass surveillance. ",
+        "bbox": [
+            174,
+            799,
+            825,
+            854
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement We thank the anonymous reviewers for the constructive comments. This material is based upon work supported by the National Science Foundation under Grant No. IIS-1845485 and IIS-2006820. ",
+        "bbox": [
+            176,
+            869,
+            825,
+            910
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Mayank Bansal, Alex Krizhevsky, and Abhijit Ogale. Chauffeurnet: Learning to drive by imitating the best and synthesizing the worst. RSS, 2019.   \n[2] Holger Caesar, Varun Bankiti, Alex H. Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. CVPR, 2020.   \n[3] Zhaowei Cai and Nuno Vasconcelos. Cascade r-cnn: Delving into high quality object detection. CVPR, 2018. [4] Qi Chen, Lin Sun, Ernest Cheung, Kui Jia, and Alan Yuille. Every view counts: Cross-view consistency in 3d object detection with hybrid-cylindrical-spherical voxelization. NeurIPS, 2020.   \n[5] Qi Chen, Lin Sun, Zhixin Wang, Kui Jia, and Alan Yuille. Object as hotspots: An anchor-free 3d object detection approach via firing of hotspots. ECCV, 2020.   \n[6] Xiaozhi Chen, Kaustav Kundu, Ziyu Zhang, Huimin Ma, Sanja Fidler, and Raquel Urtasun. Monocular 3d object detection for autonomous driving. CVPR, 2016.   \n[7] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. Multi-view 3d object detection network for autonomous driving. CVPR, 2017.   \n[8] Yilun Chen, Shu Liu, Xiaoyong Shen, and Jiaya Jia. Fast point r-cnn. ICCV, 2019.   \n[9] Bowen Cheng, Alexander G Schwing, and Alexander Kirillov. Per-pixel classification is not all you need for semantic segmentation. arXiv preprint arXiv:2107.06278, 2021.   \n[10] Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong Zhang, Han Hu, and Yichen Wei. Deformable convolutional networks. ICCV, 2017.   \n[11] Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. IJCV, 2010.   \n[12] Andreas Geiger, Philip Lenz, and Raquel Urtasun. Are we ready for autonomous driving? the kitti vision benchmark suite. CVPR, 2012.   \n[13] Ross Girshick. Fast r-cnn. ICCV, 2015.   \n[14] Benjamin Graham, Martin Engelcke, and Laurens van der Maaten. 3d semantic segmentation with submanifold sparse convolutional networks. CVPR, 2018.   \n[15] Tianrui Guan, Jun Wang, Shiyi Lan, Rohan Chandra, Zuxuan Wu, Larry Davis, and Dinesh Manocha. M3detr: Multi-representation, multi-scale, mutual-relation 3d object detection with transformers. arXiv preprint arXiv:2104.11896, 2021.   \n[16] Sylvain Gugger. The 1cycle policy. https://sgugger.github.io/the-1cycle-policy. html, 2018.   \n[17] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick. Mask r-cnn. ICCV, 2017.   \n[18] Mu Hu, Shuling Wang, Bin Li, Shiyu Ning, Li Fan, and Xiaojin Gong. Penet: Towards precise and efficient image guided depth completion. 2021.   \n[19] Peiyun Hu, Jason Ziglar, David Held, and Deva Ramanan. What you see is what you get: Exploiting visibility for 3d object detection. CVPR, 2020.   \n[20] Tengteng Huang, Zhe Liu, Xiwu Chen, and Xiang Bai. Epnet: Enhancing point features with image semantics for 3d object detection. ECCV, 2020.   \n[21] Saif Imran, Xiaoming Liu, and Daniel Morris. Depth completion with twin surface extrapolation at occlusion boundaries. pages 2583–2592, 2021.   \n[22] Jason Ku, Melissa Mozifian, Jungwook Lee, Ali Harakeh, and Steven L Waslander. Joint 3d proposal generation and object detection from view aggregation. IROS, 2018.   \n[23] Alex H. Lang, Sourabh Vora, Holger Caesar, Lubing Zhou, Jiong Yang, and Oscar Beijbom. Pointpillars: Fast encoders for object detection from point clouds. CVPR, 2019.   \n[24] Hei Law and Jia Deng. Cornernet: Detecting objects as paired keypoints. ECCV, 2018.   \n[25] Peiliang Li, Xiaozhi Chen, and Shaojie Shen. Stereo r-cnn based 3d object detection for autonomous driving. CVPR, 2019.   \n[26] Peixuan Li, Huaici Zhao, Pengfei Liu, and Feidao Cao. Rtm3d: Real-time monocular 3d detection from object keypoints for autonomous driving. arXiv preprint arXiv:2001.03343, 2, 2020.   \n[27] Ruihui Li, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng. Pu-gan: a point cloud upsampling adversarial network. pages 7203–7212, 2019.   \n[28] Ming Liang, Bin Yang, Yun Chen, Rui Hu, and Raquel Urtasun. Multi-task multi-sensor fusion for 3d object detection. CVPR, 2019.   \n[29] Ming Liang, Bin Yang, Shenlong Wang, and Raquel Urtasun. Deep continuous fusion for multi-sensor 3d object detection. ECCV, 2018.   \n[30] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollar. Focal loss for dense object detection. ICCV, 2017.   \n[31] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. ECCV, 2014.   \n[32] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. ECCV, 2014.   \n[33] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg. Ssd: Single shot multibox detector. ECCV, 2016.   \n[34] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. ICLR, 2019.   \n[35] Su Pang, Daniel Morris, and Hayder Radha. Clocs: Camera-lidar object candidates fusion for 3d object detection. arXiv preprint arXiv:2009.00784, 2020.   \n[36] Charles R. Qi, Or Litany, Kaiming He, and Leonidas Guibas. Deep hough voting for 3d object detection in point clouds. ICCV, 2019.   \n[37] Charles R. Qi, Wei Liu, Chenxia Wu, Hao Su, and Leonidas J. Guibas. Frustum pointnets for 3d object detection from rgb-d data. CVPR, 2018.   \n[38] Charles R Qi, Wei Liu, Chenxia Wu, Hao Su, and Leonidas J Guibas. Frustum pointnets for 3d object detection from rgb-d data. CVPR, 2018.   \n[39] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. NeurIPS, 2017.   \n[40] Rui Qian, Divyansh Garg, Yan Wang, Yurong You, Serge Belongie, Bharath Hariharan, Mark Campbell, Kilian Q Weinberger, and Wei-Lun Chao. End-to-end pseudo-lidar for image-based 3d object detection. CVPR, 2020.   \n[41] Cody Reading, Ali Harakeh, Julia Chae, and Steven L Waslander. Categorical depth distribution network for monocular 3d object detection. arXiv preprint arXiv:2103.01100, 2021.   \n[42] Joseph Redmon and Ali Farhadi. Yolo9000: better, faster, stronger. CVPR, 2017.   \n[43] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. NIPS, 2015.   \n[44] Hamid Rezatofighi, Nathan Tsoi, JunYoung Gwak, Amir Sadeghian, Ian Reid, and Silvio Savarese. Generalized intersection over union: A metric and a loss for bounding box regression. CVPR, 2019.   \n[45] Shaoshuai Shi, Chaoxu Guo, Li Jiang, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. Pv-rcnn: Point-voxel feature set abstraction for 3d object detection. CVPR, 2020.   \n[46] Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. Pointrcnn: 3d object proposal generation and detection from point cloud. CVPR, 2019.   \n[47] Shaoshuai Shi, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. From points to parts: 3d object detection from point cloud with part-aware and part-aggregation network. TPAMI, 2020.   \n[48] Vishwanath A Sindagi, Yin Zhou, and Oncel Tuzel. Mvx-net: Multimodal voxelnet for 3d object detection. ICRA, 2019.   \n[49] Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, et al. Scalability in perception for autonomous driving: An open dataset benchmark. CVPR, 2020.   \n[50] Shubham Tulsiani, Abhishek Kar, João Carreira, and Jitendra Malik. Learning category-specific deformable 3d models for object reconstruction. TPAMI, 2017.   \n[51] Jonas Uhrig, Nick Schneider, Lukas Schneider, Uwe Franke, Thomas Brox, and Andreas Geiger. Sparsity invariant cnns. pages 11–20, 2017.   \n[52] Sourabh Vora, Alex H Lang, Bassam Helou, and Oscar Beijbom. Pointpainting: Sequential fusion for 3d object detection. CVPR, 2020.   \n[53] Chunwei Wang, Chao Ma, Ming Zhu, and Xiaokang Yang. Pointaugmenting: Cross-modal augmentation for 3d object detection. CVPR, 2021.   \n[54] Chunwei Wang, Chao Ma, Ming Zhu, and Xiaokang Yang. Pointaugmenting: Cross-modal augmentation for 3d object detection. pages 11794–11803, June 2021.   \n[55] Dequan Wang, Coline Devin, Qi-Zhi Cai, Philipp Krähenbühl, and Trevor Darrell. Monocular plan view networks for autonomous driving. IROS, 2019.   \n[56] Yan Wang, Wei-Lun Chao, Divyansh Garg, Bharath Hariharan, Mark Campbell, and Kilian Q Weinberger. Pseudo-lidar from visual depth estimation: Bridging the gap in 3d object detection for autonomous driving. CVPR, 2019.   \n[57] Yue Wang, Alireza Fathi, Abhijit Kundu, David Ross, Caroline Pantofaru, Tom Funkhouser, and Justin Solomon. Pillar-based object detection for autonomous driving. ECCV, 2020.   \n[58] Zhixin Wang and Kui Jia. Frustum convnet: Sliding frustums to aggregate local point-wise features for amodal 3d object detection. arXiv preprint arXiv:1903.01864, 2019.   \n[59] Kelvin Wong, Shenlong Wang, Mengye Ren, Ming Liang, and Raquel Urtasun. Identifying unknown instances for autonomous driving. CORL, 2019.   \n[60] Yan Yan, Yuxing Mao, and Bo Li. Second: Sparsely embedded convolutional detection. Sensors, 2018.   \n[61] Bin Yang, Wenjie Luo, and Raquel Urtasun. Pixor: Real-time 3d object detection from point clouds. CVPR, 2018.   \n[62] Zetong Yang, Yanan Sun, Shu Liu, and Jiaya Jia. 3dssd: Point-based 3d single stage object detector. CVPR, 2020.   \n[63] Zetong Yang, Yanan Sun, Shu Liu, Xiaoyong Shen, and Jiaya Jia. Std: Sparse-to-dense 3d object detector for point cloud. ICCV, 2019.   \n[64] Wang Yifan, Shihao Wu, Hui Huang, Daniel Cohen-Or, and Olga Sorkine-Hornung. Patch-based progressive 3d point set upsampling. pages 5958–5967, 2019.   \n[65] Junbo Yin, Jianbing Shen, Chenye Guan, Dingfu Zhou, and Ruigang Yang. Lidar-based online 3d video object detection with graph-based message passing and spatiotemporal transformer attention. CVPR, 2020.   \n[66] Tianwei Yin, Xingyi Zhou, and Philipp Krähenbühl. Center-based 3d object detection and tracking. CVPR, 2021.   \n[67] Jin Hyeok Yoo, Yecheol Kim, Ji Song Kim, and Jun Won Choi. 3d-cvf: Generating joint camera and lidar features using cross-view spatial feature fusion for 3d object detection. arXiv preprint arXiv:2004.12636, 3, 2020.   \n[68] Fisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep layer aggregation. CVPR, 2018.   \n[69] Lequan Yu, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng. Pu-net: Point cloud upsampling network. pages 2790–2799, 2018.   \n[70] Wentao Yuan, Tejas Khot, David Held, Christoph Mertz, and Martial Hebert. Pcn: Point completion network. 3DV, 2018.   \n[71] Haolin Zhang, Dongfang Yang, Ekim Yurtsever, Keith A Redmill, and Ümit Özgüner. Farawayfrustum: Dealing with lidar sparsity for 3d object detection using fusion. arXiv preprint arXiv:2011.01404, 2020.   \n[72] Wenwei Zhang, Zhe Wang, and Chen Change Loy. Multi-modality cut and paste for 3d object detection. arXiv preprint arXiv:2012.12741, 2020.   \n[73] Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl. Probabilistic two-stage detection. arXiv preprint arXiv:2103.07461, 2021.   \n[74] Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Objects as points. arXiv:1904.07850, 2019.   \n[75] Yin Zhou and Oncel Tuzel. Voxelnet: End-to-end learning for point cloud based 3d object detection. CVPR, 2018.   \n[76] Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and Gang Yu. Class-balanced grouping and sampling for point cloud 3d object detection. arXiv:1908.09492, 2019. ",
+        "bbox": [
+            171,
+            78,
+            828,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            35,
+            830,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            59,
+            828,
+            916
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            85,
+            828,
+            617
+        ],
+        "page_idx": 13
+    }
+]

parse/train/MdZPf3qCF7s/MdZPf3qCF7s.md

@@ -0,0 +1,436 @@
+# Data augmentation for efficient learning from parametric experts
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We present a simple, yet powerful data-augmentation technique to enable data  
+2 efficient learning from parametric experts. Whereas behavioral cloning refers to   
+3 learning from samples of an expert, we focus here on what we refer to as the policy   
+4 cloning setting which allows for offline queries of an expert or expert policy. This   
+5 setting arises naturally in a number of problems, especially as a component of   
+6 other algorithms. We achieve a very high level of data efficiency in transferring   
+7 behavior from an expert to a student policy for high Degrees of Freedom (DoF)   
+8 control problems using our augmented policy cloning (APC) approach, which   
+9 combines conventional image-based data augmentation to build invariance to   
+10 image perturbations with an expert-aware offline data augmentation approach that   
+11 induces appropriate feedback-sensitivity in a region around expert trajectories. We   
+12 show that our method increases data-efficiency of policy cloning, enabling transfer   
+13 of complex high-DoF behaviors from just a few trajectories, and we also show   
+14 benefits of our approach in the context of algorithms in which policy cloning is a   
+15 constituent part.
+
+# 16 1 Introduction
+
+17 In various control and reinforcement learning settings, there is a need to transfer behavior from an   
+18 expert policy to a student policy. Broadly, when only samples from the expert policy are available, the   
+19 standard approach is to employ a version of regression from states to actions. This class of approaches   
+20 for producing a policy is known as behavioral cloning [Pomerleau, 1989, Michie and Sammut, 1996].   
+21 Behavioral cloning is quite flexible and supports the setting where the expert trajectories come from a   
+22 human teleoperating the relevant system directly, as well as various settings where the trajectories are   
+23 sampled from other controllers, which themselves may have been trained or scripted. However, for   
+24 any of the settings where the expert policy is actually available, rather than just samples from the   
+25 expert, it is reasonable to suspect that sampling random rollouts from the expert policy followed by   
+26 performing behavioral cloning is not the optimally efficient approach for transferring behavior from   
+27 the expert to the student. Once a trajectory has been sampled via an expert rollout, there is actually   
+28 additional information available that can be ascertained in the neighborhood of the trajectory, without   
+29 having to perform an additional rollout, via the local feedback properties of the expert.   
+30 In this work, we refer to this setting, where we want to transfer from an expert policy to a student   
+31 policy, while assuming the expert policy can be queried, as policy cloning. Naturally, there is still   
+32 often an incentive to reduce the total number of rollouts, which may require actually collecting data   
+33 in an unsafe or costly fashion, especially for real-world control problems. As such, there is an aim to   
+34 characterize any efficiency that can be gained in learning from small numbers of rollouts without as   
+35 much concern for how many offline queries are required of the expert policy. If one has primarily   
+36 encountered behavioral cloning in the context of learning from human demonstrations, policy cloning,   
+37 with an available expert policy may seem contrived. However, policy cloning naturally arises in many   
+38 settings. For example, we may have multiple experts that we wish to consolidate into a single neural   
+39 network policy or there may be memory considerations that motivate compressing a large expert   
+40 network into a smaller model with similar behavior. Perhaps most natural are settings in which an   
+41 expert policy is costly or slow to execute, for example due to running a compute intensive procedure   
+42 such as model predictive control (MPC) on specialized hardware (e.g. GPU); in such settings, the   
+43 aim is to transfer expert behavior to a parametric student policy that amortizes the cost. DAGGER is   
+44 one well known approach for efficiently transferring behavior from an expert to a student under these   
+45 kinds of constraints [Ross et al., 2011]. In a separate setting, the expert may be suboptimal and the   
+46 student needs to learn from expert while also being able to exceed the expert performance, perhaps   
+47 by continuing to learn from a task via RL. This problem has been described as kickstarting in one   
+48 incarnation [Schmitt et al., 2018], but also can arise when learning from behavioral priors [Tirumala   
+49 et al., 2020], [Galashov et al., 2019], as also happens, for example, in Distral [Teh et al., 2017].   
+50 To improve data-efficiency in supervised settings generally, including in behavioral cloning settings,   
+51 it is reasonable to consider data augmentation. Data augmentation refers to applying perturbations to   
+52 a finite training dataset to effectively amplify its diversity, usually in the hopes of producing a model   
+53 that is invariant to the class of perturbations performed. For example, in the well studied problem   
+54 of object classification from single images, it is known that applying many kinds of perturbation   
+55 should not affect the object label, so a model can be trained with many input perturbations all yielding   
+56 the same output [Shorten and Khoshgoftaar, 2019]. This setting is fairly representative, with data   
+57 augmentation usually intended to make the model “robust" to nuisance perturbations of the input.   
+58 This class of image-perturbation has also been recently demonstrated to be effective in the context of   
+59 control problems in the offline RL setting [Yarats et al., 2021, Laskin et al., 2020].   
+60 Critically, for control problems it is not the case that the action should be invariant to the input state.   
+61 Or rather, while it does make sense for a control policy to be invariant to certain classes of sensor   
+62 noise, an important class of robustness is that the policy is appropriately feedback-responsive. This is   
+63 to say that for small perturbations of the state of the control system, the optimal action is different   
+64 in precisely the way that the expert implicitly knows. This has been recognized and exploited in   
+65 previous research that has distilled feedback-control plans into controllers [Mordatch and Todorov,   
+66 2014, Mordatch et al., 2015, Merel et al., 2019]. A similar intuition also underlies schemes which   
+67 inject noise into the expert during rollouts to sample more comprehensively the space of how the   
+68 expert recovers from perturbations [Laskey et al., 2017, Merel et al., 2019].   
+69 In this work, we leverage this insight to develop a highly efficient policy cloning approach that   
+70 makes use of both classes of data augmentation. For a high-DoF control problem that operates only   
+71 from state (humanoid run task from DeepMind control suite [Tunyasuvunakool et al., 2020]), we   
+72 demonstrate the feasibility of policy cloning that employs state-based data augmentation with expert   
+73 querying to transfer the feedback-sensitive behavior of the expert in a region around a small number   
+74 of rollouts. Then on a more difficult high-DoF control problem that involves both state-derived and   
+75 egocentric image observations (humanoid running through corrdiors task from DeepMind control   
+76 suite [Tunyasuvunakool et al., 2020]), we combine the state-based expert-aware data augmentation   
+77 with a separate image augmentation intended to induce invariance to image perturbations. Essentially   
+78 our expert-aware data augmentation involves applying random perturbations to the state-derived   
+79 observations, and training the student to match the expert-queried optimal action at each perturbed   
+80 state, thereby gaining considerable knowledge from the expert without performing excessive rollouts   
+81 simply to cover the state space around existing trajectories. Our approach compares favorably to   
+82 sensible baselines, including the naive approach of attempting to perform behavioral cloning with   
+83 state perturbations, which seeks to induce invariance (as proposed in [Laskin et al., 2020]) rather than   
+84 feedback-sensitivity to state-derived observations.   
+85 In the presentation that follows, we will describe the problem setting (Section 2) as well as our   
+86 approach (Section 3), describe the domains we employ and present our initial experiments (Section 4),   
+87 show that our augmented policy cloning approach works well when used as a component of other   
+88 algorithms like DAGGER and kickstarting (Section 5), and finally close with a discussion (Section 6).
+
+# 89 2 Problem description
+
+# 2.1 Expert-driven learning
+
+91 We start by introducing a notion of expert-driven learning that will be used throughout the paper.   
+92 At first, we present a general form of the expert-driven objective and then introduce a few concrete   
+93 examples. We consider a standard Reinforcement Learning (RL) problem. We present the domain as   
+94 an MDP with continuous states for simplicity, however the problem definition is similar for a POMDP   
+95 with observations derived from the state. Formally, we describe the MDP in terms of a continuous state   
+96 space $S \in \mathcal { R } ^ { n }$ (for some $n > 0$ ), an action space $\mathcal { A }$ , transition dynamics $p ( s ^ { \prime } | s , a ) : S \times \mathcal { A }  p ( S )$ ,   
+97 and a reward function $r : \mathcal { S } \times \mathcal { A }  \mathcal { R }$ . Let $\Pi$ be a set of parametric policies, i.e. of mappings   
+98 $\pi _ { \theta } : S  p ( { \mathcal { A } } )$ from the state space $s$ to the probability distributions over actions $\mathcal { A }$ , where $\theta \in \mathcal { R } ^ { m }$   
+99 for some $m > 0$ . For simplicity of the notation, we omit the parameter in front of the policy, i.e.   
+100 $\pi = \pi _ { \theta }$ and optimizing over the set of policies would be equivalent to the optimizing over a set of   
+101 parameters. A reinforcement learning problem consists in finding such a policy $\pi$ that it maximizes   
+102 the expected discounted future reward:
+
+$$
+J ( \pi ) = \mathbb { E } _ { p ( \tau ) } \left[ \sum _ { t } \gamma ^ { t } r ( a _ { t } | s _ { t } ) \right] ,
+$$
+
+103 where $\begin{array} { r } { p ( \tau ) = p ( s _ { 0 } ) \prod _ { t } p ( a _ { t } | s _ { t } ) p ( s _ { t + 1 } | s _ { t } , a _ { t } ) } \end{array}$ is a trajectory distribution. We assume the existence   
+104 of an expert policy $\pi _ { E } ( a | s )$ . This policy could be used to simplify the learning of a new policy on the   
+105 same problem. Formally, we construct a new learning objective which aims to maximize the expected   
+106 reward of the problem in hand as well as to clone the expert policy:
+
+$$
+J ( \pi , \pi _ { E } ) = \alpha J ( \pi ) - \lambda D ( \pi , \pi _ { E } ) ,
+$$
+
+107 where $D$ is some measure of distance of $\pi$ from $\pi _ { E }$ and $\alpha \ge 0 , \lambda \ge 0$ are parameters measuring   
+108 importance of both objectives. In most of the applications, $\alpha \in \{ 0 , 1 \}$ and $\lambda \geq 0$ represents a relative   
+109 importance of cloning an expert policy with respect to the RL objective.
+
+# 2.2 Behavioral cloning (BC)
+
+One important instance of the objective (2) with $\alpha = 0 , \lambda = 1$ is behavioral cloning. In this case, the measure of distance is defined as:
+
+$$
+D _ { B C } ( \pi , \pi _ { E } ) = - \mathbb { E } _ { ( a , s ) \in \mathcal { B } _ { E } } [ \log \pi ( a | s ) ]
+$$
+
+113 Here, $\boldsymbol { B _ { E } } = \{ ( s _ { i } , a _ { i } ) , i = 1 , \ldots , N \}$ , $N > 0$ is a fixed dataset containing expert data. Minimizing   
+114 the objective (3) is be equivalent to maximizing the likelihood of the expert data under the policy $\pi$ .   
+115 The action in eqn. (3) can be replaced by $\pi _ { E } ( s )$ for deterministic policies or by the mean or the mode   
+116 for stochastic policies (e.g., by the mean $\mu _ { E } ( s )$ for Gaussian policies $\pi _ { E } ( \cdot | s ) = \mathcal { N } ( \mu _ { E } ( s ) , \sigma _ { E } ( s ) ) )$ .
+
+# 117 2.3 DAGGER
+
+118 Performance of Behavioral Cloning (BC) can be limited due to the fixed dataset, since the resulting   
+119 policy may fail to generalize to states outside the training distribution. A different approach, known   
+120 in the literature as DAGGER [Ross et al., 2011] was proposed to overcome this limitation. In this   
+121 setting, the expert is queried in states visited by the student, thus reducing distribution shift. In our   
+122 notation, this corresponds to $\alpha = 0$ , $\lambda = 1$ in eqn. (2) and the measure of distance is defined as:
+
+$$
+D _ { \mathrm { D A G G E R } } ( \pi , \pi _ { E } ) = - \mathbb { E } _ { p _ { \beta } ( \tau ) } [ \log \pi ( a _ { t } ^ { \prime } | s _ { t } ) ] ,
+$$
+
+123 where $p _ { \beta } ( \tau ) , \beta \in [ 0 , 1 ]$ is a trajectory distribution where actions are sampled according to the mixture   
+124 policy between a student and an expert:
+
+$$
+p ( a | s ) = \beta \tilde { \pi } ( a | s ) + ( 1 - \beta ) \pi _ { E } ( a | s ) ,
+$$
+
+125 The action $a _ { t } ^ { \prime }$ in eqn. (4) is resampled from the expert policy for the state $s _ { t }$ , i.e., $a _ { t } ^ { \prime } \sim \pi _ { E } ( \cdot | s _ { t } )$   
+126 As in Section 2.2, for stochastic experts this action can be replaced by the mean or mode of the   
+127 distribution in eqn. (4). The policy $\tilde { \pi } ( a | s )$ corresponds to a frozen version of student policy $\pi$ so   
+128 that the gradient $\nabla _ { \pi } D _ { \mathrm { D A G G E R } } ( \pi , \pi _ { E } )$ is not taken with respect to $p ( a | s )$ . Note that even though, in   
+129 eqn. (4) we collect data from the environment, the setting nevertheless corresponds to pure imitation   
+130 learning since expected reward is not directly maximized.   
+132 In eqn. (2), we combine both maximization of expected task reward and minimization of distance to   
+133 the expert. In literature, it is known as Kickstarting [Schmitt et al., 2018]. In this case, in the objective   
+134 from eqn. (2), $\alpha = 1$ , and $\lambda \geq 0$ . As the measure of distance, we use the cross-entropy from expert   
+135 to a student, similarly to [Schmitt et al., 2018]:
+
+$$
+J ( \pi , \pi _ { E } ) = J ( \pi ) - \lambda \mathbb { E } _ { p ( \tau ) } \left[ - \mathbb { E } _ { \pi _ { E } ( a | s ) } \log \pi ( a | s ) \right]
+$$
+
+136 where $p ( \tau )$ is a trajectory distribution, where actions are sampled according to the student policy   
+137 $\pi ( \cdot | s )$ . Usually, in the Kickstarting setting, the expert is sub-optimal and the goal is to train a policy   
+138 that eventually outperforms the expert. Thus, it is customary to reduce $\lambda$ over the course of training.   
+139 Yet, for simplicity, in our experiments we keep this coefficient fixed.
+
+# 40 3 Augmented policy cloning
+
+141 The previous section has demonstrated that the objective corresponding to the cloning behavior from   
+142 the parametric expert policy could arise in multiple scenarios. In this section we propose a new and   
+143 simple method which can significantly improve the data efficiency of the approaches described in   
+144 Section 2. We explain the basic idea for BC, but its generalization to other expert-driven learning   
+145 approaches described in Section 2 is straightforward. In Section 5 we show results for these problems.   
+146 When optimizing the objective (3), for every state $s \in \mathcal { D } _ { E }$ from the expert trajectories dataset, we   
+147 consider a small Gaussian state perturbation:
+
+$$
+\delta s \sim \mathcal N ( 0 , \sigma _ { s } ^ { 2 } )
+$$
+
+148 which produces a new virtual state:
+
+$$
+\boldsymbol { s } ^ { \prime } = \boldsymbol { s } + \delta \boldsymbol { s }
+$$
+
+149 Then, for this state we query the expert and obtain a new action
+
+$$
+a ^ { \prime } \sim \pi _ { E } ( \cdot | s + \delta s )
+$$
+
+150 We then augment the dataset $\mathcal { D } _ { E }$ with these new pairs of virtual states and actions. More explicitly   
+151 the idea can be expressed in terms of the following objective:
+
+$$
+\begin{array} { r } { D ( \pi , \pi _ { E } ) _ { A P C } = \mathbb { E } _ { ( a , s ) \in B _ { E } } [ \log \pi ( a | s ) + \mathbb { E } _ { \delta s \sim \mathcal { N } ( 0 , \sigma _ { s } ^ { 2 } ) , a ^ { \prime } \sim \pi _ { E } ( \cdot | s + \delta s ) } \log \pi ( a ^ { \prime } | s + \delta s ) ] } \end{array}
+$$
+
+152 We call this approach Augmented Policy Cloning (APC) as it queries the expert policy to augment   
+153 the training data. This approach is different from a naive data-augmentation technique, where a new   
+154 state would be generated, but associated with the original action (and not a new one). It therefore   
+155 allows to build policies which are feedback-responsive with respect to the expert. We formulate APC   
+156 algorithm for BC in Algorithm 1.
+
+# 157 4 Core Results: Evaluation of Augmented Policy Cloning
+
+# 4.1 Domains
+
+159 To study how our method performs on complex control domains, we consider two complex, high-DoF   
+160 continuous control tasks involving control of a physically simulated humanoid body. Both domains   
+161 are implemented using the MuJoCo physics engine [Todorov et al., 2012] and are available in the   
+162 dm_control repository [Tunyasuvunakool et al., 2020]. The first task is the standard control suite   
+163 Run task, where the Humanoid body needs to run at a target speed and observations are based on   
+164 proprioception. The second task is the Walls task which requires the same Humanoid body to run   
+165 along a corridor and avoid walls, using both proprioception and egocentric vision as observations.   
+166 Both of these problems are rather challenging insofar as they require stabilization and locomotion   
+167 control of a relatively complex humanoid body with 21 actuated DoFs, in one case using vision to   
+168 guide the movement. Note these environments are related to the domains that have been proposed   
+169 for use in offline RL benchmarks [Gulcehre et al., 2020]; however, the experiments we perform in   
+170 this work require availability of the expert policy, so we do not use offline data, but instead train   
+171 new experts and perform experiments in the very low data regime. For more details, please refer to   
+172 Section 1.1 in Supplementary Material.
+
+Parametric student policy: $\pi _ { \theta }$   
+Initial parameters: $\theta _ { 0 }$   
+expert policy: $\pi _ { E }$   
+Dataset $\boldsymbol { B _ { E } } = \{ ( s _ { i } , a _ { i } ) , i = 1 , \ldots , N \}$ , $N > 0$ of expert state-action pairs   
+State perturbation noise $\sigma _ { s }$   
+Learning rate $\alpha$   
+Number of augmented samples: $M$   
+Number of gradient updates: $K$   
+Size of a batch: $L$   
+for $\mathbf { k } { = } 1 , \ldots , \mathbf { K }$ do Sample a batch of pairs $\{ ( a _ { i } , s _ { i } ) \} _ { i = 1 } ^ { L } \sim B _ { E }$ For each state $s _ { i }$ , sample $M$ perturbations $\delta s _ { j } \sim \mathcal { N } ( 0 , \sigma _ { s } ) , j = 1 , \ldots , M$ Construct $M$ virtual states $s _ { i , j } ^ { \prime } = s _ { i } + \delta s _ { j } , i = 1 , \dots , L , j = 1 , \dots , M$ Resample new actions from expert $a _ { i , j } ^ { \prime } \sim \pi _ { E } ( \cdot | s _ { i , j } ^ { \prime } )$ For Gaussian experts, the action $a _ { i } = \mu _ { E } ( s _ { i } )$ and the new actions are $a _ { i , j } ^ { \prime } = \mu _ { E } ( s _ { i , j } ^ { \prime } )$ Compute the empirical negative log-likelihood: $\begin{array} { r } { \mathcal { L } = - \left[ \log \pi _ { \theta _ { k } } ^ { - } ( a _ { i } | s _ { i } ) + \frac { 1 } { M } \sum _ { j = 1 } ^ { M } \log \pi _ { \theta _ { k } } ( a _ { i , j } ^ { \prime } | s _ { i , j } ^ { \prime } ) \right] } \end{array}$ Update the parameters $\theta _ { k + 1 } = \theta _ { k } - \alpha \nabla _ { \theta } \mathcal { L }$   
+end for
+
+173 For each task, we train expert policies to convergence using the MPO algorithm Abdolmaleki et al.   
+174 [2018]. Since the expert policy essentially saturates task performance, for each task, we keep three   
+175 partially trained experts such that we can assess the ability of the kickstarting approach to outperform   
+176 sub-optimal experts. We refer to the different experts as Low, achieving approximatively $25 \%$ of the   
+177 optimal policy reward, Medium, achieving around $50 \%$ of the performance and High, corresponding   
+178 to the converged policy. Each expert is represented by a Gaussian policy. For more details, please   
+179 refer to Section 1.2 in Supplementary Material.
+
+# 80 4.2 Applying Augmented Policy Cloning
+
+81 First, we evaluate the performance of APC in fitting a fixed dataset of expert trajectories. In order   
+82 to study the data efficiency of the method, we construct datasets containing different numbers of   
+83 expert trajectories. The expert policies are represented by conditionally Gaussian distributions,   
+84 i.e. $\pi _ { E } ( \cdot | \bar { s } ) = \mathcal { N } ( \mu _ { E } ( s ) , \bar { \sigma ( s ) } )$ . Thus, to assess the robustness of our method to expert noise we   
+85 produce trajectories using the experts’ mean but adding different levels of (homoscedastic) zero-mean   
+86 Gaussian noise $\sigma _ { E }$ .
+
+$$
+\displaystyle a \sim \mathcal { N } ( \mu _ { E } ( s ) , \sigma _ { E } )
+$$
+
+187 Note that in addition to policy noise $\sigma _ { E }$ which is introduced when sampling trajectories, initial pose   
+188 and environment layout (for the Walls task) are also sampled randomly for each episode. We consider   
+189 4 levels of expert policy noise: Deterministic, which uses the Gaussian mean for the action, Low,   
+190 with $\sigma _ { E } = 0 . 2$ , Medium $\sigma _ { E } = 0 . 5$ and High $\sigma _ { E } = 1 . 0$ .   
+191 For the APC method, we rely on Algorithm 1. For baselines, we consider BC algorithm from eqn. (3)   
+192 as well as a simple modification of BC, where we apply, similar to APC, state perturbations as in   
+193 eqn. (7) and eqn. (8), but we do not produce a new action from the expert. We call this approach   
+194 Naive Augmented Behavior Cloning (Naive ABC) which essentially corresponds to robustification   
+195 of the student policies with respect to state perturbation and is similar in spirit to standard data  
+196 augmentation approaches. For vision-based tasks, we consider random crop augmentations of size   
+197 $4 8 \mathbf { x } 4 8$ (downsampled from the input image of 64x64), similar to Laskin et al. [2020]. When the   
+198 image augmentations are used we add "with image" to the method name. On top of that, we consider   
+199 a variant, where only image augmentation is used, which we call Naive ABC (image only). For   
+200 all methods, as an action in the objective from eqn. (3), we use an expert mean $\mu _ { E } ( s )$ . We train   
+201 all approaches to convergence (300K learning iterations on Walls and 13M learning iterations on   
+202 Run). Each learning iteration corresponds applying gradients to 64 trajectories, each containing   
+203 10 time steps. After each learning iteration, we evaluate the policy on both a validation set (50   
+204 random instances of the environment) and a test set (150 random instances of the environments).   
+205 We apply early-stopping based on the validation set performance to select the best model and   
+206 report corresponding performance on the test set. For more details, please refer to Section 1.3 in   
+207 Supplementary Material. As an additional evaluation, we test robustness of the obtained policies to a   
+208 fixed amount of noise during execution. For a learned student policy $\pi ( \cdot | s ) = \mathcal { N } ( \mu ( s ) , \sigma ( s ) )$ , we   
+209 evaluate it by executing an action:
+
+![](images/6ba5e847a627b5cd0f9ec953ef682d44960299536be320f0eb5349ad60ca4536.jpg)  
+Figure 1: Behavioral cloning results on Run and Walls tasks (represented by rows). The X-axis represents the number of trajectories, whereas the Y-axis corresponds to the episodic reward averaged among 150 independent evaluations. The highest point of the bar corresponds to the mean, whereas the dashed lines indicate the standard deviation. The pink dashed line indicate average expert performance. The legend describes a method which is used. On the plot on the left depicts a standard BC experiment, where dataset contains a specified number of full trajectories from the expert. The plot on the right illustrates the experiment, where a dataset contains 1 full trajectories and the rest are the short ones, containing only 200 timesteps each.
+
+$$
+a \sim { \mathcal { N } } ( \mu ( s ) , \sigma ) ,
+$$
+
+210 where $\sigma$ is the fixed amount of student noise. We consider similar noise magnitudes as for the   
+211 expert. For APC and Naive ABC, we sweep over state perturbation noise levels and choose the ones   
+212 performing the best on the validation set. For APC, we use $\sigma _ { s } = 0 . 1$ for Run and $\sigma _ { s } = 1 . 0$ for Walls.   
+213 For Naive ABC, we use $\sigma _ { s } = 0 . 0 0 1$ for Run and $\sigma _ { s } = 0 . 0 1$ for Walls. The ablation experiments   
+214 over noise levels for APC and Naive ABC are presented in Section 2 of the Supplementary Material   
+215 (Figure 1 and Figure 2).   
+216 The first of results, in Figure 1 (left) demonstrates the increased data efficiency of APC over BC and   
+217 Naive ABC in terms of number of trajectories. The noise level of the expert and student are fixed to   
+218 Low for the ease of comparison. We also see that Naive ABC performs similarly to BC. To further   
+219 push the limits of data efficiency, we conducted a variant where a dataset contains only 1 full trajectory   
+220 (1000 timesteps for Run and around $2 \mathrm { k }$ timesteps for Walls) along with multiple short trajectories   
+221 (200 time-step only). This dramatically reduces the amount of expert data available to learn from.   
+222 However, we hypothesise that in the environments considered, much of the diversity of the trajectories   
+223 arises due to initial state variation. This setting might arise in domains where execution is costly,   
+224 such as robotics applications. In such setting we might have a few longer trajectories along with a   
+225 patchwork of shorter trajectories covering more diverse parts of the state space. The results for this   
+226 experiment are given in figure 1 (right). Again, we see that APC is significantly more efficient than   
+227 BC and Naive ABC. Interesting to note that APC picks up quite a high performance after observing   
+228 10 (1 full and 9 short) trajectories for Run task and 100 (1 full and 99 short) trajectories for the Walls   
+29 task. In Section 3 in Supplementary material, we provide additional results for Walls task when we   
+230 use image-based perturbations.
+
+In the next experiment, in order to understand how robust our method to noise, we study the impact of different levels of student and expert noises on performance. For each run, we use a dataset of 100 trajectories. The results are given in figure 2, where each column corresponds to a different level of expert noise, and the X-axis represents different levels of student noise. At first, we observe that APC is consistently more robust than BC and Naive ABC for any level of expert and student noise. On top of that, we can notice that for any fixed level of expert noise, the performance degrades when a student noise increases. Finally, we see that for higher noise levels of expert, the learned student performs better in the high noise regime. It is consistent with the intuition - training on noisy trajectories leads to a more robust policy. Overall, APC leads the most robust policy.
+
+![](images/55922519d083cd5acd37c2639ce28266b4d4cbf46504ddb38df60d0a847fc02f.jpg)  
+Figure 2: Noise sensitivity results. We consider 4 levels of noise for student and expert: Deterministic, which uses the Gaussian mean for the action, Low, is the noise $\sigma = 0 . 2$ , Medium $\sigma = 0 . 5$ and High $\sigma = 1 . 0$ . Each column corresponds to a different level of expert noise. X-axis corresponds to a different level of student noise. Y-axis corresponds to the episodic reward averaged among 150 independent evaluations. The highest point of the bar corresponds to the mean, whereas the dashed lines indicate the standard deviation. The legend denotes a method and a row corresponds to a task. The pink dashed line indicate average expert performance.
+
+# 240 5 Additional Results: Augmented Policy Cloning as a subroutine
+
+# 241 5.1 DAGGER with data augmentation
+
+![](images/2496941df8315f2f080842d8e7e0f351aad02ffd4ab30021f8f43e0567433480.jpg)  
+Figure 3: DAGGER results. On the $\mathrm { X }$ -axis we report the number of environment steps. On the Y-axis we report averaged across 3 seeds episodic reward achieved by the student. We report confidence intervals in the shaded areas. For Run task, the confidence intervals are very small and are not visible. In solid line we report the performance without using expert policy during the acting. In dashed line, we report the performance of the policy which mixes $30 \%$ with the expert. All the methods use mean action during evaluation.
+
+242 As described in Section 2.3, DAGGER [Ross et al., 2011] is a more sophisticated approach where   
+243 data is collected from the real environment by executing a policy from eqn. (5), which is a mixture   
+244 between a student and an expert. In this section we study how data augmentation approaches affect   
+245 the data efficiency of the DAGGER algorithm.   
+246 We consider similar baselines for both tasks as in the previous section. For an expert policy that has   
+247 been pre-trained via MPO [Abdolmaleki et al., 2018], we perform online rollouts for two values of   
+248 the expert-student mixing coefficient, $\beta = 0$ and $\beta = 0 . 3$ (see eqn. 5). Since both student and expert   
+249 are Gaussian distributions, instead of using a $\log \pi$ in eqn. (4), we could use a state-conditional cross   
+250 entropy from an expert to a student, $\mathcal { H } [ \pi _ { E } ( \cdot | s ) | | \pi ( \cdot | s ) ]$ . Empirically, we found that it worked better   
+251 than using $\log \pi$ . We demonstrate a comparison in Section 4 in Supplementary Material. We run   
+252 the experiments in a data-restricted setup such that for every collected trajectory (10 time-steps), we   
+253 apply 10 gradient steps, using a replay-buffer to store the past experience. Additional experimental   
+254 details are given in Section 1.4 in Supplementary Material. Results are shown in Figure 3. We see   
+255 that APC and its vision variant outperform BC and Naive ABC similarly to the behavior cloning   
+256 experiments. While we observe that image augmentation can help, we see that the primary advantage   
+257 comes from the state-based augmentation for APC. For the Run task, we observe that all DAGGER   
+258 methods achieve slightly lower performance than an expert policy. We speculate that this is due to   
+259 insufficient coverage of the state space during training.
+
+# 260 5.2 Kickstarting with data augmentation
+
+![](images/de92604cbf28af517d0ab8201ecd8de84edebc9a85dfb463a974c61abfee81e9.jpg)  
+Figure 4: Kickstarting results. On the X-axis we show the number of environment steps. On the Y-axis we report averaged across 3 seeds episodic reward achieved by the student. We report confidence intervals in the shaded areas. For Run task, the confidence intervals are very small and are not visible. Each row indicates a task, whereas a column corresponds to the expert type. Dashed black line shows the expert performance.
+
+261 A similar in spirit approach is kickstarting Schmitt et al. [2018], where we solve an RL task as well   
+262 as cloning the expert policy. Similarly to previous section, we apply APC in kickstarting on the   
+263 cross entropy term in eqn. (6). We use 3 types of expert policy as described in Section 4.1. We run   
+264 experiments using a distributed setup with 64 acting policies and 1 learner, querying the batches   
+265 of trajectories (of size 10) from a replay buffer. On top of running BC methods, we also report the   
+266 performance of MPO Abdolmaleki et al. [2018] learning from scratch on the task of interest. All   
+267 details are given in Section 1.5 in Supplementary Material. The results are given in figure 4.
+
+268 We observe that APC performs better than Naive ABC on Run task and similarly on Walls task. Both   
+269 approaches perform better than BC and learning from scratch. We hypothesise that the reason of not   
+270 seeing a consistent advantage could be due to two factors. As we are in a high-data regime, since   
+271 there is no limit on relative acting / learning ratio, and acting policies are not restricted to collect   
+272 trajectories, it is unclear whether data-augmentation should help. In addition, we use reward signal   
+273 which makes the impact of expert cloning less important. Note that the resulting agent is less data   
+274 efficient in these experiments; this is because we do not control the relative ratio between acting   
+275 and learning (i.e., no rate-limiting on the learner, due to instability of kickstarting experiments when   
+276 rate-limiting was explored). Furthermore, unlike in kickstarting Schmitt et al. [2018], we do not use an annealing schedule of $\lambda$ to make the experiments simpler, but we still observe that a fixed coefficient helps to kickstart an experiment and outperform an expert policy. On top of that, we see that image-based augmentation have less of impact in this setting.
+
+# 280 6 Discussion
+
+Many expert-driven learning approaches actually have access to an expert that can be queried; however, this opportunity is rarely exploited fully. In this work we demonstrated a general scheme for more efficient transfer of expert behavior by augmenting expert trajectory data with virtual, perturbed states as well as the expert actions in these virtual states. This data augmentation technique is widely applicable and we demonstrated that it improves data efficiency when used in place of behavioral cloning both in the offline setting or when behavioral cloning is used as a step within DAGGER or kickstarting.
+
+Critically, data efficiency is generally very important in realistic applications, where new data acquisition cost could be high. In particular, settings involving deployment of policies in the real world, such as robotics applications, may benefit from an ability to efficiently transfer expert policy behavior from one neural network to another (for compression or execution speed reasons), or to combine behavior from multiple experts into a single neural network. While overall, we consider the present work to be fairly basic research with limited ethical impact, insofar as our approach decreases the amount of data which needs to be collected through processes which could potentially be unsafe or costly, there is a potential positive social value.
+
+The limitations of our approach consist in the reliance on the ability to query expert policy for the perturbed states which reduces the amount of applications where the method could be used. Another limitation is the reliance on the continuous state spaces. In discrete state spaces, it is unclear whether a small perturbation in state would result in a valid action from an expert.
+
+In future work, we plan to explore how our proposed augmentation technique can be leveraged in the context of KL-regularized RL with behavior priors.
+
+# References
+
+Abbas Abdolmaleki, Jost Tobias Springenberg, Yuval Tassa, Remi Munos, Nicolas Heess, and Martin Riedmiller. Maximum a posteriori policy optimisation, 2018.
+
+Alexandre Galashov, Siddhant M. Jayakumar, Leonard Hasenclever, Dhruva Tirumala, Jonathan Schwarz, Guillaume Desjardins, Wojciech M. Czarnecki, Yee Whye Teh, Razvan Pascanu, and Nicolas Heess. Information asymmetry in kl-regularized rl, 2019.
+
+Caglar Gulcehre, Ziyu Wang, Alexander Novikov, Thomas Paine, Sergio Gómez, Konrad Zolna, Rishabh Agarwal, Josh S Merel, Daniel J Mankowitz, Cosmin Paduraru, et al. Rl unplugged: A collection of benchmarks for offline reinforcement learning. Advances in Neural Information Processing Systems, 33, 2020.
+
+Michael Laskey, Jonathan Lee, Roy Fox, Anca Dragan, and Ken Goldberg. Dart: Noise injection for robust imitation learning. In Conference on robot learning, pages 143–156. PMLR, 2017.
+
+314 Michael Laskin, Kimin Lee, Adam Stooke, Lerrel Pinto, Pieter Abbeel, and Aravind Srinivas. Reinforcement learning with augmented data, 2020.
+
+Josh Merel, Leonard Hasenclever, Alexandre Galashov, Arun Ahuja, Vu Pham, Greg Wayne, Yee Whye Teh, and Nicolas Heess. Neural probabilistic motor primitives for humanoid control, 2019.   
+Donald Michie and Claude Sammut. Behavioural Clones and Cognitive Skill Models, page 387–395. Oxford University Press, Inc., USA, 1996. ISBN 019853860X.   
+Igor Mordatch and Emo Todorov. Combining the benefits of function approximation and trajectory optimization. In Robotics: Science and Systems, volume 4, 2014.   
+Igor Mordatch, Kendall Lowrey, Galen Andrew, Zoran Popovic, and Emanuel V Todorov. Interactive control of diverse complex characters with neural networks. Advances in Neural Information Processing Systems, 28:3132–3140, 2015.   
+Dean A Pomerleau. Alvinn: An autonomous land vehicle in a neural network. Technical report, Carnegie-Mellon, 1989.   
+Stephane Ross, Geoffrey J. Gordon, and J. Andrew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning, 2011.   
+Simon Schmitt, Jonathan J. Hudson, Augustin Zidek, Simon Osindero, Carl Doersch, Wojciech M. Czarnecki, Joel Z. Leibo, Heinrich Kuttler, Andrew Zisserman, Karen Simonyan, and S. M. Ali Eslami. Kickstarting deep reinforcement learning, 2018.   
+Connor Shorten and Taghi Khoshgoftaar. A survey on image data augmentation for deep learning. Journal of Big Data, 6, 07 2019. doi: 10.1186/s40537-019-0197-0.   
+Yee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. arXiv preprint arXiv:1707.04175, 2017.   
+Dhruva Tirumala, Alexandre Galashov, Hyeonwoo Noh, Leonard Hasenclever, Razvan Pascanu, Jonathan Schwarz, Guillaume Desjardins, Wojciech Marian Czarnecki, Arun Ahuja, Yee Whye Teh, et al. Behavior priors for efficient reinforcement learning. arXiv preprint arXiv:2010.14274, 2020.   
+Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026–5033. IEEE, 2012.   
+Saran Tunyasuvunakool, Alistair Muldal, Yotam Doron, Siqi Liu, Steven Bohez, Josh Merel, Tom Erez, Timothy Lillicrap, Nicolas Heess, and Yuval Tassa. dm_control: Software and tasks for continuous control. Software Impacts, 6:100022, 2020.   
+Denis Yarats, Ilya Kostrikov, and Rob Fergus. Image augmentation is all you need: Regularizing deep reinforcement learning from pixels. In 9th International Conference on Learning Representations, ICLR, volume 2021, 2021.
+
+# 51 7 NeuRIPS 2021 checklist
+
+• Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Yes, we believe that the claims in the abstract and introduction are accurately reflected by the paper’s contributions and scope.   
+• Have you read the ethics review guidelines and ensured that your paper conforms to them? Yes.   
+• Did you discuss any potential negative societal impacts of your work? Yes. We included this in the Discussion section.   
+• Did you describe the limitations of your work? Yes, we discuss these in the "Discussion" section.   
+• Did you state the full set of assumptions of all theoretical results Not applicable.   
+• Did you include complete proofs of all theoretical results Not applicable.   
+• Did you include the code, data, and instructions needed to reproduce the main experimental results ? We included complete details in the supplementary material which could be used to reproduce the experimental results.   
+• Did you specify all the training details? Yes, in the main paper and supplementary material.   
+• Did you report error bars? We did. For every plot we reported the mean and standard deviations averaged across a number of independent random evaluations.   
+• Did you include the amount of compute and the type of resources used? Yes, we did, in the supplementary material.   
+• If your work uses existing assests, did you cite the creators? Yes. We use DeepMind control suite tasks and we cite the appropriate publications. Did you mention the license of the assets? We did not explicitly mention the licence as it is clear from the reference. Did you include any new assets either in the supplementary material os as a URL? No. Did you discuss whether and how consent was obtained from people whose data you’re using / curating? Since the assets and references are in the public access, no consent was required. Did you discuss whether the data you are using / curating contains personally identifiable information or offensive content? The assets which we used did not include any personally identifiable information nor offensive content.   
+• Did you include the full text of instructions given to participants and screenshots, if applicable? Not applicable.   
+Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable Not applicable.   
+• Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? Not applicable.

parse/train/MdZPf3qCF7s/MdZPf3qCF7s_content_list.json

@@ -0,0 +1,994 @@
+[
+    {
+        "type": "text",
+        "text": "Data augmentation for efficient learning from parametric experts ",
+        "text_level": 1,
+        "bbox": [
+            223,
+            122,
+            774,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            580,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 We present a simple, yet powerful data-augmentation technique to enable data  \n2 efficient learning from parametric experts. Whereas behavioral cloning refers to   \n3 learning from samples of an expert, we focus here on what we refer to as the policy   \n4 cloning setting which allows for offline queries of an expert or expert policy. This   \n5 setting arises naturally in a number of problems, especially as a component of   \n6 other algorithms. We achieve a very high level of data efficiency in transferring   \n7 behavior from an expert to a student policy for high Degrees of Freedom (DoF)   \n8 control problems using our augmented policy cloning (APC) approach, which   \n9 combines conventional image-based data augmentation to build invariance to   \n10 image perturbations with an expert-aware offline data augmentation approach that   \n11 induces appropriate feedback-sensitivity in a region around expert trajectories. We   \n12 show that our method increases data-efficiency of policy cloning, enabling transfer   \n13 of complex high-DoF behaviors from just a few trajectories, and we also show   \n14 benefits of our approach in the context of algorithms in which policy cloning is a   \n15 constituent part. ",
+        "bbox": [
+            148,
+            349,
+            766,
+            558
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "16 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            584,
+            312,
+            602
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "17 In various control and reinforcement learning settings, there is a need to transfer behavior from an   \n18 expert policy to a student policy. Broadly, when only samples from the expert policy are available, the   \n19 standard approach is to employ a version of regression from states to actions. This class of approaches   \n20 for producing a policy is known as behavioral cloning [Pomerleau, 1989, Michie and Sammut, 1996].   \n21 Behavioral cloning is quite flexible and supports the setting where the expert trajectories come from a   \n22 human teleoperating the relevant system directly, as well as various settings where the trajectories are   \n23 sampled from other controllers, which themselves may have been trained or scripted. However, for   \n24 any of the settings where the expert policy is actually available, rather than just samples from the   \n25 expert, it is reasonable to suspect that sampling random rollouts from the expert policy followed by   \n26 performing behavioral cloning is not the optimally efficient approach for transferring behavior from   \n27 the expert to the student. Once a trajectory has been sampled via an expert rollout, there is actually   \n28 additional information available that can be ascertained in the neighborhood of the trajectory, without   \n29 having to perform an additional rollout, via the local feedback properties of the expert.   \n30 In this work, we refer to this setting, where we want to transfer from an expert policy to a student   \n31 policy, while assuming the expert policy can be queried, as policy cloning. Naturally, there is still   \n32 often an incentive to reduce the total number of rollouts, which may require actually collecting data   \n33 in an unsafe or costly fashion, especially for real-world control problems. As such, there is an aim to   \n34 characterize any efficiency that can be gained in learning from small numbers of rollouts without as   \n35 much concern for how many offline queries are required of the expert policy. If one has primarily   \n36 encountered behavioral cloning in the context of learning from human demonstrations, policy cloning,   \n37 with an available expert policy may seem contrived. However, policy cloning naturally arises in many   \n38 settings. For example, we may have multiple experts that we wish to consolidate into a single neural   \n39 network policy or there may be memory considerations that motivate compressing a large expert   \n40 network into a smaller model with similar behavior. Perhaps most natural are settings in which an   \n41 expert policy is costly or slow to execute, for example due to running a compute intensive procedure   \n42 such as model predictive control (MPC) on specialized hardware (e.g. GPU); in such settings, the   \n43 aim is to transfer expert behavior to a parametric student policy that amortizes the cost. DAGGER is   \n44 one well known approach for efficiently transferring behavior from an expert to a student under these   \n45 kinds of constraints [Ross et al., 2011]. In a separate setting, the expert may be suboptimal and the   \n46 student needs to learn from expert while also being able to exceed the expert performance, perhaps   \n47 by continuing to learn from a task via RL. This problem has been described as kickstarting in one   \n48 incarnation [Schmitt et al., 2018], but also can arise when learning from behavioral priors [Tirumala   \n49 et al., 2020], [Galashov et al., 2019], as also happens, for example, in Distral [Teh et al., 2017].   \n50 To improve data-efficiency in supervised settings generally, including in behavioral cloning settings,   \n51 it is reasonable to consider data augmentation. Data augmentation refers to applying perturbations to   \n52 a finite training dataset to effectively amplify its diversity, usually in the hopes of producing a model   \n53 that is invariant to the class of perturbations performed. For example, in the well studied problem   \n54 of object classification from single images, it is known that applying many kinds of perturbation   \n55 should not affect the object label, so a model can be trained with many input perturbations all yielding   \n56 the same output [Shorten and Khoshgoftaar, 2019]. This setting is fairly representative, with data   \n57 augmentation usually intended to make the model “robust\" to nuisance perturbations of the input.   \n58 This class of image-perturbation has also been recently demonstrated to be effective in the context of   \n59 control problems in the offline RL setting [Yarats et al., 2021, Laskin et al., 2020].   \n60 Critically, for control problems it is not the case that the action should be invariant to the input state.   \n61 Or rather, while it does make sense for a control policy to be invariant to certain classes of sensor   \n62 noise, an important class of robustness is that the policy is appropriately feedback-responsive. This is   \n63 to say that for small perturbations of the state of the control system, the optimal action is different   \n64 in precisely the way that the expert implicitly knows. This has been recognized and exploited in   \n65 previous research that has distilled feedback-control plans into controllers [Mordatch and Todorov,   \n66 2014, Mordatch et al., 2015, Merel et al., 2019]. A similar intuition also underlies schemes which   \n67 inject noise into the expert during rollouts to sample more comprehensively the space of how the   \n68 expert recovers from perturbations [Laskey et al., 2017, Merel et al., 2019].   \n69 In this work, we leverage this insight to develop a highly efficient policy cloning approach that   \n70 makes use of both classes of data augmentation. For a high-DoF control problem that operates only   \n71 from state (humanoid run task from DeepMind control suite [Tunyasuvunakool et al., 2020]), we   \n72 demonstrate the feasibility of policy cloning that employs state-based data augmentation with expert   \n73 querying to transfer the feedback-sensitive behavior of the expert in a region around a small number   \n74 of rollouts. Then on a more difficult high-DoF control problem that involves both state-derived and   \n75 egocentric image observations (humanoid running through corrdiors task from DeepMind control   \n76 suite [Tunyasuvunakool et al., 2020]), we combine the state-based expert-aware data augmentation   \n77 with a separate image augmentation intended to induce invariance to image perturbations. Essentially   \n78 our expert-aware data augmentation involves applying random perturbations to the state-derived   \n79 observations, and training the student to match the expert-queried optimal action at each perturbed   \n80 state, thereby gaining considerable knowledge from the expert without performing excessive rollouts   \n81 simply to cover the state space around existing trajectories. Our approach compares favorably to   \n82 sensible baselines, including the naive approach of attempting to perform behavioral cloning with   \n83 state perturbations, which seeks to induce invariance (as proposed in [Laskin et al., 2020]) rather than   \n84 feedback-sensitivity to state-derived observations.   \n85 In the presentation that follows, we will describe the problem setting (Section 2) as well as our   \n86 approach (Section 3), describe the domains we employ and present our initial experiments (Section 4),   \n87 show that our augmented policy cloning approach works well when used as a component of other   \n88 algorithms like DAGGER and kickstarting (Section 5), and finally close with a discussion (Section 6). ",
+        "bbox": [
+            145,
+            617,
+            825,
+            796
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            803,
+            825,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            92,
+            825,
+            270
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            277,
+            825,
+            416
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            421,
+            825,
+            546
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            554,
+            825,
+            773
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            780,
+            825,
+            835
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "89 2 Problem description ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            89,
+            375,
+            107
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Expert-driven learning ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            119,
+            374,
+            135
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "91 We start by introducing a notion of expert-driven learning that will be used throughout the paper.   \n92 At first, we present a general form of the expert-driven objective and then introduce a few concrete   \n93 examples. We consider a standard Reinforcement Learning (RL) problem. We present the domain as   \n94 an MDP with continuous states for simplicity, however the problem definition is similar for a POMDP   \n95 with observations derived from the state. Formally, we describe the MDP in terms of a continuous state   \n96 space $S \\in \\mathcal { R } ^ { n }$ (for some $n > 0$ ), an action space $\\mathcal { A }$ , transition dynamics $p ( s ^ { \\prime } | s , a ) : S \\times \\mathcal { A }  p ( S )$ ,   \n97 and a reward function $r : \\mathcal { S } \\times \\mathcal { A }  \\mathcal { R }$ . Let $\\Pi$ be a set of parametric policies, i.e. of mappings   \n98 $\\pi _ { \\theta } : S  p ( { \\mathcal { A } } )$ from the state space $s$ to the probability distributions over actions $\\mathcal { A }$ , where $\\theta \\in \\mathcal { R } ^ { m }$   \n99 for some $m > 0$ . For simplicity of the notation, we omit the parameter in front of the policy, i.e.   \n100 $\\pi = \\pi _ { \\theta }$ and optimizing over the set of policies would be equivalent to the optimizing over a set of   \n101 parameters. A reinforcement learning problem consists in finding such a policy $\\pi$ that it maximizes   \n102 the expected discounted future reward: ",
+        "bbox": [
+            142,
+            145,
+            826,
+            311
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffd2940e2c8cdba6cb857074d5bf8571b21fc9fb334d93625c02b964dbec1de3.jpg",
+        "text": "$$\nJ ( \\pi ) = \\mathbb { E } _ { p ( \\tau ) } \\left[ \\sum _ { t } \\gamma ^ { t } r ( a _ { t } | s _ { t } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            314,
+            606,
+            358
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "103 where $\\begin{array} { r } { p ( \\tau ) = p ( s _ { 0 } ) \\prod _ { t } p ( a _ { t } | s _ { t } ) p ( s _ { t + 1 } | s _ { t } , a _ { t } ) } \\end{array}$ is a trajectory distribution. We assume the existence   \n104 of an expert policy $\\pi _ { E } ( a | s )$ . This policy could be used to simplify the learning of a new policy on the   \n105 same problem. Formally, we construct a new learning objective which aims to maximize the expected   \n106 reward of the problem in hand as well as to clone the expert policy: ",
+        "bbox": [
+            140,
+            361,
+            825,
+            417
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b954363af042f0f644d4a64768b811c1ae0f0d6ef19d57f6a6d9ecc43651eef.jpg",
+        "text": "$$\nJ ( \\pi , \\pi _ { E } ) = \\alpha J ( \\pi ) - \\lambda D ( \\pi , \\pi _ { E } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            421,
+            611,
+            439
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "107 where $D$ is some measure of distance of $\\pi$ from $\\pi _ { E }$ and $\\alpha \\ge 0 , \\lambda \\ge 0$ are parameters measuring   \n108 importance of both objectives. In most of the applications, $\\alpha \\in \\{ 0 , 1 \\}$ and $\\lambda \\geq 0$ represents a relative   \n109 importance of cloning an expert policy with respect to the RL objective. ",
+        "bbox": [
+            142,
+            443,
+            825,
+            486
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Behavioral cloning (BC) ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            500,
+            380,
+            515
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One important instance of the objective (2) with $\\alpha = 0 , \\lambda = 1$ is behavioral cloning. In this case, the measure of distance is defined as: ",
+        "bbox": [
+            166,
+            525,
+            823,
+            554
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1f804cdc3d12257fc66ccf1481ffac9b898d0848dc2cf84d6938c1e8d0acaf2.jpg",
+        "text": "$$\nD _ { B C } ( \\pi , \\pi _ { E } ) = - \\mathbb { E } _ { ( a , s ) \\in \\mathcal { B } _ { E } } [ \\log \\pi ( a | s ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            558,
+            632,
+            577
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "113 Here, $\\boldsymbol { B _ { E } } = \\{ ( s _ { i } , a _ { i } ) , i = 1 , \\ldots , N \\}$ , $N > 0$ is a fixed dataset containing expert data. Minimizing   \n114 the objective (3) is be equivalent to maximizing the likelihood of the expert data under the policy $\\pi$ .   \n115 The action in eqn. (3) can be replaced by $\\pi _ { E } ( s )$ for deterministic policies or by the mean or the mode   \n116 for stochastic policies (e.g., by the mean $\\mu _ { E } ( s )$ for Gaussian policies $\\pi _ { E } ( \\cdot | s ) = \\mathcal { N } ( \\mu _ { E } ( s ) , \\sigma _ { E } ( s ) ) )$ . ",
+        "bbox": [
+            142,
+            580,
+            826,
+            637
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "117 2.3 DAGGER ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            651,
+            279,
+            666
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "118 Performance of Behavioral Cloning (BC) can be limited due to the fixed dataset, since the resulting   \n119 policy may fail to generalize to states outside the training distribution. A different approach, known   \n120 in the literature as DAGGER [Ross et al., 2011] was proposed to overcome this limitation. In this   \n121 setting, the expert is queried in states visited by the student, thus reducing distribution shift. In our   \n122 notation, this corresponds to $\\alpha = 0$ , $\\lambda = 1$ in eqn. (2) and the measure of distance is defined as: ",
+        "bbox": [
+            140,
+            676,
+            825,
+            747
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/70e5448924fadbd3b123de81161e2348116925d1b97b016da34d21234e2025ae.jpg",
+        "text": "$$\nD _ { \\mathrm { D A G G E R } } ( \\pi , \\pi _ { E } ) = - \\mathbb { E } _ { p _ { \\beta } ( \\tau ) } [ \\log \\pi ( a _ { t } ^ { \\prime } | s _ { t } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            751,
+            638,
+            770
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "123 where $p _ { \\beta } ( \\tau ) , \\beta \\in [ 0 , 1 ]$ is a trajectory distribution where actions are sampled according to the mixture   \n124 policy between a student and an expert: ",
+        "bbox": [
+            147,
+            775,
+            831,
+            803
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cddb8850672c2abf0cd61aa43c62562cc49c03201701415a5b017030212a8530.jpg",
+        "text": "$$\np ( a | s ) = \\beta \\tilde { \\pi } ( a | s ) + ( 1 - \\beta ) \\pi _ { E } ( a | s ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            806,
+            622,
+            824
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "125 The action $a _ { t } ^ { \\prime }$ in eqn. (4) is resampled from the expert policy for the state $s _ { t }$ , i.e., $a _ { t } ^ { \\prime } \\sim \\pi _ { E } ( \\cdot | s _ { t } )$   \n126 As in Section 2.2, for stochastic experts this action can be replaced by the mean or mode of the   \n127 distribution in eqn. (4). The policy $\\tilde { \\pi } ( a | s )$ corresponds to a frozen version of student policy $\\pi$ so   \n128 that the gradient $\\nabla _ { \\pi } D _ { \\mathrm { D A G G E R } } ( \\pi , \\pi _ { E } )$ is not taken with respect to $p ( a | s )$ . Note that even though, in   \n129 eqn. (4) we collect data from the environment, the setting nevertheless corresponds to pure imitation   \n130 learning since expected reward is not directly maximized.   \n132 In eqn. (2), we combine both maximization of expected task reward and minimization of distance to   \n133 the expert. In literature, it is known as Kickstarting [Schmitt et al., 2018]. In this case, in the objective   \n134 from eqn. (2), $\\alpha = 1$ , and $\\lambda \\geq 0$ . As the measure of distance, we use the cross-entropy from expert   \n135 to a student, similarly to [Schmitt et al., 2018]: ",
+        "bbox": [
+            140,
+            827,
+            825,
+            912
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            116,
+            825,
+            172
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db81e070f3dbc7cd708fd7e25282dc625e24205fa66f4e4857f116352ae50b7c.jpg",
+        "text": "$$\nJ ( \\pi , \\pi _ { E } ) = J ( \\pi ) - \\lambda \\mathbb { E } _ { p ( \\tau ) } \\left[ - \\mathbb { E } _ { \\pi _ { E } ( a | s ) } \\log \\pi ( a | s ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            176,
+            668,
+            196
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "136 where $p ( \\tau )$ is a trajectory distribution, where actions are sampled according to the student policy   \n137 $\\pi ( \\cdot | s )$ . Usually, in the Kickstarting setting, the expert is sub-optimal and the goal is to train a policy   \n138 that eventually outperforms the expert. Thus, it is customary to reduce $\\lambda$ over the course of training.   \n139 Yet, for simplicity, in our experiments we keep this coefficient fixed. ",
+        "bbox": [
+            140,
+            202,
+            825,
+            258
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "40 3 Augmented policy cloning ",
+        "text_level": 1,
+        "bbox": [
+            155,
+            276,
+            423,
+            295
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "141 The previous section has demonstrated that the objective corresponding to the cloning behavior from   \n142 the parametric expert policy could arise in multiple scenarios. In this section we propose a new and   \n143 simple method which can significantly improve the data efficiency of the approaches described in   \n144 Section 2. We explain the basic idea for BC, but its generalization to other expert-driven learning   \n145 approaches described in Section 2 is straightforward. In Section 5 we show results for these problems.   \n146 When optimizing the objective (3), for every state $s \\in \\mathcal { D } _ { E }$ from the expert trajectories dataset, we   \n147 consider a small Gaussian state perturbation: ",
+        "bbox": [
+            140,
+            308,
+            825,
+            378
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            383,
+            825,
+            412
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/879240cc09a37a72659299d47b0def4db435f1dd0a1600a2f9d65bce01ee7c05.jpg",
+        "text": "$$\n\\delta s \\sim \\mathcal N ( 0 , \\sigma _ { s } ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            447,
+            416,
+            550,
+            435
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "148 which produces a new virtual state: ",
+        "bbox": [
+            150,
+            440,
+            405,
+            455
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a33731927c7736c07c23d4d87dd7c7cdf2411038f23c26be16389deba783d84.jpg",
+        "text": "$$\n\\boldsymbol { s } ^ { \\prime } = \\boldsymbol { s } + \\delta \\boldsymbol { s }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            457,
+            453,
+            540,
+            468
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "149 Then, for this state we query the expert and obtain a new action ",
+        "bbox": [
+            145,
+            473,
+            591,
+            488
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f5cbd5f8d9738c56d4260eb0b9c85e2c4279de8210c6fe80029987948693854.jpg",
+        "text": "$$\na ^ { \\prime } \\sim \\pi _ { E } ( \\cdot | s + \\delta s )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            492,
+            560,
+            511
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "150 We then augment the dataset $\\mathcal { D } _ { E }$ with these new pairs of virtual states and actions. More explicitly   \n151 the idea can be expressed in terms of the following objective: ",
+        "bbox": [
+            138,
+            516,
+            826,
+            545
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efa2c9a2f5a5e3d7774a3d11a47265fc2bbb4209d87ab7c3dad4954a87af4cc5.jpg",
+        "text": "$$\n\\begin{array} { r } { D ( \\pi , \\pi _ { E } ) _ { A P C } = \\mathbb { E } _ { ( a , s ) \\in B _ { E } } [ \\log \\pi ( a | s ) + \\mathbb { E } _ { \\delta s \\sim \\mathcal { N } ( 0 , \\sigma _ { s } ^ { 2 } ) , a ^ { \\prime } \\sim \\pi _ { E } ( \\cdot | s + \\delta s ) } \\log \\pi ( a ^ { \\prime } | s + \\delta s ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            549,
+            766,
+            569
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "152 We call this approach Augmented Policy Cloning (APC) as it queries the expert policy to augment   \n153 the training data. This approach is different from a naive data-augmentation technique, where a new   \n154 state would be generated, but associated with the original action (and not a new one). It therefore   \n155 allows to build policies which are feedback-responsive with respect to the expert. We formulate APC   \n156 algorithm for BC in Algorithm 1. ",
+        "bbox": [
+            142,
+            571,
+            825,
+            642
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "157 4 Core Results: Evaluation of Augmented Policy Cloning ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            661,
+            665,
+            679
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Domains ",
+        "text_level": 1,
+        "bbox": [
+            165,
+            693,
+            274,
+            707
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "159 To study how our method performs on complex control domains, we consider two complex, high-DoF   \n160 continuous control tasks involving control of a physically simulated humanoid body. Both domains   \n161 are implemented using the MuJoCo physics engine [Todorov et al., 2012] and are available in the   \n162 dm_control repository [Tunyasuvunakool et al., 2020]. The first task is the standard control suite   \n163 Run task, where the Humanoid body needs to run at a target speed and observations are based on   \n164 proprioception. The second task is the Walls task which requires the same Humanoid body to run   \n165 along a corridor and avoid walls, using both proprioception and egocentric vision as observations.   \n166 Both of these problems are rather challenging insofar as they require stabilization and locomotion   \n167 control of a relatively complex humanoid body with 21 actuated DoFs, in one case using vision to   \n168 guide the movement. Note these environments are related to the domains that have been proposed   \n169 for use in offline RL benchmarks [Gulcehre et al., 2020]; however, the experiments we perform in   \n170 this work require availability of the expert policy, so we do not use offline data, but instead train   \n171 new experts and perform experiments in the very low data regime. For more details, please refer to   \n172 Section 1.1 in Supplementary Material. ",
+        "bbox": [
+            140,
+            718,
+            826,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Parametric student policy: $\\pi _ { \\theta }$   \nInitial parameters: $\\theta _ { 0 }$   \nexpert policy: $\\pi _ { E }$   \nDataset $\\boldsymbol { B _ { E } } = \\{ ( s _ { i } , a _ { i } ) , i = 1 , \\ldots , N \\}$ , $N > 0$ of expert state-action pairs   \nState perturbation noise $\\sigma _ { s }$   \nLearning rate $\\alpha$   \nNumber of augmented samples: $M$   \nNumber of gradient updates: $K$   \nSize of a batch: $L$   \nfor $\\mathbf { k } { = } 1 , \\ldots , \\mathbf { K }$ do Sample a batch of pairs $\\{ ( a _ { i } , s _ { i } ) \\} _ { i = 1 } ^ { L } \\sim B _ { E }$ For each state $s _ { i }$ , sample $M$ perturbations $\\delta s _ { j } \\sim \\mathcal { N } ( 0 , \\sigma _ { s } ) , j = 1 , \\ldots , M$ Construct $M$ virtual states $s _ { i , j } ^ { \\prime } = s _ { i } + \\delta s _ { j } , i = 1 , \\dots , L , j = 1 , \\dots , M$ Resample new actions from expert $a _ { i , j } ^ { \\prime } \\sim \\pi _ { E } ( \\cdot | s _ { i , j } ^ { \\prime } )$ For Gaussian experts, the action $a _ { i } = \\mu _ { E } ( s _ { i } )$ and the new actions are $a _ { i , j } ^ { \\prime } = \\mu _ { E } ( s _ { i , j } ^ { \\prime } )$ Compute the empirical negative log-likelihood: $\\begin{array} { r } { \\mathcal { L } = - \\left[ \\log \\pi _ { \\theta _ { k } } ^ { - } ( a _ { i } | s _ { i } ) + \\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } \\log \\pi _ { \\theta _ { k } } ( a _ { i , j } ^ { \\prime } | s _ { i , j } ^ { \\prime } ) \\right] } \\end{array}$ Update the parameters $\\theta _ { k + 1 } = \\theta _ { k } - \\alpha \\nabla _ { \\theta } \\mathcal { L }$   \nend for ",
+        "bbox": [
+            183,
+            109,
+            771,
+            386
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "173 For each task, we train expert policies to convergence using the MPO algorithm Abdolmaleki et al.   \n174 [2018]. Since the expert policy essentially saturates task performance, for each task, we keep three   \n175 partially trained experts such that we can assess the ability of the kickstarting approach to outperform   \n176 sub-optimal experts. We refer to the different experts as Low, achieving approximatively $25 \\%$ of the   \n177 optimal policy reward, Medium, achieving around $50 \\%$ of the performance and High, corresponding   \n178 to the converged policy. Each expert is represented by a Gaussian policy. For more details, please   \n179 refer to Section 1.2 in Supplementary Material. ",
+        "bbox": [
+            142,
+            414,
+            825,
+            512
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "80 4.2 Applying Augmented Policy Cloning ",
+        "text_level": 1,
+        "bbox": [
+            155,
+            527,
+            465,
+            544
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "81 First, we evaluate the performance of APC in fitting a fixed dataset of expert trajectories. In order   \n82 to study the data efficiency of the method, we construct datasets containing different numbers of   \n83 expert trajectories. The expert policies are represented by conditionally Gaussian distributions,   \n84 i.e. $\\pi _ { E } ( \\cdot | \\bar { s } ) = \\mathcal { N } ( \\mu _ { E } ( s ) , \\bar { \\sigma ( s ) } )$ . Thus, to assess the robustness of our method to expert noise we   \n85 produce trajectories using the experts’ mean but adding different levels of (homoscedastic) zero-mean   \n86 Gaussian noise $\\sigma _ { E }$ . ",
+        "bbox": [
+            150,
+            553,
+            825,
+            637
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ae4dadac8338f60587aa4c8480b702ee4326f20d696915a68e1e5336e573398.jpg",
+        "text": "$$\n\\displaystyle a \\sim \\mathcal { N } ( \\mu _ { E } ( s ) , \\sigma _ { E } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            433,
+            636,
+            565,
+            654
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "187 Note that in addition to policy noise $\\sigma _ { E }$ which is introduced when sampling trajectories, initial pose   \n188 and environment layout (for the Walls task) are also sampled randomly for each episode. We consider   \n189 4 levels of expert policy noise: Deterministic, which uses the Gaussian mean for the action, Low,   \n190 with $\\sigma _ { E } = 0 . 2$ , Medium $\\sigma _ { E } = 0 . 5$ and High $\\sigma _ { E } = 1 . 0$ .   \n191 For the APC method, we rely on Algorithm 1. For baselines, we consider BC algorithm from eqn. (3)   \n192 as well as a simple modification of BC, where we apply, similar to APC, state perturbations as in   \n193 eqn. (7) and eqn. (8), but we do not produce a new action from the expert. We call this approach   \n194 Naive Augmented Behavior Cloning (Naive ABC) which essentially corresponds to robustification   \n195 of the student policies with respect to state perturbation and is similar in spirit to standard data  \n196 augmentation approaches. For vision-based tasks, we consider random crop augmentations of size   \n197 $4 8 \\mathbf { x } 4 8$ (downsampled from the input image of 64x64), similar to Laskin et al. [2020]. When the   \n198 image augmentations are used we add \"with image\" to the method name. On top of that, we consider   \n199 a variant, where only image augmentation is used, which we call Naive ABC (image only). For   \n200 all methods, as an action in the objective from eqn. (3), we use an expert mean $\\mu _ { E } ( s )$ . We train   \n201 all approaches to convergence (300K learning iterations on Walls and 13M learning iterations on   \n202 Run). Each learning iteration corresponds applying gradients to 64 trajectories, each containing   \n203 10 time steps. After each learning iteration, we evaluate the policy on both a validation set (50   \n204 random instances of the environment) and a test set (150 random instances of the environments).   \n205 We apply early-stopping based on the validation set performance to select the best model and   \n206 report corresponding performance on the test set. For more details, please refer to Section 1.3 in   \n207 Supplementary Material. As an additional evaluation, we test robustness of the obtained policies to a   \n208 fixed amount of noise during execution. For a learned student policy $\\pi ( \\cdot | s ) = \\mathcal { N } ( \\mu ( s ) , \\sigma ( s ) )$ , we   \n209 evaluate it by executing an action: ",
+        "bbox": [
+            142,
+            655,
+            826,
+            712
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            718,
+            826,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6ba5e847a627b5cd0f9ec953ef682d44960299536be320f0eb5349ad60ca4536.jpg",
+        "image_caption": [
+            "Figure 1: Behavioral cloning results on Run and Walls tasks (represented by rows). The X-axis represents the number of trajectories, whereas the Y-axis corresponds to the episodic reward averaged among 150 independent evaluations. The highest point of the bar corresponds to the mean, whereas the dashed lines indicate the standard deviation. The pink dashed line indicate average expert performance. The legend describes a method which is used. On the plot on the left depicts a standard BC experiment, where dataset contains a specified number of full trajectories from the expert. The plot on the right illustrates the experiment, where a dataset contains 1 full trajectories and the rest are the short ones, containing only 200 timesteps each. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            87,
+            759,
+            422
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            569,
+            825,
+            638
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ae0536b5e79d953b93aef718d373cfc0b07f0ddb2551c6ba79e487121acb932.jpg",
+        "text": "$$\na \\sim { \\mathcal { N } } ( \\mu ( s ) , \\sigma ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            637,
+            553,
+            654
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "210 where $\\sigma$ is the fixed amount of student noise. We consider similar noise magnitudes as for the   \n211 expert. For APC and Naive ABC, we sweep over state perturbation noise levels and choose the ones   \n212 performing the best on the validation set. For APC, we use $\\sigma _ { s } = 0 . 1$ for Run and $\\sigma _ { s } = 1 . 0$ for Walls.   \n213 For Naive ABC, we use $\\sigma _ { s } = 0 . 0 0 1$ for Run and $\\sigma _ { s } = 0 . 0 1$ for Walls. The ablation experiments   \n214 over noise levels for APC and Naive ABC are presented in Section 2 of the Supplementary Material   \n215 (Figure 1 and Figure 2).   \n216 The first of results, in Figure 1 (left) demonstrates the increased data efficiency of APC over BC and   \n217 Naive ABC in terms of number of trajectories. The noise level of the expert and student are fixed to   \n218 Low for the ease of comparison. We also see that Naive ABC performs similarly to BC. To further   \n219 push the limits of data efficiency, we conducted a variant where a dataset contains only 1 full trajectory   \n220 (1000 timesteps for Run and around $2 \\mathrm { k }$ timesteps for Walls) along with multiple short trajectories   \n221 (200 time-step only). This dramatically reduces the amount of expert data available to learn from.   \n222 However, we hypothesise that in the environments considered, much of the diversity of the trajectories   \n223 arises due to initial state variation. This setting might arise in domains where execution is costly,   \n224 such as robotics applications. In such setting we might have a few longer trajectories along with a   \n225 patchwork of shorter trajectories covering more diverse parts of the state space. The results for this   \n226 experiment are given in figure 1 (right). Again, we see that APC is significantly more efficient than   \n227 BC and Naive ABC. Interesting to note that APC picks up quite a high performance after observing   \n228 10 (1 full and 9 short) trajectories for Run task and 100 (1 full and 99 short) trajectories for the Walls   \n29 task. In Section 3 in Supplementary material, we provide additional results for Walls task when we   \n230 use image-based perturbations. ",
+        "bbox": [
+            140,
+            655,
+            825,
+            739
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            138,
+            744,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            150,
+            90,
+            826,
+            133
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the next experiment, in order to understand how robust our method to noise, we study the impact of different levels of student and expert noises on performance. For each run, we use a dataset of 100 trajectories. The results are given in figure 2, where each column corresponds to a different level of expert noise, and the X-axis represents different levels of student noise. At first, we observe that APC is consistently more robust than BC and Naive ABC for any level of expert and student noise. On top of that, we can notice that for any fixed level of expert noise, the performance degrades when a student noise increases. Finally, we see that for higher noise levels of expert, the learned student performs better in the high noise regime. It is consistent with the intuition - training on noisy trajectories leads to a more robust policy. Overall, APC leads the most robust policy. ",
+        "bbox": [
+            163,
+            138,
+            825,
+            265
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/55922519d083cd5acd37c2639ce28266b4d4cbf46504ddb38df60d0a847fc02f.jpg",
+        "image_caption": [
+            "Figure 2: Noise sensitivity results. We consider 4 levels of noise for student and expert: Deterministic, which uses the Gaussian mean for the action, Low, is the noise $\\sigma = 0 . 2$ , Medium $\\sigma = 0 . 5$ and High $\\sigma = 1 . 0$ . Each column corresponds to a different level of expert noise. X-axis corresponds to a different level of student noise. Y-axis corresponds to the episodic reward averaged among 150 independent evaluations. The highest point of the bar corresponds to the mean, whereas the dashed lines indicate the standard deviation. The legend denotes a method and a row corresponds to a task. The pink dashed line indicate average expert performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            279,
+            761,
+            468
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "240 5 Additional Results: Augmented Policy Cloning as a subroutine ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            607,
+            727,
+            626
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "241 5.1 DAGGER with data augmentation ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            641,
+            449,
+            656
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2496941df8315f2f080842d8e7e0f351aad02ffd4ab30021f8f43e0567433480.jpg",
+        "image_caption": [
+            "Figure 3: DAGGER results. On the $\\mathrm { X }$ -axis we report the number of environment steps. On the Y-axis we report averaged across 3 seeds episodic reward achieved by the student. We report confidence intervals in the shaded areas. For Run task, the confidence intervals are very small and are not visible. In solid line we report the performance without using expert policy during the acting. In dashed line, we report the performance of the policy which mixes $30 \\%$ with the expert. All the methods use mean action during evaluation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            674,
+            759,
+            818
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "242 As described in Section 2.3, DAGGER [Ross et al., 2011] is a more sophisticated approach where   \n243 data is collected from the real environment by executing a policy from eqn. (5), which is a mixture   \n244 between a student and an expert. In this section we study how data augmentation approaches affect   \n245 the data efficiency of the DAGGER algorithm.   \n246 We consider similar baselines for both tasks as in the previous section. For an expert policy that has   \n247 been pre-trained via MPO [Abdolmaleki et al., 2018], we perform online rollouts for two values of   \n248 the expert-student mixing coefficient, $\\beta = 0$ and $\\beta = 0 . 3$ (see eqn. 5). Since both student and expert   \n249 are Gaussian distributions, instead of using a $\\log \\pi$ in eqn. (4), we could use a state-conditional cross   \n250 entropy from an expert to a student, $\\mathcal { H } [ \\pi _ { E } ( \\cdot | s ) | | \\pi ( \\cdot | s ) ]$ . Empirically, we found that it worked better   \n251 than using $\\log \\pi$ . We demonstrate a comparison in Section 4 in Supplementary Material. We run   \n252 the experiments in a data-restricted setup such that for every collected trajectory (10 time-steps), we   \n253 apply 10 gradient steps, using a replay-buffer to store the past experience. Additional experimental   \n254 details are given in Section 1.4 in Supplementary Material. Results are shown in Figure 3. We see   \n255 that APC and its vision variant outperform BC and Naive ABC similarly to the behavior cloning   \n256 experiments. While we observe that image augmentation can help, we see that the primary advantage   \n257 comes from the state-based augmentation for APC. For the Run task, we observe that all DAGGER   \n258 methods achieve slightly lower performance than an expert policy. We speculate that this is due to   \n259 insufficient coverage of the state space during training. ",
+        "bbox": [
+            140,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            138,
+            154,
+            825,
+            348
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "260 5.2 Kickstarting with data augmentation ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            367,
+            470,
+            381
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/de92604cbf28af517d0ab8201ecd8de84edebc9a85dfb463a974c61abfee81e9.jpg",
+        "image_caption": [
+            "Figure 4: Kickstarting results. On the X-axis we show the number of environment steps. On the Y-axis we report averaged across 3 seeds episodic reward achieved by the student. We report confidence intervals in the shaded areas. For Run task, the confidence intervals are very small and are not visible. Each row indicates a task, whereas a column corresponds to the expert type. Dashed black line shows the expert performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            400,
+            761,
+            717
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "261 A similar in spirit approach is kickstarting Schmitt et al. [2018], where we solve an RL task as well   \n262 as cloning the expert policy. Similarly to previous section, we apply APC in kickstarting on the   \n263 cross entropy term in eqn. (6). We use 3 types of expert policy as described in Section 4.1. We run   \n264 experiments using a distributed setup with 64 acting policies and 1 learner, querying the batches   \n265 of trajectories (of size 10) from a replay buffer. On top of running BC methods, we also report the   \n266 performance of MPO Abdolmaleki et al. [2018] learning from scratch on the task of interest. All   \n267 details are given in Section 1.5 in Supplementary Material. The results are given in figure 4. ",
+        "bbox": [
+            140,
+            814,
+            825,
+            912
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "268 We observe that APC performs better than Naive ABC on Run task and similarly on Walls task. Both   \n269 approaches perform better than BC and learning from scratch. We hypothesise that the reason of not   \n270 seeing a consistent advantage could be due to two factors. As we are in a high-data regime, since   \n271 there is no limit on relative acting / learning ratio, and acting policies are not restricted to collect   \n272 trajectories, it is unclear whether data-augmentation should help. In addition, we use reward signal   \n273 which makes the impact of expert cloning less important. Note that the resulting agent is less data   \n274 efficient in these experiments; this is because we do not control the relative ratio between acting   \n275 and learning (i.e., no rate-limiting on the learner, due to instability of kickstarting experiments when   \n276 rate-limiting was explored). Furthermore, unlike in kickstarting Schmitt et al. [2018], we do not use an annealing schedule of $\\lambda$ to make the experiments simpler, but we still observe that a fixed coefficient helps to kickstart an experiment and outperform an expert policy. On top of that, we see that image-based augmentation have less of impact in this setting. ",
+        "bbox": [
+            140,
+            90,
+            825,
+            257
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "280 6 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            280,
+            294,
+            296
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Many expert-driven learning approaches actually have access to an expert that can be queried; however, this opportunity is rarely exploited fully. In this work we demonstrated a general scheme for more efficient transfer of expert behavior by augmenting expert trajectory data with virtual, perturbed states as well as the expert actions in these virtual states. This data augmentation technique is widely applicable and we demonstrated that it improves data efficiency when used in place of behavioral cloning both in the offline setting or when behavioral cloning is used as a step within DAGGER or kickstarting. ",
+        "bbox": [
+            173,
+            313,
+            825,
+            410
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Critically, data efficiency is generally very important in realistic applications, where new data acquisition cost could be high. In particular, settings involving deployment of policies in the real world, such as robotics applications, may benefit from an ability to efficiently transfer expert policy behavior from one neural network to another (for compression or execution speed reasons), or to combine behavior from multiple experts into a single neural network. While overall, we consider the present work to be fairly basic research with limited ethical impact, insofar as our approach decreases the amount of data which needs to be collected through processes which could potentially be unsafe or costly, there is a potential positive social value. ",
+        "bbox": [
+            173,
+            415,
+            825,
+            526
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The limitations of our approach consist in the reliance on the ability to query expert policy for the perturbed states which reduces the amount of applications where the method could be used. Another limitation is the reliance on the continuous state spaces. In discrete state spaces, it is unclear whether a small perturbation in state would result in a valid action from an expert. ",
+        "bbox": [
+            176,
+            532,
+            825,
+            588
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In future work, we plan to explore how our proposed augmentation technique can be leveraged in the context of KL-regularized RL with behavior priors. ",
+        "bbox": [
+            171,
+            594,
+            820,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            646,
+            266,
+            661
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Abbas Abdolmaleki, Jost Tobias Springenberg, Yuval Tassa, Remi Munos, Nicolas Heess, and Martin Riedmiller. Maximum a posteriori policy optimisation, 2018. ",
+        "bbox": [
+            171,
+            671,
+            823,
+            700
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alexandre Galashov, Siddhant M. Jayakumar, Leonard Hasenclever, Dhruva Tirumala, Jonathan Schwarz, Guillaume Desjardins, Wojciech M. Czarnecki, Yee Whye Teh, Razvan Pascanu, and Nicolas Heess. Information asymmetry in kl-regularized rl, 2019. ",
+        "bbox": [
+            171,
+            713,
+            823,
+            756
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Caglar Gulcehre, Ziyu Wang, Alexander Novikov, Thomas Paine, Sergio Gómez, Konrad Zolna, Rishabh Agarwal, Josh S Merel, Daniel J Mankowitz, Cosmin Paduraru, et al. Rl unplugged: A collection of benchmarks for offline reinforcement learning. Advances in Neural Information Processing Systems, 33, 2020. ",
+        "bbox": [
+            173,
+            770,
+            825,
+            825
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Michael Laskey, Jonathan Lee, Roy Fox, Anca Dragan, and Ken Goldberg. Dart: Noise injection for robust imitation learning. In Conference on robot learning, pages 143–156. PMLR, 2017. ",
+        "bbox": [
+            160,
+            840,
+            823,
+            869
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "314 Michael Laskin, Kimin Lee, Adam Stooke, Lerrel Pinto, Pieter Abbeel, and Aravind Srinivas. Reinforcement learning with augmented data, 2020. ",
+        "bbox": [
+            147,
+            883,
+            826,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Josh Merel, Leonard Hasenclever, Alexandre Galashov, Arun Ahuja, Vu Pham, Greg Wayne, Yee Whye Teh, and Nicolas Heess. Neural probabilistic motor primitives for humanoid control, 2019.   \nDonald Michie and Claude Sammut. Behavioural Clones and Cognitive Skill Models, page 387–395. Oxford University Press, Inc., USA, 1996. ISBN 019853860X.   \nIgor Mordatch and Emo Todorov. Combining the benefits of function approximation and trajectory optimization. In Robotics: Science and Systems, volume 4, 2014.   \nIgor Mordatch, Kendall Lowrey, Galen Andrew, Zoran Popovic, and Emanuel V Todorov. Interactive control of diverse complex characters with neural networks. Advances in Neural Information Processing Systems, 28:3132–3140, 2015.   \nDean A Pomerleau. Alvinn: An autonomous land vehicle in a neural network. Technical report, Carnegie-Mellon, 1989.   \nStephane Ross, Geoffrey J. Gordon, and J. Andrew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning, 2011.   \nSimon Schmitt, Jonathan J. Hudson, Augustin Zidek, Simon Osindero, Carl Doersch, Wojciech M. Czarnecki, Joel Z. Leibo, Heinrich Kuttler, Andrew Zisserman, Karen Simonyan, and S. M. Ali Eslami. Kickstarting deep reinforcement learning, 2018.   \nConnor Shorten and Taghi Khoshgoftaar. A survey on image data augmentation for deep learning. Journal of Big Data, 6, 07 2019. doi: 10.1186/s40537-019-0197-0.   \nYee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. arXiv preprint arXiv:1707.04175, 2017.   \nDhruva Tirumala, Alexandre Galashov, Hyeonwoo Noh, Leonard Hasenclever, Razvan Pascanu, Jonathan Schwarz, Guillaume Desjardins, Wojciech Marian Czarnecki, Arun Ahuja, Yee Whye Teh, et al. Behavior priors for efficient reinforcement learning. arXiv preprint arXiv:2010.14274, 2020.   \nEmanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026–5033. IEEE, 2012.   \nSaran Tunyasuvunakool, Alistair Muldal, Yotam Doron, Siqi Liu, Steven Bohez, Josh Merel, Tom Erez, Timothy Lillicrap, Nicolas Heess, and Yuval Tassa. dm_control: Software and tasks for continuous control. Software Impacts, 6:100022, 2020.   \nDenis Yarats, Ilya Kostrikov, and Rob Fergus. Image augmentation is all you need: Regularizing deep reinforcement learning from pixels. In 9th International Conference on Learning Representations, ICLR, volume 2021, 2021. ",
+        "bbox": [
+            161,
+            93,
+            828,
+            722
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "51 7 NeuRIPS 2021 checklist ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            739,
+            406,
+            756
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Yes, we believe that the claims in the abstract and introduction are accurately reflected by the paper’s contributions and scope.   \n• Have you read the ethics review guidelines and ensured that your paper conforms to them? Yes.   \n• Did you discuss any potential negative societal impacts of your work? Yes. We included this in the Discussion section.   \n• Did you describe the limitations of your work? Yes, we discuss these in the \"Discussion\" section.   \n• Did you state the full set of assumptions of all theoretical results Not applicable.   \n• Did you include complete proofs of all theoretical results Not applicable.   \n• Did you include the code, data, and instructions needed to reproduce the main experimental results ? We included complete details in the supplementary material which could be used to reproduce the experimental results.   \n• Did you specify all the training details? Yes, in the main paper and supplementary material.   \n• Did you report error bars? We did. For every plot we reported the mean and standard deviations averaged across a number of independent random evaluations.   \n• Did you include the amount of compute and the type of resources used? Yes, we did, in the supplementary material.   \n• If your work uses existing assests, did you cite the creators? Yes. We use DeepMind control suite tasks and we cite the appropriate publications. Did you mention the license of the assets? We did not explicitly mention the licence as it is clear from the reference. Did you include any new assets either in the supplementary material os as a URL? No. Did you discuss whether and how consent was obtained from people whose data you’re using / curating? Since the assets and references are in the public access, no consent was required. Did you discuss whether the data you are using / curating contains personally identifiable information or offensive content? The assets which we used did not include any personally identifiable information nor offensive content.   \n• Did you include the full text of instructions given to participants and screenshots, if applicable? Not applicable.   \nDid you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable Not applicable.   \n• Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? Not applicable. ",
+        "bbox": [
+            215,
+            767,
+            826,
+            912
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            214,
+            84,
+            826,
+            746
+        ],
+        "page_idx": 10
+    }
+]

parse/train/POWv6hDd9XH/POWv6hDd9XH.md

@@ -0,0 +1,415 @@
+# BRECQ: PUSHING THE LIMIT OF POST-TRAININGQUANTIZATION BY BLOCK RECONSTRUCTION
+
+Yuhang $\mathbf { L i } ^ { 1 2 } \hat { \mathbf { \Omega } }$ ∗, Ruihao $\mathbf { G o n g ^ { 2 * } }$ , $\mathbf { X } \mathbf { u } \ \mathbf { T a n } ^ { 2 }$ , Yang Yang2, Peng $\mathbf { H } \mathbf { u } ^ { 2 }$ , Qi Zhang2, Fengwei $\mathbf { Y } \mathbf { u } ^ { 2 }$ , Wei Wang, Shi $\mathbf { G u } ^ { \bar { 1 } \boxtimes }$ 1University of Electronic Science and Technology of China, 2SenseTime Research liyuhang699@gmail.com, gongruihao@sensetime.com, gus@uestc.edu.cn
+
+# ABSTRACT
+
+We study the challenging task of neural network quantization without end-toend retraining, called Post-training Quantization (PTQ). PTQ usually requires a small subset of training data but produces less powerful quantized models than Quantization-Aware Training (QAT). In this work, we propose a novel PTQ framework, dubbed BRECQ, which pushes the limits of bitwidth in PTQ down to INT2 for the first time. BRECQ leverages the basic building blocks in neural networks and reconstructs them one-by-one. In a comprehensive theoretical study of the second-order error, we show that BRECQ achieves a good balance between crosslayer dependency and generalization error. To further employ the power of quantization, the mixed precision technique is incorporated in our framework by approximating the inter-layer and intra-layer sensitivity. Extensive experiments on various handcrafted and searched neural architectures are conducted for both image classification and object detection tasks. And for the first time we prove that, without bells and whistles, PTQ can attain 4-bit ResNet and MobileNetV2 comparable with QAT and enjoy $2 4 0 \times$ faster production of quantized models. Codes are available at https://github.com/yhhhli/BRECQ.
+
+# 1 INTRODUCTION
+
+The past decade has witnessed the rapid development of deep learning in many tasks, such as computer vision, autonomous driving, etc. However, the issue of huge computation cost and memory footprint requirements in deep learning has received considerable attention. Some works such as neural architecture search (Zoph & Le, 2016) try to design and search a tiny network, while others, like quantization (Hubara et al., 2017), and network pruning (Han et al., 2015) are designed to compress and accelerate off-the-shelf well-trained redundant networks.
+
+Many popular quantization and network pruning methods follow a simple pipeline: training the original model and then finetune the quantized/pruned model. However, this pipeline requires a full training dataset and many computation resources to perform end-to-end backpropagation, which will greatly delay the production cycle of compressed models. Besides, not all training data are always ready-to-use considering the privacy problem. Therefore, there is more demand in industry for quantizing the neural networks without retraining, which is called Post-training Quantization. Although PTQ is fast and light, it suffers from severe accuracy degeneration when the quantization precision is low. For example, DFQ (Nagel et al., 2019) can quantize ResNet-18 to 8-bit without accuracy loss $6 9 . 7 \%$ top-1 accuracy) but in 4-bit quantization, it can only achieve $39 \%$ top-1 accuracy. The primary reason is the approximation in the parameter space is not equivalent to the approximation in model space thus we cannot assure the optimal minimization on the final task loss.
+
+Recent works like (Nagel et al., 2020) recognized the problem and analyzed the loss degradation by Taylor series expansion. Analysis of the second-order error term indicates we can reconstruct each layer output to approximate the task loss degeneration. However, their work cannot further quantize the weights into INT2 because the cross-layer dependency in the Hessian matrix cannot be ignored when the perturbation on weight is not small enough. In this work, we analyze the second-order error based on the Gauss-Newton matrix. We show that the second-order error can be transformed into network final outputs but suffer from bad generalization. To achieve the best tradeoff, we adopt an intermediate choice, block reconstruction. In addition, our contributions are threefold:
+
+1. Based on the second-order analysis, we define a set of reconstruction units and show that block reconstruction is the best choice with the support from theoretical and empirical evidence. We also use Fisher Information Matrix to assign each pre-activation with an importance measure during reconstruction.   
+2. We incorporate genetic algorithm and the well-defined intra-block sensitivity measure to generate latency and size guaranteed mixed precision quantized neural networks, which fulfills a general improvement on both specialized hardware (FPGA) and general hardware (ARM CPU).   
+3. We conduct extensive experiments to verify our proposed methods. We find that our method is applicable to a large variety of tasks and models. Moreover, we show that post-training quantization can quantize weights to INT2 without significant accuracy loss for the first time.
+
+# 2 PRELIMINARIES
+
+Notations Vectors are denoted by small bold letters and matrices (or tensors) are denoted by capital bold letters. For instance, $\mathbf { W }$ and w represent the weight tensor and its flattened version. Bar accent denotes the expectation over data points, e.g. a¯. Bracketed superscript $\mathbf { w } ^ { ( \ell ) }$ indicates the layer index. For a convolutional or a fully-connected layer, we mark its input and output vectors by $\mathbf { X }$ and $\mathbf { z }$ . Thus given a feedforward neural network with $n$ layers, we can denote the forward process by
+
+$$
+\begin{array} { r } { \mathbf { x } ^ { ( \ell + 1 ) } = h ( \mathbf { z } ^ { ( \ell ) } ) = h ( \mathbf { W } ^ { ( \ell ) } \mathbf { x } ^ { ( \ell ) } + \mathbf { b } ^ { ( \ell ) } ) , ~ 1 \leq \ell \leq n , } \end{array}
+$$
+
+where $h ( \cdot )$ indicates the activation function (ReLU in this paper). For simplicity, we omit the analysis of bias $\mathbf { b } ^ { ( \ell ) }$ as it can be merged into activation. $| | \cdot | | _ { F }$ denotes the Frobenius norm.
+
+Quantization Background Uniform symmetric quantization maps the floating-point numbers to several fixed-points. These points (or grids) have the same interval and are symmetrically distributed. We denote the set that contains these grids as $\mathcal { Q } _ { b } ^ { \mathrm { u , s y m } } = s \times \{ - 2 ^ { b - 1 } , \ldots , \mathsf { \bar { 0 } } , \ldots , 2 ^ { b - 1 } - 1 \}$ . Here, $q ( \cdot ) : \mathcal { R } \to \mathcal { Q } _ { b } ^ { \mathrm { u , s y } \mathrm { \hat { m } } }$ $s$ is the step size between two grids and , is generally designed to minimize the quantization error: $b$ is the bit-width. Quantization function, denoted by
+
+$$
+\operatorname* { m i n } | | \hat { \mathbf { w } } - \mathbf { w } | | _ { F } ^ { 2 } . \mathrm { ~ s . t . ~ } \hat { \mathbf { w } } \in \mathcal { Q } _ { b } ^ { \mathrm { u , s y m } }
+$$
+
+Solving this minimization problem, one can easily get the $q ( \cdot )$ by leveraging the rounding-to-nearest operation $\lfloor \cdot \rceil$ . Rounding-to-nearest is a prevalent method to perform quantization, e.g. PACT (Choi et al., 2018). However, recently some empirical or theoretical evidence supports that simply minimizing the quantization error in parameter space does not bring optimal task performances. Specifically, Esser et al. (2020) propose to learn the step size $s$ by gradient descent in quantization-aware training (QAT). LAPQ (Nahshan et al., 2019) finds the optimal step size when the loss function is minimized without re-training the weights. Their motivations are all towards minimizing a final objective, which is the task loss, i.e.,
+
+$$
+\operatorname* { m i n } \mathbb { E } [ L ( \hat { \mathbf { w } } ) ] , \mathrm { s . t . } \hat { \mathbf { w } } \in \mathcal { Q } _ { b } ^ { \mathsf { u , s y m } } .
+$$
+
+While this optimization objective is simple and can be well-optimized in QAT scenarios, it is not easy to learn the quantized weight without end-to-end finetuning as well as sufficient training data and computing resources. In post-training quantization settings, we only have full precision weights that $\mathbf { w } ^ { \star } = \arg \operatorname* { m i n } _ { \mathbf { w } } \mathbb { E } [ L ( \mathbf { w } ) ]$ where $\mathbf { w } \in \mathcal { R }$ and a small subset of training data to do calibration.
+
+Taylor Expansion It turns out that the quantization imposed on weights could be viewed as a special case of weight perturbation. To quantitatively analyze the loss degradation caused by quantization, Nagel et al. (2020) use Taylor series expansions and approximates the loss degradation by
+
+$$
+\mathbb { E } [ L ( \mathbf { w } + \Delta \mathbf { w } ) ] - \mathbb { E } [ L ( \mathbf { w } ) ] \approx \Delta \mathbf { w } ^ { \mathsf { T } } \bar { \mathbf { g } } ^ { ( \mathbf { w } ) } + \frac { 1 } { 2 } \Delta \mathbf { w } ^ { \mathsf { T } } \bar { \mathbf { H } } ^ { ( \mathbf { w } ) } \Delta \mathbf { w } ,
+$$
+
+where $\bar { \bf g } ^ { ( \mathbf { w } ) } = \mathbb { E } [ \nabla _ { \mathbf { w } } L ]$ and $\bar { \bf H } ^ { ( \bf w ) } = \mathbb { E } [ \nabla _ { \bf w } ^ { 2 } L ]$ are the gradients and the Hessian matrix and $\Delta \mathbf { w }$ is the weight perturbation. Given the pre-trained model is converged to a minimum, the gradients can be safely thought to be close to 0. However, optimizing with the large-scale full Hessian is memoryinfeasible on many devices as the full Hessian requires terabytes of memory space. To tackle this problem, they make two assumptions:
+
+1. Layers are mutual-independent, thus the Hessian is in the form of layer-diagonal1 and Kroneckerfactored, i.e., $\bar { \mathbf { H } } ^ { ( \mathbf { w } ^ { ( \ell ) } ) } = \mathbb { E } [ \mathbf { x } ^ { ( \ell ) } \mathbf { x } ^ { ( \ell ) , \top } \otimes \mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) } ]$ , where $\otimes$ is the Kronecker product.
+
+2. The second-order derivatives of pre-activations are constant diagonal matrix $( \mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) } = c \times \mathbf { I } )$ which is independent of input data points.
+
+At last, the objective is transformed into a practical proxy signal, the change in feature-maps $( \mathbf { z } = \mathbf { W } \mathbf { x } ,$ ), and the quantized model can be obtained by a layer-by-layer feature map reconstruction algorithm (with few calibration images). Recent works, like Bit-Split (Wang et al., 2020) and AdaQuant (Hubara et al., 2020), also take this layer-wise objective to improve the post-training quantization. However, they failed to quantize weights to INT2. We think the inherent reason is that when $\Delta \mathbf { w }$ grows higher, the former assumptions do not hold and an accurate signal is required.
+
+# 3 PROPOSED METHOD
+
+# 3.1 CROSS-LAYER DEPENDENCY
+
+Denote the neural network output $\mathbf { z } ^ { ( n ) } = f ( \theta )$ , the loss function can be represented by $L ( f ( \theta ) )$ where $\boldsymbol { \theta } = \operatorname { v e c } [ \mathbf { w } ^ { ( 1 ) , \top } , \dots , \mathbf { w } ^ { ( n ) , \top } ] ^ { \top }$ is the stacked vector of weights in all $n$ layers. The Hessian matrix can be computed by
+
+$$
+\frac { \partial ^ { 2 } L } { \partial \theta _ { i } \partial \theta _ { j } } = \frac { \partial } { \partial \theta _ { j } } \left( \sum _ { k = 1 } ^ { m } \frac { \partial L } { \partial \mathbf { z } _ { k } ^ { ( n ) } } \frac { \partial \mathbf { z } _ { k } ^ { ( n ) } } { \partial \theta _ { i } } \right) = \sum _ { k = 1 } ^ { m } \frac { \partial L } { \partial \mathbf { z } _ { k } ^ { ( n ) } } \frac { \partial ^ { 2 } \mathbf { z } _ { k } ^ { ( n ) } } { \partial \theta _ { i } \partial \theta _ { j } } + \sum _ { k , l = 1 } ^ { m } \frac { \partial \mathbf { z } _ { k } ^ { ( n ) } } { \partial \theta _ { i } } \frac { \partial ^ { 2 } L } { \partial \mathbf { z } _ { k } ^ { ( n ) } \partial \mathbf { z } _ { l } ^ { ( n ) } } \frac { \partial \mathbf { z } _ { l } ^ { ( n ) } } { \partial \theta _ { j } } ,
+$$
+
+where $\mathbf { z } ^ { ( n ) } \in \mathcal { R } ^ { m }$ . Since the pretrained full precision model is converged to a local minimum, we can assume the Hessian is positive-semidefinite (PSD). Specifically, the converged model has $\nabla _ { \mathbf { z } ^ { ( n ) } } L$ close to 0 so the first term in Eq. (5) is neglected and Hessian becomes the Gauss-Newton (GN) matrix $\mathbf { G } ^ { ( \theta ) }$ . GN matrix can be written in matrix form (Botev et al., 2017) as
+
+$$
+{ \bf H } ^ { ( \theta ) } \approx { \bf G } ^ { ( \theta ) } = { \bf J } _ { { \bf z } ^ { ( n ) } } ( \theta ) ^ { \top } { \bf H } ^ { ( { \bf z } ^ { ( n ) } ) } { \bf J } _ { { \bf z } ^ { ( n ) } } ( \theta ) ,
+$$
+
+where $\mathbf { J } _ { \mathbf { z } ^ { ( n ) } } ( \theta )$ is the Jacobian matrix of the network output with respect to the network parameters. However, in practice, we cannot explicitly compute and store the Jacobian for each input data point in such a raw form. To reduce the computation and memory budget, we will transform the secondorder error into the network output, as shown in the following theorem.
+
+Theorem 3.1. Consider an $n$ -layer feedforward neural network with ReLU activation function. Assuming all weights are quantized, the second-order error optimization can be transformed by:
+
+$$
+\underset { \hat { \theta } } { \arg \operatorname* { m i n } } \Delta \theta ^ { \mathsf { T } } \bar { \mathbf { H } } ^ { ( \theta ) } \Delta \theta \approx \underset { \hat { \theta } } { \arg \operatorname* { m i n } } \mathbb { E } \left[ \Delta \mathbf { z } ^ { ( n ) , \mathsf { T } } \mathbf { H } ^ { ( \mathbf { z } ^ { ( n ) } ) } \Delta \mathbf { z } ^ { ( n ) } \right] .
+$$
+
+Remark 3.1. The same transformation is also applicable for activation quantization. The quadratic loss is defined as $\mathbb { E } [ \Delta \gamma ^ { \mathsf { T } } \mathbf { H } ^ { ( \gamma ) } \Delta \gamma ]$ where $\Delta \gamma = \mathrm { v e c } [ \Delta \mathbf { x } ^ { ( 1 ) , \top } , \allowbreak . . . , \Delta \mathbf { x } ^ { ( n ) , \top } ] ^ { \top }$ .
+
+We prove the theorem using the quadratic form, details can be found in Appendix A.1. Here we provide a sketch of the proof by matrix form. The product of perturbation and Jacobian can be thought as the first-order Taylor approximation of the change in network output $\Delta \mathbf { z } ^ { ( n ) }$ :
+
+$$
+\Delta \mathbf { z } ^ { ( n ) } = \hat { \mathbf { z } } ^ { ( n ) } - \mathbf { z } ^ { ( n ) } \approx \mathbf { J } _ { \mathbf { z } ^ { ( n ) } } ( \theta ) \Delta \theta .
+$$
+
+Therefore, combining Eq. (8) and Eq. (6) we can transform the large-scale second-order error into the change in network outputs characterized by the output Hessian $\mathbf { \bar { H } } ^ { ( \mathbf { z } ^ { ( n ) } ) }$ . The theorem indicates a simple observation, suppose a well-trained teacher model and an initialized student model, we can minimize their discrepancy by reconstructing the network’s final output $\mathbf { z } ^ { ( n ) }$ , which coincides with and generalizes the distillation (Hinton et al., 2015; Polino et al., 2018). LAPQ (Nahshan et al., 2019) also considers the dependency but their optimization does not rely on second-order information. However, we should emphasize that distillation requires the same computation and data resources as in normal training procedure, which is impractical for PTQ with limited data.
+
+![](images/2c4d2943862772d6549f28456ee661ad08a9183ada453acc84ab94973276f6a2.jpg)
+
+![](images/2995b947da6ca9943ecc8d8c03f90225445c2812e8c7189ddd2d8a96ff030fe0.jpg)
+
+(a) A typical structure of CNN (taken from Radosavovic et al. (2020)). Network is composed of a stem layer (first convolution on input images), a body and a head layer (average pooling with a fully connected layer). A body contains several stages, and a stage contains several blocks. A representative block is the bottleneck block with residual path.
+
+(b) An example illustration of Hessian (or Fisher) matrix. Blue sub-block means the layer-diagonal and each layer are mutualindependent; orange sub-block consider the dependency inside a building block and green parts measure all dependencies.
+
+Figure 1: We define 4 kinds of reconstruction granularity, namely net-wise, stage-wise, block-wise and layerwise optimization, each of them corresponds an essential component of CNN.
+
+# 3.2 BLOCK RECONSTRUCTION
+
+Although the network output reconstruction has an accurate estimation of the second-order error, we find in practice it is worse than the layer-by-layer reconstruction in PTQ. The primary reason for this is optimizing the whole networks over 1024 calibration data samples leads to over-fitting easily. As Jakubovitz et al. (2019) explained, the networks can have perfect expressivity when the number of parameters exceeds the number of data samples during training, but lower training error does not ensure lower test error. We find layer-wise reconstruction acts like a regularizer which reduces the generalization error by matching each layer’s output distribution. In other words, both layer-wise and network-wise output reconstruction has their own drawbacks. And there should be a better bias-variance trade-off choice to conduct reconstruction at an intermediate granularity.
+
+The layer-wise optimization corresponds to layer-diagonal Hessian (Fig. 1b blue parts) and the network-wise optimization corresponds to full Hessian (Fig. 1b green parts). Similarly, we can define an intermediate block-diagonal Hessian. Formally, if layer $k$ to layer $\ell$ (where $1 \leq k < \ell \leq n$ ) form a block, the weight vector is defined as $\tilde { \theta } = \sec [ \mathbf { w } ^ { ( k ) , \top } , \dots , \mathbf { w } ^ { ( \ell ) , \top } ] ^ { \top }$ and the Hessian can be also transformed by $\Delta \tilde { { \boldsymbol { \theta } } } ^ { \mathsf { T } } \bar { \mathbf { H } } ^ { ( \tilde { \theta } ) } \Delta \tilde { { \boldsymbol { \theta } } } = \mathbb { E } [ \Delta \mathbf { z } ^ { ( \ell ) , \mathsf { T } } \mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) } \Delta \mathbf { z } ^ { ( \ell ) } ]$ . Such block-diagonal Hessian ignores the inter-block dependency and considers the intra-block dependency but it produces less generalization error. Then we can block-by-block reconstruct the intermediate output.
+
+To this end, we define 2 extra kinds of intermediate reconstruction granularity: Stage-wise reconstruction and Block-wise reconstruction. These 4 reconstruction granularities are described below:
+
+1. Layer-wise Reconstruction: Assume the Hessian matrix is layer-diagonal and optimize the layer output one-by-one. It does not consider cross-layer dependency and resemble existing methods (Nagel et al., 2020; Hubara et al., 2020; Wang et al., 2020).   
+2. Block-wise Reconstruction: A block is the core component in modern CNN, such as the Residual Bottleneck Block as shown in Fig. 1a. This method assumes the Hessian matrix is blockdiagonal and block-by-block perform reconstruction, which ignores inter-block dependencies.   
+3. Stage-wise Reconstruction: A stage is where the featuremaps will be downsampled and generate more channels, which is believed to produce higher-level features. Typical CNN in ImageNet dataset contains 4 or 5 different stages. This method simultaneously optimizes all layers within a stage and thus considers more dependencies than the block-wise method.   
+4. Network-wise Reconstruction: Optimize the whole quantized network by reconstructing the output of the final layers. This method resembles distillation but does not result in good performances with few images because of high generalization error.
+
+The relationship between network, stage, block, and layer is illustrated in Fig. 1a. We test these 4 kinds of reconstruction granularity and find that block-wise optimization outperforms others. We think this is because the main off-diagonal loss in the Hessian is concentrated in each block, as Fig. 1b orange part illustrated, while the inter-block loss is small and can be ignored in the opti
+
+# Algorithm 1: BRECQ optimization
+
+Input: Pretrained FP model; Calibration dataset, iteration $T$ for all $i = 1 , 2 , \dots , N$ -th block in the FP model do
+
+Collect input data to the block $\mathbf { x } ^ { ( i ) }$ , the FP output $\mathbf { z } ^ { ( i ) }$ and its gradient $\mathbf { g } ^ { ( \mathbf { z } ^ { ( i ) } ) }$ ;   
+for all $j = 1 , 2 , \dots , T$ -iteration do Get quantized output $\hat { \mathbf { z } } ^ { ( i ) }$ and compute $\Delta \mathbf { z } ^ { ( i ) } = \mathbf { z } ^ { ( i ) } - \hat { \mathbf { z } } ^ { ( i ) }$ ; Descend Eq. (10) and update the rounding of all the weights in this block (Eq. (16)); if Activation Quantization is triggered then Update the activation quantization step size (Eq. (18)).
+
+After optimization, compute the sensitivity for each layer and between layers (2-bit only);
+
+return Quantized model, Sensitivities for mixed precision;
+
+mization. The shortcut connections, which is proposed in (He et al., 2016), may also increase the dependencies within a block. Also, the stage-wise or net-wise optimization suffer from the bad generalization on the validation set and degenerate the final performances. We report the quantitative comparison in Sec. 4.1. We name our algorithm BRECQ, because we choose block as our base reconstruction unit. It is necessary to point out that our analysis does not give the optimal configuration of the reconstruction granularity. The choice of block-wise optimization comes from our experiments and we find this choice has two merits. (1) No hyper-parameters included and (2) applicable for all models and all tasks we tested.
+
+# 3.3 APPROXIMATING PRE-ACTIVATION HESSIAN
+
+With block-diagonal approximated Hessian matrix, we can measure the cross-layer dependency inside each block and transform any block’s second-order error to the output of this block $\mathbb { E } [ \Delta \mathbf { z } ^ { ( \ell ) , \mathsf { T } } \mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) } \Delta \mathbf { z } ^ { ( \ell ) } ]$ . This objective requires the further computation of the knowledge in the rest of the network, i.e., pre-activation Hessian $\mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) }$ . One way is to follow Nagel et al. (2020) and assume $\mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) } = c \times \mathbf { I }$ . Therefore the quadratic loss becomes $| | \Delta \mathbf { z } ^ { ( \ell ) } | | ^ { 2 }$ . This method might be easy to implement but lose too much information.
+
+We use the diagonal Fisher Information Matrix (FIM) to replace the pre-activation Hessian. Formally, given a probabilistic model $p ( x | \theta )$ , the FIM is defined as:
+
+$$
+\bar { \mathbf { F } } ^ { ( \theta ) } = \mathbb { E } \left[ \nabla _ { \theta } \log p _ { \theta } ( y | x ) \nabla _ { \theta } \log p _ { \theta } ( y | x ) ^ { \top } \right] = - \mathbb { E } \left[ \nabla _ { \theta } ^ { 2 } \log p _ { \theta } ( y | x ) \right] = - \bar { \mathbf { H } } _ { \log p ( x | \theta ) } ^ { ( \theta ) } .
+$$
+
+The FIM is equal to the negative expected Hessian of the log-likelihood function, therefore, a simple corollary is that the Hessian of task loss will become FIM if the model distribution matches the true data distribution (LeCun et al., 2012). Although matching true data distribution seems impossible, this is the best we can do since the pretrained model is converged.
+
+The diagonal of the pre-activation FIM is equal to the squared gradients of each elements, which is successfully used in Adam (Kingma & Ba, 2014) for the second momentum. The optimization objective becomes
+
+$$
+\operatorname* { m i n } _ { \hat { \mathbf { w } } } \mathbb { E } \left[ \Delta \mathbf { z } ^ { ( \ell ) , \top } \mathbf { H } ^ { ( \mathbf { z } ^ { ( \ell ) } ) } \Delta \mathbf { z } ^ { ( \ell ) } \right] = \operatorname* { m i n } _ { \hat { \mathbf { w } } } \mathbb { E } \left[ \Delta \mathbf { z } ^ { ( \ell ) , \top } \mathrm { d i a g } \left( ( \frac { \partial L } { \partial \mathbf { z } _ { 1 } ^ { ( \ell ) } } ) ^ { 2 } , \dots , ( \frac { \partial L } { \partial \mathbf { z } _ { a } ^ { ( \ell ) } } ) ^ { 2 } \right) \Delta \mathbf { z } ^ { ( \ell ) } \right] .
+$$
+
+Compared with the MSE minimization, the above minimization incorporates the squared gradient information. If the output has higher absolute gradients, it will receive more attention when being reconstructed. A similar method for pruning the pre-activation has been proposed in Theis et al. (2018).
+
+Note that BRECQ is compatible with any optimization method, like STE (Hubara et al., 2017). Here we adopt adaptive rounding (Nagel et al., 2020) for weights and learned step size (Esser et al., 2020) for activation step size because we observe they generally perform better in PTQ, see details in Appendix B.4.1. We formulate the overall calibration algorithm for a unified precision model in algorithm 1. We should emphasize that we only need a small subset (1024 in our experiments) of the whole training dataset to calibrate the quantized model. And we can obtain a quantized ResNet-18 within 20 minutes on a single GTX 1080TI GPU.
+
+# 3.4 MIXED PRECISION
+
+To further push the limit of post-training quantization, we employ mixed precision techniques, which can be formulated by
+
+$$
+\operatorname* { m i n } _ { \mathbf { c } } L ( \hat { \mathbf { w } } , \mathbf { c } ) , \ \mathrm { s . t . } \ H ( \mathbf { c } ) \leq \delta , \ \mathbf { c } \in \{ 2 , 4 , 8 \} ^ { n } .
+$$
+
+Here, c is the bit-width vector with the shape of number of layers. $H ( \cdot )$ is a hardware performance measurement function, which is used to ensure the mixed precision model has the same or lower hardware performance (e.g., memory and speed) than a predefined threshold $\delta$ . We choose 2, 4, 8-bit for mixed precision because they are most common in practical deployment.
+
+Regarding the training loss $L$ , we find that nearly all existing literature (Cai et al., 2020; Hubara et al., 2020; Dong et al., 2019) uses layer-wise measurement. They all assume the sensitivity within a layer is independent and can be summed together. Therefore, the mixed precision problem becomes an integer programming problem. However, we argue that the loss measurement should contain two parts: diagonal loss and off-diagonal loss, the first is the same with previous works and measure the sensitivity of each layer independently, while the off-diagonal loss is used to measure the cross-layer sensitivity. Theoretically, we should examine all permutations, which results in $3 ^ { n }$ possibilities and prohibits the search algorithm. Our first attempt is to reduce the off-diagonal loss into the blocklevel as we mentioned that the Hessian can be approximated to a block-diagonal matrix. Granted, we still find the search space is large, for example, if a block has four layers, then we have to consider the $3 ^ { 4 } ~ = ~ 8 1$ permutations for a single block. Based on our preliminary experiments, we find that 4-bit and 8-bit quantization nearly do not drop the final accuracy. Hence we only take 2-bit permutations into consideration and drastically reduce the search space. We use genetic algorithm (Guo et al., 2020) to search the optimal bitwidth configuration with hardware performance threshold, the algorithm is located in algorithm 2. Due to space limits, we put related works in Appendix 5. Readers can refer to related works for a brief discussion on quantization and secondorder analysis.
+
+# 4 EXPERIMENTS
+
+In this section, we report experimental results for the ImageNet classification task and MS COCO object detection task. The detailed implementation of the experiments can be found in the Appendix B.4.4. The rest of this section will contain ablation study on reconstruction granularity, classification and detection results, mixed precision results and comparison with quantization-aware training. In Appendix B, we conduct more experiments, including the impact of the first and the last layer, the impact of calibration dataset size and data source.
+
+# 4.1 ABLATION STUDY
+
+We test four kinds of reconstruction granularity: Net-wise, Stage-wise, Block-wise, and Layer-wise reconstruction. We conduct ImageNet experiments using MobileNetV2 and ResNet-18 with 2- bit weight quantization for all layers except for the first and the last layer. It can be seen from Table 1 that Block-wise optimization outperforms other methods. This result implies that the generalization error in net-wise and stage-wise optimization outweighs their off-diagonal loss. In ResNet
+
+Table 1: Ablation study.   
+
+<table><tr><td>Model</td><td>Layer</td><td>Block</td><td>Stage</td><td>Net</td></tr><tr><td>ResNet-18</td><td>65.19</td><td>66.39</td><td>66.01</td><td>54.15</td></tr><tr><td>MobileNetV2</td><td>52.13</td><td>59.67</td><td>54.23</td><td>40.76</td></tr></table>
+
+18, we find the difference is not significant, this can be potentially attributed to that ResNet-18 only has 19 layers in the body and the block size, as well as the stage size, is small, therefore leading to indistinct results.
+
+# 4.2 IMAGENET
+
+We conduct experiments on a variety of modern deep learning architectures, including ResNet (He et al., 2016) with normal convolution, MobileNetV2 (Sandler et al., 2018) with depthwise separable convolution and RegNet (Radosavovic et al., 2020) with group convolution. Last but not least important, we also investigate the neural architecture searched (NAS) models, MNasNet (Tan et al., 2019). In Table 2, we only quantize weights into low-bit integers and keep activations full precision. We compare with strong baselines including Bias Correction, optimal MSE, AdaRound, AdaQuant, and Bit-split. Note that the first and the last layer are kept with 8-bit. While most of the existing methods have good performances in 4-bit quantization, they cannot successfully quantize the model into 2-bit. Our method consistently achieves the lowest accuracy degradation for ResNets (within $5 \%$ ) and other compact models. We further quantize activations into 4-bit to make the quantized model run on integer-arithmetic hardware platforms. We find that 4-bit activation quantization can have a huge impact on RegNet and MobileNet. Nonetheless, our methods produce higher performance than other state-of-the-arts. To be noted, BRECQ is the first to promote the 2W4A accuracy of PTQ to a usable level while all other existing methods crashed.
+
+Table 2: Accuracy comparison on weight-only quantized post-training models. Activations here are unquantized and kept full precision. We also conduct variance study for our experiments. Bold values indicates best results. \* indicates our implementation based on open-source codes.   
+
+<table><tr><td>Methods</td><td>Bits (W/A)</td><td>ResNet-18</td><td>ResNet-50</td><td>MobileNetV2</td><td>RegNet-600MF</td><td>RegNet-3.2GF</td><td>MnasNet-2.0</td></tr><tr><td>Full Prec.</td><td>32/32</td><td>71.08</td><td>77.00</td><td>72.49</td><td>73.71</td><td>78.36</td><td>76.68</td></tr><tr><td>Bias Correction*</td><td>4/32</td><td>50.43</td><td>64.64</td><td>62.82</td><td>67.09</td><td>71.73</td><td>72.31</td></tr><tr><td>OMSE (Choukroun et al.,2019)</td><td>4/32</td><td>67.12</td><td>74.67</td><td>-</td><td></td><td>-</td><td>-</td></tr><tr><td>AdaRound (Nagel et al.,2020)</td><td>4/32</td><td>68.71</td><td>75.23</td><td>69.78</td><td>71.97*</td><td>77.12*</td><td>74.87*</td></tr><tr><td>AdaQuant (Hubara et al., 2020)</td><td>4/32</td><td>68.82</td><td>75.22</td><td>44.78</td><td></td><td></td><td></td></tr><tr><td>Bit-Split (Wang et al.,2020)</td><td>4/32</td><td>69.11</td><td>75.58</td><td>=</td><td>=</td><td>=</td><td>=</td></tr><tr><td>BRECQ (Ours)</td><td>4/32</td><td>70.70±0.07</td><td>76.29±0.04</td><td>71.66±0.04</td><td>73.02±0.09</td><td>78.04±0.04</td><td>76.00±0.02</td></tr><tr><td>Bias Correction*</td><td>3/32</td><td>12.85</td><td>7.97</td><td>10.89</td><td>28.82</td><td>17.95</td><td>40.72</td></tr><tr><td>AdaRound (Nagel et al., 2020)*</td><td>3/32</td><td>68.07</td><td>73.42</td><td>64.33</td><td>67.71</td><td>72.31</td><td>69.33</td></tr><tr><td>AdaQuant (Hubara et al.,2020)*</td><td>3/32</td><td>58.12</td><td>67.61</td><td>12.56</td><td>=</td><td></td><td>=</td></tr><tr><td>Bit-Split (Wang et al., 2020)</td><td>3/32</td><td>66.75</td><td>73.24</td><td>-</td><td>=</td><td>=</td><td></td></tr><tr><td>BRECQ (Ours)</td><td>3/32</td><td>69.81±0.05</td><td>75.61±0.09</td><td>69.50±0.12</td><td>71.48±0.07</td><td>77.22±0.04</td><td>74.58±0.08</td></tr><tr><td>Bias Correction*</td><td>2/32</td><td>0.13</td><td>0.12</td><td>0.14</td><td>0.18</td><td>0.11</td><td>0.11</td></tr><tr><td>AdaRound (Nagel etal., 2020)*</td><td>2/32</td><td>55.96</td><td>47.95</td><td>32.54</td><td>25.66</td><td>24.70</td><td>30.60</td></tr><tr><td>AdaQuant (Hubara et al.,2020)*</td><td>2/32</td><td>0.30</td><td>0.49</td><td>0.11</td><td>=</td><td>=</td><td>=</td></tr><tr><td>BRECQ (Ours)</td><td>2/32</td><td>66.30±0.12</td><td>72.40±0.12</td><td>59.67±0.13</td><td>65.83±0.13</td><td>73.88±0.14</td><td>67.13±0.13</td></tr></table>
+
+Table 3: Accuracy comparison on fully quantized post-training models. Activations here are quantized to 4-bit. Notations follows the upper table.   
+
+<table><tr><td>Methods</td><td>Bits (W/A)</td><td>ResNet-18</td><td>ResNet-50</td><td>MobileNetV2</td><td>RegNet-600MF</td><td>RegNet-3.2GF</td><td>MNasNet-2.0</td></tr><tr><td>Full Prec.</td><td>32/32</td><td>71.08</td><td>77.00</td><td>72.49</td><td>73.71</td><td>78.36</td><td>76.68</td></tr><tr><td>ACIQ-Mix (Banner et al.,2019)</td><td>4/4</td><td>67.0</td><td>73.8</td><td>-</td><td>·</td><td>-</td><td>·</td></tr><tr><td>ZeroQ(Cai et al.,2020)*</td><td>4/4</td><td>21.71</td><td>2.94</td><td>26.24</td><td>28.54</td><td>12.24</td><td>3.89</td></tr><tr><td>LAPQ (Nahshan et al.,2019)</td><td>4/4</td><td>60.3</td><td>70.0</td><td>49.7</td><td>57.71*</td><td>55.89*</td><td>65.32*</td></tr><tr><td>AdaQuant (Hubara et al., 2020)</td><td>4/4</td><td>67.5</td><td>73.7</td><td>34.95*</td><td>-</td><td></td><td></td></tr><tr><td>Bit-Split (Wang et al.,2020)</td><td>4/4</td><td>67.56</td><td>73.71</td><td>=</td><td>-</td><td>=</td><td></td></tr><tr><td>BRECQ (Ours)</td><td>4/4</td><td>69.60±0.04</td><td>75.05±0.09</td><td>66.57±0.67</td><td>68.33±0.28</td><td>74.21±0.19</td><td>73.56±0.24</td></tr><tr><td>ZeroQ(Cai et al.,2020)*</td><td>2/4</td><td>0.08</td><td>0.08</td><td>0.10</td><td>0.10</td><td>0.05</td><td>0.12</td></tr><tr><td>LAPQ (Nahshan et al., 2019)*</td><td>2/4</td><td>0.18</td><td>0.14</td><td>0.13</td><td>0.17</td><td>0.12</td><td>0.18</td></tr><tr><td>AdaQuant (Hubara et al., 2020)*</td><td>2/4</td><td>0.21</td><td>0.12</td><td>0.10</td><td>=</td><td></td><td></td></tr><tr><td>BRECQ (Ours)</td><td>2/4</td><td>64.80±0.08</td><td>70.29±0.23</td><td>53.34±0.15</td><td>59.31±0.49</td><td>67.15±0.11</td><td>63.01±0.35</td></tr></table>
+
+# 4.3 COMPARISON WITH QUANTIZATION-AWARE TRAINING
+
+Table 4: Performance as well as training cost comparison with quantization-aware training (QAT).   
+
+<table><tr><td>Models</td><td>Methods</td><td>Precision</td><td>Accuracy</td><td>Model Size</td><td>Training Data</td><td>GPU hours</td></tr><tr><td rowspan="6">ResNet-18 FP: 71.08</td><td>ZEROQ(CAI ET AL., 2020))</td><td>4/4</td><td>21.20</td><td>5.81MB</td><td>0</td><td>0.008</td></tr><tr><td>BRECQ(OURS)</td><td>4/4</td><td>69.60</td><td>5.81MB</td><td>1024</td><td>0.4</td></tr><tr><td>BRECQ (W/DISTILLED DATA)</td><td>4/4</td><td>69.32</td><td>5.81MB</td><td>0</td><td>0.4</td></tr><tr><td>PACT(CHOI ET AL., 2018)</td><td>4/4</td><td>69.2</td><td>5.81MB</td><td>1.2 M</td><td>100</td></tr><tr><td>DSQ (GONG ET AL., 2019)</td><td>4/4</td><td>69.56</td><td>5.81MB</td><td>1.2 M</td><td>100</td></tr><tr><td>LSQ (ESSER ET AL., 2020)</td><td>4/4</td><td>71.1</td><td>5.81MB</td><td>1.2 M</td><td>100</td></tr><tr><td rowspan="5">MobileNetV2 FP: 72.49</td><td>BRECQ (OURS)</td><td>4/4</td><td>66.57</td><td>2.26MB</td><td>1024</td><td>0.8</td></tr><tr><td>PACT (CHOI ET AL., 2018)</td><td>4/4</td><td>61.40</td><td>2.26MB</td><td>1.2 M</td><td>192</td></tr><tr><td>DSQ (GONG ET AL., 2019)</td><td>4/4</td><td>64.80</td><td>2.26MB</td><td>1.2 M</td><td>192</td></tr><tr><td>BRECQ (OURS)</td><td>Mixed/8</td><td>70.74</td><td>1.38 MB</td><td>1024</td><td>3.2</td></tr><tr><td>HAQ (WANG ET AL., 2019)</td><td>Mixed/8</td><td>70.90</td><td>1.38 MB</td><td>1.2 M</td><td>384</td></tr></table>
+
+Table 5: Objection detection task (MS COCO) comparison on fully quantized post-training models. Activations here are quantized to 8-bit. We report the bounding box mean Average Precision (mAP) metric.   
+
+<table><tr><td rowspan="2">Models</td><td rowspan="2">Backbone</td><td>Full Prec.</td><td colspan="2">Bias Correction*</td><td colspan="2">AdaRound*</td><td>ZeroQ</td><td colspan="3">BRECQ (Ours)</td></tr><tr><td>32/32</td><td>8/8</td><td>4/8</td><td>4/8</td><td>2/8</td><td>4MP/8</td><td>8/8</td><td>4/8</td><td>2/8</td></tr><tr><td rowspan="3">Faster RCNN (Ren et al.,</td><td>ResNet-18</td><td>34.55</td><td>34.30</td><td>0.84</td><td>33.96</td><td>23.01</td><td>-</td><td>34.53</td><td>34.34</td><td>31.82</td></tr><tr><td>ResNet-50</td><td>38.55</td><td>38.25</td><td>0.25</td><td>37.58</td><td>19.63</td><td>-</td><td>38.54</td><td>38.29</td><td>34.23</td></tr><tr><td>MobileNetV2</td><td>33.44</td><td>33.24</td><td>18.39</td><td>32.77</td><td>16.35</td><td>-</td><td>33.40</td><td>33.18</td><td>27.54</td></tr><tr><td rowspan="3">RetinaNet (Lin et al.,</td><td>ResNet-18</td><td>33.20</td><td>33.00</td><td>0.04</td><td>32.59</td><td>19.93</td><td>-</td><td>33.14</td><td>33.01</td><td>31.42</td></tr><tr><td>ResNet-50</td><td>36.82</td><td>36.68</td><td>0.07</td><td>36.00</td><td>19.97</td><td>33.7</td><td>36.73</td><td>36.65</td><td>34.75</td></tr><tr><td>MobileNetV2</td><td>32.63</td><td>32.60</td><td>18.47</td><td>31.89</td><td>14.10</td><td>-</td><td>32.57</td><td>32.31</td><td>27.59</td></tr></table>
+
+![](images/a2a7f48a153c4cc829a025e4c6f2842d218853db061e313bc45e5f17f971bfba.jpg)  
+Figure 2: Mixed Precision results.
+
+In this section, we compare our algorithm (post-training quantization) with some quantization-aware training methods, including PACT (Choi et al., 2018), DSQ (Gong et al., 2019), LSQ (Esser et al., 2020), and a mixed precision technique HAQ (Wang et al., 2019). Table 4 shows that although BRECQ is a PTQ method with limited available data, it can achieve comparable accuracy results with existing quantization-aware training models. In addition, our method can surpass them in 4- bit MobileNetV2 while using less than one training GPU hours. Our method also has comparable accuracy with HAQ, which is a training-based mixed precision method. Note that our GPU hours include 3 unified precision training (2-, 4-, 8-bit respectively) and further mixed-precision training only needs to check the lookup table. Instead, HAQ would end-to-end search for each hardware performance threshold from scratch.
+
+# 4.4 MS COCO
+
+To validate the effectiveness of BRECQ on other tasks, we conduct object detection on the twostage Faster-RCNN (Ren et al., 2015) and the one-stage RetinaNet (Lin et al., 2017). ResNet-18, 50 as well as MobileNetV2 are adopted as backbones for the detection model. Results in Table 5 demonstrate our method nearly does not drop the performance in 4-bit weight quantization and 8- bit activation. In particular, BRECQ only decreases ${ \bf 0 . 2 1 \% }$ mAP performance on 4-bit ResNet-18 backboned Faster RCNN. On 4-bit ResNet-50 backboned RetinaNet, our method is outperforms the mixed precision based ZeroQ model by $3 \%$ mAP. Even when the weight bit decreases to 2, the model still achieves near-to-original mAP.
+
+# 4.5 MIXED PRECISION
+
+In this section, we test (1) model-size guaranteed mixed precision and (2) FPGA latency guaranteed mixed precision2 to unleash the potential of mixed precision and further push the limit of PTQ. We choose ResNet-18, MobileNetV2, and RegNetX-600MF to validate the efficacy of our algorithm. Note that in this section, we keep activation in 8-bit because we only compare the discrepancy between the unified and mixed precision in weights. We omit 3-bit weight quantization in unified precision because it is usually unfriendly to the hardware. Latency settings can be found in Appendix B.4.3. From Fig. 2 we find that (1) mixed precision consistently outperforms unified precision, especially when using extremely low-bit, e.g., up to $10 \%$ accuracy increase with the same latency as the 2-bit model. (2) mixed precision can produce many bit configurations that can adapt to plenty of hardware requirements while unified precision can only have 2 fixed models.
+
+# 5 RELATED WORKS
+
+Quantization Model quantization can be classified into two categories: Quantization-aware Training (QAT) and Post-training Quantization (PTQ). Rounding floating-point numbers to fixed-points numbers will produce 0 gradients almost everywhere. Therefore, most QAT methods employ the Straight-Through Estimator (STE) for gradients approximation. Gong et al. (2019) uses a differentiable tanh function to gradually approach the step function. Choi et al. (2018); Esser et al. (2020) introduces parameterized clipping thresholds to learn it by STE. Apart from uniform quantization, some works like Li et al. (2019) argue that non-uniform quantization has better performance than uniform quantization while keeping its efficiency. Despite the promising results given by QAT methods, they usually need more than 100 GPU hours to get it. In that case, PTQ plays an important role which is what we focus on in this paper. Generally, most deep learning models can be safely quantized to 8-bit without re-training. Data-Free Quantization Nagel et al. (2019) even do layer-wise 8-bit PTQ without any data. However, in 4-bit quantization, most parameter space-based methods cannot obtain good performances. Recently, Nagel et al. (2020) propose to do layer-wise calibration and made huge progress in 4-bit quantization. Our work continues its analysis on Taylor expansion and considers the off-diagonal loss. Another perspective of quantification is the precision allocation scheme. Hardware-aware Quantization (HAQ Wang et al. (2019)) leverages reinforcement learning to search the optimal bitwidth configuration. Hessian-aware Weight Quantization (HAWQ) (Dong et al., 2019) utilizes the second-order information to decide the bitwidth. Mixed precision also appears in PTQ, such as the Pareto frontier method in ZeroQ (Cai et al., 2020) and the Integer Programming method in AdaQuant (Hubara et al., 2020).
+
+Second-order Analysis and Optimization The history of second-order information in perturbation analysis can be traced to the 1990s like Optimal Brain Surgeon (Hassibi & Stork, 1993; Dong et al., 2017). The Hessian matrix is essential for pruning and quantization. As aforementioned, HAWQ uses the largest eigenvalue of Hessian to determine the sensitivity. Hessian matrix is also important for second-order optimization like Newton’s method as it consists of the curvature information. However, calculating the real full Hessian is prohibitive on today’s deep learning architectures. Therefore, approximations are made to simplify the calculation and make the storage more flexible, e.g., Gauss-Newton optimization with Kronecker-factored recursive approximation Botev et al. (2017). Hessian-Free optimization (Martens, 2010) avoids the explicit computation of the Hessian matrix by solving the linear system $g = H v$ . Second-order optimization with FIM is called Natural Gradient Descent (Amari, 1998). K-FAC (Martens & Grosse, 2015) utilizes the layer-diagonal FIM and the approximation of the expected Kronecker product to compute the curvature information.
+
+# 6 CONCLUSION
+
+In this paper, we propose BRECQ, a post-training quantization framework by analyzing the secondorder error. We show that the reconstruction of quantization at the block granularity arrives at a good balance of cross-layer dependency and first order approximation, especially in 2-bit weight quantization where no prior works succeed to quantize. BRECQ is compatible with mixed precision and can reduce the search cost. To our best knowledge, BRECQ reaches the highest performance in post-training quantization and is the first to be on a par with quantization-aware training using 4-bit.
+
+# ACKNOWLEDGMENT
+
+We thank Markus Nagel and anonymous reviewers for their kind help of this work. This project is primarily supported by NSFC 61876032.
+
+# REFERENCES
+
+Shun-Ichi Amari. Natural gradient works efficiently in learning. Neural computation, 10(2):251– 276, 1998.
+
+Haoli Bai, Jiaxiang Wu, Irwin King, and Michael Lyu. Few shot network compression via cross distillation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 3203–3210, 2020.
+
+Ron Banner, Yury Nahshan, and Daniel Soudry. Post training 4-bit quantization of convolutional networks for rapid-deployment. In Advances in Neural Information Processing Systems, 2019.
+
+Aleksandar Botev, Hippolyt Ritter, and David Barber. Practical gauss-newton optimisation for deep learning. arXiv preprint arXiv:1706.03662, 2017.
+
+Yaohui Cai, Zhewei Yao, Zhen Dong, Amir Gholami, Michael W Mahoney, and Kurt Keutzer. Zeroq: A novel zero shot quantization framework. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13169–13178, 2020.
+
+Jungwook Choi, Zhuo Wang, Swagath Venkataramani, Pierce I-Jen Chuang, Vijayalakshmi Srinivasan, and Kailash Gopalakrishnan. Pact: Parameterized clipping activation for quantized neural networks. arXiv preprint arXiv:1805.06085, 2018.
+
+Yoni Choukroun, Eli Kravchik, Fan Yang, and Pavel Kisilev. Low-bit quantization of neural networks for efficient inference. In 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pp. 3009–3018. IEEE, 2019.
+
+Xin Dong, Shangyu Chen, and Sinno Pan. Learning to prune deep neural networks via layer-wise optimal brain surgeon. In Advances in Neural Information Processing Systems, 2017.
+
+Zhen Dong, Zhewei Yao, Amir Gholami, Michael W Mahoney, and Kurt Keutzer. Hawq: Hessian aware quantization of neural networks with mixed-precision. In Proceedings of the IEEE International Conference on Computer Vision, pp. 293–302, 2019.
+
+Steven K. Esser, Jeffrey L. McKinstry, Deepika Bablani, Rathinakumar Appuswamy, and Dharmendra S. Modha. Learned step size quantization. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ rkgO66VKDS.
+
+Ruihao Gong, Xianglong Liu, Shenghu Jiang, Tianxiang Li, Peng Hu, Jiazhen Lin, Fengwei Yu, and Junjie Yan. Differentiable soft quantization: Bridging full-precision and low-bit neural networks. arXiv preprint arXiv:1908.05033, 2019.
+
+Zichao Guo, Xiangyu Zhang, Haoyuan Mu, Wen Heng, Zechun Liu, Yichen Wei, and Jian Sun. Single path one-shot neural architecture search with uniform sampling. In European Conference on Computer Vision, pp. 544–560. Springer, 2020.
+
+Qingchang Han, Yongmin Hu, Fengwei Yu, Hailong Yang, Bing Liu, Peng Hu, Ruihao Gong, Yanfei Wang, Rui Wang, Zhongzhi Luan, et al. Extremely low-bit convolution optimization for quantized neural network on modern computer architectures. In 49th International Conference on Parallel Processing-ICPP, pp. 1–12, 2020.
+
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015.
+
+Babak Hassibi and David G Stork. Second order derivatives for network pruning: Optimal brain surgeon. In Advances in neural information processing systems, pp. 164–171, 1993.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
+
+Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Quantized neural networks: Training neural networks with low precision weights and activations. The Journal of Machine Learning Research, 18(1):6869–6898, 2017.
+
+Itay Hubara, Yury Nahshan, Yair Hanani, Ron Banner, and Daniel Soudry. Improving post training neural quantization: Layer-wise calibration and integer programming. arXiv preprint arXiv:2006.10518, 2020.
+
+Daniel Jakubovitz, Raja Giryes, and Miguel RD Rodrigues. Generalization error in deep learning. In Compressed Sensing and Its Applications, pp. 153–193. Springer, 2019.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Yann A LeCun, Leon Bottou, Genevieve B Orr, and Klaus-Robert M ´ uller. Efficient backprop. In ¨ Neural networks: Tricks of the trade, pp. 9–48. Springer, 2012.
+
+Yuhang Li, Xin Dong, and Wei Wang. Additive powers-of-two quantization: An efficient nonuniform discretization for neural networks. In International Conference on Learning Representations, 2019.
+
+Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollar. Focal loss for dense ´ object detection. In Proceedings of the IEEE international conference on computer vision, pp. 2980–2988, 2017.
+
+James Martens. Deep learning via hessian-free optimization. 2010.
+
+James Martens and Roger Grosse. Optimizing neural networks with kronecker-factored approximate curvature. In International conference on machine learning, pp. 2408–2417, 2015.
+
+Markus Nagel, Mart van Baalen, Tijmen Blankevoort, and Max Welling. Data-free quantization through weight equalization and bias correction. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1325–1334, 2019.
+
+Markus Nagel, Rana Ali Amjad, Mart van Baalen, Christos Louizos, and Tijmen Blankevoort. Up or down? adaptive rounding for post-training quantization. arXiv preprint arXiv:2004.10568, 2020.
+
+Yury Nahshan, Brian Chmiel, Chaim Baskin, Evgenii Zheltonozhskii, Ron Banner, Alex M Bronstein, and Avi Mendelson. Loss aware post-training quantization. arXiv preprint arXiv:1911.07190, 2019.
+
+Antonio Polino, Razvan Pascanu, and Dan Alistarh. Model compression via distillation and quantization. arXiv preprint arXiv:1802.05668, 2018.
+
+Ilija Radosavovic, Raj Prateek Kosaraju, Ross Girshick, Kaiming He, and Piotr Dollar. Designing ´ network design spaces. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10428–10436, 2020.
+
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pp. 91–99, 2015.
+
+Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4510–4520, 2018.
+
+Hardik Sharma, Jongse Park, Naveen Suda, Liangzhen Lai, Benson Chau, Vikas Chandra, and Hadi Esmaeilzadeh. Bit fusion: Bit-level dynamically composable architecture for accelerating deep neural network. In 2018 ACM/IEEE 45th Annual International Symposium on Computer Architecture (ISCA), pp. 764–775. IEEE, 2018.
+
+Mingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, Mark Sandler, Andrew Howard, and Quoc V Le. Mnasnet: Platform-aware neural architecture search for mobile. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2820–2828, 2019.
+
+Lucas Theis, Iryna Korshunova, Alykhan Tejani, and Ferenc Huszar. Faster gaze prediction with ´ dense networks and fisher pruning. arXiv preprint arXiv:1801.05787, 2018.
+
+Kuan Wang, Zhijian Liu, Yujun Lin, Ji Lin, and Song Han. Haq: Hardware-aware automated quantization with mixed precision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8612–8620, 2019.
+
+Peisong Wang, Qiang Chen, Xiangyu He, and Jian Cheng. Towards accurate post-training network quantization via bit-split and stitching. In Proc. 37nd Int. Conf. Mach. Learn.(ICML), 2020.
+
+Barret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. arXiv preprint arXiv:1611.01578, 2016.
+
+# A MAIN PROOFS
+
+# A.1 PROOF OF THEOREM 3.1
+
+Proof. We will prove the theorem using quadratic form. Denote the weight vector shape as $\theta \in \mathbb { R } ^ { d }$ , and the network output vector shape as $\mathbf { z } ^ { ( n ) } \in \mathbb { R } ^ { m }$ . The quadratic form of the $\Delta \boldsymbol { \theta } ^ { \top } \bar { \mathbf { H } ^ { ( \theta ) } } \Delta \boldsymbol { \theta }$ can be represented by:
+
+$$
+\Delta \theta ^ { \mathsf { T } } \mathbf { H } ^ { ( \theta ) } \Delta \theta = \sum _ { i = 1 } ^ { d } \Delta \theta _ { i } ^ { 2 } \left( \frac { \partial ^ { 2 } L } { \partial \theta _ { i } ^ { 2 } } \right) + 2 \sum _ { i < j } ^ { d } \Delta \theta _ { i } \Delta \theta _ { j } \frac { \partial L } { \partial \theta _ { i } \theta _ { j } } = \sum _ { i = 1 } ^ { d } \sum _ { j = 1 } ^ { d } \left( \Delta \theta _ { i } \Delta \theta _ { j } \frac { \partial L } { \partial \theta _ { i } \theta _ { j } } \right) ,
+$$
+
+where $L$ is the cross-entropy loss. Based on Eq. (5), we have
+
+$$
+\frac { \partial ^ { 2 } L } { \partial \theta _ { i } \theta _ { j } } = \sum _ { k , l } ^ { m } \frac { \partial \mathbf { z } _ { k } ^ { ( n ) } } { \partial \theta _ { l } } \frac { \partial ^ { 2 } L } { \partial \mathbf { z } _ { k } ^ { ( n ) } \mathbf { z } _ { l } ^ { ( n ) } } \frac { \partial \mathbf { z } _ { l } ^ { ( n ) } } { \partial \theta _ { j } }
+$$
+
+Substituting above equation in Eq. (12), we have
+
+$$
+\begin{array} { l } { { \displaystyle \Delta \theta ^ { \top } { \bf H } ^ { ( \theta ) } \Delta \theta = \sum _ { i = 1 } ^ { d } \sum _ { j = 1 } ^ { d } \Delta \theta _ { i } \Delta \theta _ { j } \left( \displaystyle \sum _ { k = 1 } ^ { m } \frac { m } { i \omega _ { i } } \frac { \partial { \bf z } _ { k } ^ { ( n ) } } { \partial \theta _ { i } } \frac { \partial ^ { 2 } L } { \partial { \bf z } _ { k } ^ { ( n ) } { \bf z } _ { i } ^ { ( n ) } } \frac { \partial { \bf z } _ { l } ^ { ( n ) } } { \partial \theta _ { j } } \right) } } \\ { { \displaystyle \ = \sum _ { i = 1 } ^ { d } \sum _ { j = 1 } ^ { d } \sum _ { k = 1 } ^ { m } \sum _ { l = 1 } ^ { m } \left( \Delta \theta _ { i } \Delta \theta _ { j } \frac { \partial { \bf z } _ { k } ^ { ( n ) } } { \partial \theta _ { i } } \frac { \partial ^ { 2 } L } { \partial { \bf z } _ { k } ^ { ( n ) } { \bf z } _ { l } ^ { ( n ) } } \frac { \partial { \bf z } _ { l } ^ { ( n ) } } { \partial \theta _ { j } } \right) } } \\ { { \displaystyle \ = \sum _ { k = 1 } ^ { m } \sum _ { l = 1 } ^ { m } \left( \frac { \partial ^ { 2 } L } { \partial { \bf z } _ { k } ^ { ( n ) } { \bf z } _ { l } ^ { ( n ) } } \right) \left( \sum _ { i = 1 } ^ { d } \Delta \theta _ { i } \frac { \partial { \bf z } _ { k } ^ { ( n ) } } { \partial \theta _ { i } } \right) \left( \sum _ { j = 1 } ^ { d } \Delta \theta _ { j } \frac { \partial { \bf z } _ { k } ^ { ( n ) } } { \partial \theta _ { j } } \right) } } \\ { { \displaystyle \ = ( \Delta \theta { \bf J } \left[ \frac { { \bf z } ^ { ( n ) } } { \theta } \right] ) ^ { \top } { \bf H } ^ { ( \bf z } ^ { ( n ) } ) \left( \Delta \theta { \bf J } \left[ \frac { { \bf z } ^ { ( n ) } } { \theta } \right] \right) , } } \end{array}
+$$
+
+where we define the $\mathbf { J } [ \textstyle { \frac { x } { y } } ]$ is the Jacobian matrix of $x$ w.r.t. $y$ . To this end, we use the first-order Taylor expansion as we did in Eq. (8) to approximate the change in network output, i.e.,
+
+$$
+\Delta \mathbf { z } ^ { ( n ) } \approx \Delta \theta \mathbf { J } \left[ \frac { \mathbf { z } ^ { ( n ) } } { \theta } \right]
+$$
+
+Therefore, the final objective is transformed to $\Delta \mathbf { z } ^ { ( n ) , \mathsf { T } } \mathbf { H } ^ { ( \mathbf { z } ^ { ( n ) } ) } \Delta \mathbf { z } ^ { ( n ) }$ .
+
+# B EXPERIMENTS
+
+# B.1 EFFECT OF THE FIRST AND THE LAST LAYER
+
+Many papers claim that the first and the last layer can have a huge impact on the final accuracy. In this section, we investigate this phenomenon as well as the impact of the first and the last layer on hardware performances. We test ResNet-18, MobileNetV2 as well as RegNet-600MF. Our observations include:
+
+1. In terms of accuracy, the 4-bit quantization is essentially good, both of these two layers won’t drop too much accuracy (with $0 . 2 \%$ ). But in 2-bit quantization, the last fully connected layer is far more important than the first layer. We also observe that the first layer in MobileNetV2 and RegNet $3 { \times } 3$ kernels, 32 channels) is slightly more sensitive than that in ResNet-18 $7 \times 7$ kernel, 64 channels).   
+2. In terms of model size, the first layer merely has a minor impact because the input images only have 3 channels, while the last layer contains many weight parameters and greatly affects the memory footprint. We should point out that just the model size in the first layer is low doesn’t mean the memory burden is low, because the input image will cost huge memory space.   
+3. In terms of latency, the situation depends on the architecture. For example, in ResNet-18 the first layer has a huge impact on the latency, while in MobileNetV2 and RegNet-600MF the last layer is more important than the first layer. This is because the latency is affected by multiple factors, such as the input size of the featuremap, the FLOPs, and the weight memory size. The arithmetic intensity (OPs/byte) greatly affects the latency. We find that the operations with high arithmetic intensity, i.e., shallow layers in the network, generate a less latency gap between different bitwidths.
+
+In conclusion, we find that keeping the first and the last layer 8-bit is unnecessary. Especially in ResNet-18, we find that setting all layers to 4-bit results in $5 3 . 3 ~ \mathrm { m s }$ latency and is faster than the $5 9 . 8 ~ \mathrm { m s }$ in 2-bit quantization (with first and last layer 8-bit), but the accuracy is even $4 \%$ higher. Such phenomenon indicates the potential power of the mixed precision.
+
+# B.2 EFFECT OF DATA
+
+We evaluated the influence of the size of calibration dataset and the source of the data on ResNet-18. We test different numbers of input data point and find that the improvement in 4-bit quantization is trivial. Yet in 2-bit quantization we can see that the accuracy increases $5 \%$ when #data points increase. We also test the distilled data introduced in ZeroQ (Cai et al., 2020). Distilled data is learned from pretrained models’ BN statistics, i.e., $\begin{array} { r } { \mathbf { x } _ { \mathrm { d i s t i l l e d } } ^ { 1 } = \arg \operatorname* { m i n } _ { \mathbf { x } \in \mathcal { R } } \sum _ { i = 1 } ^ { n } ( ( \mu _ { i } - \hat { \mu } _ { i } ) ^ { 2 } + ( \varsigma _ { i } - } \end{array}$ $\hat { \varsigma } _ { i } )$ ) where $\mu _ { i }$ and $\varsigma _ { i }$ is the original mean and variance in BN statistics of the $( i )$ -th layer. We find that distilled data performs good in 4-bit quantization but still has a large margin with the original ImageNet dataset in 2-bit quantization. We also find the final accuracy does not benefit much from the increase of number of distilled data, this might because the distilled data are minimized by a same objective and has low diversity.
+
+Table 6: Impact of the first and the last layer   
+
+<table><tr><td rowspan="2">Models</td><td colspan="2">No Quantization</td><td colspan="3">Precision 4/8</td><td colspan="3">Precision 2/8</td></tr><tr><td>First</td><td>Last</td><td>Accuracy</td><td>Model Size</td><td>Latency</td><td>Accuracy</td><td>Model Size</td><td>Latency</td></tr><tr><td rowspan="4">ResNet-18 FP: 71.08</td><td>√</td><td>√</td><td>70.76</td><td>5.81MB</td><td>70.72 ms</td><td>66.30</td><td>3.15 MB</td><td>59.84 ms</td></tr><tr><td>X</td><td>√</td><td>70.66</td><td>5.81 MB</td><td>53.76 ms</td><td>65.95</td><td>3.15MB</td><td>31.20 ms</td></tr><tr><td>√</td><td>X</td><td>70.64</td><td>5.57MB</td><td>70.08 ms</td><td>64.87</td><td>2.79 MB</td><td>58.72 ms</td></tr><tr><td>X</td><td>X</td><td>70.58</td><td>5.56 MB</td><td>53.28 ms</td><td>64.53</td><td>2.78 MB</td><td>30.88 ms</td></tr><tr><td rowspan="4">MobileNetV2 FP: 72.49</td><td>√</td><td>√</td><td>71.80</td><td>2.26MB</td><td>32.80 ms</td><td>59.59</td><td>1.74 MB</td><td>30.40 ms</td></tr><tr><td>X</td><td>���</td><td>71.69</td><td>2.26 MB</td><td>32.64 ms</td><td>59.13</td><td>1.74 MB</td><td>30.24 ms</td></tr><tr><td>√</td><td>×</td><td>71.42</td><td>1.65MB</td><td>31.52 ms</td><td>56.29</td><td>0.83MB</td><td>28.48 ms</td></tr><tr><td>×</td><td>×</td><td>71.42</td><td>1.65MB</td><td>31.36 ms</td><td>55.58</td><td>0.82 MB</td><td>28.32 ms</td></tr><tr><td rowspan="4">RegNet-600MF FP: 73.71</td><td>√</td><td>√</td><td>72.98</td><td>3.19MB</td><td>31.84 ms</td><td>65.66</td><td>1.84 MB</td><td>23.20ms</td></tr><tr><td>X</td><td>√</td><td>72.89</td><td>3.19 MB</td><td>31.68 ms</td><td>65.83</td><td>1.85 MB</td><td>22.88 ms</td></tr><tr><td>√</td><td>X</td><td>72.69</td><td>2.94 MB</td><td>31.20 ms</td><td>62.93</td><td>1.47 MB</td><td>22.40 ms</td></tr><tr><td>×</td><td>X</td><td>72.73</td><td>2.94 MB</td><td>31.04 ms</td><td>63.08</td><td>1.47 MB</td><td>22.08 ms</td></tr></table>
+
+![](images/59c4631a56e6fcef3e7851a2ffbf0201f57db5f4221bb579520373e5836a23d2.jpg)  
+Figure 3: Effect of #data points and data source.
+
+B.3 MOBILE CPU LATENCY GUARANTEED MIXED PRECISION
+
+![](images/905a2edc6b76eb7f1bee8d3489609c1999c6d297ce9f43851193a791e4a0d9a5.jpg)  
+Figure 4: Mixed precision results on ResNet-18 and 50.
+
+In this section, we test the mobile CPU latency guaranteed mixed precision. The latency lookup table is tested using the technique in Gong et al. (2019). We only validate it on ResNet-18 and ResNet-50 because the current low-bit General Matrix Multiply (GEMM) implementation only supports normal convolution. The results concur with Fig. 2. Below 4-bit, the mixed precision can achieve better task performance than the unified precision models. For ResNet-50, the improvement is lower than that for ResNet-18 and any other mixed precision models. We think this is because the sensitivity in ResNet-50 is not distinct and therefore the improvement brought by mixed precision is trivial.
+
+# B.4 IMPLEMENTATION
+
+# B.4.1 LEARNING STRATEGIES
+
+In this work, we mainly focus on developing optimization objective rather than optimization strategies. We observe adaptive rounding performs well in post-training quantization. A brief introduction on AdaRound is given below, the detailed algorithm can be found in Nagel et al. (2020).
+
+Traditional quantization function is performed by rounding-to-nearest operation: $\hat { \textbf { w } } = \textbf { \em s } \times \textbf {  { n } }$ $\mathrm { c l i p } ( \lfloor \mathbf { w } / s \rceil , \bar { n } , p )$ . AdaRound optimizes the rounding policy in post-training quantization. Specifically, all weights are initially rounded by floor operation, and a learnable variable $\mathbf { v }$ determines the final rounding result to be flooring or ceiling. A sigmoid-like function $\sigma ( \cdot )$ keeps the learnable variable $\mathbf { v }$ moving between 0 and 1 and a regularization term assures the $\sigma ( \mathbf { v } )$ can converged to either 0 or 1. The formulation is given by
+
+$$
+\hat { \mathbf { w } } = s \times \mathrm { c l i p } \left( \lfloor \frac { \mathbf { w } } { s } \rfloor + \sigma ( \mathbf { v } ) , n , p \right)
+$$
+
+The minimization problem together with the regularization is given by
+
+$$
+\underset { \mathbf { v } } { \arg \operatorname* { m i n } } \mathbb { E } \left[ \Delta \mathbf { z } ^ { ( \ell ) , \top } \mathrm { d i a g } ( ( \frac { \partial L } { \partial \mathbf { z } _ { 1 } ^ { ( \ell ) } } ) ^ { 2 } , \dots , ( \frac { \partial L } { \partial \mathbf { z } _ { a } ^ { ( \ell ) } } ) ^ { 2 } ) \Delta \mathbf { z } ^ { ( \ell ) } \right] + \lambda \sum _ { i } \left( 1 - | 2 \sigma ( \mathbf { v } _ { i } ) - 1 | ^ { \beta } \right) ,
+$$
+
+where progressively decreasing $\beta$ in the calibration ensures the $\sigma ( \mathbf { v } )$ converged to binary values. The activations cannot be quantized using adaptive rounding because they vary with different input data points. Thus, we can only adjust its quantization step size Esser et al. (2020). Denoting the quadratic loss in above equation as $L _ { q }$ , the gradients of step size is given by
+
+$$
+\frac { \partial L _ { q } } { \partial s } = \left\{ \begin{array} { l l } { \displaystyle \frac { \partial L _ { q } } { \partial \hat { \mathbf { x } } } } & { \mathrm { i f } \mathbf { x } > n } \\ { \displaystyle \frac { \partial L _ { q } } { \partial \hat { \mathbf { x } } } \left( \frac { \hat { \mathbf { x } } } { s } - \frac { \mathbf { x } } { s } \right) } & { \mathrm { i f } 0 \leq \mathbf { x } < \alpha } \\ { 0 } & { \mathrm { i f } \mathbf { x } \leq 0 } \end{array} \right. ,
+$$
+
+where all step size in the block will be optimized.
+
+# Algorithm 2: Genetic algorithm
+
+B.4.2 GENETIC ALGORITHM FOR MIXED PRECISION   
+
+<table><tr><td>Input:Random initialized population Po with population size S; Iteration T, mutation probability p; Hardware performance threshold δ; Hardware measurement function H()</td></tr><tr><td>TopK= ; for all t = 1,2,...,T-th iteration do</td></tr><tr><td>Evaluate fitness value (Eq.(11)) for each individual ;</td></tr><tr><td>Update and sort TopK based on fitness function;</td></tr><tr><td>repeat</td></tr><tr><td>New bitwidth configuration by crossover Ccross = Crossover(TopK);</td></tr><tr><td>Pcrossover := Pcrossover + Ccross if H(Ccross)&lt; 8;</td></tr><tr><td>until Size of Pcrossver equal to S/2;</td></tr><tr><td>repeat New bitwidth configuration by mutation Cmutate = Mutate(TopK, probability = p);</td></tr><tr><td></td></tr><tr><td>Pmutate := Pmutate + Cmutate if H(Cmutate) &lt; δ;</td></tr><tr><td></td></tr><tr><td>until Size of Pmutate equal to S/2;</td></tr><tr><td></td></tr><tr><td>Pt = Pcrossover U Pmutate;</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>Pmutate = O, Pcrossover = Ω;</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></table>
+
+Get the best fitted entry and then do the overall block reconstruction (cf. algorithm 1); return mixed precision model
+
+# B.4.3 LATENCY ACQUISITION
+
+We test the latency of quantized neural networks on a self-developed simulator of a precisionvariable accelerator for NN. The basic architecture of this accelerator is inspired by typical systolicmatrix multiplication. The accelerator supports the per-channel quantization parameter. The precision of each layer of a NN is highly configurable in this accelerator, supporting 9 types of precision combination: activation: 2-, 4-, 8-bit $\times$ weight: 2-, 4-, 8-bit, see Fig. 5a. With the support of scalable function-unit (Sharma et al., 2018), the peak performance of the accelerator is able to achieve corresponding linear improvement as the precision decreases. For example, the peak performance of this accelerator is 256 GMAC/s in $\mathsf { 8 - b i t } \times \mathsf { 8 }$ -bit precision, and it scales to 512 GMAC/s in 8- bit $\times 4$ -bit precision and 4 TMAC/s in $2 { \cdot } \mathrm { b i t } \times 2$ -bit precision. Although this accelerator provides considerable computation resources especially in low precision, the parallelism of the specific layer (like depthwise convolution) and the bandwidth of on-chip buffer is limited. Consequently, actual performance may not scale accurately along with the peak performance, and the final performance differs according to the size and type of layers. The simulator performs cycle-accurate simulation and evaluation for a given NN executed on the accelerator, so we can get an equivalent evaluation by using this simulator. The simulator is available in the provided source codes.
+
+For the acquisition of mobile ARM CPU latency, we adopt the redesigned low-bit GEMM implementation in Han et al. (2020). Fig. 5b shows a brief overview of the low-bit GEMM implementation. Since there is no instruction supporting the bit-width below 8 on ARM architecture, we can not get a higher computation efficiency for extremely low-bit such as 2-bit and 4-bit. But we can acquire a better memory access efficiency. The primary speedup comes from the reduction of data movement. Specifically, we can conduct more times of addition in the same 8-bit register before we have to move it to a 16-bit register to avoid overflow. The lower bit-width is used, the less movement is needed. Together with the optimized data packing and data padding, we can run mixed precision quantization on Raspberry Pi 3B, which has a 1.2 GHz 64-bit quad-core ARM Cortex-A53. Note that this implementation is not optimized for depthwise separable or group convolution, therefore we only verify the latency on ResNets.
+
+![](images/2894ef4faf7d055a2760f2fb94fbabdf97d56ea93cf2a3a37c54bab69f9f8a73.jpg)  
+Figure 5: FPGA-based and mobile CPU-based latency acquisition.
+
+# B.4.4 IMPLEMENTATION DETAILS
+
+The ImageNet dataset consists of 1.2M training images and 50K test images. We follows standard pre-process (He et al., 2016) to get 1024 $2 2 4 \times 2 2 4$ input images as the calibration dataset. We fold the batch normalization layer into convolution and freeze the BN statistics before post-training quantization. We use Adam optimizer (Kingma & Ba, 2014) to learn the weight rounding and activation range to reconstruct the block output. Note that some layers are not a component of any block, such as the first convolutional layer and the last fully connected layer and the last convolutional layer in the MobileNetV2. These layers use naive layer reconstruction. The batch size of learning is set to 32 and each block will be optimized for $2 \times 1 \dot { 0 } ^ { 4 }$ iterations. The learning rate is set to $1 0 ^ { - \bar { 3 } }$ during the whole learning process. Other hyper-parameters such as the temperature $\beta$ are kept the same with Nagel et al. (2020). For activation step size, we also use Adam optimizer and set the learning rate to 4e-5. Note that we do not implement the gradient scale as introduced in the original paper (Esser et al., 2020). After reconstruction, we will store the sensitivity measured on the calibration dataset. Note that we will store intra-block sensitivity in 2-bit quantization. The sensitivity, as well as hardware performances for each layer, will be stored in a lookup table. When calculating the fitness value and determining the hardware performances in a genetic algorithm, we will check the lookup table. For genetic algorithm, we set the population size to 50 and evolve 100 iterations to obtain the best individual. The first population is initialized by Gaussian distribution and we round the samples to integers in [0, 1, 2], corresponding to bit-width [2, 4, 8]. The mutation probability is set to 0.1. The genetic algorithm usually completes the evolution in only about 3 seconds.
+
+For object detection tasks, we use 256 training images taken from the MS COCO dataset for calibration. The image resolution is set to 800 (max size 1333) for ResNet-18 and ResNet-50, while the image resolution for MobileNetV2 is set to 600 (max size 1000). Note that we only apply block reconstruction in the backbone because other parts of the architecture, such as Feature Pyramid Net, do not have the block structure. Therefore a naive layer reconstruction is applied to the rest of the network. Learning hyper-parameters are kept the same with ImageNet experiments.

parse/train/d20KTY2VrNk/d20KTY2VrNk.md

@@ -0,0 +1,250 @@
+# Instance-Dependent Label-Noise Learning under Structural Causal Models
+
+Yu Yao1 Tongliang Liu1† Mingming Gong2 Bo Han3 Gang Niu4 Kun Zhang5
+
+1TML Lab, University of Sydney; 2University of Melbourne; 3Hong Kong Baptist University; 4RIKEN AIP; 5Carnegie Mellon University
+
+# Abstract
+
+Label noise generally degenerates the performance of deep learning algorithms because deep neural networks easily overfit label errors. Let $X$ and $Y$ denote the instance and clean label, respectively. When $Y$ is a cause of $X$ , according to which many datasets have been constructed, e.g., SVHN and CIFAR, the distributions of $P ( X )$ and $P ( { Y \vert } X )$ are generally entangled. This means that the unsupervised instances are helpful to learn the classifier and thus reduce the side effect of label noise. However, it remains elusive on how to exploit the causal information to handle the label-noise problem. We propose to model and make use of the causal process in order to correct the label-noise effect. Empirically, the proposed method outperforms all state-of-the-art methods on both synthetic and real-world labelnoise datasets.
+
+# 1 Introduction
+
+Learning with noisy labels can be dated back to [1] and has recently drawn a lot of attention [15, 19, 11, 10, 28]. In real life, large-scale datasets are likely to contain label noise. It is partly because that many cheap but imperfect data collection methods such as crowd-sourcing and web crawling are widely used to build large-scale datasets. Training with such data usually lead to poor generalization abilities of deep neural networks because they can memorize noisy labels [2, 33].
+
+To improve the generalization ability of training with noisy labels, one family of existing label-noise learning methods is to model how the label noise was generated [17, 19, 27, 34, 14]. Specifically, these methods try to reveal the transition relationship from clean labels to noisy labels of instances, i.e., the distribution $P ( \tilde { Y } | Y , X )$ , where $\tilde { Y }$ , $Y$ and $X$ are the random variables for the noisy label, latent clean label, and instance, respectively. The advantage of modelling label noise is that given only the noisy data, when the transition relationship is identifiable, classifiers can be learned to converge to the optimal ones defined by the clean data, with theoretical guarantees. However, the transition relationship is not identifiable in general. To make it identifiable, various assumptions have been made on the transition relationship. For example, Natarajan et al. [17] assume that the transition relationship is instance independent, i.e., $P ( \tilde { Y } | Y , X ) = \bar { P } ( \tilde { Y } | Y )$ ; Xia et al. [30] assume that the $P ( { \tilde { Y } } | Y , X )$ is dependent on different parts of an instance. Cheng et al. [7] assume that the label noise rates are upper bounded. In practice, these assumptions may not be satisfied and are generally hard to be verified given noisy data alone.
+
+Inspired by causal learning [20, 25, 21, 23], we provide a causal perspective of label-noise learning method named CausalNL. We exploit the causal information to help identifiability of the transition matrix $P ( { \tilde { Y } } | Y , X )$ other than making assumptions directly on the transition relationship.
+
+Specifically, we assume that the data containing instancedependent label noise is generated according to the causal graph in Fig. 1. For example, for the Street View House Numbers (SVHN) dataset [18], $X$ represents the image containing the digit; $Y$ represents the clean label of the digit shown on the plate; $Z$ represents the latent variable that captures the information affecting the generation of the images, e.g., orientation, lighting, and font style. Here $Y$ is naturally a cause of $X$ and the causal generative process can be described in the following way. First, the house plate is generated according to the street number and attached to the front door. Then, the house plate is captured by a camera (installed in a Google street view car) to form $X$ , taking into account of other factors such as illumination and viewpoint. Finally, the images containing house numbers are collected and relabeled to form the dataset. Let us denote the annotated label by the noisy label $\tilde { Y }$ as the annotator may not be always reliable, especially when the dataset is very large but the budget is limited. During the annotation process, the noisy labels were generated according to both the images and the range of predefined digit numbers. Hence, both $X$ and $Y$ are causes of $\tilde { Y }$ . Note that most existing image datasets are collected with the causal relationship that $Y$ causes $X$ . For example, see the widely used FashionMNIST and CIFAR. When we synthesize instance-dependent label noise based on them, we will have the causal graph illustrated in Fig. 1. Note also that some datasets are generated with the causal relationship that $X$ causes $Y$ . Other than using domain knowledge, the different causal relationships can be verified by employing causal discovery [26, 25, 21, 37].
+
+![](images/11a3256e8647b5d8a82c80d99a5dcb468f5604150dd14b34f720132d9fb55de5.jpg)  
+Figure 1: A graphical causal model which reveals a generative process of the data which contains instancedependent label noise, where the shaded variables are observable and the unshaded variables are latent.
+
+When the latent clean label $Y$ is a cause of $X$ , $P ( X )$ will generally contain some information about $P ( { Y \vert } X )$ . This is because, under such a generative process, the distributions of $P ( X )$ and $P ( { Y \vert } X )$ are entangled [22, 39]. To help estimate $P ( { Y \vert } X )$ with $P ( X )$ , we make use of the causal generative process to estimate $P ( X | Y )$ , which directly benefits from $P ( X )$ by generative modeling. The modeling of $P ( X | Y )$ in turn encourages the identifiability of the transition relationship and helps learn $P ( { Y \vert } X )$ . For example, in Fig. 2(a), we have added instance-dependent label-noise with a rate $45 \%$ (i.e., $\mathrm { I D L N } { - } 4 5 \% )$ to the MOON dataset and employed different methods [10, 36] to solve the label-noise learning problem. As illustrated in Fig. 2(b) and Fig. 2(c), previous methods fail to infer clean labels. In contrast, by constraining the conditional distribution of the instances, i.e., restricting the data of each class to be on a manifold by setting the dimension of the latent variable $Z$ to be 1-dimensional, the label transition as well as the clean labels can be successfully recovered (by the proposed method), which is showed in Fig. 2(d). It is worth noting that the idea of finding $P ( { Y \vert } X )$ by modeling $P ( Y )$ and $P ( X | Y )$ instead has been exploited in the context of domain adaptation; for instance, it inspires target shift, (generalized) conditional shift and other settings for domain adaptation [38, 9]
+
+Specifically, to make use of the causal graph to contribute to the identifiability of the transition matrix, we propose a causally inspired deep generative method, which models the causal structure with all the observable and latent variables, i.e., the instance $X$ , the noisy label $\tilde { Y }$ , the latent feature $Z$ , and the latent clean label $Y$ . The proposed generative model captures the variables’ relationship indicated by the causal graph. Furthermore, built on the variational autoencoder (VAE) framework [12], we build an inference network which could efficiently infer the latent variables $Z$ and $Y$ when maximising the marginal likelihood $p ( X , { \tilde { Y } } )$ on the given noisy data. In the decoder phase, the data will be reconstructed by exploiting the conditional distribution of instances $P ( X | Y , Z )$ and the transition relationship $P ( { \tilde { Y } } | Y , X )$ , i.e.,
+
+$$
+p _ { \theta } ( X , \tilde { Y } ) = \int _ { z , y } P ( Z = z ) P ( Y = y ) p _ { \theta _ { 1 } } ( X | Y = y , Z = z ) p _ { \theta _ { 2 } } ( \tilde { Y } | Y = y , X ) \mathrm { d } z \mathrm { d } y
+$$
+
+will be exploited, where $\theta : = \left( \theta _ { 1 } , \theta _ { 2 } \right)$ are the parameters of the causal generative model (more details can be found in Section 3). Ay a high level, according to the equation, given the noisy data and the distributions of $Z$ and $Y$ , constraining $p _ { \theta _ { 1 } } ( X | Y , Z )$ will also greatly reduce the uncertainty of $p _ { \theta _ { 2 } } ( { \tilde { Y } } | Y , X )$ and thus contribute to the identifiability of the transition matrix. Note that adding a constraint on $p _ { \theta _ { 1 } } ( X | Y , Z )$ is natural; for example, images often have a low-dimensional manifold [3]. We can restrict $P ( Z )$ to fulfill the constraint on $p _ { \theta _ { 1 } } ( X | Y , Z )$ . By exploiting the causal structure and the constraint on instances to better model label noise, the proposed method significantly outperforms the baselines. When the label noise rate is large, the superiority is evidenced by a large gain in the classification performance.
+
+![](images/8edd33b129b52da12f985bc53180e31caf4a896b594844f7855f0560017d3e3c.jpg)  
+Figure 2: (a) An illustration of the MOON training dataset which contains $4 5 \%$ of instance-dependent label noise. Different instances have different noise rates which are randomly generated according to Xia et al. [30]. (b)-(d) The illustration of the classification performance of co-teaching, mixup, and our method, respectively.
+
+The rest of the paper is organized as follows. In Section 2, we briefly review the background knowledge of label-noise learning and causality. In Section 3, we formulate our method, named CausalNL, and discuss how it helps to learn a clean classifier, followed by the implementation details. Experimental validations are provided in Section 4. Section 5 concludes the paper.
+
+# 2 Noisy Labels and Causality
+
+In this section, firstly, we introduce how to model label noise. Then, we introduce the structural causal model and discuss how to exploit the model to encourage the identifiability of the transition relationship and help learn the classifier.
+
+Transition Relationship To build a statistically consistent classifier that converges to the optimal classifier defined on clean data by only employing noisy data, the transition relationship $P ( \tilde { Y } | Y , X )$ has to be identified. Given an instance, the conditional distribution can be written in an $C \times C$ matrix which is called the transition matrix [19, 29, 30], where $C$ represents the number of classes. Specifically, for each instance $x$ , there is a transition matrix $T ( x )$ . The $i j$ -th entry of the transition matrix is $T _ { i j } ( x ) = P ( \tilde { Y } = i | Y = j , X = x )$ , which represents the probability that the instance $x$ with the clean label $Y = j$ will have a noisy label $\tilde { Y } = i$ .
+
+The transition matrix has been widely studied to build statistically consistent classifiers, because the clean class posterior distribution $P ( \pmb { Y } | \pmb { x } ) = [ P ( \pmb { Y } = 1 | \pmb { X } = \pmb { x } ) , \dots , P ( \pmb { Y } = C | \pmb { X } = \pmb { x } ) ] ^ { \top }$ can be inferred by using the transition matrix and the noisy class posterior $P ( \tilde { Y } | x ) = [ P ( \tilde { Y } =$ $1 | X = x ) , \ldots , P ( \tilde { Y } = C | X = x ) ] ^ { \top }$ , i.e., we have $P ( \tilde { \mathbf { Y } } | x ) = T ( x ) P ( \mathbf { Y } | x )$ . Specifically, the transition matrix has been used to modify loss functions to build risk-consistent estimators; see, e.g., [8, 19, 35, 28], and has been used to correct hypotheses to build classifier-consistent algorithms; see, e.g., [17, 24, 19]. Moreover, the state-of-the-art statically inconsistent algorithms [11, 10] also use diagonal entries of the transition matrix to help select reliable examples used for training.
+
+However, the distribution $P ( { \tilde { Y } } | Y , X )$ is not generally identifiable [28]. To make it identifiable, one has to resort to additional assumptions. The most widely used assumption is that given clean label $Y$ , the noisy label $\tilde { Y }$ is conditionally independent of instance $X$ , i.e., $\mathbf { \bar { \xi } } P ( \tilde { Y } | Y , X ) = P ( \tilde { Y } | Y )$ . Under such an assumption, the transition relationship $P ( \tilde { Y } | Y )$ can be successfully identified with the anchor point assumption [15, 34, 14]. However, in the real-world scenarios, this assumption may be hard to satisfied. Although $P ( \tilde { Y } | Y )$ can be used to approximate $P ( { \tilde { Y } } | Y , X )$ , the approximation error can be large in many cases. As for the efforts to model the instance-dependent transition matrix directly, existing methods rely on rather strong assumptions, e.g., the bounded noise rate assumption [7], the part-dependent label noise assumption [30], and the requirement of additional information about the transition matrix [4]. Although the assumptions help the methods achieve superior performance empirically, they are generally difficult to verify or fulfill, limiting their applications in practice.
+
+Structural Causal Model Motivated by the limitation of the current methods, we provide a new causal perspective to learn the identifiable of instance-dependent label noise model. Here we briefly introduce some background knowledge of causality [25] used in this paper. A structural causal model (SCM) consists of a set of variables connected by a set of functions. It represents a flow of information and reveals causal relationships among all the variables, providing a fine-grained description of the data generation process. The causal structure encoded by SCMs can be represented as a graphical casual model as shown in Fig. 1, where each node is a variable and each edge is a function involving noise. The SCM corresponding to the graph in Fig. 1 can be written as
+
+![](images/8870323952342d39d1beb50c1a7bee11d402367d8f4504321c5813b9c4f8fe5c.jpg)  
+Figure 3: A working flow of our method.
+
+$$
+Z = \epsilon _ { Z } , Y = \epsilon _ { Y } , X = f ( Z , Y , \epsilon _ { X } ) , \tilde { Y } = f ( X , Y , \epsilon _ { \tilde { Y } } ) ,
+$$
+
+where $\epsilon _ { Z } , \epsilon _ { Y } ,$ $\epsilon _ { X }$ and $\epsilon _ { \tilde { Y } }$ are independent exogenous variables, and they sometimes also called error variables. For example, $\epsilon _ { X }$ are an error variable for $X$ , which is responsible for any difference between the actual value of $X$ and the value predicted on the basis of $Z$ and $Y$ alone. Each equation specifies a distribution of a variable conditioned on its parents (which are an empty set for root cause variables in the graph).
+
+By making use of the SCM, the benefit of the instances to learning the classifier can be clearly explained. Specifically, the instance $X$ is a function of its label $Y$ and latent feature $Z$ , which means that the instance $X$ is generated from $Y$ and $Z$ . Therefore $X$ must contains information about its clean label $Y$ and latent feature $Z$ . That is the reason that $P ( X )$ can help identify $P ( { Y \vert } X )$ and also $P ( Z | X )$ . However, since we do not have clean labels, it is hard to fully identify $P ( { Y \vert } X )$ from $P ( X )$ in the unsupervised setting. For example, on the MOON dataset shown in Fig. 2 , we can possibly discover the two clusters by enforcing the manifold constraint, but it is impossible to see which class each cluster belongs to. We show in the following that we can make use of the property of $P ( X | Y )$ to help model label noise, i.e., encourage the identifiability of the transition relationship, thereby learning a better classifier.
+
+Specifically, under the Markov condition [20], which intuitively means the independence of exogenous variables, the joint distribution $P ( \tilde { Y } , X , Y , Z )$ specified by the SCM can be factorized as follows.
+
+$$
+P ( X , \tilde { Y } , Y , Z ) = P ( Y ) P ( Z ) P ( X | Y , Z ) P ( \tilde { Y } | Y , X ) .
+$$
+
+This motivates us to extend VAE [12] to perform inference in our causal model to fit the noisy data in the next section. In the decoder phase, given the noisy data and the distributions of $Z$ and $Y$ , adding a constraint on $P ( X | Y , Z )$ will reduce the uncertainty in the distribution $P ( { \tilde { Y } } | Y , X )$ . In other words, modeling of $P ( X | Y , Z )$ will encourage the identifiability of the transition relationship and thus better model label noise. Since $P ( { \tilde { Y } } | Y , X )$ functions as a bridge to connect the noisy labels to clean labels, we accordingly can better learn $P ( { Y \vert } X )$ or the classifier by only using the noisy data.
+
+There are normally two ways to add constraints on the instances, i.e., assuming a specific parametric generative model or introducing prior knowledge of the instances. In this paper, since we mainly study the image classification problem with noisy labels, we focus on the manifold property of images and apply the low-dimensional manifold constraint to the instances.
+
+# 3 Causality Captured Instance-Dependent Label-Noise Learning
+
+In this section, we propose a structural generative method which captures the causal relationship and utilizes $P ( X )$ to help identify the label-noise transition matrix, and therefore, our method leads to a better classifier that assigns more accurate labels.
+
+# 3.1 Variational Inference under the Structural Causal Model
+
+To model the generation process of noisy data and to approximate the distribution of the noisy data, our method is designed to follow the causal factorization (see Eq. 2). Specifically, our model contains two decoder networks which jointly model a distribution $p _ { \theta } ( X , { \tilde { Y } } | Y , Z )$ and two encoder (inference) networks which jointly model the posterior distribution $q _ { \phi } ( Z , Y | X )$ . Here we discuss each component of our model in detail.
+
+Let the two decoder networks model the distributions $p _ { \theta _ { 1 } } ( X | Y , Z )$ and $p _ { \theta _ { 2 } } ( { \tilde { Y } } | Y , X )$ , respectively. Let $\theta _ { 1 }$ and $\theta _ { 2 }$ be learnable parameters of the distributions. Without loss of generality, we set $p ( Z )$ to a standard normal distribution and $p ( Y )$ to a uniform distribution. Then, modeling the joint distribution in Eq. 2 boils down to modeling the distribution $p _ { \theta } ( X , { \tilde { Y } } | Y , Z )$ , which is decomposed as follows:
+
+$$
+\begin{array} { r } { p _ { \theta } ( X , \tilde { Y } | Y , Z ) = p _ { \theta _ { 1 } } ( X | Y , Z ) p _ { \theta _ { 2 } } ( \tilde { Y } | Y , X ) . } \end{array}
+$$
+
+To infer latent variables $Z$ and $Y$ with only observable variables $X$ and $\tilde { Y }$ , we design an inference network which model the variational distribution $q _ { \phi } ( Z , Y | \tilde { Y } , X )$ . Specifically, let $q _ { \phi _ { 2 } } ( Z | Y , X )$ and $q _ { \phi _ { 1 } } ( Y | \tilde { Y } , X )$ be the distributions parameterized by learnable parameters $\phi _ { 1 }$ and $\phi _ { 2 }$ , and then the posterior distribution can be decomposed as follows:
+
+$$
+q _ { \phi } ( Z , Y | \tilde { Y } , X ) = q _ { \phi _ { 2 } } ( Z | Y , X ) q _ { \phi _ { 1 } } ( Y | \tilde { Y } , X ) ,
+$$
+
+where we do not include $\tilde { Y }$ as a conditioning variable in $q _ { \phi _ { 2 } } ( Z | Y , X )$ because the causal graph implies $Z \perp \tilde { Y } | X , Y$ . One problem with this posterior form is that we cannot directly employ $q _ { \phi _ { 1 } } ( Y | \tilde { Y } , X )$ to predict labels on the test data, on which $\tilde { Y }$ is absent.
+
+To reduce computational costs and to allow our method efficiently infer clean labels, we approximate $q _ { \phi _ { 1 } } ( Y | \tilde { Y } , X )$ by assuming that given the instance $X$ , the clean label $Y$ is conditionally independent from the noisy label $\tilde { Y }$ , i.e., $q _ { \phi _ { 1 } } ( Y | \tilde { Y } , X ) = q _ { \phi _ { 1 } } ( Y | X )$ . This approximation is expected not to have very large approximation error because the images contain sufficient information to predict the clean labels. Thus, we could simplify Eq. 4 as follows
+
+$$
+q _ { \phi } ( Z , Y | X ) = q _ { \phi _ { 2 } } ( Z | Y , X ) q _ { \phi _ { 1 } } ( Y | X ) ,
+$$
+
+such that our encoder networks model $q _ { \phi _ { 2 } } ( Z | Y , X )$ and $q _ { \phi _ { 1 } } ( Y | X )$ , respectively. This way, $q _ { \phi _ { 1 } } ( Y | X )$ can be used to infer clean labels efficiently.2 We also found that the encoder network modelling $q _ { \phi _ { 1 } } ( Y | X )$ can directly act as a regularizer, which helps to identify $p _ { \theta _ { 2 } } ( { \tilde { Y } } | Y , X )$ . Moreover, be benefited from this, our method can serve as a general framework which can be easily integrated with the current discriminative label-noise methods [28, 16, 10], and we will showcase it by collaborating co-teaching [10] with our method.
+
+Optimization of Parameters Because the marginal distribution $p _ { \theta } ( X , { \tilde { Y } } )$ is usually intractable, to learn the set of parameters $\{ \theta _ { 1 } , \theta _ { 2 } , \phi _ { 1 } , \phi _ { 2 } \}$ given only noisy data, we follow the variational inference framework [5] to maximize the negative evidence lower-bound $\operatorname { E L B O } ( x , \tilde { y } )$ of the marginal likelihood of each datapoint $( x , \tilde { y } )$ instead of maximizing the marginal likelihood itself. By ensembling our decoder and encoder networks, $\operatorname { E L B O } ( x , \tilde { y } )$ is derived as follows:
+
+$$
+\begin{array} { r l } & { \mathrm { E L B O } ( x , \tilde { y } ) = \mathbb { E } _ { ( z , y ) \sim q _ { \phi } ( Z , Y \mid x ) } \left[ \log p _ { \theta _ { 1 } } ( x \vert y , z ) \right] + \mathbb { E } _ { y \sim q _ { \phi _ { 1 } } ( Y \mid x ) } \left[ \log p _ { \theta _ { 2 } } ( \tilde { y } \vert y , x ) \right] } \\ & { \phantom { = \ } - k l ( q _ { \phi _ { 1 } } ( Y \vert x ) \| p ( Y ) ) - \mathbb { E } _ { y \sim q _ { \phi _ { 1 } } ( Y \mid x ) } \left[ k l ( q _ { \phi } ( Z \vert y , x ) \| p ( Z ) ) \right] , } \end{array}
+$$
+
+where $k l ( \cdot )$ is the Kullback–Leibler divergence between two distributions. The derivation details are left out in Appendix A. Our model learns the class-conditional distribution $P ( X | Y )$ by maximizing the first expectation in ELBO, which is equivalent to minimizing the reconstruction loss [12]. By learning $P ( X )$ , the inference network $q _ { \phi _ { 1 } } ( Y | X )$ has to select a suitable parameter $\phi ^ { * }$ which samples the $y$ and $z$ to minimize the reconstruction loss $\mathbb { E } _ { ( z , y ) \sim q _ { \phi } ( Z , Y | x ) } \left[ \log p _ { \theta _ { 1 } } ( x | y , z ) \right]$ . When the dimension of $Z$ is chosen to be much smaller than that of $X$ , to obtain a smaller reconstruction error, the decoder has to utilize the information provided by $Y$ , and force the value of $Y$ to be useful for
+
+# Algorithm 1 CausalNL
+
+Input: A noisy sample $S$ , Average noise rate $\rho$ , Total epoch $T _ { m a x }$ , Batch size $N$ .   
+1: For $\mathrm { T } = 1 , \dots , T _ { m a x }$ :   
+2: For mini-batch $\bar { S } = \{ x _ { i } \} _ { i = 0 } ^ { N } , \tilde { L } = \{ \tilde { y } _ { i } \} _ { i = 0 } ^ { N }$ in $S$ :   
+3: Feed $\bar { S }$ to encoders $\hat { q } _ { \phi _ { 1 } ^ { 1 } }$ and $\hat { q } _ { \phi _ { 1 } ^ { 2 } }$ to get clean label sets $L _ { 1 }$ and $L _ { 2 }$ , respectively;   
+4: Feed $( \bar { S } , L _ { 1 } )$ to encoder $\hat { q } _ { \phi _ { 2 } ^ { 1 } }$ to get a representation set $H _ { 1 }$ , feed $( \bar { S } , L _ { 2 } )$ to $\hat { q } _ { \phi _ { 2 } ^ { 2 } }$ to get $H _ { 2 }$ ;   
+5: Update $\hat { q } _ { \phi _ { 2 } ^ { 1 } }$ and $\hat { q } _ { \phi _ { 2 } ^ { 2 } }$ with co-teaching loss;   
+6: Feed $( L _ { 1 } , H _ { 1 } )$ to decoder $\hat { p } _ { \theta _ { 1 } ^ { 1 } }$ to get reconstructed dataset $\bar { S } _ { 1 }$ , feed $( L _ { 2 } , H _ { 2 } )$ to $\hat { p } _ { \theta _ { 1 } ^ { 2 } }$ to get $\bar { S } _ { 2 }$ ;   
+7: Feed $( { \bar { S } } _ { 1 } , L _ { 1 } )$ to decoder $\hat { p } _ { \theta _ { 2 } ^ { 1 } }$ to get predicted noisy labels ${ \tilde { L } } _ { 1 }$ , feed $( \bar { S } _ { 2 } , L _ { 2 } )$ to $\hat { p } _ { \theta _ { 2 } ^ { 2 } }$ to get ${ \tilde { L } } _ { 2 }$ ;   
+8: Update networks $\hat { q } _ { \phi _ { 1 } ^ { 1 } }$ , $\hat { q } _ { \phi _ { 2 } ^ { 1 } }$ , $\hat { p } _ { \theta _ { 1 } ^ { 1 } }$ and $\hat { p } _ { \theta _ { 2 } ^ { 1 } }$ by calculating ELBO on $( \bar { S } , \bar { S } _ { 1 } , \tilde { L } , \tilde { L } _ { 1 } )$ , update   
+networks $\hat { q } _ { \phi _ { 1 } ^ { 2 } }$ , $\hat { q } _ { \phi _ { 2 } ^ { 2 } }$ , $\hat { p } _ { \theta _ { 1 } ^ { 2 } }$ and $\hat { p } _ { \theta _ { 2 } ^ { 2 } }$ by calculating ELBO on $( \bar { S } , \bar { S } _ { 2 } , \tilde { L } , \tilde { L } _ { 2 } )$ ;
+
+Output: The inference network $\hat { q } _ { \phi _ { 1 } ^ { 1 } }$
+
+prediction. Furthermore, we constrain the $Y$ to be a one-hot vector, and then $Y$ could be a cluster ID of which the manifold of the $X$ belongs.
+
+So far, the latent variable $Y$ can be inferred as a cluster ID instead of a clean class ID. To further link the clusters to clean labels, a naive approach is to select some reliable examples and keep the cluster numbers to be consistent with the noisy labels on these examples. In such a way, the latent representation $Z$ and clean label $Y$ can be effectively inferred, and therefore, it encourages the identifiability of the transition relationship $p _ { \theta _ { 2 } } ( { \tilde { Y } } | Y , { \tilde { X } } )$ . To achieve this, instead of explicitly selecting the reliable example in advance, our method is trained end-to-end, i.e., reliable examples are selected dynamically during the update of parameters of our model by using the co-teaching technique [10]. The advantage of this approach is that the selection bias of the reliable example [6] can be greatly reduced. Intuitively, the accurately selected reliable examples can encourage the identifiability of $p _ { \theta _ { 2 } } ( { \tilde { Y } } | Y , X )$ and $p _ { \theta _ { 1 } } ( X | Y , Z )$ , and the accurately estimated $p _ { \theta _ { 2 } } ( { \tilde { Y } } | Y , X )$ and $p _ { \theta _ { 1 } } ( X | Y , Z )$ will encourage the network to select more reliable examples.
+
+# 3.2 Practical Implementation
+
+Our method is summarized in Algorithm 1 and illustrated in Fig. 3. Here we introduce the structure of our model and loss functions.
+
+Model Structure Because we incorporate co-teaching in our model training, we need to add a copy of the decoder and encoders in our method. As the two branches share the same architectures, we first present the details of the first branch and then briefly introduce the second branch.
+
+For the first branch, we need a set of encoders and decoders to model the distributions in Eq. 3 and 5. Specifically, we have two encoder networks
+
+$$
+Y _ { 1 } = \hat { q } _ { \phi _ { 1 } ^ { 1 } } ( X ) , ~ Z _ { 1 } \sim \hat { q } _ { \phi _ { 2 } ^ { 1 } } ( X , Y _ { 1 } )
+$$
+
+for Eq. 5 and two decoder networks
+
+$$
+X _ { 1 } = \hat { p } _ { \theta _ { 1 } ^ { 1 } } ( Y _ { 1 } , Z _ { 1 } ) , \ \tilde { Y } _ { 1 } = \hat { p } _ { \theta _ { 2 } ^ { 1 } } ( X _ { 1 } , Y _ { 1 } )
+$$
+
+for Eq. 3. The first encoder ${ \hat { q } } _ { \phi _ { 1 } ^ { 1 } } ( X )$ takes an instance $X$ as input ${ \hat { q } } _ { \phi _ { 1 } ^ { 1 } } ( X )$ and output a predicted clean label $Y _ { 1 }$ . The second encoder ${ \hat { q } } _ { \phi _ { 2 } ^ { 1 } } ( X , Y _ { 1 } )$ takes both the instance $X$ and the generated label $Y _ { 1 }$ as input and outputs a latent feature $Z _ { 1 }$ . Then the generated $Y _ { 1 }$ and $Z _ { 1 }$ are passed through the decoder $\hat { p } _ { \theta _ { 1 } ^ { 1 } } ( Y _ { 1 } , Z _ { 1 } )$ which will generate a reconstructed image $X _ { 1 }$ . Finally, the generated $X _ { 1 }$ and $Y _ { 1 }$ are the input to another decoder $\hat { p } _ { \theta _ { 2 } ^ { 1 } } ( X _ { 1 } , Y _ { 1 } )$ which returns predicted noisy labels $\tilde { Y } _ { 1 }$ . It is worth mentioning that the reparameterization trick [12] is used for sampling, so as to allow backpropagation in $\hat { q } _ { \phi _ { 2 } ^ { 1 } } ( X , Y _ { 1 } )$ .
+
+Similarly, the encoder and decoder networks in the second branch are defined as follows:
+
+$$
+Y _ { 2 } = \hat { q } _ { \phi _ { 1 } ^ { 2 } } ( X ) , Z _ { 2 } \sim \hat { q } _ { \phi _ { 2 } ^ { 2 } } ( X , Y _ { 2 } ) , X _ { 2 } = \hat { p } _ { \theta _ { 1 } ^ { 2 } } ( Y _ { 2 } , Z _ { 2 } ) , \tilde { Y } _ { 2 } = \hat { p } _ { \theta _ { 2 } ^ { 2 } } ( X _ { 2 } , Y _ { 2 } ) .
+$$
+
+During training, we let two encoders ${ \hat { q } } _ { \phi _ { 1 } ^ { 1 } } ( X )$ and $\hat { q } _ { \phi _ { 1 } ^ { 2 } } ( X )$ teach each other given every mini-batch.
+
+Loss Functions We divide the loss functions into two parts. The first part is the negative ELBO in Eq. 6, and the second part is a co-teaching loss. The detailed formulation is leaved in Appendix B.
+
+For the negative ELBO, the first term $- \mathbb { E } _ { ( z , y ) \sim q _ { \phi } ( Z , Y \mid x ) } \left[ \log p _ { \theta _ { 1 } } ( x \mid y , z ) \right]$ is a reconstruction loss, and we use the $\ell { 1 }$ loss for reconstruction. The second term is $- \mathbb { E } _ { y \sim q _ { \phi _ { 1 } } ( Y | x ) } \left[ \log p _ { \theta _ { 2 } } ( \tilde { y } | y , x ) \right]$ , which aims to learn noisy labels given inference $y$ and $x$ , and can be simply replaced by the cross-entropy loss on outputs of both decoders $\hat { p } _ { \theta _ { 2 } ^ { 1 } } ( X _ { 1 } , Y _ { 1 } )$ and $\hat { p } _ { \theta _ { 2 } ^ { 2 } } ( X _ { 2 } , Y _ { 2 } )$ with the noisy labels contained in the training data. The additional two terms are two regularizers. To calculate $k l ( q _ { \phi _ { 1 } } ( Y | x ) | | p ( Y ) )$ , we assume that the prior $P ( Y )$ is a uniform distribution. Then minimizing $k l ( q _ { \phi _ { 1 } } ( Y | x ) | | p ( Y ) )$ is equivalent to maximizing the entropy of $q _ { \phi _ { 1 } } ( Y | x )$ for each instance $x$ , i.e., $\begin{array} { r } { - \sum _ { y } q _ { \phi _ { 1 } } ( y | x ) \log q _ { \phi _ { 1 } } ( y | x ) } \end{array}$ . The benefit for having this term is that it could reduce the overfiting problem of the inference network. For $\mathbb { E } _ { y \sim q _ { \phi _ { 1 } } ( Y | x ) } \left[ k l ( q _ { \phi } ( Z | y , x ) | | p ( Z ) ) \right]$ , we let $p ( Z )$ be a standard multivariate Gaussian distribution. Empirically, $q _ { \phi } ( Z | y , x )$ is modeled by the encoders ${ \hat { q } } _ { \phi _ { 1 } ^ { 1 } } ( X )$ and $\hat { q } _ { \phi _ { 1 } ^ { 2 } } ( X )$ which are designed to be deterministic mappings, therefore, the expectation can be removed, and only the $k l$ term $k l ( q _ { \phi } ( Z | y , x ) | | p ( Z ) )$ is left. When $p ( Z )$ is a Gaussian distribution, the $k l$ term nicely has a closed form solution [12], i.e., $\begin{array} { r l r } { \mathrm { ~ } } & { { } } & { - \frac { 1 } { 2 } \sum _ { j = 1 } ^ { J } ( 1 + \log ( ( \sigma _ { j } ) ^ { 2 } ) - ( \mu _ { j } ) ^ { 2 } - ( \sigma _ { j } ) ^ { 2 } ) } \end{array}$ , where $J$ is the dimension of a latent representation $z$ , and $\sigma _ { j }$ and $\mu _ { j }$ are the encoder outputs.
+
+For the co-teaching loss, we follow the work of Han et al. [10]. Intuitively, two encoders ${ \hat { q } } _ { \phi _ { 1 } ^ { 1 } } ( X )$ and ${ \hat { q } } _ { \phi _ { 1 } ^ { 2 } } ( X )$ feed all data forward and selects some data of possibly clean labels. Then, two networks communicate with each other to select possible clean data in this mini-batch and use them for training. Finally, each encoder backpropagates over the data selected by its peer network and updates itself by cross-entropy loss.
+
+# 4 Experiments
+
+In this section, we compare the classification accuracy of proposed method with popular label-noise learning algorithms [15, 19, 11, 10, 28, 36, 16] on both synthetic and real-world datasets.
+
+# 4.1 Experimental Setup
+
+Datasets We examine the efficacy of our approach on manually corrupted versions of four datasets, i.e., FashionMNIST [31], SVHN [18], CIFAR10, CIFAR100 [13], and one real-world noisy dataset, i.e., Clothing1M [32]. FashionMNIST contains 60,000 training images and 10,000 test images with 10 classes; SVHN contains 73,257 training images and 26,032 test images with 10 classes. CIFAR10 contains 50,000 training images and 10,000 test images. CIFAR10 and CIFAR100 both contain 50,000 training images and 10,000 test images but the former has 10 classes of images, and the latter has 10 classes of images. The four datasets contain clean data. We add instance-dependent label noise to the training sets manually according to Xia et al. [30]. Clothing1M has 1M images with real-world noisy labels and $1 0 \mathrm { k }$ images with clean labels for testing. For all the synthetic noisy datasets, the experiments are repeated 5 times.
+
+Network structure and optimization For a fair comparison, all experiments are conducted on NVIDIA Tesla V100, and all methods are implemented by PyTorch. Dimension of the latent representation $Z$ is set to 25 for all synthetic noisy datasets. For encoder networks ${ \hat { q } } _ { \phi _ { 1 } ^ { 1 } } ( X )$ and $\hat { q } _ { \phi _ { 1 } ^ { 2 } } ( X )$ , we use the same network structures with the baseline method. Specially, we use a ResNet-18 network for FashionMNIST, a ResNet-34 network for SVHN and CIFAR10, a ResNet-50 network for CIFAR100 without pretraining. For Clothing1M, we use ResNet-50 networks pre-trained on ImageNet. The data-augmentation methods random crop and horizontal flip are used for our method. For Clothing1M, we use a ResNet-50 network pre-trained on ImageNet, and the clean training data is not used. The dimensionality of the latent representation $Z$ is set to 100. Due to limited space, we leave the detailed structure of other decoders and encoders in Appendix C.
+
+Baselines and measurements We compare the proposed method with the following state-of-the-art approaches: (i). CE, which trains the standard deep network with the cross-entropy loss on noisy datasets. (ii). Decoupling [16], which trains two networks on samples whose predictions from the two networks are different. (iii). MentorNet [11] and Co-teaching [10], which mainly handle noisy labels by training on instances with small loss values. (iv). Forward [19], Reweight [15], and T-Revision [28]. These approaches utilize a class-dependent transition matrix $T$ to correct the loss function. For these baselines, we follow the experiments settings of the original papers. We report average test accuracy on over the last ten epochs of each model on the clean test set. Higher classification accuracy means that the algorithm is more robust to the label noise.
+
+Table 1: Means and standard deviations (percentage) of classification accuracy on FashionMNIST with different label noise levels.   
+
+<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>88.54±0.32</td><td>88.38±0.42</td><td>84.22±0.35</td><td>69.72±0.72</td><td>52.32±0.68</td></tr><tr><td>Co-teaching</td><td>91.21±0.31</td><td>90.30±0.42</td><td>89.10±0.29</td><td>86.78±0.90</td><td>63.22±1.56</td></tr><tr><td>Decoupling</td><td>90.70±0.28</td><td>90.34±0.36</td><td>88.78±0.44</td><td>87.54±0.53</td><td>68.32±1.77</td></tr><tr><td>MentorNet</td><td>91.57±0.29</td><td>90.52±0.41</td><td>88.14±0.76</td><td>85.12±0.76</td><td>61.62±1.42</td></tr><tr><td>Mixup</td><td>88.68±0.37</td><td>88.02±0.37</td><td>85.47±0.55</td><td>79.57±0.75</td><td>66.02±2.58</td></tr><tr><td>Forward</td><td>90.05±0.43</td><td>88.65±0.43</td><td>86.27±0.48</td><td>73.35±1.03</td><td>58.23±3.14</td></tr><tr><td>Reweight</td><td>90.27±0.27</td><td>89.58±0.37</td><td>87.04±0.32</td><td>80.69±0.89</td><td>64.13±1.23</td></tr><tr><td>T-Revision</td><td>91.58±0.31</td><td>90.11±0.61</td><td>89.46±0.42</td><td>84.01±1.14</td><td>68.99±1.04</td></tr><tr><td>CausalNL</td><td>90.84±0.31</td><td>90.68±0.37</td><td>90.01±0.45</td><td>88.75±0.81</td><td>78.19±1.01</td></tr></table>
+
+Table 2: Means and standard deviations (percentage) of classification accuracy on SVHN with different label noise levels.   
+
+<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>91.51±0.45</td><td>91.21±0.43</td><td>87.87±1.12</td><td>67.15±1.65</td><td>51.01±3.62</td></tr><tr><td>Co-teaching</td><td>93.93±0.31</td><td>92.06±0.31</td><td>91.93±0.81</td><td>89.33±0.71</td><td>67.62±1.99</td></tr><tr><td>Decoupling</td><td>90.02±0.25</td><td>91.59±0.25</td><td>88.27±0.42</td><td>84.57±0.89</td><td>65.14±2.79</td></tr><tr><td>MentorNet</td><td>94.08±0.12</td><td>92.73±0.37</td><td>90.41±0.49</td><td>87.45±0.75</td><td>61.23±2.82</td></tr><tr><td>Mixup</td><td>89.73±0.37</td><td>90.02±0.35</td><td>85.47±0.63</td><td>82.41±0.62</td><td>68.95±2.58</td></tr><tr><td>Forward</td><td>91.89±0.31</td><td>91.59±0.23</td><td>89.33±0.53</td><td>80.15±1.91</td><td>62.53±3.35</td></tr><tr><td>Reweight</td><td>92.44±0.34</td><td>92.32±0.51</td><td>91.31±0.67</td><td>85.93±0.84</td><td>64.13±3.75</td></tr><tr><td>T-Revision</td><td>93.14±0.53</td><td>93.51±0.74</td><td>92.65±0.76</td><td>88.54±1.58</td><td>64.51±3.42</td></tr><tr><td>CausalNL</td><td>94.06±0.23</td><td>93.86±0.65</td><td>93.82±0.64</td><td>93.19±0.93</td><td>85.41±2.95</td></tr></table>
+
+Table 3: Means and standard deviations (percentage) of classification accuracy on CIFAR10 with different label noise levels.   
+
+<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>75.81±0.26</td><td>69.15±0.65</td><td>62.45±0.86</td><td>51.72±1.34</td><td>39.42±2.52</td></tr><tr><td>Co-teaching</td><td>80.96±0.31</td><td>78.56±0.61</td><td>73.41±0.78</td><td>71.60±0.79</td><td>45.92±2.21</td></tr><tr><td>Decoupling</td><td>78.71±0.15</td><td>75.17±0.58</td><td>61.73±0.34</td><td>58.61±1.73</td><td>50.43±2.19</td></tr><tr><td>MentorNet</td><td>81.03±0.24</td><td>77.22±0.47</td><td>71.83±0.49</td><td>66.18±0.64</td><td>47.89±2.03</td></tr><tr><td>Mixup</td><td>73.17±0.34</td><td>70.02±0.31</td><td>61.56±0.71</td><td>56.45±0.67</td><td>48.95±2.58</td></tr><tr><td>Forward</td><td>74.64±0.26</td><td>69.75±0.56</td><td>60.21±0.75</td><td>48.81±2.59</td><td>46.27±1.30</td></tr><tr><td>Reweight</td><td>76.23±0.25</td><td>70.12±0.72</td><td>62.58±0.46</td><td>51.54±0.92</td><td>45.46±2.56</td></tr><tr><td>T-Revision</td><td>76.15±0.37</td><td>70.36±0.54</td><td>64.09±0.37</td><td>52.42±1.01</td><td>49.02±2.13</td></tr><tr><td>CausalNL</td><td>81.47±0.32</td><td>80.38±0.44</td><td>77.53±0.45</td><td>78.60±1.06</td><td>67.39±1.24</td></tr></table>
+
+# 4.2 Classification Accuracy Evaluation
+
+Results on synthetic noisy datasets Tables 1, 2, 3, and 4 report the classification accuracy on the datasets of $F$ -MNIST, SVHN, CIFAR-10, and CIFAR100, respectively. The synthetic experiments reveal that our method is powerful in handling instance-dependent label noise particularly in the situation of high noise rates. Specifically, for all datasets, the classification accuracy of our method decrease much slower than that of baseline methods. Additionally, the classification accuracies on these datasets are improved by using CausalNL, which implies that our method should capture the underlying data generation process, and then $Y$ should be a cause of $X$ for all these datasets.
+
+Table 4: Means and standard deviations (percentage) of classification accuracy on CIFAR100 with different label noise levels.   
+
+<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>30.42±0.44</td><td>24.15±0.78</td><td>21.45±0.70</td><td>15.23±1.32</td><td>14.42±2.21</td></tr><tr><td>Co-teaching</td><td>37.96±0.53</td><td>33.43±0.74</td><td>28.04±1.43</td><td>25.60±0.93</td><td>23.97±1.91</td></tr><tr><td>Decoupling</td><td>36.53±0.49</td><td>30.93±0.88</td><td>27.85±0.91</td><td>23.81±1.31</td><td>19.59±2.12</td></tr><tr><td>MentorNet</td><td>38.91±0.54</td><td>34.23±0.73</td><td>31.89±1.19</td><td>27.53��1.23</td><td>24.15±2.31</td></tr><tr><td>Mixup</td><td>32.92±0.76</td><td>29.76±0.87</td><td>25.92±1.26</td><td>23.13±2.15</td><td>21.31±1.32</td></tr><tr><td>Forward</td><td>36.38±0.92</td><td>33.17±0.73</td><td>26.75±0.93</td><td>21.93±1.29</td><td>19.27±2.11</td></tr><tr><td>Reweight</td><td>36.73±0.72</td><td>31.91±0.91</td><td>28.39±1.46</td><td>24.12±1.41</td><td>20.23±1.23</td></tr><tr><td>T-Revision</td><td>37.24±0.85</td><td>36.54±0.79</td><td>27.23±1.13</td><td>25.53±1.94</td><td>22.54±1.95</td></tr><tr><td>CausalNL</td><td>41.47±0.43</td><td>40.98±0.62</td><td>34.02±0.95</td><td>33.34±1.13</td><td>32.129±2.23</td></tr></table>
+
+Table 5: Classification accuracy on Clothing1M. In the experiments, only noisy samples are exploited to train and validate the deep model.   
+
+<table><tr><td>CE</td><td>Decoupling</td><td>MentorNet</td><td>Co-teaching</td><td>Forward</td><td>Reweight</td><td>T-Revision</td><td>caualNL</td></tr><tr><td>68.88</td><td>54.53</td><td>56.79</td><td>60.15</td><td>69.91</td><td>70.40</td><td>70.97</td><td>72.24</td></tr></table>
+
+For noisy $F$ -MNIST, SVHN and CIFAR-10, in the easy case IDN- $20 \%$ , almost all methods work well. When the noise rate is $30 \%$ , the advantages of causalNL begin to show. We surpassed all methods obviously. When the noise rate raises, all the baselines are gradually defeated. Finally, in the hardest case, i.e., IDN- $50 \%$ , the superiority of causalNL widens the gap of performance. The classification accuracy of causalNL is at least over $10 \%$ higher than the best baseline method. For noisy CIFAR-100, none of the methods works well. However, causalNL still overtakes the other methods with clear gaps for all different levels of noise rate.
+
+Results on the real-world noisy dataset On the real-world noisy dataset Clothing1M, our method causalNL outperforms all the baselines, as shown in Table 5. The experimental results also show that the noise type in Clothing1M is more likely to be instance-dependent label noise, suggesting that the instance-independent assumption on the transition matrix sometimes can be strong.
+
+# 5 Conclusion
+
+In this paper, we have investigated how to use $P ( X )$ to help learn instance-dependent label noise. Specifically, previous assumptions are made on the transition matrix, and the assumptions are hard to be verified and might be violated on real-world datasets. From inspired by a causal perspective, when $Y$ is a cause of $X$ , then $P ( X )$ should contain useful information to infer the clean label $Y$ . We propose a novel generative approach called causalNL for instance-dependent label-noise learning. Our model makes use of the causal graph to contribute to the identifiability of the transition matrix, and therefore helps learn clean labels. In order to learn $P ( X )$ , compared to the previous methods, our method contains more parameters. But experiments on both synthetic and real-world noisy datasets show that a bit sacrifice on computational efficiency is worth it, i.e., the classification accuracy of casualNL significantly outperforms all the state-of-the-art methods. Additionally, the results also indicates that in classification problems, $Y$ can usually be considered as a cause of $X$ , and suggests that the understanding and modeling of the data generation process can help leverage additional information that is useful in solving advanced machine learning problems concerning the relationship between different modules of the data joint distribution. In our future work, we will study the theoretical properties of our method and establish identifiability results under certain assumptions on the data-generative process.
+
+# Acknowledgments
+
+TL was partially supported by Australian Research Council Projects DP-180103424, DE-190101473, and IC-190100031. GM was supported by Australian Research Council Project DE210101624. BH was supported by supported by the RGC Early Career Scheme No. 22200720 and NSFC Young Scientists Fund No. 62006202. GN was supported by JST AIP Acceleration Research Grant Number JPMJCR20U3, Japan. KZ was supported in part by the National Institutes of Health (NIH) under Contract R01HL159805, by the United States Air Force under Contract No. FA8650-17-C7715, by the NSF-Convergence Accelerator Track-D award #2134901, and by a grant from Apple. The NIH or NSF is not responsible for the views reported in this article. The authors thank the reviewers and the meta-reviewer for their helpful and constructive comments on this work.
+
+# References
+
+[1] Dana Angluin and Philip Laird. Learning from noisy examples. Machine Learning, 2(4): 343–370, 1988. [2] Devansh Arpit, Stanisław Jastrz˛ebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, et al. A closer look at memorization in deep networks. In International Conference on Machine Learning, pages 233–242. PMLR, 2017.   
+[3] Mikhail Belkin, Partha Niyogi, and Vikas Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of machine learning research, 7(11), 2006.   
+[4] Antonin Berthon, Bo Han, Gang Niu, Tongliang Liu, and Masashi Sugiyama. Confidence scores make instance-dependent label-noise learning possible. In ICML, 2021.   
+[5] David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859–877, 2017.   
+[6] Hao Cheng, Zhaowei Zhu, Xingyu Li, Yifei Gong, Xing Sun, and Yang Liu. Learning with instance-dependent label noise: A sample sieve approach. In ICLR, 2021.   
+[7] Jiacheng Cheng, Tongliang Liu, Kotagiri Ramamohanarao, and Dacheng Tao. Learning with bounded instance and label-dependent label noise. In ICML, 2020.   
+[8] Jacob Goldberger and Ehud Ben-Reuven. Training deep neural-networks using a noise adaptation layer. In ICLR, 2017.   
+[9] Mingming Gong, Kun Zhang, Tongliang Liu, Dacheng Tao, Clark Glymour, and Bernhard Schölkopf. Domain adaptation with conditional transferable components. In International conference on machine learning, pages 2839–2848. PMLR, 2016.   
+[10] Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In NeurIPS, pages 8527–8537, 2018.   
+[11] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-Fei. MentorNet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In ICML, pages 2309–2318, 2018.   
+[12] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   
+[13] Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009.   
+[14] Xuefeng Li, Tongliang Liu, Bo Han, Gang Niu, and Masashi Sugiyama. Provably end-to-end label-noise learning without anchor points. In ICML, 2021.   
+[15] Tongliang Liu and Dacheng Tao. Classification with noisy labels by importance reweighting. IEEE Transactions on pattern analysis and machine intelligence, 38(3):447–461, 2016.   
+[16] Eran Malach and Shai Shalev-Shwartz. Decoupling" when to update" from" how to update". In NeurIPS, pages 960–970, 2017.   
+[17] Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and Ambuj Tewari. Learning with noisy labels. In NeurIPS, pages 1196–1204, 2013.   
+[18] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew $\mathrm { Y . N g }$ . Reading digits in natural images with unsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.   
+[19] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In CVPR, pages 1944–1952, 2017.   
+[20] Judea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, New York, NY, USA, 2000. ISBN 0-521-77362-8.   
+[21] Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.   
+[22] B Schölkopf, D Janzing, J Peters, E Sgouritsa, K Zhang, and J Mooij. On causal and anticausal learning. In 29th International Conference on Machine Learning (ICML 2012), pages 1255– 1262. International Machine Learning Society, 2012.   
+[23] Bernhard Schölkopf. Causality for machine learning. arXiv preprint arXiv:1911.10500, 2019.   
+[24] Clayton Scott. A rate of convergence for mixture proportion estimation, with application to learning from noisy labels. In AISTATS, pages 838–846, 2015.   
+[25] Peter Spirtes and Kun Zhang. Causal discovery and inference: concepts and recent methodological advances. In Applied informatics, volume 3, pages 1–28. SpringerOpen, 2016.   
+[26] Peter Spirtes, Clark N Glymour, Richard Scheines, David Heckerman, Christopher Meek, Gregory Cooper, and Thomas Richardson. Causation, prediction, and search. MIT press, 2000.   
+[27] Songhua Wu, Xiaobo Xia, Tongliang Liu, Bo Han, Mingming Gong, Nannan Wang, Haifeng Liu, and Gang Niu. Class2simi: A noise reduction perspective on learning with noisy labels. In International Conference on Machine Learning, pages 11285–11295. PMLR, 2021.   
+[28] Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In NeurIPS, pages 6835–6846, 2019.   
+[29] Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In NeurIPS, pages 6838–6849, 2019.   
+[30] Xiaobo Xia, Tongliang Liu, Bo Han, Nannan Wang, Mingming Gong, Haifeng Liu, Gang Niu, Dacheng Tao, and Masashi Sugiyama. Part-dependent label noise: Towards instance-dependent label noise. In NeurIPS, 2020.   
+[31] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.   
+[32] Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang. Learning from massive noisy labeled data for image classification. In CVPR, pages 2691–2699, 2015.   
+[33] Quanming Yao, Hansi Yang, Bo Han, Gang Niu, and James T Kwok. Searching to exploit memorization effect in learning with noisy labels. In ICML, 2020.   
+[34] Yu Yao, Tongliang Liu, Bo Han, Mingming Gong, Jiankang Deng, Gang Niu, and Masashi Sugiyama. Dual t: Reducing estimation error for transition matrix in label-noise learning. In NeurIPS, 2020.   
+[35] Xiyu Yu, Tongliang Liu, Mingming Gong, and Dacheng Tao. Learning with biased complementary labels. In ECCV, pages 68–83, 2018.   
+[36] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In ICLR, 2018.   
+[37] Kun Zhang and Aapo Hyvarinen. On the identifiability of the post-nonlinear causal model. arXiv preprint arXiv:1205.2599, 2012.   
+[38] Kun Zhang, Mingming Gong, and Bernhard Schölkopf. Multi-source domain adaptation: A causal view. In Twenty-ninth AAAI conference on artificial intelligence, 2015.   
+[39] Kun Zhang, Jiji Zhang, and Bernhard Schölkopf. Distinguishing cause from effect based on exogeneity. arXiv preprint arXiv:1504.05651, 2015.

parse/train/d20KTY2VrNk/d20KTY2VrNk_content_list.json

@@ -0,0 +1,1107 @@
+[
+    {
+        "type": "text",
+        "text": "Instance-Dependent Label-Noise Learning under Structural Causal Models ",
+        "text_level": 1,
+        "bbox": [
+            204,
+            122,
+            795,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yu Yao1 Tongliang Liu1† Mingming Gong2 Bo Han3 Gang Niu4 Kun Zhang5 ",
+        "bbox": [
+            336,
+            220,
+            661,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1TML Lab, University of Sydney; 2University of Melbourne; 3Hong Kong Baptist University; 4RIKEN AIP; 5Carnegie Mellon University ",
+        "bbox": [
+            187,
+            251,
+            803,
+            280
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            314,
+            535,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Label noise generally degenerates the performance of deep learning algorithms because deep neural networks easily overfit label errors. Let $X$ and $Y$ denote the instance and clean label, respectively. When $Y$ is a cause of $X$ , according to which many datasets have been constructed, e.g., SVHN and CIFAR, the distributions of $P ( X )$ and $P ( { Y \\vert } X )$ are generally entangled. This means that the unsupervised instances are helpful to learn the classifier and thus reduce the side effect of label noise. However, it remains elusive on how to exploit the causal information to handle the label-noise problem. We propose to model and make use of the causal process in order to correct the label-noise effect. Empirically, the proposed method outperforms all state-of-the-art methods on both synthetic and real-world labelnoise datasets. ",
+        "bbox": [
+            233,
+            348,
+            766,
+            501
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            530,
+            312,
+            547
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning with noisy labels can be dated back to [1] and has recently drawn a lot of attention [15, 19, 11, 10, 28]. In real life, large-scale datasets are likely to contain label noise. It is partly because that many cheap but imperfect data collection methods such as crowd-sourcing and web crawling are widely used to build large-scale datasets. Training with such data usually lead to poor generalization abilities of deep neural networks because they can memorize noisy labels [2, 33]. ",
+        "bbox": [
+            174,
+            564,
+            825,
+            632
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To improve the generalization ability of training with noisy labels, one family of existing label-noise learning methods is to model how the label noise was generated [17, 19, 27, 34, 14]. Specifically, these methods try to reveal the transition relationship from clean labels to noisy labels of instances, i.e., the distribution $P ( \\tilde { Y } | Y , X )$ , where $\\tilde { Y }$ , $Y$ and $X$ are the random variables for the noisy label, latent clean label, and instance, respectively. The advantage of modelling label noise is that given only the noisy data, when the transition relationship is identifiable, classifiers can be learned to converge to the optimal ones defined by the clean data, with theoretical guarantees. However, the transition relationship is not identifiable in general. To make it identifiable, various assumptions have been made on the transition relationship. For example, Natarajan et al. [17] assume that the transition relationship is instance independent, i.e., $P ( \\tilde { Y } | Y , X ) = \\bar { P } ( \\tilde { Y } | Y )$ ; Xia et al. [30] assume that the $P ( { \\tilde { Y } } | Y , X )$ is dependent on different parts of an instance. Cheng et al. [7] assume that the label noise rates are upper bounded. In practice, these assumptions may not be satisfied and are generally hard to be verified given noisy data alone. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            824
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Inspired by causal learning [20, 25, 21, 23], we provide a causal perspective of label-noise learning method named CausalNL. We exploit the causal information to help identifiability of the transition matrix $P ( { \\tilde { Y } } | Y , X )$ other than making assumptions directly on the transition relationship. ",
+        "bbox": [
+            176,
+            832,
+            825,
+            876
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we assume that the data containing instancedependent label noise is generated according to the causal graph in Fig. 1. For example, for the Street View House Numbers (SVHN) dataset [18], $X$ represents the image containing the digit; $Y$ represents the clean label of the digit shown on the plate; $Z$ represents the latent variable that captures the information affecting the generation of the images, e.g., orientation, lighting, and font style. Here $Y$ is naturally a cause of $X$ and the causal generative process can be described in the following way. First, the house plate is generated according to the street number and attached to the front door. Then, the house plate is captured by a camera (installed in a Google street view car) to form $X$ , taking into account of other factors such as illumination and viewpoint. Finally, the images containing house numbers are collected and relabeled to form the dataset. Let us denote the annotated label by the noisy label $\\tilde { Y }$ as the annotator may not be always reliable, especially when the dataset is very large but the budget is limited. During the annotation process, the noisy labels were generated according to both the images and the range of predefined digit numbers. Hence, both $X$ and $Y$ are causes of $\\tilde { Y }$ . Note that most existing image datasets are collected with the causal relationship that $Y$ causes $X$ . For example, see the widely used FashionMNIST and CIFAR. When we synthesize instance-dependent label noise based on them, we will have the causal graph illustrated in Fig. 1. Note also that some datasets are generated with the causal relationship that $X$ causes $Y$ . Other than using domain knowledge, the different causal relationships can be verified by employing causal discovery [26, 25, 21, 37]. ",
+        "bbox": [
+            174,
+            92,
+            576,
+            297
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/11a3256e8647b5d8a82c80d99a5dcb468f5604150dd14b34f720132d9fb55de5.jpg",
+        "image_caption": [
+            "Figure 1: A graphical causal model which reveals a generative process of the data which contains instancedependent label noise, where the shaded variables are observable and the unshaded variables are latent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            607,
+            92,
+            805,
+            185
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            297,
+            826,
+            425
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "When the latent clean label $Y$ is a cause of $X$ , $P ( X )$ will generally contain some information about $P ( { Y \\vert } X )$ . This is because, under such a generative process, the distributions of $P ( X )$ and $P ( { Y \\vert } X )$ are entangled [22, 39]. To help estimate $P ( { Y \\vert } X )$ with $P ( X )$ , we make use of the causal generative process to estimate $P ( X | Y )$ , which directly benefits from $P ( X )$ by generative modeling. The modeling of $P ( X | Y )$ in turn encourages the identifiability of the transition relationship and helps learn $P ( { Y \\vert } X )$ . For example, in Fig. 2(a), we have added instance-dependent label-noise with a rate $45 \\%$ (i.e., $\\mathrm { I D L N } { - } 4 5 \\% )$ to the MOON dataset and employed different methods [10, 36] to solve the label-noise learning problem. As illustrated in Fig. 2(b) and Fig. 2(c), previous methods fail to infer clean labels. In contrast, by constraining the conditional distribution of the instances, i.e., restricting the data of each class to be on a manifold by setting the dimension of the latent variable $Z$ to be 1-dimensional, the label transition as well as the clean labels can be successfully recovered (by the proposed method), which is showed in Fig. 2(d). It is worth noting that the idea of finding $P ( { Y \\vert } X )$ by modeling $P ( Y )$ and $P ( X | Y )$ instead has been exploited in the context of domain adaptation; for instance, it inspires target shift, (generalized) conditional shift and other settings for domain adaptation [38, 9] ",
+        "bbox": [
+            173,
+            430,
+            825,
+            637
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Specifically, to make use of the causal graph to contribute to the identifiability of the transition matrix, we propose a causally inspired deep generative method, which models the causal structure with all the observable and latent variables, i.e., the instance $X$ , the noisy label $\\tilde { Y }$ , the latent feature $Z$ , and the latent clean label $Y$ . The proposed generative model captures the variables’ relationship indicated by the causal graph. Furthermore, built on the variational autoencoder (VAE) framework [12], we build an inference network which could efficiently infer the latent variables $Z$ and $Y$ when maximising the marginal likelihood $p ( X , { \\tilde { Y } } )$ on the given noisy data. In the decoder phase, the data will be reconstructed by exploiting the conditional distribution of instances $P ( X | Y , Z )$ and the transition relationship $P ( { \\tilde { Y } } | Y , X )$ , i.e., ",
+        "bbox": [
+            173,
+            643,
+            826,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/458a397fc9a6ff6d29adb6b73c24fddb3927682b00748b96e9448b3a32eeb775.jpg",
+        "text": "$$\np _ { \\theta } ( X , \\tilde { Y } ) = \\int _ { z , y } P ( Z = z ) P ( Y = y ) p _ { \\theta _ { 1 } } ( X | Y = y , Z = z ) p _ { \\theta _ { 2 } } ( \\tilde { Y } | Y = y , X ) \\mathrm { d } z \\mathrm { d } y\n$$",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            782,
+            777,
+            818
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "will be exploited, where $\\theta : = \\left( \\theta _ { 1 } , \\theta _ { 2 } \\right)$ are the parameters of the causal generative model (more details can be found in Section 3). Ay a high level, according to the equation, given the noisy data and the distributions of $Z$ and $Y$ , constraining $p _ { \\theta _ { 1 } } ( X | Y , Z )$ will also greatly reduce the uncertainty of $p _ { \\theta _ { 2 } } ( { \\tilde { Y } } | Y , X )$ and thus contribute to the identifiability of the transition matrix. Note that adding a constraint on $p _ { \\theta _ { 1 } } ( X | Y , Z )$ is natural; for example, images often have a low-dimensional manifold [3]. We can restrict $P ( Z )$ to fulfill the constraint on $p _ { \\theta _ { 1 } } ( X | Y , Z )$ . By exploiting the causal structure and the constraint on instances to better model label noise, the proposed method significantly outperforms the baselines. When the label noise rate is large, the superiority is evidenced by a large gain in the classification performance. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            912
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/8edd33b129b52da12f985bc53180e31caf4a896b594844f7855f0560017d3e3c.jpg",
+        "image_caption": [
+            "Figure 2: (a) An illustration of the MOON training dataset which contains $4 5 \\%$ of instance-dependent label noise. Different instances have different noise rates which are randomly generated according to Xia et al. [30]. (b)-(d) The illustration of the classification performance of co-teaching, mixup, and our method, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            97,
+            805,
+            200
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            277,
+            821,
+            320
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The rest of the paper is organized as follows. In Section 2, we briefly review the background knowledge of label-noise learning and causality. In Section 3, we formulate our method, named CausalNL, and discuss how it helps to learn a clean classifier, followed by the implementation details. Experimental validations are provided in Section 4. Section 5 concludes the paper. ",
+        "bbox": [
+            174,
+            325,
+            826,
+            382
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Noisy Labels and Causality ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            404,
+            431,
+            421
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, firstly, we introduce how to model label noise. Then, we introduce the structural causal model and discuss how to exploit the model to encourage the identifiability of the transition relationship and help learn the classifier. ",
+        "bbox": [
+            174,
+            436,
+            825,
+            479
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transition Relationship To build a statistically consistent classifier that converges to the optimal classifier defined on clean data by only employing noisy data, the transition relationship $P ( \\tilde { Y } | Y , X )$ has to be identified. Given an instance, the conditional distribution can be written in an $C \\times C$ matrix which is called the transition matrix [19, 29, 30], where $C$ represents the number of classes. Specifically, for each instance $x$ , there is a transition matrix $T ( x )$ . The $i j$ -th entry of the transition matrix is $T _ { i j } ( x ) = P ( \\tilde { Y } = i | Y = j , X = x )$ , which represents the probability that the instance $x$ with the clean label $Y = j$ will have a noisy label $\\tilde { Y } = i$ . ",
+        "bbox": [
+            174,
+            484,
+            825,
+            589
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The transition matrix has been widely studied to build statistically consistent classifiers, because the clean class posterior distribution $P ( \\pmb { Y } | \\pmb { x } ) = [ P ( \\pmb { Y } = 1 | \\pmb { X } = \\pmb { x } ) , \\dots , P ( \\pmb { Y } = C | \\pmb { X } = \\pmb { x } ) ] ^ { \\top }$ can be inferred by using the transition matrix and the noisy class posterior $P ( \\tilde { Y } | x ) = [ P ( \\tilde { Y } =$ $1 | X = x ) , \\ldots , P ( \\tilde { Y } = C | X = x ) ] ^ { \\top }$ , i.e., we have $P ( \\tilde { \\mathbf { Y } } | x ) = T ( x ) P ( \\mathbf { Y } | x )$ . Specifically, the transition matrix has been used to modify loss functions to build risk-consistent estimators; see, e.g., [8, 19, 35, 28], and has been used to correct hypotheses to build classifier-consistent algorithms; see, e.g., [17, 24, 19]. Moreover, the state-of-the-art statically inconsistent algorithms [11, 10] also use diagonal entries of the transition matrix to help select reliable examples used for training. ",
+        "bbox": [
+            173,
+            594,
+            825,
+            712
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, the distribution $P ( { \\tilde { Y } } | Y , X )$ is not generally identifiable [28]. To make it identifiable, one has to resort to additional assumptions. The most widely used assumption is that given clean label $Y$ , the noisy label $\\tilde { Y }$ is conditionally independent of instance $X$ , i.e., $\\mathbf { \\bar { \\xi } } P ( \\tilde { Y } | Y , X ) = P ( \\tilde { Y } | Y )$ . Under such an assumption, the transition relationship $P ( \\tilde { Y } | Y )$ can be successfully identified with the anchor point assumption [15, 34, 14]. However, in the real-world scenarios, this assumption may be hard to satisfied. Although $P ( \\tilde { Y } | Y )$ can be used to approximate $P ( { \\tilde { Y } } | Y , X )$ , the approximation error can be large in many cases. As for the efforts to model the instance-dependent transition matrix directly, existing methods rely on rather strong assumptions, e.g., the bounded noise rate assumption [7], the part-dependent label noise assumption [30], and the requirement of additional information about the transition matrix [4]. Although the assumptions help the methods achieve superior performance empirically, they are generally difficult to verify or fulfill, limiting their applications in practice. ",
+        "bbox": [
+            173,
+            717,
+            826,
+            877
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Structural Causal Model Motivated by the limitation of the current methods, we provide a new causal perspective to learn the identifiable of instance-dependent label noise model. Here we briefly introduce some background knowledge of causality [25] used in this paper. A structural causal model (SCM) consists of a set of variables connected by a set of functions. It represents a flow of information and reveals causal relationships among all the variables, providing a fine-grained description of the data generation process. The causal structure encoded by SCMs can be represented as a graphical casual model as shown in Fig. 1, where each node is a variable and each edge is a function involving noise. The SCM corresponding to the graph in Fig. 1 can be written as ",
+        "bbox": [
+            174,
+            882,
+            820,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/8870323952342d39d1beb50c1a7bee11d402367d8f4504321c5813b9c4f8fe5c.jpg",
+        "image_caption": [
+            "Figure 3: A working flow of our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            89,
+            761,
+            257
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            291,
+            825,
+            376
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e51b57396e26c522284c7a0739d4c18475057d9158f1e0c962cfc9619181836.jpg",
+        "text": "$$\nZ = \\epsilon _ { Z } , Y = \\epsilon _ { Y } , X = f ( Z , Y , \\epsilon _ { X } ) , \\tilde { Y } = f ( X , Y , \\epsilon _ { \\tilde { Y } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            377,
+            691,
+            396
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon _ { Z } , \\epsilon _ { Y } ,$ $\\epsilon _ { X }$ and $\\epsilon _ { \\tilde { Y } }$ are independent exogenous variables, and they sometimes also called error variables. For example, $\\epsilon _ { X }$ are an error variable for $X$ , which is responsible for any difference between the actual value of $X$ and the value predicted on the basis of $Z$ and $Y$ alone. Each equation specifies a distribution of a variable conditioned on its parents (which are an empty set for root cause variables in the graph). ",
+        "bbox": [
+            173,
+            398,
+            825,
+            468
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By making use of the SCM, the benefit of the instances to learning the classifier can be clearly explained. Specifically, the instance $X$ is a function of its label $Y$ and latent feature $Z$ , which means that the instance $X$ is generated from $Y$ and $Z$ . Therefore $X$ must contains information about its clean label $Y$ and latent feature $Z$ . That is the reason that $P ( X )$ can help identify $P ( { Y \\vert } X )$ and also $P ( Z | X )$ . However, since we do not have clean labels, it is hard to fully identify $P ( { Y \\vert } X )$ from $P ( X )$ in the unsupervised setting. For example, on the MOON dataset shown in Fig. 2 , we can possibly discover the two clusters by enforcing the manifold constraint, but it is impossible to see which class each cluster belongs to. We show in the following that we can make use of the property of $P ( X | Y )$ to help model label noise, i.e., encourage the identifiability of the transition relationship, thereby learning a better classifier. ",
+        "bbox": [
+            173,
+            473,
+            825,
+            613
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Specifically, under the Markov condition [20], which intuitively means the independence of exogenous variables, the joint distribution $P ( \\tilde { Y } , X , Y , Z )$ specified by the SCM can be factorized as follows. ",
+        "bbox": [
+            176,
+            617,
+            821,
+            648
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/50f008e26c208b60b80da24bd8045bc14dade9cf1b079d54a2a1ffe2e9d4f2b1.jpg",
+        "text": "$$\nP ( X , \\tilde { Y } , Y , Z ) = P ( Y ) P ( Z ) P ( X | Y , Z ) P ( \\tilde { Y } | Y , X ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            650,
+            674,
+            670
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This motivates us to extend VAE [12] to perform inference in our causal model to fit the noisy data in the next section. In the decoder phase, given the noisy data and the distributions of $Z$ and $Y$ , adding a constraint on $P ( X | Y , Z )$ will reduce the uncertainty in the distribution $P ( { \\tilde { Y } } | Y , X )$ . In other words, modeling of $P ( X | Y , Z )$ will encourage the identifiability of the transition relationship and thus better model label noise. Since $P ( { \\tilde { Y } } | Y , X )$ functions as a bridge to connect the noisy labels to clean labels, we accordingly can better learn $P ( { Y \\vert } X )$ or the classifier by only using the noisy data. ",
+        "bbox": [
+            173,
+            671,
+            825,
+            760
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There are normally two ways to add constraints on the instances, i.e., assuming a specific parametric generative model or introducing prior knowledge of the instances. In this paper, since we mainly study the image classification problem with noisy labels, we focus on the manifold property of images and apply the low-dimensional manifold constraint to the instances. ",
+        "bbox": [
+            174,
+            765,
+            825,
+            820
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Causality Captured Instance-Dependent Label-Noise Learning ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            838,
+            728,
+            856
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose a structural generative method which captures the causal relationship and utilizes $P ( X )$ to help identify the label-noise transition matrix, and therefore, our method leads to a better classifier that assigns more accurate labels. ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Variational Inference under the Structural Causal Model ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            90,
+            609,
+            106
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To model the generation process of noisy data and to approximate the distribution of the noisy data, our method is designed to follow the causal factorization (see Eq. 2). Specifically, our model contains two decoder networks which jointly model a distribution $p _ { \\theta } ( X , { \\tilde { Y } } | Y , Z )$ and two encoder (inference) networks which jointly model the posterior distribution $q _ { \\phi } ( Z , Y | X )$ . Here we discuss each component of our model in detail. ",
+        "bbox": [
+            173,
+            116,
+            825,
+            188
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let the two decoder networks model the distributions $p _ { \\theta _ { 1 } } ( X | Y , Z )$ and $p _ { \\theta _ { 2 } } ( { \\tilde { Y } } | Y , X )$ , respectively. Let $\\theta _ { 1 }$ and $\\theta _ { 2 }$ be learnable parameters of the distributions. Without loss of generality, we set $p ( Z )$ to a standard normal distribution and $p ( Y )$ to a uniform distribution. Then, modeling the joint distribution in Eq. 2 boils down to modeling the distribution $p _ { \\theta } ( X , { \\tilde { Y } } | Y , Z )$ , which is decomposed as follows: ",
+        "bbox": [
+            173,
+            195,
+            825,
+            255
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0c9224bc88c55b7bbcdae487a13b43bbc2db90c1df30bf95490e5d4946a5ec9.jpg",
+        "text": "$$\n\\begin{array} { r } { p _ { \\theta } ( X , \\tilde { Y } | Y , Z ) = p _ { \\theta _ { 1 } } ( X | Y , Z ) p _ { \\theta _ { 2 } } ( \\tilde { Y } | Y , X ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            261,
+            647,
+            280
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To infer latent variables $Z$ and $Y$ with only observable variables $X$ and $\\tilde { Y }$ , we design an inference network which model the variational distribution $q _ { \\phi } ( Z , Y | \\tilde { Y } , X )$ . Specifically, let $q _ { \\phi _ { 2 } } ( Z | Y , X )$ and $q _ { \\phi _ { 1 } } ( Y | \\tilde { Y } , X )$ be the distributions parameterized by learnable parameters $\\phi _ { 1 }$ and $\\phi _ { 2 }$ , and then the posterior distribution can be decomposed as follows: ",
+        "bbox": [
+            173,
+            287,
+            825,
+            349
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/272dd2b0a201dd17ba4eb8069d8398e7b68877ca80af7b9cbb2e37e2542e428b.jpg",
+        "text": "$$\nq _ { \\phi } ( Z , Y | \\tilde { Y } , X ) = q _ { \\phi _ { 2 } } ( Z | Y , X ) q _ { \\phi _ { 1 } } ( Y | \\tilde { Y } , X ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            356,
+            648,
+            376
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we do not include $\\tilde { Y }$ as a conditioning variable in $q _ { \\phi _ { 2 } } ( Z | Y , X )$ because the causal graph implies $Z \\perp \\tilde { Y } | X , Y$ . One problem with this posterior form is that we cannot directly employ $q _ { \\phi _ { 1 } } ( Y | \\tilde { Y } , X )$ to predict labels on the test data, on which $\\tilde { Y }$ is absent. ",
+        "bbox": [
+            173,
+            383,
+            825,
+            433
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To reduce computational costs and to allow our method efficiently infer clean labels, we approximate $q _ { \\phi _ { 1 } } ( Y | \\tilde { Y } , X )$ by assuming that given the instance $X$ , the clean label $Y$ is conditionally independent from the noisy label $\\tilde { Y }$ , i.e., $q _ { \\phi _ { 1 } } ( Y | \\tilde { Y } , X ) = q _ { \\phi _ { 1 } } ( Y | X )$ . This approximation is expected not to have very large approximation error because the images contain sufficient information to predict the clean labels. Thus, we could simplify Eq. 4 as follows ",
+        "bbox": [
+            173,
+            436,
+            825,
+            512
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4acac89c1170e45fa5c1e9516f98e210a7b924717b5b7d2ce317d2045653d7e8.jpg",
+        "text": "$$\nq _ { \\phi } ( Z , Y | X ) = q _ { \\phi _ { 2 } } ( Z | Y , X ) q _ { \\phi _ { 1 } } ( Y | X ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            518,
+            629,
+            536
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "such that our encoder networks model $q _ { \\phi _ { 2 } } ( Z | Y , X )$ and $q _ { \\phi _ { 1 } } ( Y | X )$ , respectively. This way, $q _ { \\phi _ { 1 } } ( Y | X )$ can be used to infer clean labels efficiently.2 We also found that the encoder network modelling $q _ { \\phi _ { 1 } } ( Y | X )$ can directly act as a regularizer, which helps to identify $p _ { \\theta _ { 2 } } ( { \\tilde { Y } } | Y , X )$ . Moreover, be benefited from this, our method can serve as a general framework which can be easily integrated with the current discriminative label-noise methods [28, 16, 10], and we will showcase it by collaborating co-teaching [10] with our method. ",
+        "bbox": [
+            173,
+            541,
+            826,
+            630
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Optimization of Parameters Because the marginal distribution $p _ { \\theta } ( X , { \\tilde { Y } } )$ is usually intractable, to learn the set of parameters $\\{ \\theta _ { 1 } , \\theta _ { 2 } , \\phi _ { 1 } , \\phi _ { 2 } \\}$ given only noisy data, we follow the variational inference framework [5] to maximize the negative evidence lower-bound $\\operatorname { E L B O } ( x , \\tilde { y } )$ of the marginal likelihood of each datapoint $( x , \\tilde { y } )$ instead of maximizing the marginal likelihood itself. By ensembling our decoder and encoder networks, $\\operatorname { E L B O } ( x , \\tilde { y } )$ is derived as follows: ",
+        "bbox": [
+            174,
+            636,
+            825,
+            708
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/22054737de41d93c7ececa868b7162ef8796476b9db32db2357facf940827825.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { E L B O } ( x , \\tilde { y } ) = \\mathbb { E } _ { ( z , y ) \\sim q _ { \\phi } ( Z , Y \\mid x ) } \\left[ \\log p _ { \\theta _ { 1 } } ( x \\vert y , z ) \\right] + \\mathbb { E } _ { y \\sim q _ { \\phi _ { 1 } } ( Y \\mid x ) } \\left[ \\log p _ { \\theta _ { 2 } } ( \\tilde { y } \\vert y , x ) \\right] } \\\\ & { \\phantom { = \\ } - k l ( q _ { \\phi _ { 1 } } ( Y \\vert x ) \\| p ( Y ) ) - \\mathbb { E } _ { y \\sim q _ { \\phi _ { 1 } } ( Y \\mid x ) } \\left[ k l ( q _ { \\phi } ( Z \\vert y , x ) \\| p ( Z ) ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            713,
+            764,
+            756
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $k l ( \\cdot )$ is the Kullback–Leibler divergence between two distributions. The derivation details are left out in Appendix A. Our model learns the class-conditional distribution $P ( X | Y )$ by maximizing the first expectation in ELBO, which is equivalent to minimizing the reconstruction loss [12]. By learning $P ( X )$ , the inference network $q _ { \\phi _ { 1 } } ( Y | X )$ has to select a suitable parameter $\\phi ^ { * }$ which samples the $y$ and $z$ to minimize the reconstruction loss $\\mathbb { E } _ { ( z , y ) \\sim q _ { \\phi } ( Z , Y | x ) } \\left[ \\log p _ { \\theta _ { 1 } } ( x | y , z ) \\right]$ . When the dimension of $Z$ is chosen to be much smaller than that of $X$ , to obtain a smaller reconstruction error, the decoder has to utilize the information provided by $Y$ , and force the value of $Y$ to be useful for ",
+        "bbox": [
+            173,
+            760,
+            826,
+            858
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 CausalNL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            330,
+            106
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            186,
+            109,
+            728,
+            125
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: A noisy sample $S$ , Average noise rate $\\rho$ , Total epoch $T _ { m a x }$ , Batch size $N$ .   \n1: For $\\mathrm { T } = 1 , \\dots , T _ { m a x }$ :   \n2: For mini-batch $\\bar { S } = \\{ x _ { i } \\} _ { i = 0 } ^ { N } , \\tilde { L } = \\{ \\tilde { y } _ { i } \\} _ { i = 0 } ^ { N }$ in $S$ :   \n3: Feed $\\bar { S }$ to encoders $\\hat { q } _ { \\phi _ { 1 } ^ { 1 } }$ and $\\hat { q } _ { \\phi _ { 1 } ^ { 2 } }$ to get clean label sets $L _ { 1 }$ and $L _ { 2 }$ , respectively;   \n4: Feed $( \\bar { S } , L _ { 1 } )$ to encoder $\\hat { q } _ { \\phi _ { 2 } ^ { 1 } }$ to get a representation set $H _ { 1 }$ , feed $( \\bar { S } , L _ { 2 } )$ to $\\hat { q } _ { \\phi _ { 2 } ^ { 2 } }$ to get $H _ { 2 }$ ;   \n5: Update $\\hat { q } _ { \\phi _ { 2 } ^ { 1 } }$ and $\\hat { q } _ { \\phi _ { 2 } ^ { 2 } }$ with co-teaching loss;   \n6: Feed $( L _ { 1 } , H _ { 1 } )$ to decoder $\\hat { p } _ { \\theta _ { 1 } ^ { 1 } }$ to get reconstructed dataset $\\bar { S } _ { 1 }$ , feed $( L _ { 2 } , H _ { 2 } )$ to $\\hat { p } _ { \\theta _ { 1 } ^ { 2 } }$ to get $\\bar { S } _ { 2 }$ ;   \n7: Feed $( { \\bar { S } } _ { 1 } , L _ { 1 } )$ to decoder $\\hat { p } _ { \\theta _ { 2 } ^ { 1 } }$ to get predicted noisy labels ${ \\tilde { L } } _ { 1 }$ , feed $( \\bar { S } _ { 2 } , L _ { 2 } )$ to $\\hat { p } _ { \\theta _ { 2 } ^ { 2 } }$ to get ${ \\tilde { L } } _ { 2 }$ ;   \n8: Update networks $\\hat { q } _ { \\phi _ { 1 } ^ { 1 } }$ , $\\hat { q } _ { \\phi _ { 2 } ^ { 1 } }$ , $\\hat { p } _ { \\theta _ { 1 } ^ { 1 } }$ and $\\hat { p } _ { \\theta _ { 2 } ^ { 1 } }$ by calculating ELBO on $( \\bar { S } , \\bar { S } _ { 1 } , \\tilde { L } , \\tilde { L } _ { 1 } )$ , update   \nnetworks $\\hat { q } _ { \\phi _ { 1 } ^ { 2 } }$ , $\\hat { q } _ { \\phi _ { 2 } ^ { 2 } }$ , $\\hat { p } _ { \\theta _ { 1 } ^ { 2 } }$ and $\\hat { p } _ { \\theta _ { 2 } ^ { 2 } }$ by calculating ELBO on $( \\bar { S } , \\bar { S } _ { 2 } , \\tilde { L } , \\tilde { L } _ { 2 } )$ ; ",
+        "bbox": [
+            179,
+            114,
+            823,
+            277
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Output: The inference network $\\hat { q } _ { \\phi _ { 1 } ^ { 1 } }$ ",
+        "bbox": [
+            187,
+            279,
+            431,
+            295
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "prediction. Furthermore, we constrain the $Y$ to be a one-hot vector, and then $Y$ could be a cluster ID of which the manifold of the $X$ belongs. ",
+        "bbox": [
+            174,
+            327,
+            821,
+            354
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "So far, the latent variable $Y$ can be inferred as a cluster ID instead of a clean class ID. To further link the clusters to clean labels, a naive approach is to select some reliable examples and keep the cluster numbers to be consistent with the noisy labels on these examples. In such a way, the latent representation $Z$ and clean label $Y$ can be effectively inferred, and therefore, it encourages the identifiability of the transition relationship $p _ { \\theta _ { 2 } } ( { \\tilde { Y } } | Y , { \\tilde { X } } )$ . To achieve this, instead of explicitly selecting the reliable example in advance, our method is trained end-to-end, i.e., reliable examples are selected dynamically during the update of parameters of our model by using the co-teaching technique [10]. The advantage of this approach is that the selection bias of the reliable example [6] can be greatly reduced. Intuitively, the accurately selected reliable examples can encourage the identifiability of $p _ { \\theta _ { 2 } } ( { \\tilde { Y } } | Y , X )$ and $p _ { \\theta _ { 1 } } ( X | Y , Z )$ , and the accurately estimated $p _ { \\theta _ { 2 } } ( { \\tilde { Y } } | Y , X )$ and $p _ { \\theta _ { 1 } } ( X | Y , Z )$ will encourage the network to select more reliable examples. ",
+        "bbox": [
+            173,
+            361,
+            825,
+            517
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Practical Implementation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            534,
+            392,
+            549
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our method is summarized in Algorithm 1 and illustrated in Fig. 3. Here we introduce the structure of our model and loss functions. ",
+        "bbox": [
+            174,
+            559,
+            823,
+            587
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Model Structure Because we incorporate co-teaching in our model training, we need to add a copy of the decoder and encoders in our method. As the two branches share the same architectures, we first present the details of the first branch and then briefly introduce the second branch. ",
+        "bbox": [
+            174,
+            593,
+            825,
+            636
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the first branch, we need a set of encoders and decoders to model the distributions in Eq. 3 and 5. Specifically, we have two encoder networks ",
+        "bbox": [
+            173,
+            641,
+            825,
+            670
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36e7f10596c3a863280078546af513ed91e6a6ba9afab07a4a3cf40d1347fe34.jpg",
+        "text": "$$\nY _ { 1 } = \\hat { q } _ { \\phi _ { 1 } ^ { 1 } } ( X ) , ~ Z _ { 1 } \\sim \\hat { q } _ { \\phi _ { 2 } ^ { 1 } } ( X , Y _ { 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            676,
+            609,
+            696
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for Eq. 5 and two decoder networks ",
+        "bbox": [
+            174,
+            702,
+            410,
+            717
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d10f444025a98b23627729385dbbd0082887d87e606c41633ab457a617524ee2.jpg",
+        "text": "$$\nX _ { 1 } = \\hat { p } _ { \\theta _ { 1 } ^ { 1 } } ( Y _ { 1 } , Z _ { 1 } ) , \\ \\tilde { Y } _ { 1 } = \\hat { p } _ { \\theta _ { 2 } ^ { 1 } } ( X _ { 1 } , Y _ { 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            723,
+            627,
+            744
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for Eq. 3. The first encoder ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 1 } } ( X )$ takes an instance $X$ as input ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 1 } } ( X )$ and output a predicted clean label $Y _ { 1 }$ . The second encoder ${ \\hat { q } } _ { \\phi _ { 2 } ^ { 1 } } ( X , Y _ { 1 } )$ takes both the instance $X$ and the generated label $Y _ { 1 }$ as input and outputs a latent feature $Z _ { 1 }$ . Then the generated $Y _ { 1 }$ and $Z _ { 1 }$ are passed through the decoder $\\hat { p } _ { \\theta _ { 1 } ^ { 1 } } ( Y _ { 1 } , Z _ { 1 } )$ which will generate a reconstructed image $X _ { 1 }$ . Finally, the generated $X _ { 1 }$ and $Y _ { 1 }$ are the input to another decoder $\\hat { p } _ { \\theta _ { 2 } ^ { 1 } } ( X _ { 1 } , Y _ { 1 } )$ which returns predicted noisy labels $\\tilde { Y } _ { 1 }$ . It is worth mentioning that the reparameterization trick [12] is used for sampling, so as to allow backpropagation in $\\hat { q } _ { \\phi _ { 2 } ^ { 1 } } ( X , Y _ { 1 } )$ . ",
+        "bbox": [
+            173,
+            751,
+            826,
+            861
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similarly, the encoder and decoder networks in the second branch are defined as follows: ",
+        "bbox": [
+            176,
+            866,
+            758,
+            882
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17e232c7c39bc42a67a9b51256080d1c6d3f283a1a3afd43284c9e3c720769e2.jpg",
+        "text": "$$\nY _ { 2 } = \\hat { q } _ { \\phi _ { 1 } ^ { 2 } } ( X ) , Z _ { 2 } \\sim \\hat { q } _ { \\phi _ { 2 } ^ { 2 } } ( X , Y _ { 2 } ) , X _ { 2 } = \\hat { p } _ { \\theta _ { 1 } ^ { 2 } } ( Y _ { 2 } , Z _ { 2 } ) , \\tilde { Y } _ { 2 } = \\hat { p } _ { \\theta _ { 2 } ^ { 2 } } ( X _ { 2 } , Y _ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            888,
+            767,
+            910
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "During training, we let two encoders ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 1 } } ( X )$ and $\\hat { q } _ { \\phi _ { 1 } ^ { 2 } } ( X )$ teach each other given every mini-batch. ",
+        "bbox": [
+            173,
+            90,
+            820,
+            108
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Loss Functions We divide the loss functions into two parts. The first part is the negative ELBO in Eq. 6, and the second part is a co-teaching loss. The detailed formulation is leaved in Appendix B. ",
+        "bbox": [
+            173,
+            113,
+            820,
+            142
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the negative ELBO, the first term $- \\mathbb { E } _ { ( z , y ) \\sim q _ { \\phi } ( Z , Y \\mid x ) } \\left[ \\log p _ { \\theta _ { 1 } } ( x \\mid y , z ) \\right]$ is a reconstruction loss, and we use the $\\ell { 1 }$ loss for reconstruction. The second term is $- \\mathbb { E } _ { y \\sim q _ { \\phi _ { 1 } } ( Y | x ) } \\left[ \\log p _ { \\theta _ { 2 } } ( \\tilde { y } | y , x ) \\right]$ , which aims to learn noisy labels given inference $y$ and $x$ , and can be simply replaced by the cross-entropy loss on outputs of both decoders $\\hat { p } _ { \\theta _ { 2 } ^ { 1 } } ( X _ { 1 } , Y _ { 1 } )$ and $\\hat { p } _ { \\theta _ { 2 } ^ { 2 } } ( X _ { 2 } , Y _ { 2 } )$ with the noisy labels contained in the training data. The additional two terms are two regularizers. To calculate $k l ( q _ { \\phi _ { 1 } } ( Y | x ) | | p ( Y ) )$ , we assume that the prior $P ( Y )$ is a uniform distribution. Then minimizing $k l ( q _ { \\phi _ { 1 } } ( Y | x ) | | p ( Y ) )$ is equivalent to maximizing the entropy of $q _ { \\phi _ { 1 } } ( Y | x )$ for each instance $x$ , i.e., $\\begin{array} { r } { - \\sum _ { y } q _ { \\phi _ { 1 } } ( y | x ) \\log q _ { \\phi _ { 1 } } ( y | x ) } \\end{array}$ . The benefit for having this term is that it could reduce the overfiting problem of the inference network. For $\\mathbb { E } _ { y \\sim q _ { \\phi _ { 1 } } ( Y | x ) } \\left[ k l ( q _ { \\phi } ( Z | y , x ) | | p ( Z ) ) \\right]$ , we let $p ( Z )$ be a standard multivariate Gaussian distribution. Empirically, $q _ { \\phi } ( Z | y , x )$ is modeled by the encoders ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 1 } } ( X )$ and $\\hat { q } _ { \\phi _ { 1 } ^ { 2 } } ( X )$ which are designed to be deterministic mappings, therefore, the expectation can be removed, and only the $k l$ term $k l ( q _ { \\phi } ( Z | y , x ) | | p ( Z ) )$ is left. When $p ( Z )$ is a Gaussian distribution, the $k l$ term nicely has a closed form solution [12], i.e., $\\begin{array} { r l r } { \\mathrm { ~ } } & { { } } & { - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { J } ( 1 + \\log ( ( \\sigma _ { j } ) ^ { 2 } ) - ( \\mu _ { j } ) ^ { 2 } - ( \\sigma _ { j } ) ^ { 2 } ) } \\end{array}$ , where $J$ is the dimension of a latent representation $z$ , and $\\sigma _ { j }$ and $\\mu _ { j }$ are the encoder outputs. ",
+        "bbox": [
+            173,
+            147,
+            825,
+            358
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the co-teaching loss, we follow the work of Han et al. [10]. Intuitively, two encoders ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 1 } } ( X )$ and ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 2 } } ( X )$ feed all data forward and selects some data of possibly clean labels. Then, two networks communicate with each other to select possible clean data in this mini-batch and use them for training. Finally, each encoder backpropagates over the data selected by its peer network and updates itself by cross-entropy loss. ",
+        "bbox": [
+            174,
+            363,
+            825,
+            438
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            458,
+            312,
+            476
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare the classification accuracy of proposed method with popular label-noise learning algorithms [15, 19, 11, 10, 28, 36, 16] on both synthetic and real-world datasets. ",
+        "bbox": [
+            174,
+            489,
+            823,
+            518
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Experimental Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            536,
+            351,
+            551
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets We examine the efficacy of our approach on manually corrupted versions of four datasets, i.e., FashionMNIST [31], SVHN [18], CIFAR10, CIFAR100 [13], and one real-world noisy dataset, i.e., Clothing1M [32]. FashionMNIST contains 60,000 training images and 10,000 test images with 10 classes; SVHN contains 73,257 training images and 26,032 test images with 10 classes. CIFAR10 contains 50,000 training images and 10,000 test images. CIFAR10 and CIFAR100 both contain 50,000 training images and 10,000 test images but the former has 10 classes of images, and the latter has 10 classes of images. The four datasets contain clean data. We add instance-dependent label noise to the training sets manually according to Xia et al. [30]. Clothing1M has 1M images with real-world noisy labels and $1 0 \\mathrm { k }$ images with clean labels for testing. For all the synthetic noisy datasets, the experiments are repeated 5 times. ",
+        "bbox": [
+            173,
+            561,
+            825,
+            700
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Network structure and optimization For a fair comparison, all experiments are conducted on NVIDIA Tesla V100, and all methods are implemented by PyTorch. Dimension of the latent representation $Z$ is set to 25 for all synthetic noisy datasets. For encoder networks ${ \\hat { q } } _ { \\phi _ { 1 } ^ { 1 } } ( X )$ and $\\hat { q } _ { \\phi _ { 1 } ^ { 2 } } ( X )$ , we use the same network structures with the baseline method. Specially, we use a ResNet-18 network for FashionMNIST, a ResNet-34 network for SVHN and CIFAR10, a ResNet-50 network for CIFAR100 without pretraining. For Clothing1M, we use ResNet-50 networks pre-trained on ImageNet. The data-augmentation methods random crop and horizontal flip are used for our method. For Clothing1M, we use a ResNet-50 network pre-trained on ImageNet, and the clean training data is not used. The dimensionality of the latent representation $Z$ is set to 100. Due to limited space, we leave the detailed structure of other decoders and encoders in Appendix C. ",
+        "bbox": [
+            173,
+            707,
+            825,
+            849
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines and measurements We compare the proposed method with the following state-of-the-art approaches: (i). CE, which trains the standard deep network with the cross-entropy loss on noisy datasets. (ii). Decoupling [16], which trains two networks on samples whose predictions from the two networks are different. (iii). MentorNet [11] and Co-teaching [10], which mainly handle noisy labels by training on instances with small loss values. (iv). Forward [19], Reweight [15], and T-Revision [28]. These approaches utilize a class-dependent transition matrix $T$ to correct the loss function. For these baselines, we follow the experiments settings of the original papers. We report average test accuracy on over the last ten epochs of each model on the clean test set. Higher classification accuracy means that the algorithm is more robust to the label noise. ",
+        "bbox": [
+            174,
+            856,
+            823,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/9ba0e724333e14ebf5ae74c6a7c9fcffd045f11c0d42cea2d12a7c6be488b0bd.jpg",
+        "table_caption": [
+            "Table 1: Means and standard deviations (percentage) of classification accuracy on FashionMNIST with different label noise levels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>88.54±0.32</td><td>88.38±0.42</td><td>84.22±0.35</td><td>69.72±0.72</td><td>52.32±0.68</td></tr><tr><td>Co-teaching</td><td>91.21±0.31</td><td>90.30±0.42</td><td>89.10±0.29</td><td>86.78±0.90</td><td>63.22±1.56</td></tr><tr><td>Decoupling</td><td>90.70±0.28</td><td>90.34±0.36</td><td>88.78±0.44</td><td>87.54±0.53</td><td>68.32±1.77</td></tr><tr><td>MentorNet</td><td>91.57±0.29</td><td>90.52±0.41</td><td>88.14±0.76</td><td>85.12±0.76</td><td>61.62±1.42</td></tr><tr><td>Mixup</td><td>88.68±0.37</td><td>88.02±0.37</td><td>85.47±0.55</td><td>79.57±0.75</td><td>66.02±2.58</td></tr><tr><td>Forward</td><td>90.05±0.43</td><td>88.65±0.43</td><td>86.27±0.48</td><td>73.35±1.03</td><td>58.23±3.14</td></tr><tr><td>Reweight</td><td>90.27±0.27</td><td>89.58±0.37</td><td>87.04±0.32</td><td>80.69±0.89</td><td>64.13±1.23</td></tr><tr><td>T-Revision</td><td>91.58±0.31</td><td>90.11±0.61</td><td>89.46±0.42</td><td>84.01±1.14</td><td>68.99±1.04</td></tr><tr><td>CausalNL</td><td>90.84±0.31</td><td>90.68±0.37</td><td>90.01±0.45</td><td>88.75±0.81</td><td>78.19±1.01</td></tr></table>",
+        "bbox": [
+            200,
+            125,
+            794,
+            286
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/ea50641a5d11a024c1aae14c120f56a435fe0586fe2b0d4c445a44ef957857f9.jpg",
+        "table_caption": [
+            "Table 2: Means and standard deviations (percentage) of classification accuracy on SVHN with different label noise levels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>91.51±0.45</td><td>91.21±0.43</td><td>87.87±1.12</td><td>67.15±1.65</td><td>51.01±3.62</td></tr><tr><td>Co-teaching</td><td>93.93±0.31</td><td>92.06±0.31</td><td>91.93±0.81</td><td>89.33±0.71</td><td>67.62±1.99</td></tr><tr><td>Decoupling</td><td>90.02±0.25</td><td>91.59±0.25</td><td>88.27±0.42</td><td>84.57±0.89</td><td>65.14±2.79</td></tr><tr><td>MentorNet</td><td>94.08±0.12</td><td>92.73±0.37</td><td>90.41±0.49</td><td>87.45±0.75</td><td>61.23±2.82</td></tr><tr><td>Mixup</td><td>89.73±0.37</td><td>90.02±0.35</td><td>85.47±0.63</td><td>82.41±0.62</td><td>68.95±2.58</td></tr><tr><td>Forward</td><td>91.89±0.31</td><td>91.59±0.23</td><td>89.33±0.53</td><td>80.15±1.91</td><td>62.53±3.35</td></tr><tr><td>Reweight</td><td>92.44±0.34</td><td>92.32±0.51</td><td>91.31±0.67</td><td>85.93±0.84</td><td>64.13±3.75</td></tr><tr><td>T-Revision</td><td>93.14±0.53</td><td>93.51±0.74</td><td>92.65±0.76</td><td>88.54±1.58</td><td>64.51±3.42</td></tr><tr><td>CausalNL</td><td>94.06±0.23</td><td>93.86±0.65</td><td>93.82±0.64</td><td>93.19±0.93</td><td>85.41±2.95</td></tr></table>",
+        "bbox": [
+            200,
+            338,
+            794,
+            500
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f05e68eb5a96cf6ab58ea4fe6e8097deeb3e3ac436dd36fc71c67920ca7be966.jpg",
+        "table_caption": [
+            "Table 3: Means and standard deviations (percentage) of classification accuracy on CIFAR10 with different label noise levels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>75.81±0.26</td><td>69.15±0.65</td><td>62.45±0.86</td><td>51.72±1.34</td><td>39.42±2.52</td></tr><tr><td>Co-teaching</td><td>80.96±0.31</td><td>78.56±0.61</td><td>73.41±0.78</td><td>71.60±0.79</td><td>45.92±2.21</td></tr><tr><td>Decoupling</td><td>78.71±0.15</td><td>75.17±0.58</td><td>61.73±0.34</td><td>58.61±1.73</td><td>50.43±2.19</td></tr><tr><td>MentorNet</td><td>81.03±0.24</td><td>77.22±0.47</td><td>71.83±0.49</td><td>66.18±0.64</td><td>47.89±2.03</td></tr><tr><td>Mixup</td><td>73.17±0.34</td><td>70.02±0.31</td><td>61.56±0.71</td><td>56.45±0.67</td><td>48.95±2.58</td></tr><tr><td>Forward</td><td>74.64±0.26</td><td>69.75±0.56</td><td>60.21±0.75</td><td>48.81±2.59</td><td>46.27±1.30</td></tr><tr><td>Reweight</td><td>76.23±0.25</td><td>70.12±0.72</td><td>62.58±0.46</td><td>51.54±0.92</td><td>45.46±2.56</td></tr><tr><td>T-Revision</td><td>76.15±0.37</td><td>70.36±0.54</td><td>64.09±0.37</td><td>52.42±1.01</td><td>49.02±2.13</td></tr><tr><td>CausalNL</td><td>81.47±0.32</td><td>80.38±0.44</td><td>77.53±0.45</td><td>78.60±1.06</td><td>67.39±1.24</td></tr></table>",
+        "bbox": [
+            200,
+            551,
+            794,
+            712
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            739,
+            826,
+            809
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 Classification Accuracy Evaluation ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            829,
+            457,
+            844
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on synthetic noisy datasets Tables 1, 2, 3, and 4 report the classification accuracy on the datasets of $F$ -MNIST, SVHN, CIFAR-10, and CIFAR100, respectively. The synthetic experiments reveal that our method is powerful in handling instance-dependent label noise particularly in the situation of high noise rates. Specifically, for all datasets, the classification accuracy of our method decrease much slower than that of baseline methods. Additionally, the classification accuracies on these datasets are improved by using CausalNL, which implies that our method should capture the underlying data generation process, and then $Y$ should be a cause of $X$ for all these datasets. ",
+        "bbox": [
+            174,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/91690d05aa3c888aec60dcd983db2c0ed2061768144ec8cbb4d8e4e0aa8faab0.jpg",
+        "table_caption": [
+            "Table 4: Means and standard deviations (percentage) of classification accuracy on CIFAR100 with different label noise levels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>IDN-20%</td><td>IDN-30%</td><td>IDN-40%</td><td>IDN-45%</td><td>IDN-50%</td></tr><tr><td>CE</td><td>30.42±0.44</td><td>24.15±0.78</td><td>21.45±0.70</td><td>15.23±1.32</td><td>14.42±2.21</td></tr><tr><td>Co-teaching</td><td>37.96±0.53</td><td>33.43±0.74</td><td>28.04±1.43</td><td>25.60±0.93</td><td>23.97±1.91</td></tr><tr><td>Decoupling</td><td>36.53±0.49</td><td>30.93±0.88</td><td>27.85±0.91</td><td>23.81±1.31</td><td>19.59±2.12</td></tr><tr><td>MentorNet</td><td>38.91±0.54</td><td>34.23±0.73</td><td>31.89±1.19</td><td>27.53±1.23</td><td>24.15±2.31</td></tr><tr><td>Mixup</td><td>32.92±0.76</td><td>29.76±0.87</td><td>25.92±1.26</td><td>23.13±2.15</td><td>21.31±1.32</td></tr><tr><td>Forward</td><td>36.38±0.92</td><td>33.17±0.73</td><td>26.75±0.93</td><td>21.93±1.29</td><td>19.27±2.11</td></tr><tr><td>Reweight</td><td>36.73±0.72</td><td>31.91±0.91</td><td>28.39±1.46</td><td>24.12±1.41</td><td>20.23±1.23</td></tr><tr><td>T-Revision</td><td>37.24±0.85</td><td>36.54±0.79</td><td>27.23±1.13</td><td>25.53±1.94</td><td>22.54±1.95</td></tr><tr><td>CausalNL</td><td>41.47±0.43</td><td>40.98±0.62</td><td>34.02±0.95</td><td>33.34±1.13</td><td>32.129±2.23</td></tr></table>",
+        "bbox": [
+            197,
+            125,
+            797,
+            285
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/461c0afd8fb5d0ee1f1b24e82bbef26f85c263df8d189f77840de1a550ac9dc6.jpg",
+        "table_caption": [
+            "Table 5: Classification accuracy on Clothing1M. In the experiments, only noisy samples are exploited to train and validate the deep model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CE</td><td>Decoupling</td><td>MentorNet</td><td>Co-teaching</td><td>Forward</td><td>Reweight</td><td>T-Revision</td><td>caualNL</td></tr><tr><td>68.88</td><td>54.53</td><td>56.79</td><td>60.15</td><td>69.91</td><td>70.40</td><td>70.97</td><td>72.24</td></tr></table>",
+        "bbox": [
+            173,
+            342,
+            838,
+            385
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            420,
+            825,
+            462
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For noisy $F$ -MNIST, SVHN and CIFAR-10, in the easy case IDN- $20 \\%$ , almost all methods work well. When the noise rate is $30 \\%$ , the advantages of causalNL begin to show. We surpassed all methods obviously. When the noise rate raises, all the baselines are gradually defeated. Finally, in the hardest case, i.e., IDN- $50 \\%$ , the superiority of causalNL widens the gap of performance. The classification accuracy of causalNL is at least over $10 \\%$ higher than the best baseline method. For noisy CIFAR-100, none of the methods works well. However, causalNL still overtakes the other methods with clear gaps for all different levels of noise rate. ",
+        "bbox": [
+            174,
+            468,
+            825,
+            565
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results on the real-world noisy dataset On the real-world noisy dataset Clothing1M, our method causalNL outperforms all the baselines, as shown in Table 5. The experimental results also show that the noise type in Clothing1M is more likely to be instance-dependent label noise, suggesting that the instance-independent assumption on the transition matrix sometimes can be strong. ",
+        "bbox": [
+            174,
+            571,
+            825,
+            628
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            654,
+            299,
+            671
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have investigated how to use $P ( X )$ to help learn instance-dependent label noise. Specifically, previous assumptions are made on the transition matrix, and the assumptions are hard to be verified and might be violated on real-world datasets. From inspired by a causal perspective, when $Y$ is a cause of $X$ , then $P ( X )$ should contain useful information to infer the clean label $Y$ . We propose a novel generative approach called causalNL for instance-dependent label-noise learning. Our model makes use of the causal graph to contribute to the identifiability of the transition matrix, and therefore helps learn clean labels. In order to learn $P ( X )$ , compared to the previous methods, our method contains more parameters. But experiments on both synthetic and real-world noisy datasets show that a bit sacrifice on computational efficiency is worth it, i.e., the classification accuracy of casualNL significantly outperforms all the state-of-the-art methods. Additionally, the results also indicates that in classification problems, $Y$ can usually be considered as a cause of $X$ , and suggests that the understanding and modeling of the data generation process can help leverage additional information that is useful in solving advanced machine learning problems concerning the relationship between different modules of the data joint distribution. In our future work, we will study the theoretical properties of our method and establish identifiability results under certain assumptions on the data-generative process. ",
+        "bbox": [
+            174,
+            689,
+            826,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            328,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "TL was partially supported by Australian Research Council Projects DP-180103424, DE-190101473, and IC-190100031. GM was supported by Australian Research Council Project DE210101624. BH was supported by supported by the RGC Early Career Scheme No. 22200720 and NSFC Young Scientists Fund No. 62006202. GN was supported by JST AIP Acceleration Research Grant Number JPMJCR20U3, Japan. KZ was supported in part by the National Institutes of Health (NIH) under Contract R01HL159805, by the United States Air Force under Contract No. FA8650-17-C7715, by the NSF-Convergence Accelerator Track-D award #2134901, and by a grant from Apple. The NIH or NSF is not responsible for the views reported in this article. The authors thank the reviewers and the meta-reviewer for their helpful and constructive comments on this work. ",
+        "bbox": [
+            173,
+            121,
+            826,
+            244
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            265,
+            266,
+            281
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] Dana Angluin and Philip Laird. Learning from noisy examples. Machine Learning, 2(4): 343–370, 1988. [2] Devansh Arpit, Stanisław Jastrz˛ebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, et al. A closer look at memorization in deep networks. In International Conference on Machine Learning, pages 233–242. PMLR, 2017.   \n[3] Mikhail Belkin, Partha Niyogi, and Vikas Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of machine learning research, 7(11), 2006.   \n[4] Antonin Berthon, Bo Han, Gang Niu, Tongliang Liu, and Masashi Sugiyama. Confidence scores make instance-dependent label-noise learning possible. In ICML, 2021.   \n[5] David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859–877, 2017.   \n[6] Hao Cheng, Zhaowei Zhu, Xingyu Li, Yifei Gong, Xing Sun, and Yang Liu. Learning with instance-dependent label noise: A sample sieve approach. In ICLR, 2021.   \n[7] Jiacheng Cheng, Tongliang Liu, Kotagiri Ramamohanarao, and Dacheng Tao. Learning with bounded instance and label-dependent label noise. In ICML, 2020.   \n[8] Jacob Goldberger and Ehud Ben-Reuven. Training deep neural-networks using a noise adaptation layer. In ICLR, 2017.   \n[9] Mingming Gong, Kun Zhang, Tongliang Liu, Dacheng Tao, Clark Glymour, and Bernhard Schölkopf. Domain adaptation with conditional transferable components. In International conference on machine learning, pages 2839–2848. PMLR, 2016.   \n[10] Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In NeurIPS, pages 8527–8537, 2018.   \n[11] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-Fei. MentorNet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In ICML, pages 2309–2318, 2018.   \n[12] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   \n[13] Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009.   \n[14] Xuefeng Li, Tongliang Liu, Bo Han, Gang Niu, and Masashi Sugiyama. Provably end-to-end label-noise learning without anchor points. In ICML, 2021.   \n[15] Tongliang Liu and Dacheng Tao. Classification with noisy labels by importance reweighting. IEEE Transactions on pattern analysis and machine intelligence, 38(3):447–461, 2016.   \n[16] Eran Malach and Shai Shalev-Shwartz. Decoupling\" when to update\" from\" how to update\". In NeurIPS, pages 960–970, 2017.   \n[17] Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and Ambuj Tewari. Learning with noisy labels. In NeurIPS, pages 1196–1204, 2013.   \n[18] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew $\\mathrm { Y . N g }$ . Reading digits in natural images with unsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.   \n[19] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In CVPR, pages 1944–1952, 2017.   \n[20] Judea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, New York, NY, USA, 2000. ISBN 0-521-77362-8.   \n[21] Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.   \n[22] B Schölkopf, D Janzing, J Peters, E Sgouritsa, K Zhang, and J Mooij. On causal and anticausal learning. In 29th International Conference on Machine Learning (ICML 2012), pages 1255– 1262. International Machine Learning Society, 2012.   \n[23] Bernhard Schölkopf. Causality for machine learning. arXiv preprint arXiv:1911.10500, 2019.   \n[24] Clayton Scott. A rate of convergence for mixture proportion estimation, with application to learning from noisy labels. In AISTATS, pages 838–846, 2015.   \n[25] Peter Spirtes and Kun Zhang. Causal discovery and inference: concepts and recent methodological advances. In Applied informatics, volume 3, pages 1–28. SpringerOpen, 2016.   \n[26] Peter Spirtes, Clark N Glymour, Richard Scheines, David Heckerman, Christopher Meek, Gregory Cooper, and Thomas Richardson. Causation, prediction, and search. MIT press, 2000.   \n[27] Songhua Wu, Xiaobo Xia, Tongliang Liu, Bo Han, Mingming Gong, Nannan Wang, Haifeng Liu, and Gang Niu. Class2simi: A noise reduction perspective on learning with noisy labels. In International Conference on Machine Learning, pages 11285–11295. PMLR, 2021.   \n[28] Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In NeurIPS, pages 6835–6846, 2019.   \n[29] Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In NeurIPS, pages 6838–6849, 2019.   \n[30] Xiaobo Xia, Tongliang Liu, Bo Han, Nannan Wang, Mingming Gong, Haifeng Liu, Gang Niu, Dacheng Tao, and Masashi Sugiyama. Part-dependent label noise: Towards instance-dependent label noise. In NeurIPS, 2020.   \n[31] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.   \n[32] Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang. Learning from massive noisy labeled data for image classification. In CVPR, pages 2691–2699, 2015.   \n[33] Quanming Yao, Hansi Yang, Bo Han, Gang Niu, and James T Kwok. Searching to exploit memorization effect in learning with noisy labels. In ICML, 2020.   \n[34] Yu Yao, Tongliang Liu, Bo Han, Mingming Gong, Jiankang Deng, Gang Niu, and Masashi Sugiyama. Dual t: Reducing estimation error for transition matrix in label-noise learning. In NeurIPS, 2020.   \n[35] Xiyu Yu, Tongliang Liu, Mingming Gong, and Dacheng Tao. Learning with biased complementary labels. In ECCV, pages 68–83, 2018.   \n[36] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In ICLR, 2018.   \n[37] Kun Zhang and Aapo Hyvarinen. On the identifiability of the post-nonlinear causal model. arXiv preprint arXiv:1205.2599, 2012.   \n[38] Kun Zhang, Mingming Gong, and Bernhard Schölkopf. Multi-source domain adaptation: A causal view. In Twenty-ninth AAAI conference on artificial intelligence, 2015.   \n[39] Kun Zhang, Jiji Zhang, and Bernhard Schölkopf. Distinguishing cause from effect based on exogeneity. arXiv preprint arXiv:1504.05651, 2015. ",
+        "bbox": [
+            173,
+            287,
+            826,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            34,
+            828,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            90,
+            828,
+            270
+        ],
+        "page_idx": 11
+    }
+]

parse/train/rk49Mg-CW/rk49Mg-CW.md

@@ -0,0 +1,267 @@
+# STOCHASTIC VARIATIONAL VIDEO PREDICTION
+
+Mohammad Babaeizadeh1, Chelsea Finn2, Dumitru Erhan3, Roy Campbell1, and Sergey Levine2,3
+
+1University of Illinois at Urbana-Champaign 2University of California, Berkeley 3Google Brain
+
+mb2@uiuc.edu, cbfinn@eecs.berkeley.edu, dumitru@google.com, rhc@illinois.edu, svlevine@eecs.berkeley.edu
+
+# ABSTRACT
+
+Predicting the future in real-world settings, particularly from raw sensory observations such as images, is exceptionally challenging. Real-world events can be stochastic and unpredictable, and the high dimensionality and complexity of natural images require the predictive model to build an intricate understanding of the natural world. Many existing methods tackle this problem by making simplifying assumptions about the environment. One common assumption is that the outcome is deterministic and there is only one plausible future. This can lead to low-quality predictions in real-world settings with stochastic dynamics. In this paper, we develop a stochastic variational video prediction (SV2P) method that predicts a different possible future for each sample of its latent variables. To the best of our knowledge, our model is the first to provide effective stochastic multi-frame prediction for real-world videos. We demonstrate the capability of the proposed method in predicting detailed future frames of videos on multiple real-world datasets, both action-free and action-conditioned. We find that our proposed method produces substantially improved video predictions when compared to the same model without stochasticity, and to other stochastic video prediction methods. Our SV2P implementation will be open sourced upon publication.
+
+# 1 INTRODUCTION
+
+Understanding the interaction dynamics of objects and predicting what happens next is one of the key capabilities of humans which we heavily rely on to make decisions in everyday life (Bubic et al., 2010). A model that can accurately predict future observations of complex sensory modalities such as vision must internally represent the complex dynamics of real-world objects and people, and therefore is more likely to acquire a representation that can be used for a variety of visual perception tasks, such as object tracking and action recognition (Srivastava et al., 2015; Lotter et al., 2017; Denton & Birodkar, 2017). Furthermore, such models can be inherently useful themselves, for example, to allow an autonomous agent or robot to decide how to interact with the world to bring about a desired outcome (Oh et al., 2015; Finn & Levine, 2017).
+
+However, modeling future distributions over images is a challenging task, given the high dimensionality of the data and the complex dynamics of the environment. Hence, it is common to make various simplifying assumptions. One particularly common assumption is that the environment is deterministic and that there is only one possible future (Chiappa et al., 2017; Srivastava et al., 2015; Boots et al., 2014; Lotter et al., 2017). Models conditioned on the actions of an agent frequently make this assumption, since the world is more deterministic in these settings (Oh et al., 2015; Finn et al., 2016). However, most real-world prediction tasks, including the action-conditioned settings, are in fact not deterministic, and a deterministic model can lose many of the nuances that are present in real physical interactions. Given the stochastic nature of video prediction, any deterministic model is obliged to predict a statistic of all the possible outcomes. For example, deterministic models trained with a mean squared error loss function generate the expected value of all the possibilities for each pixel independently, which is inherently blurry (Mathieu et al., 2016).
+
+![](images/aba4881bdcfe439c582ea1ff7e0e21c025bd9dddb8b5b58fb4b62ba499e9a6db.jpg)  
+Figure 1: Importance of stochasticity in video prediction. In each video, a random shape follows a random direction (first row). Given only the first frame, the deterministic model from Finn et al. (2016) predicts the average of all the possibilities. The third row is the output of SV2P with latent sampled from approximated posterior which predicts the correct motion. Last two rows are stochastic outcomes using random latent values sampled from assumed prior. As observed, these outcomes are random but within the range of possible futures. Second sample of Figure 1c shows a case where the model predicts the average of more than one outcome.
+
+Our main contribution in this paper is a stochastic variational method for video prediction, named SV2P, that predicts a different plausible future for each sample of its latent random variables. We also provide a stable training procedure for training a neural network based implementation of this method. To the extent of our knowledge, SV2P is the first latent variable model to successfully predict multiple frames in real-world settings. Our model also supports action-conditioned predictions, while still being able to predict stochastic outcomes of ambiguous actions, as exemplified in our experiments. We evaluate SV2P on multiple real-world video datasets, as well as a carefully designed toy dataset that highlights the importance of stochasticity in video prediction (see Figure 1). In both our qualitative and quantitative comparisons, SV2P produces substantially improved video predictions when compared to the same model without stochasticity, with respect to standard metrics such as PSNR and SSIM. The stochastic nature of SV2P is most apparent when viewing the predicted videos. Therefore, we highly encourage the reader to check the project website https://goo.gl/iywUHc to view the actual videos of the experiments. The TensorFlow (Abadi et al., 2016) implementation of this project will be open sourced upon publication.
+
+# 2 RELATED WORK
+
+A number of prior works have addressed video frame prediction while assuming deterministic environments (Ranzato et al., 2014; Srivastava et al., 2015; Vondrick et al., 2015; Xingjian et al., 2015; Boots et al., 2014; Lotter et al., 2017). In this work, we build on the deterministic video prediction model proposed by Finn et al. (2016), which generates the future frames by predicting the motion flow of dynamically masked out objects extracted from the previous frames. Similar transformationbased models were also proposed by De Brabandere et al. (2016); Liu et al. (2017). Prior work has also considered alternative objectives for deterministic video prediction models to mitigate the blurriness of the predicted frames and produce sharper predictions (Mathieu et al., 2016; Vondrick & Torralba, 2017). Despite the adversarial objective, Mathieu et al. (2016) found that injecting noise did not lead to stochastic predictions, even for predicting a single frame. Oh et al. (2015); Chiappa et al. (2017) make sharp video predictions by assuming deterministic outcomes in video games given the actions of the agents. However, this assumption does not hold in real-world settings, which almost always have stochastic dynamics.
+
+Auto-regressive models have been proposed for modeling the joint distribution of the raw pixels (Kalchbrenner et al., 2017). Although these models predict sharp images of the future, their training and inference time is extremely high, making them difficult to use in practice. Reed et al. (2017) proposed a parallelized multi-scale algorithm that significantly improves the training and prediction time but still requires more than a minute to generate one second of $6 4 \times 6 4$ video on a GPU. Our comparisons suggest that the predictions from these models are sharp, but noisy, and that our method produces substantially better predictions, especially for longer horizons.
+
+Another approach for stochastic prediction uses generative adversarial networks (GANs) (Goodfellow et al., 2014), which have been used for video generation and prediction (Tulyakov et al., 2017; Li et al., 2017). Vondrick et al. (2016); Chen et al. (2017) applied adversarial training to predict video from a single image. Although GANs generate sharp images, they tend to suffer from modecollapse (Goodfellow, 2016), particularly in conditional generation settings (Zhu et al., 2017).
+
+Variational auto-encoders (VAEs) (Kingma & Welling, 2014) also have been explored for stochastic prediction tasks. Walker et al. (2016) uses conditional VAEs to predict dense trajectories from pixels. Xue et al. (2016) predicts a single stochastic frame using cross convolutional networks in a VAElike architecture. Shu et al. (2016) uses conditional VAEs and Gaussian mixture priors for stochastic prediction. Both of these works have been evaluated solely on synthetic datasets with simple moving sprites and no object interaction. Real images significantly complicate video prediction due to the diversity and variety of stochastic events that can occur. Fragkiadaki et al. (2017) compared various architectures for multimodal motion forecasting and one-frame video prediction, including variational inference and straightforward sampling from the prior. Unlike these prior models, our focus is on designing a multi-frame video prediction model to produce stochastic predictions of the future. Multi-frame prediction is dramatically harder than single-frame prediction, since complex events such as collisions require multiple frames to fully resolve, and single-frame predictions can simply ignore this complexity. We believe, our approach is the first latent variable model to successfully demonstrate stochastic multi-frame video prediction on real world datasets.
+
+# 3 STOCHASTIC VARIATIONAL VIDEO PREDICTION (SV2P)
+
+In order to construct our stochastic variational video prediction model, we first formulate a probabilistic graphical model that explains the stochasticity in the video. Since our goal is to perform conditional video prediction, the predictions are conditioned on a set of $c$ context frames $\mathbf { x } _ { 0 } , \ldots , \mathbf { x } _ { c - 1 }$ (e.g., if we are conditioning on one frame, $c = 1 \AA$ ), and our goal is to sample from $p ( \mathbf { x } _ { c : T } | \mathbf { x } _ { 0 : c - 1 } )$ , where $\mathbf { x } _ { i }$ denotes the $\mathrm { i ^ { \mathrm { t h } } }$ frame of the video (Figure 2).
+
+Video prediction is stochastic as a consequence of the latent events that are not observable from the context frames alone. For example, when a robot’s arm pushes a toy on a table, the unknown weight of that toy affects how it moves. We therefore introduce a vector of latent variables $\mathbf { z }$ into our model, distributed according to a prior $\mathbf { z } \sim p ( \mathbf { z } )$ , and build a model $p ( \mathbf { x } _ { c : T } | \mathbf { x } _ { 0 : c - 1 } , \mathbf { z } )$ . This model is still stochastic but uses a more general representation, such as a conditional Gaussian, to explain just the noise in the image, while $\mathbf { z }$ accounts for the more complex stochastic phenomena. We can then factorize this model to $\begin{array} { r } { \prod _ { t = c } ^ { T } p _ { \theta } \big ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 : t - 1 } , \mathbf { z } \big ) } \end{array}$ Learning then involves training the parameters of these factors $\theta$ , which we assume to be shared between all the time steps.
+
+At inference time we need to estimate values for the true posterior $p ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ , which is intractable due its dependency on $p ( \mathbf { x } _ { 0 : T } )$ . We
+
+![](images/d9426d145ac0b3d1886124cb14d9334930e47f47b7fcfc7179483e9be5edd844.jpg)  
+Figure 2: Probabilistic graphical model of stochastic variational video prediction, assuming time-invariant latent. The generative model predicts the next frame conditioned on the previous frames and latent variables (solid lines), while the variational inference model approximates the posterior given all the frames (dotted lines).
+
+overcome this problem by approximating the posterior with an inference network $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ that outputs the parameters of a conditionally Gaussian distribution $\mathcal { N } ( \mu _ { \phi } ( \mathbf { x } _ { 0 : T } ) , \sigma _ { \phi } ( \mathbf { x } _ { 0 : T } ) )$ . This network is trained using the reparameterization trick (Kingma & Welling, 2014), according to:
+
+![](images/16adbaba343d53be1ef1ee5168ad849ee9ab134fbaab97ce502f9893b6db2c7c.jpg)  
+Figure 3: Architecture of SV2P. At training time, the inference network (top) estimates the posterior $\bar { q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } ) } = \mathcal { N } \big ( \mu ( \mathbf { x } _ { 0 : T } ) , \sigma ( \mathbf { x } _ { 0 : T } ) \big )$ . The latent value ${ \mathbf z } \sim q _ { \phi } ( { \mathbf z } | { \mathbf x } _ { 0 : T } )$ is passed to the generative network along with the (optional) action. The generative network (from Finn et al. (2016)) predicts the next frame given the previous frames, latent values, and actions. At test time, $\mathbf { z }$ is sampled from the assumed prior $\mathcal { N } ( \mathbf { 0 } , \bar { \mathbf { I } } )$ .
+
+$$
+\mathbf { z } = \mu _ { \phi } ( \mathbf { x } _ { 0 : T } ) + \sigma _ { \phi } ( \mathbf { x } _ { 0 : T } ) \times \epsilon , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )
+$$
+
+Here, $\theta$ and $\phi$ are the parameters of the generative model and inference network, respectively. To learn these parameters, we can optimize the variational lower bound, as in the variational autoencoder (VAE) (Kingma & Welling, 2014):
+
+$$
+\mathcal { L } ( \mathbf { x } ) = - \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } ) } \left[ \log p _ { \theta } ( \mathbf { x } _ { t : T } | \mathbf { x } _ { 0 : t - 1 } , \mathbf { z } ) \right] + D _ { K L } \big ( q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } ) | | p ( \mathbf { z } ) \big )
+$$
+
+where $D _ { K L }$ is the Kullback-Leibler divergence between the approximated posterior and assumed prior $p ( \mathbf { z } )$ which in our case is the standard Gaussian $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ .
+
+In Equation 2, the first term on the RHS represents the reconstruction loss while the second term represents the divergence of the variational posterior from the prior on the latent variable. It is important to emphasize that the approximated posterior is conditioned on all of the frames, including the future frames $\mathbf { x } _ { t : T }$ . This is feasible during training, since $\mathbf { x } _ { t : T }$ is available at the training time, while at test time we can sample the latents from the assumed prior. Since the aim in our method is to recover latent variables that correspond to events which might explain the variability in the videos, we found that it is in fact crucial to condition the inference network on future frames. At test time, the latent variables are simply sampled from the prior which corresponds to a smoothing-like inference process. In principle, we could also perform a filtering-like inference procedure of the form $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : t - 1 } )$ for time step $t$ to infer the most likely latent variables based only on the conditioning frames, instead of sampling from the prior, which could produce more accurate predictions at test time. However, it would be undesirable to use a filtering process at training time: in order to incentivize the forward prediction network to make use of the latent variables, they must contain some information that is useful for predicting future frames that is not already present in the context frames. If they are predicted entirely from the context frames, no such information is present, and indeed we found that a purely filtering-based model simply ignores the latent variables.
+
+So far, we’ve assumed that the latent events are constant over the entire video. We can relax this assumption by conditioning prediction on a time-variant latent variable $\mathbf { z } _ { t }$ that is sampled at every time step from $p ( \mathbf { z } )$ . The generative model then becomes $\begin{array} { r } { p ( \mathbf { z } _ { t } ) \prod _ { t = c } ^ { T } p _ { \theta } \big ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 : t - 1 } , \mathbf { z } _ { t } \big ) } \end{array}$ and, assuming a fixed posterior, the inference model will be approximated by $q _ { \phi } \big ( \mathbf { z } _ { t } | \mathbf { x } _ { 0 : T } \big )$ , where the model parameters $\phi$ are shared across time. In practice, the only difference between these two formulations is the frequency of sampling $\mathbf { z }$ from $p ( \mathbf { z } )$ and $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ . In the time-invariant version, we sample $\mathbf { z }$ once per video, whereas with the time-variant latent, sampling happens every frame. The main benefit of time-variant latent variable is better generalization beyond $T$ , since the model does not have to encode all the events of the video in one vector $\mathbf { z }$ . We provide an empirical comparison of these formulations in Section 5.2.
+
+![](images/4bfa96253e877aba226a73c223f94d0e5e06c8cc5ba6268162a30dfba292d162.jpg)  
+Figure 4: Three phases of training. In the first phase, the inference network is turned off and only the generative network is being trained, resulting in deterministic predictions. The inference network is used in the second phase without a KL-loss. The last phase includes $D _ { K L } \big ( q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } ) | | p ( \mathbf { z } ) \big )$ to enable accurate sampling latent from $p ( \mathbf { z } )$ . (a) the KL-loss $( b )$ the reconstruction loss (c) Training stability. This graph compares reconstruction loss at the end of five training sessions on the BAIR robot pushing dataset, with and without following all the steps of the training procedure. The proposed training is quite stable and results in lower error compared to na¨ıve training.
+
+In action-conditioned settings, we modify the generative model to be conditioned on action vector $\mathbf { a } _ { t }$ . This results in $\begin{array} { r } { p ( \mathbf { z } _ { t } ) \prod _ { t = c } ^ { T } p _ { \boldsymbol \theta } ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 : t - 1 } , \mathbf { z } _ { t } , \mathbf { a } _ { t } ) } \end{array}$ as generative model while keeping the posterior approximation intact. Conditioning the outcome on actions can decrease future variability; however it will not eliminate it if the environment is inherently stochastic or the actions are ambiguous. In this case, the model is still capable of predicting stochastic outcomes in a narrower range of possibilities.
+
+# 3.1 MODEL ARCHITECTURE
+
+To model the approximated posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ we used a deep convolutional neural network as shown in the top row of Figure 3. Since we assumed a diagonal Gaussian distribution for $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ , this network outputs the mean $\mu _ { \phi } \big ( \mathbf { x } _ { 0 : T } \big )$ and standard deviation $\log \sigma _ { \phi } ( \mathbf { x } _ { 0 : T } )$ of the approximated posterior. Hence the entire inference network is convolutional, the predicted parameters are $8 \times 8$ single channel response maps. We assume each entry in this response maps is pairwise independent, forming the latent vector $\mathbf { z }$ . The latent value is then sampled using Equation 1. As discussed before, this sampling happens every frame for time-varying latent, and once per video in time-invariant case.
+
+For $p ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 : t - 1 } , \mathbf { z } )$ , we used the CDNA architecture proposed by Finn et al. (2016), which is a deterministic convolutional recurrent network that predicts the next frame $\mathbf { x } _ { t }$ given the previous frame $\mathbf { x } _ { t - 1 }$ and an optional action $\mathbf { a } _ { t }$ . This model constructs the next frames by predicting the motions of segments of the image (i.e., objects) and then merging these predictions via masking. Although this model directly outputs pixels, it is partially-appearance invariant and can generalize to unseen objects (Finn et al., 2016). To condition on the latent value, we modify the CDNA architecture by stacking $\mathbf { z } _ { t }$ as an additional channel on tiled action $\mathbf { a } _ { t }$ .
+
+# 3.2 TRAINING PROCEDURE
+
+Our model can be trained end-to-end. However, our experiments show that na¨ıve training usually results in the model ignoring the latent variables and converging to a suboptimal deterministic solution (Figure 4). Therefore, we train the model end-to-end in three phases, as follows:
+
+1. Training the generative network: In this phase, the inference network has been disabled and the latent value $\mathbf { z }$ will be randomly sampled from $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ . The intuition behind this phase is to train the generative model to predict the future frames deterministically (i.e. modeling $\mathbf { \bar { \rho } } _ { p _ { \theta } ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 : t - 1 } ) } ) $ .   
+2. Training the inference network: In the second phase, the inference network is trained to estimate the approximate posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ ; however, the KL-loss is set to 0. This means that the model can use the latent value without being penalized for diverging from $p ( \mathbf { z } )$ . As seen in Figure 4, this phase results in very low reconstruction error, however it is not usable at the test time since $D _ { K L } \mathbf { \bar { ( } } q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } ) | | p ( \mathbf { z } ) \rrangle \gg 0$ and sampling $\mathbf { z }$ from the assumed prior will be inaccurate.
+
+3. Divergence reduction: In the last phase, the KL-loss is added, resulting in a sudden drop of KLdivergence and an increase of reconstruction error. The reconstruction loss converging to a value lower than the first phase and KL-loss converging to zero are indicators of successful training. This means that $\mathbf { z }$ can be sampled from $p ( \mathbf { z } )$ at test time for effective stochastic prediction.
+
+To gradually transition from the second phase to the third, we add a multiplier to KL-loss that is set to zero during the first two phases and then increased slowly in the last phase. This is similar to the $\beta$ hyper-parameter in Higgins et al. (2016) and Bowman et al. (2016) that is used to balance latent channel capacity and independence constraints with reconstruction accuracy.
+
+We found that this training procedure is quite stable and the model almost always converges to the desired parameters. To demonstrate this stability, we trained the model with and without the proposed training procedure, five times each. Figure 4 shows the average and standard deviation of reconstruction loss at the end of these training sessions. Na¨ıve training results in a slightly better error compared to Finn et al. (2016), but with high variance. When following the proposed training algorithm, the model consistently converges to a much lower reconstruction error.
+
+# 4 STOCHASTIC MOVEMENT DATASET
+
+To highlight the importance of stochasticity in video prediction, we created a toy video dataset with intentionally stochastic motion. Each video in this dataset is four frames long. The first frame contains a random shape (triangle, rectangle or circle) with random size and color, centered in the frame, which then randomly moves to one of the eight directions (up, down, left, right, up-left, upright, down-left, down-right). Each frame is $6 4 \times 6 4 \times 3$ and the background is static gray. The main intuition behind this design is that, given only the first frame, a model can figure out the shape, color, and size of the moving object, but not its movement direction.
+
+We train Finn et al. (2016) and SV2P to predict the future frames, given only the first frame. Figure 1 shows the video predictions from these two models. Since Finn et al. (2016) is a deterministic model with mean squared error as loss, it predicts the average of all possible outcomes, as expected. In contrast, SV2P predicts different possible futures for each sample of the latent variable $\mathbf { z } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ . In our experiments, all the videos predicted by SV2P are within the range of plausible futures (e.g. we never saw the shape moves in any direction other than the original eight). However, in some cases, SV2P still predicts the average of more than one future, as it can be seen in the first random sample of Figure 1c. The main reason for this problem seems to be overlapping posterior distributions in latent space which can cause some latent values (sampled from $p ( \mathbf { z } ) _ { , } ^ { \dag }$ ) to be ambiguous.
+
+To demonstrate that the inference network is working properly and that the latent variable does indeed learn to store the information necessary for stochastic prediction (i.e., the direction of movement), we include predicted futures when ${ \mathbf z } \sim q _ { \phi } ( { \mathbf x } _ { 0 : T } )$ . By estimating the correct parameters of the latent distribution, using the inference network, the model always generates the right outcome. However, this cannot be used in practice, since the inference network requires access to all the frames, including the ones in the future. Instead, $\mathbf { z }$ will be sampled from assumed prior $p ( \mathbf { z } )$ .
+
+# 5 EXPERIMENTS
+
+To evaluate SV2P, we test it on three real-world video datasets by comparing it to the CDNA model (Finn et al., 2016), as a deterministic baseline, as well as a baseline that outputs the last seen frame as the prediction. We compare SV2P with an auto-regressive stochastic model, video pixel networks (VPN) (Kalchbrenner et al., 2017). We use the parallel multi-resolution implementation of VPN from Reed et al. (2017), which is an order of magnitude faster than the original VPN, but still requires more than a minute to generate one second of $6 4 \times 6 4$ video. In all of these experiments, we plot the results of sampling the latent once per video (SV2P time-invariant latent) and once per frame (SV2P time-variant latent). We strongly encourage readers to view https://goo.gl/iywUHc for videos of the results which are more illustrative than printed frames.
+
+# 5.1 DATASETS
+
+We quantitatively and qualitatively evaluate SV2P on following real-world datasets:
+
+• BAIR robot pushing dataset (Ebert et al., 2017): This dataset contains action-conditioned videos collected by a Sawyer robotic arm pushing a variety of objects. All of the videos in this datasets have similar table top settings with static background. Each video also has recorded actions taken by the robotic arm which correspond to the commanded gripper pose. An interesting property of this dataset is the fact that the arm movements are quite unpredictable in the absence of actions (compared to the robot pushing dataset (Finn et al., 2016) which the arm moves to the center of the bin). For this dataset, we train the models to predict the next ten frames given the first two, both in action-conditioned and action-free settings.
+
+• Human3.6M (Ionescu et al., 2014): Humans and animals are one of the most interesting sources of stochasticity in natural videos, which behave in complex ways as a consequence of unpredictable intentions. To study human motion prediction, we use the Human3.6M dataset which consists of actors performing various actions in a room. We used the pre-processing and testing format of Finn et al. (2016): a $1 0 \ : \mathrm { H z }$ frame rate and 10-frame prediction given the previous ten. The videos from this datasets contains various actions performed by humans (walking, talking on the phone, . . . ). Similar to Finn et al. (2016), we included videos from all the performed actions in training dataset while keeping all the videos from an specific actor out for testing.
+
+• Robotic pushing prediction (Finn et al., 2016): We use the robot pushing prediction dataset to compare SV2P with another stochastic prediction method, video pixel networks (VPNs) (Kalchbrenner et al., 2017). VPNs demonstrated excellent results on this dataset in prior work, and therefore robot pushing dataset provides a strong point of comparison. However, in contrast to our method, VPNs do not include latent stochastic variables that represent random events, and rely on an expensive auto-regressive architecture. In this experiment, the models have been trained to predict the next ten frames, given the first two. Similar to BAIR robot pushing dataset, this dataset also contains actions taken by the robotic arm which are the pose of the commanded gripper.
+
+# 5.2 QUANTITATIVE COMPARISON
+
+In our quantitative evaluation, we aim to understand whether the range of possible futures captured by our stochastic model includes the true future. Models that are more stochastic do not necessarily score better on average standard metrics such as PSNR (Huynh-Thu & Ghanbari, 2008) and SSIM (Wang et al., 2004). However, if we are interested primarily in understanding whether the true outcome is within the set of predictions, we can instead evaluate the score of the best sample from multiple random priors. We argue that this is a better metric for stochastic models, since it allows us to understand if uncertain futures contain the true outcome. Figure 5 illustrates how this metric changes with different numbers of samples. By predicting more possible futures, the probability of predicting the true outcome increases, and therefore it is more likely to get a sample with higher PSNR compared to the ground truth. Of course, as with all video prediction metrics, it is imperfect, and is only suitable for understanding the performance of the model when combined with a visual examination of the qualitative results in Section 5.3.
+
+![](images/78b52c592b0095218a20984840158d2ab406420ba33d49820e2bca9795b24d83.jpg)  
+Figure 5: Stochasticity of SV2P predictions on the action-free BAIR dataset. Each line presents the sample with highest PSNR compared to ground truth, after multiple sampling. The number on the right indicates the number of random samples. As can be seen, SV2P predicts highly stochastic videos and, on average, only three samples is enough to predict outcomes with higher quality compared to Finn et al. (2016).
+
+To use this metric, we sample 100 latent values from prior $\mathbf { z } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ and use them to predict 100 videos and show the result of the sample with
+
+highest PSNR. For a fair comparison to VPN, we use the same best out of 100 samples for our stochastic baseline. Since even the fast implementation of VPN is quite slow, we limit the comparison with VPN to only last dataset with 256 test samples.
+
+![](images/82a2595a17bf365c56d24b74d6c233663fec9ac13b964cf26182ff353185ecb9.jpg)  
+Figure 6: Quantitative comparison of the prediction methods. The stochastic models have been sampled 100 times and the results with the best PSNR have been displayed. For SV2P, we demonstrate the results of both time-variant and time-invariant latent sampling. Repeat shows the results of the lower bound prediction by repeating the last seen frame as the prediction. In the last column, we compare the results of video pixel networks (VPN). All the models, including Finn et al. (2016), have been trained up to the frame marked by vertical separator and the results beyond this line display their generalization. The plots are the average SSIM and PSNR over the test set and shadow is the $9 5 \%$ confidence interval. In all of these graphs, higher is better.
+
+Figure 6 displays the quantitative comparison of the predictions on all of the datasets. In this graph, the top row is a PSNR comparison and the bottom row is SSIM, while each column represents a different dataset. To evaluate the generalization of the models beyond what they have been trained for, we generate more frames than what the models observed during training time. The length of the training sequences is marked by a vertical separator in all of the graphs, and the results beyond this line represent extrapolation to longer sequences.
+
+Overall, SV2P with both time-variant and time-invariant latent sampling outperform all of the other baselines, by predicting higher quality videos with higher PSNR and SSIM. Time-varying latent sampling is more stable beyond the time horizon used during training (Figure 6b). One possible explanation for this behaviour is that the time-invariant latent has to include the information required for predicting all the frames and therefore, beyond training time, it collapses. This issue is mitigated by a time-variant latent variable which takes a different value at each time step. However, this stability is not always the case as it is more evident in late frames of Figure 6a.
+
+One other interesting observation is that the time-invariant model outperforms the time-variant model in the Human3.6M dataset. In this dataset, the most important latent event – the action performed by the actor – is consistent across the whole video which is easier to capture using timeinvariant latent.
+
+# 5.3 QUALITATIVE COMPARISON
+
+We can better understand the performance of the proposed model by visual examination of the qualitative results. We highlight some of the most important and observable differences in predictions by different models in Figures 8-11 1. In all of these figures, the $\mathbf { X }$ -axis is time (i.e., each row is one video). The first row is the ground truth video, and the second row is the result of Finn et al. (2016). The result of sampling the latent from approximated posterior is provided in the third row. For stochastic methods, we show the best (highest PSNR) and worst (lowest PSNR) predictions out of 100 samples (as discussed in Section 5.2), as well as two random predicted videos from our model.
+
+Figure 8 illustrates two examples from the BAIR robot pushing dataset in the action-free setting. As a consequence of the high stochasticity in the movement of the arm in absence of actions, Finn et al. (2016) only blurs the arm out, while SV2P predicts varied but coherent movements of the arm. Note that, although each predicted movements of the arm is random, it is still in the valid range of possible outcomes (i.e., there is no sudden jump of the arm nor random movement of the objects). The proposed model also learned how to move objects in cases where they have been pushed by the predicted movements of the arm, as can be seen in the zoomed images of both samples.
+
+In the action-conditioned setting (Figure 9), the differences are more subtle: the range of possible outcomes is narrower, but we can still observe stochasticity in the behavior of the pushed objects. Interactions between the arm and objects are uncertain due to ambiguity in depth, friction, and mass, and SV2P is able to capture some of this variation. Since these variations are subtle and occupy a smaller part of the images, we illustrate this with zoomed insets in Figure 9. Some examples of varied object movements can be found in last three rows of right example of Figure 9. SV2P also generates sharper outputs, compared to Finn et al. (2016) as is evident in the left example of Figure 9.
+
+![](images/4ef3813f7c9049d9bb6445c5d0f95258bf228837402cb7c08d1cb69e202cc394.jpg)  
+Figure 7: Quantitative comparison of the predicted frames on Human3.6M dataset using confidence of object detection as quality metric. The y-axis demonstrates the average confidence of Huang et al. (2016) in detecting humans in predicted frames. Based on this metric, SV2P predicts images with more meaningful semantics compared to to Finn et al. (2016).
+
+Please note that the approximate posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ is still trained with the evidence lower bound (ELBO), which means that the posterior must compress the information of the future events. Perfect reconstruction of high-quality images from posterior distributions over latent states is an open problem, and the results in our experiments compare favorably to those typically observed even in single-image VAEs (e.g. see Xue et al. (2016)). This is why the model cannot reconstruct all the future frames perfectly, even though when latent values are sampled from $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { 0 : T } )$ .
+
+Figure 10 displays two examples from the Human3.6M dataset. In absence of actions, Finn et al. (2016) manages to separate the foreground from background, but cannot predict what happens next accurately. This results in distorted or blurred foregrounds. On the other hand, SV2P predicts a variety of different outcomes, and moves the actor accordingly. Note that PSNR and SSIM are measuring reconstruction loss with respect to the ground truth and they may not generally present a better prediction. For some applications, a prediction with lower PSNR/SSIM might have higher quality and be more interesting. A good example is the prediction with the worst PSNR in Figure 10- right, where the model predicts that the actor is spinning in his chair with relatively high quality. However, this output has the lowest PSNR compared to the ground truth.
+
+However, pixel-wise metrics such as PSNR and SSIM may not be the best measures for semantic evaluation of predicted frames. Therefore, we use the confidence of an object detector to show the predicted frames contain useful semantic information. For this purpose, we use the open-sourced implementation of Huang et al. (2016) to compare the quality of predicted frames in Human3.6M dataset. As it can be seen in Figure 7, SV2P predicted frames which the human inside can be detected with higher confidence, compared to Finn et al. (2016).
+
+Finally, Figure 11 demonstrates results on the Google robot pushing dataset. The qualitative and quantitative results in Figure 11 and 6 both indicate that SV2P produces substantially better predictions than VPNs. The quantitative results suggest that our best-of-100 metric is a reasonable measure of performance: the VPN predictions are more noisy, but simply increasing noise is not sufficient to increase the quality of the best sample. The stochasticity in our predictions is more coherent, corresponding to differences in object or arm motion, while much of the stochasticity in the VPN predictions resembles noise in the image, as well as visible artifacts when predicting for substantially longer time horizons.
+
+# 6 CONCLUSION
+
+We proposed stochastic variational video prediction (SV2P), an approach for multi-step video prediction based on variational inference. Our primary contributions include an effective stochastic prediction method with latent variables, a network architecture that succeeds on natural videos, and a training procedure that provides for stable optimization. The source code for our method will be released upon acceptance. We evaluated our proposed method on three real-world datasets in actionconditioned and action-free settings, as well as one toy dataset which has been carefully designed to highlight the importance of the stochasticity in video prediction. Both qualitative and quantitative results indicate higher quality predictions compared to other deterministic and stochastic baselines.
+
+SV2P can be expanded in numerous ways. First, the current inference network design is fully convolutional, which exposes multiple limitations, such as unmodeled spatial correlations between the latent variables. The model could be improved by incorporating the spatial correlation induced by the convolutions into the prior, using a learned structured prior in place of the standard spherical Gaussian. Time-variant posterior approximation to reflect the new information that is revealed as the video progresses, is another possible SV2P improvement. However, as discussed in Section 3, this requires incentivizing the inference network to incorporate the latent information at training time. This would allow time-variant latent distributions which is more aligned with generative neural models for time-series(Johnson et al., 2016; Gao et al., 2016; Krishnan et al., 2017).
+
+Another exciting direction for future research would be to study how stochastic predictions can be used to act in the real world, producing model-based reinforcement learning methods that can execute risk-sensitive behaviors from raw image observations. Accounting for risk in this way could be especially important in safety-critical settings, such as robotics.
+
+# ACKNOWLEDGEMENT
+
+The authors would like to thank Matt Johnson for providing feedback on an early draft of the paper, and Alex Lee for fixing bugs in the deterministic version of the model. This material is based upon work supported by the National Science Foundation under award no. 1725729 and was partially done while author was interning at Google Brain.
+
+# REFERENCES
+
+Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for largescale machine learning. In OSDI, volume 16, pp. 265–283, 2016.
+
+Byron Boots, Arunkumar Byravan, and Dieter Fox. Learning predictive models of a depth camera & manipulator from raw execution traces. In International Conference on Robotics and Automation (ICRA), 2014.
+
+Samuel R. Bowman, Luke Vilnis, Oriol Vinyals, Andrew M. Dai, Rafal Jozefowicz, and Samy ´ Bengio. Generating sentences from a continuous space. In Proceedings of the 20th SIGNLL Conference on Computational Natural Language Learning, CoNLL 2016, Berlin, Germany, August 11-12, 2016, pp. 10–21, 2016. URL http://aclweb.org/anthology/K/K16/ K16-1002.pdf.
+
+Andreja Bubic, D Yves Von Cramon, and Ricarda I Schubotz. Prediction, cognition and the brain. Frontiers in human neuroscience, 4, 2010.
+
+Baoyang Chen, Wenmin Wang, Jinzhuo Wang, Xiongtao Chen, and Weimian Li. Video imagination from a single image with transformation generation. arXiv preprint arXiv:1706.04124, 2017.
+
+![](images/a8457f3b2981417099f8ec0f8472dc0520463d5a03763affd471257be50b7a32.jpg)  
+Figure 8: Comparing the results of SV2P with Finn et al. (2016) (second row) on action-free BAIR robot pushing dataset. Fourth and fifth rows are the predictions with minimum and maximum PSNR out of 100 random outputs with time-invariant latent sampling. The last two rows are random predicted outcomes. The numbers on top indicate the predicted frame number. In lack of actions and therefore high stochasticity, Finn et al. (2016) only blurs the robotic arm out while the proposed method predicts sharper frames on each sampling. SV2P also predicts the interaction dynamics between random movements of the arm and the objects.
+
+![](images/aeffc9ce522fc01f59fc532bfeee8dd2cf2e3e965715578d204755eb3faf00fc.jpg)  
+Figure 9: Similar comparison as Figure 8 this time action-conditioned with time-variant latent sampling. SV2P predicts sharper and slightly variant outcomes compared to Finn et al. (2016). This is mostly evident in zoomed in objects which have been pushed by the arm.
+
+![](images/01075ef00723db057ec71d5426419f9a4646c302104026d6443451ee9d56ea2f.jpg)  
+Figure 10: Prediction results on the action-free Human3.6M dataset. SV2P predicts a different outcome on each sampling given the latent. In the left example, the model predicts walking as well as stopping which result in different outputs in predicted future frames. Similarly, the right example demonstrates various outcomes including spinning.
+
+![](images/c92dfa1707968d59cea926015c300dfbf4b9777833a4dc8ab341f2d5f1a65bed.jpg)  
+Figure 11: Comparing the results of video pixel networks (VPN) (Kalchbrenner et al., 2017; Reed et al., 2017) with SV2P on the robotic pushing dataset. We use the same best PSNR out of 100 random samples for both methods. Besides stochastic movements of the pushed objects, another source of stochasticity is the starting lag in movements of the robotic arm. SV2P generates sharper images compared to Finn et al. (2016) (notice the pushed objects in zoomed images) with less noise compared to Reed et al. (2017) (look at the accumulated noise in later frames).
+
+Silvia Chiappa, Sebastien Racani ´ ere, Daan Wierstra, and Shakir Mohamed. Recurrent environment \` simulators. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.
+
+Bert De Brabandere, Xu Jia, Tinne Tuytelaars, and Luc Van Gool. Dynamic filter networks. In Neural Information Processing Systems (NIPS), 2016.
+
+Emily Denton and Vighnesh Birodkar. Unsupervised learning of disentangled representations from video. arXiv preprint arXiv:1705.10915, 2017.
+
+Frederik Ebert, Chelsea Finn, Alex X. Lee, and Sergey Levine. Self-Supervised Visual Planning with Temporal Skip Connections. Conference on Robot Learning (CoRL), 2017.
+
+Chelsea Finn and Sergey Levine. Deep visual foresight for planning robot motion. In International Conference on Robotics and Automation (ICRA), 2017.
+
+Chelsea Finn, Ian Goodfellow, and Sergey Levine. Unsupervised learning for physical interaction through video prediction. In Advances in Neural Information Processing Systems, 2016.
+
+Katerina Fragkiadaki, Jonathan Huang, Alex Alemi, Sudheendra Vijayanarasimhan, Susanna Ricco, and Rahul Sukthankar. Motion prediction under multimodality with conditional stochastic networks. CoRR, abs/1705.02082, 2017.
+
+Yuanjun Gao, Evan W Archer, Liam Paninski, and John P Cunningham. Linear dynamical neural population models through nonlinear embeddings. In Advances in Neural Information Processing Systems, pp. 163–171, 2016.
+
+Ian Goodfellow. Nips 2016 tutorial: Generative adversarial networks. arXiv preprint arXiv:1701.00160, 2016.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, 2014.
+
+Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. International Conference on Learning Representations (ICLR), 2016.
+
+Jonathan Huang, Vivek Rathod, Chen Sun, Menglong Zhu, Anoop Korattikara, Alireza Fathi, Ian Fischer, Zbigniew Wojna, Yang Song, Sergio Guadarrama, et al. Speed/accuracy trade-offs for modern convolutional object detectors. arXiv preprint arXiv:1611.10012, 2016.
+
+Quan Huynh-Thu and Mohammed Ghanbari. Scope of validity of psnr in image/video quality assessment. Electronics letters, 2008.
+
+Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu. Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments. IEEE transactions on pattern analysis and machine intelligence, 36(7), 2014.
+
+Matthew Johnson, David K Duvenaud, Alex Wiltschko, Ryan P Adams, and Sandeep R Datta. Composing graphical models with neural networks for structured representations and fast inference. In Advances in neural information processing systems, pp. 2946–2954, 2016.
+
+Nal Kalchbrenner, Aaron van den Oord, Karen Simonyan, Ivo Danihelka, Oriol Vinyals, Alex ¨ Graves, and Koray Kavukcuoglu. Video pixel networks. International Conference on Machine Learning (ICML), 2017.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. International Conference on Learning Representations (ICLR), 2014.
+
+Rahul G Krishnan, Uri Shalit, and David Sontag. Structured inference networks for nonlinear state space models. In AAAI, pp. 2101–2109, 2017.
+
+Yitong Li, Martin Renqiang Min, Dinghan Shen, David Carlson, and Lawrence Carin. Video generation from text. arXiv preprint arXiv:1710.00421, 2017.
+
+Ziwei Liu, Raymond Yeh, Xiaoou Tang, Yiming Liu, and Aseem Agarwala. Video frame synthesis using deep voxel flow. International Conference on Computer Vision (ICCV), 2017.
+
+William Lotter, Gabriel Kreiman, and David Cox. Deep predictive coding networks for video prediction and unsupervised learning. International Conference on Learning Representations (ICLR), 2017.
+
+Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. International Conference on Learning Representations (ICLR), 2016.
+
+Junhyuk Oh, Xiaoxiao Guo, Honglak Lee, Richard L Lewis, and Satinder Singh. Action-conditional video prediction using deep networks in atari games. In Advances in Neural Information Processing Systems, 2015.
+
+MarcAurelio Ranzato, Arthur Szlam, Joan Bruna, Michael Mathieu, Ronan Collobert, and Sumit Chopra. Video (language) modeling: a baseline for generative models of natural videos. arXiv preprint arXiv:1412.6604, 2014.
+
+Scott E. Reed, Aaron van den Oord, Nal Kalchbrenner, Sergio Gomez Colmenarejo, Ziyu Wang, ¨ Yutian Chen, Dan Belov, and Nando de Freitas. Parallel multiscale autoregressive density estimation. International Conference on Machine Learning (ICML), 2017.
+
+Rui Shu, James Brofos, Frank Zhang, Hung Hai Bui, Mohammad Ghavamzadeh, and Mykel Kochenderfer. Stochastic video prediction with conditional density estimation. In ECCV Workshop on Action and Anticipation for Visual Learning, 2016.
+
+Nitish Srivastava, Elman Mansimov, and Ruslan Salakhudinov. Unsupervised learning of video representations using lstms. In International Conference on Machine Learning, 2015.
+
+Sergey Tulyakov, Ming-Yu Liu, Xiaodong Yang, and Jan Kautz. Mocogan: Decomposing motion and content for video generation. arXiv preprint arXiv:1707.04993, 2017.
+
+Carl Vondrick and Antonio Torralba. Generating the future with adversarial transformers. In Computer Vision and Pattern Recognition (CVPR), 2017.
+
+Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Anticipating the future by watching unlabeled video. arXiv preprint arXiv:1504.08023, 2015.
+
+Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Generating videos with scene dynamics. In Advances In Neural Information Processing Systems, 2016.
+
+Jacob Walker, Carl Doersch, Abhinav Gupta, and Martial Hebert. An uncertain future: Forecasting from static images using variational autoencoders. In European Conference on Computer Vision, pp. 835–851. Springer, 2016.
+
+Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 2004.
+
+SHI Xingjian, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional lstm network: A machine learning approach for precipitation nowcasting. In Advances in Neural Information Processing Systems, 2015.
+
+Tianfan Xue, Jiajun Wu, Katherine Bouman, and Bill Freeman. Visual dynamics: Probabilistic future frame synthesis via cross convolutional networks. In Advances in Neural Information Processing Systems, 2016.
+
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. International Conference on Computer Vision (ICCV), 2017.
+
+# A TRAINING DETAILS
+
+Figure 3 contains details of the network architectures used as generative and inference models. In all of the experiments we used the same set of hyper-parameters which can be found in Table 1.
+
+Table 1: Hyper-parameters used for experiments.   
+
+<table><tr><td colspan="2">Generative Network</td></tr><tr><td>model type batch size learning rate scheduled sampling (k) #of masks</td><td>CDNA 16 0.001 900.0 10</td></tr><tr><td># of iterations InferenceNetwork</td><td>200000</td></tr><tr><td>latent minimumo starting β final β # of latent channels # step 1 iterations</td><td>-5.0 0.0 0.001 1 50000</td></tr><tr><td># step 2 iterations # step 3 iterations</td><td>50000 100000</td></tr><tr><td>Optimization</td><td>ADAM</td></tr><tr><td>Method β1</td><td>0.9</td></tr><tr><td>β2 E</td><td>0.999 1e-8</td></tr></table>
+
+In the first step of training, we disable the inference network and instead sample latent values from $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ . In step 2, the latent values will be sampled from the approximated posterior $q _ { \phi } ( { \bf z } | { \bf x } _ { 0 : T } ) =$ $\mathcal { N } \big ( \mu ( \mathbf { x } _ { 0 : T } ) , \sigma ( \mathbf { x } _ { 0 : T } ) \big )$ . Please note that the inference network approximates $\log ( \sigma )$ instead of $\sigma$ for numerical stability. To gradually switch from Step 2 of training procedure to Step 3, we increase $\beta$ linearly from its starting value to its end value over the length of training.

parse/train/rk49Mg-CW/rk49Mg-CW_content_list.json

@@ -0,0 +1,1501 @@
+[
+    {
+        "type": "text",
+        "text": "STOCHASTIC VARIATIONAL VIDEO PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            751,
+            122
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Babaeizadeh1, Chelsea Finn2, Dumitru Erhan3, Roy Campbell1, and Sergey Levine2,3 ",
+        "bbox": [
+            178,
+            150,
+            816,
+            166
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1University of Illinois at Urbana-Champaign 2University of California, Berkeley 3Google Brain ",
+        "bbox": [
+            352,
+            176,
+            645,
+            219
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "mb2@uiuc.edu, cbfinn@eecs.berkeley.edu, dumitru@google.com, rhc@illinois.edu, svlevine@eecs.berkeley.edu ",
+        "bbox": [
+            238,
+            234,
+            758,
+            261
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            303,
+            544,
+            318
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Predicting the future in real-world settings, particularly from raw sensory observations such as images, is exceptionally challenging. Real-world events can be stochastic and unpredictable, and the high dimensionality and complexity of natural images require the predictive model to build an intricate understanding of the natural world. Many existing methods tackle this problem by making simplifying assumptions about the environment. One common assumption is that the outcome is deterministic and there is only one plausible future. This can lead to low-quality predictions in real-world settings with stochastic dynamics. In this paper, we develop a stochastic variational video prediction (SV2P) method that predicts a different possible future for each sample of its latent variables. To the best of our knowledge, our model is the first to provide effective stochastic multi-frame prediction for real-world videos. We demonstrate the capability of the proposed method in predicting detailed future frames of videos on multiple real-world datasets, both action-free and action-conditioned. We find that our proposed method produces substantially improved video predictions when compared to the same model without stochasticity, and to other stochastic video prediction methods. Our SV2P implementation will be open sourced upon publication. ",
+        "bbox": [
+            233,
+            333,
+            764,
+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            593,
+            336,
+            608
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding the interaction dynamics of objects and predicting what happens next is one of the key capabilities of humans which we heavily rely on to make decisions in everyday life (Bubic et al., 2010). A model that can accurately predict future observations of complex sensory modalities such as vision must internally represent the complex dynamics of real-world objects and people, and therefore is more likely to acquire a representation that can be used for a variety of visual perception tasks, such as object tracking and action recognition (Srivastava et al., 2015; Lotter et al., 2017; Denton & Birodkar, 2017). Furthermore, such models can be inherently useful themselves, for example, to allow an autonomous agent or robot to decide how to interact with the world to bring about a desired outcome (Oh et al., 2015; Finn & Levine, 2017). ",
+        "bbox": [
+            174,
+            625,
+            825,
+            750
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, modeling future distributions over images is a challenging task, given the high dimensionality of the data and the complex dynamics of the environment. Hence, it is common to make various simplifying assumptions. One particularly common assumption is that the environment is deterministic and that there is only one possible future (Chiappa et al., 2017; Srivastava et al., 2015; Boots et al., 2014; Lotter et al., 2017). Models conditioned on the actions of an agent frequently make this assumption, since the world is more deterministic in these settings (Oh et al., 2015; Finn et al., 2016). However, most real-world prediction tasks, including the action-conditioned settings, are in fact not deterministic, and a deterministic model can lose many of the nuances that are present in real physical interactions. Given the stochastic nature of video prediction, any deterministic model is obliged to predict a statistic of all the possible outcomes. For example, deterministic models trained with a mean squared error loss function generate the expected value of all the possibilities for each pixel independently, which is inherently blurry (Mathieu et al., 2016). ",
+        "bbox": [
+            174,
+            757,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/aba4881bdcfe439c582ea1ff7e0e21c025bd9dddb8b5b58fb4b62ba499e9a6db.jpg",
+        "image_caption": [
+            "Figure 1: Importance of stochasticity in video prediction. In each video, a random shape follows a random direction (first row). Given only the first frame, the deterministic model from Finn et al. (2016) predicts the average of all the possibilities. The third row is the output of SV2P with latent sampled from approximated posterior which predicts the correct motion. Last two rows are stochastic outcomes using random latent values sampled from assumed prior. As observed, these outcomes are random but within the range of possible futures. Second sample of Figure 1c shows a case where the model predicts the average of more than one outcome. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            95,
+            813,
+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contribution in this paper is a stochastic variational method for video prediction, named SV2P, that predicts a different plausible future for each sample of its latent random variables. We also provide a stable training procedure for training a neural network based implementation of this method. To the extent of our knowledge, SV2P is the first latent variable model to successfully predict multiple frames in real-world settings. Our model also supports action-conditioned predictions, while still being able to predict stochastic outcomes of ambiguous actions, as exemplified in our experiments. We evaluate SV2P on multiple real-world video datasets, as well as a carefully designed toy dataset that highlights the importance of stochasticity in video prediction (see Figure 1). In both our qualitative and quantitative comparisons, SV2P produces substantially improved video predictions when compared to the same model without stochasticity, with respect to standard metrics such as PSNR and SSIM. The stochastic nature of SV2P is most apparent when viewing the predicted videos. Therefore, we highly encourage the reader to check the project website https://goo.gl/iywUHc to view the actual videos of the experiments. The TensorFlow (Abadi et al., 2016) implementation of this project will be open sourced upon publication. ",
+        "bbox": [
+            173,
+            458,
+            825,
+            651
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            674,
+            344,
+            690
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A number of prior works have addressed video frame prediction while assuming deterministic environments (Ranzato et al., 2014; Srivastava et al., 2015; Vondrick et al., 2015; Xingjian et al., 2015; Boots et al., 2014; Lotter et al., 2017). In this work, we build on the deterministic video prediction model proposed by Finn et al. (2016), which generates the future frames by predicting the motion flow of dynamically masked out objects extracted from the previous frames. Similar transformationbased models were also proposed by De Brabandere et al. (2016); Liu et al. (2017). Prior work has also considered alternative objectives for deterministic video prediction models to mitigate the blurriness of the predicted frames and produce sharper predictions (Mathieu et al., 2016; Vondrick & Torralba, 2017). Despite the adversarial objective, Mathieu et al. (2016) found that injecting noise did not lead to stochastic predictions, even for predicting a single frame. Oh et al. (2015); Chiappa et al. (2017) make sharp video predictions by assuming deterministic outcomes in video games given the actions of the agents. However, this assumption does not hold in real-world settings, which almost always have stochastic dynamics. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Auto-regressive models have been proposed for modeling the joint distribution of the raw pixels (Kalchbrenner et al., 2017). Although these models predict sharp images of the future, their training and inference time is extremely high, making them difficult to use in practice. Reed et al. (2017) proposed a parallelized multi-scale algorithm that significantly improves the training and prediction time but still requires more than a minute to generate one second of $6 4 \\times 6 4$ video on a GPU. Our comparisons suggest that the predictions from these models are sharp, but noisy, and that our method produces substantially better predictions, especially for longer horizons. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Another approach for stochastic prediction uses generative adversarial networks (GANs) (Goodfellow et al., 2014), which have been used for video generation and prediction (Tulyakov et al., 2017; Li et al., 2017). Vondrick et al. (2016); Chen et al. (2017) applied adversarial training to predict video from a single image. Although GANs generate sharp images, they tend to suffer from modecollapse (Goodfellow, 2016), particularly in conditional generation settings (Zhu et al., 2017). ",
+        "bbox": [
+            174,
+            180,
+            825,
+            251
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Variational auto-encoders (VAEs) (Kingma & Welling, 2014) also have been explored for stochastic prediction tasks. Walker et al. (2016) uses conditional VAEs to predict dense trajectories from pixels. Xue et al. (2016) predicts a single stochastic frame using cross convolutional networks in a VAElike architecture. Shu et al. (2016) uses conditional VAEs and Gaussian mixture priors for stochastic prediction. Both of these works have been evaluated solely on synthetic datasets with simple moving sprites and no object interaction. Real images significantly complicate video prediction due to the diversity and variety of stochastic events that can occur. Fragkiadaki et al. (2017) compared various architectures for multimodal motion forecasting and one-frame video prediction, including variational inference and straightforward sampling from the prior. Unlike these prior models, our focus is on designing a multi-frame video prediction model to produce stochastic predictions of the future. Multi-frame prediction is dramatically harder than single-frame prediction, since complex events such as collisions require multiple frames to fully resolve, and single-frame predictions can simply ignore this complexity. We believe, our approach is the first latent variable model to successfully demonstrate stochastic multi-frame video prediction on real world datasets. ",
+        "bbox": [
+            174,
+            257,
+            825,
+            450
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 STOCHASTIC VARIATIONAL VIDEO PREDICTION (SV2P) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            488,
+            674,
+            506
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to construct our stochastic variational video prediction model, we first formulate a probabilistic graphical model that explains the stochasticity in the video. Since our goal is to perform conditional video prediction, the predictions are conditioned on a set of $c$ context frames $\\mathbf { x } _ { 0 } , \\ldots , \\mathbf { x } _ { c - 1 }$ (e.g., if we are conditioning on one frame, $c = 1 \\AA$ ), and our goal is to sample from $p ( \\mathbf { x } _ { c : T } | \\mathbf { x } _ { 0 : c - 1 } )$ , where $\\mathbf { x } _ { i }$ denotes the $\\mathrm { i ^ { \\mathrm { t h } } }$ frame of the video (Figure 2). ",
+        "bbox": [
+            174,
+            531,
+            825,
+            602
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Video prediction is stochastic as a consequence of the latent events that are not observable from the context frames alone. For example, when a robot’s arm pushes a toy on a table, the unknown weight of that toy affects how it moves. We therefore introduce a vector of latent variables $\\mathbf { z }$ into our model, distributed according to a prior $\\mathbf { z } \\sim p ( \\mathbf { z } )$ , and build a model $p ( \\mathbf { x } _ { c : T } | \\mathbf { x } _ { 0 : c - 1 } , \\mathbf { z } )$ . This model is still stochastic but uses a more general representation, such as a conditional Gaussian, to explain just the noise in the image, while $\\mathbf { z }$ accounts for the more complex stochastic phenomena. We can then factorize this model to $\\begin{array} { r } { \\prod _ { t = c } ^ { T } p _ { \\theta } \\big ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 : t - 1 } , \\mathbf { z } \\big ) } \\end{array}$ Learning then involves training the parameters of these factors $\\theta$ , which we assume to be shared between all the time steps. ",
+        "bbox": [
+            174,
+            609,
+            483,
+            847
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "At inference time we need to estimate values for the true posterior $p ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ , which is intractable due its dependency on $p ( \\mathbf { x } _ { 0 : T } )$ . We ",
+        "bbox": [
+            174,
+            854,
+            483,
+            896
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9426d145ac0b3d1886124cb14d9334930e47f47b7fcfc7179483e9be5edd844.jpg",
+        "image_caption": [
+            "Figure 2: Probabilistic graphical model of stochastic variational video prediction, assuming time-invariant latent. The generative model predicts the next frame conditioned on the previous frames and latent variables (solid lines), while the variational inference model approximates the posterior given all the frames (dotted lines). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            532,
+            625,
+            789,
+            760
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "overcome this problem by approximating the posterior with an inference network $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ that outputs the parameters of a conditionally Gaussian distribution $\\mathcal { N } ( \\mu _ { \\phi } ( \\mathbf { x } _ { 0 : T } ) , \\sigma _ { \\phi } ( \\mathbf { x } _ { 0 : T } ) )$ . This network is trained using the reparameterization trick (Kingma & Welling, 2014), according to: ",
+        "bbox": [
+            174,
+            897,
+            826,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/16adbaba343d53be1ef1ee5168ad849ee9ab134fbaab97ce502f9893b6db2c7c.jpg",
+        "image_caption": [
+            "Figure 3: Architecture of SV2P. At training time, the inference network (top) estimates the posterior $\\bar { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } ) } = \\mathcal { N } \\big ( \\mu ( \\mathbf { x } _ { 0 : T } ) , \\sigma ( \\mathbf { x } _ { 0 : T } ) \\big )$ . The latent value ${ \\mathbf z } \\sim q _ { \\phi } ( { \\mathbf z } | { \\mathbf x } _ { 0 : T } )$ is passed to the generative network along with the (optional) action. The generative network (from Finn et al. (2016)) predicts the next frame given the previous frames, latent values, and actions. At test time, $\\mathbf { z }$ is sampled from the assumed prior $\\mathcal { N } ( \\mathbf { 0 } , \\bar { \\mathbf { I } } )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            99,
+            794,
+            301
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            406,
+            771,
+            422
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1828bfb76bdbc6193417975ffce2f5bf110875361ec1307e4a143a6822185a7d.jpg",
+        "text": "$$\n\\mathbf { z } = \\mu _ { \\phi } ( \\mathbf { x } _ { 0 : T } ) + \\sigma _ { \\phi } ( \\mathbf { x } _ { 0 : T } ) \\times \\epsilon , \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            428,
+            668,
+            445
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\theta$ and $\\phi$ are the parameters of the generative model and inference network, respectively. To learn these parameters, we can optimize the variational lower bound, as in the variational autoencoder (VAE) (Kingma & Welling, 2014): ",
+        "bbox": [
+            174,
+            449,
+            825,
+            491
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef1cc35f5ce2a989e98ee5ec4321924bf5caf70fa9a3d9f731c041e8cd0d1623.jpg",
+        "text": "$$\n\\mathcal { L } ( \\mathbf { x } ) = - \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } ) } \\left[ \\log p _ { \\theta } ( \\mathbf { x } _ { t : T } | \\mathbf { x } _ { 0 : t - 1 } , \\mathbf { z } ) \\right] + D _ { K L } \\big ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } ) | | p ( \\mathbf { z } ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            494,
+            743,
+            515
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { K L }$ is the Kullback-Leibler divergence between the approximated posterior and assumed prior $p ( \\mathbf { z } )$ which in our case is the standard Gaussian $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ . ",
+        "bbox": [
+            173,
+            518,
+            823,
+            546
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Equation 2, the first term on the RHS represents the reconstruction loss while the second term represents the divergence of the variational posterior from the prior on the latent variable. It is important to emphasize that the approximated posterior is conditioned on all of the frames, including the future frames $\\mathbf { x } _ { t : T }$ . This is feasible during training, since $\\mathbf { x } _ { t : T }$ is available at the training time, while at test time we can sample the latents from the assumed prior. Since the aim in our method is to recover latent variables that correspond to events which might explain the variability in the videos, we found that it is in fact crucial to condition the inference network on future frames. At test time, the latent variables are simply sampled from the prior which corresponds to a smoothing-like inference process. In principle, we could also perform a filtering-like inference procedure of the form $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : t - 1 } )$ for time step $t$ to infer the most likely latent variables based only on the conditioning frames, instead of sampling from the prior, which could produce more accurate predictions at test time. However, it would be undesirable to use a filtering process at training time: in order to incentivize the forward prediction network to make use of the latent variables, they must contain some information that is useful for predicting future frames that is not already present in the context frames. If they are predicted entirely from the context frames, no such information is present, and indeed we found that a purely filtering-based model simply ignores the latent variables. ",
+        "bbox": [
+            173,
+            553,
+            825,
+            775
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "So far, we’ve assumed that the latent events are constant over the entire video. We can relax this assumption by conditioning prediction on a time-variant latent variable $\\mathbf { z } _ { t }$ that is sampled at every time step from $p ( \\mathbf { z } )$ . The generative model then becomes $\\begin{array} { r } { p ( \\mathbf { z } _ { t } ) \\prod _ { t = c } ^ { T } p _ { \\theta } \\big ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 : t - 1 } , \\mathbf { z } _ { t } \\big ) } \\end{array}$ and, assuming a fixed posterior, the inference model will be approximated by $q _ { \\phi } \\big ( \\mathbf { z } _ { t } | \\mathbf { x } _ { 0 : T } \\big )$ , where the model parameters $\\phi$ are shared across time. In practice, the only difference between these two formulations is the frequency of sampling $\\mathbf { z }$ from $p ( \\mathbf { z } )$ and $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ . In the time-invariant version, we sample $\\mathbf { z }$ once per video, whereas with the time-variant latent, sampling happens every frame. The main benefit of time-variant latent variable is better generalization beyond $T$ , since the model does not have to encode all the events of the video in one vector $\\mathbf { z }$ . We provide an empirical comparison of these formulations in Section 5.2. ",
+        "bbox": [
+            173,
+            782,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/4bfa96253e877aba226a73c223f94d0e5e06c8cc5ba6268162a30dfba292d162.jpg",
+        "image_caption": [
+            "Figure 4: Three phases of training. In the first phase, the inference network is turned off and only the generative network is being trained, resulting in deterministic predictions. The inference network is used in the second phase without a KL-loss. The last phase includes $D _ { K L } \\big ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } ) | | p ( \\mathbf { z } ) \\big )$ to enable accurate sampling latent from $p ( \\mathbf { z } )$ . (a) the KL-loss $( b )$ the reconstruction loss (c) Training stability. This graph compares reconstruction loss at the end of five training sessions on the BAIR robot pushing dataset, with and without following all the steps of the training procedure. The proposed training is quite stable and results in lower error compared to na¨ıve training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            102,
+            816,
+            236
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In action-conditioned settings, we modify the generative model to be conditioned on action vector $\\mathbf { a } _ { t }$ . This results in $\\begin{array} { r } { p ( \\mathbf { z } _ { t } ) \\prod _ { t = c } ^ { T } p _ { \\boldsymbol \\theta } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 : t - 1 } , \\mathbf { z } _ { t } , \\mathbf { a } _ { t } ) } \\end{array}$ as generative model while keeping the posterior approximation intact. Conditioning the outcome on actions can decrease future variability; however it will not eliminate it if the environment is inherently stochastic or the actions are ambiguous. In this case, the model is still capable of predicting stochastic outcomes in a narrower range of possibilities. ",
+        "bbox": [
+            173,
+            376,
+            825,
+            449
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 MODEL ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            469,
+            383,
+            483
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To model the approximated posterior $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ we used a deep convolutional neural network as shown in the top row of Figure 3. Since we assumed a diagonal Gaussian distribution for $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ , this network outputs the mean $\\mu _ { \\phi } \\big ( \\mathbf { x } _ { 0 : T } \\big )$ and standard deviation $\\log \\sigma _ { \\phi } ( \\mathbf { x } _ { 0 : T } )$ of the approximated posterior. Hence the entire inference network is convolutional, the predicted parameters are $8 \\times 8$ single channel response maps. We assume each entry in this response maps is pairwise independent, forming the latent vector $\\mathbf { z }$ . The latent value is then sampled using Equation 1. As discussed before, this sampling happens every frame for time-varying latent, and once per video in time-invariant case. ",
+        "bbox": [
+            173,
+            496,
+            825,
+            594
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For $p ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 : t - 1 } , \\mathbf { z } )$ , we used the CDNA architecture proposed by Finn et al. (2016), which is a deterministic convolutional recurrent network that predicts the next frame $\\mathbf { x } _ { t }$ given the previous frame $\\mathbf { x } _ { t - 1 }$ and an optional action $\\mathbf { a } _ { t }$ . This model constructs the next frames by predicting the motions of segments of the image (i.e., objects) and then merging these predictions via masking. Although this model directly outputs pixels, it is partially-appearance invariant and can generalize to unseen objects (Finn et al., 2016). To condition on the latent value, we modify the CDNA architecture by stacking $\\mathbf { z } _ { t }$ as an additional channel on tiled action $\\mathbf { a } _ { t }$ . ",
+        "bbox": [
+            174,
+            601,
+            825,
+            698
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 TRAINING PROCEDURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            718,
+            377,
+            733
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our model can be trained end-to-end. However, our experiments show that na¨ıve training usually results in the model ignoring the latent variables and converging to a suboptimal deterministic solution (Figure 4). Therefore, we train the model end-to-end in three phases, as follows: ",
+        "bbox": [
+            174,
+            746,
+            823,
+            787
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. Training the generative network: In this phase, the inference network has been disabled and the latent value $\\mathbf { z }$ will be randomly sampled from $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ . The intuition behind this phase is to train the generative model to predict the future frames deterministically (i.e. modeling $\\mathbf { \\bar { \\rho } } _ { p _ { \\theta } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 : t - 1 } ) } ) $ .   \n2. Training the inference network: In the second phase, the inference network is trained to estimate the approximate posterior $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ ; however, the KL-loss is set to 0. This means that the model can use the latent value without being penalized for diverging from $p ( \\mathbf { z } )$ . As seen in Figure 4, this phase results in very low reconstruction error, however it is not usable at the test time since $D _ { K L } \\mathbf { \\bar { ( } } q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } ) | | p ( \\mathbf { z } ) \\rrangle \\gg 0$ and sampling $\\mathbf { z }$ from the assumed prior will be inaccurate. ",
+        "bbox": [
+            173,
+            804,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3. Divergence reduction: In the last phase, the KL-loss is added, resulting in a sudden drop of KLdivergence and an increase of reconstruction error. The reconstruction loss converging to a value lower than the first phase and KL-loss converging to zero are indicators of successful training. This means that $\\mathbf { z }$ can be sampled from $p ( \\mathbf { z } )$ at test time for effective stochastic prediction. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            160
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To gradually transition from the second phase to the third, we add a multiplier to KL-loss that is set to zero during the first two phases and then increased slowly in the last phase. This is similar to the $\\beta$ hyper-parameter in Higgins et al. (2016) and Bowman et al. (2016) that is used to balance latent channel capacity and independence constraints with reconstruction accuracy. ",
+        "bbox": [
+            174,
+            171,
+            825,
+            228
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We found that this training procedure is quite stable and the model almost always converges to the desired parameters. To demonstrate this stability, we trained the model with and without the proposed training procedure, five times each. Figure 4 shows the average and standard deviation of reconstruction loss at the end of these training sessions. Na¨ıve training results in a slightly better error compared to Finn et al. (2016), but with high variance. When following the proposed training algorithm, the model consistently converges to a much lower reconstruction error. ",
+        "bbox": [
+            174,
+            234,
+            825,
+            318
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 STOCHASTIC MOVEMENT DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            338,
+            496,
+            353
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To highlight the importance of stochasticity in video prediction, we created a toy video dataset with intentionally stochastic motion. Each video in this dataset is four frames long. The first frame contains a random shape (triangle, rectangle or circle) with random size and color, centered in the frame, which then randomly moves to one of the eight directions (up, down, left, right, up-left, upright, down-left, down-right). Each frame is $6 4 \\times 6 4 \\times 3$ and the background is static gray. The main intuition behind this design is that, given only the first frame, a model can figure out the shape, color, and size of the moving object, but not its movement direction. ",
+        "bbox": [
+            174,
+            369,
+            825,
+            468
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We train Finn et al. (2016) and SV2P to predict the future frames, given only the first frame. Figure 1 shows the video predictions from these two models. Since Finn et al. (2016) is a deterministic model with mean squared error as loss, it predicts the average of all possible outcomes, as expected. In contrast, SV2P predicts different possible futures for each sample of the latent variable $\\mathbf { z } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ . In our experiments, all the videos predicted by SV2P are within the range of plausible futures (e.g. we never saw the shape moves in any direction other than the original eight). However, in some cases, SV2P still predicts the average of more than one future, as it can be seen in the first random sample of Figure 1c. The main reason for this problem seems to be overlapping posterior distributions in latent space which can cause some latent values (sampled from $p ( \\mathbf { z } ) _ { , } ^ { \\dag }$ ) to be ambiguous. ",
+        "bbox": [
+            174,
+            474,
+            825,
+            599
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate that the inference network is working properly and that the latent variable does indeed learn to store the information necessary for stochastic prediction (i.e., the direction of movement), we include predicted futures when ${ \\mathbf z } \\sim q _ { \\phi } ( { \\mathbf x } _ { 0 : T } )$ . By estimating the correct parameters of the latent distribution, using the inference network, the model always generates the right outcome. However, this cannot be used in practice, since the inference network requires access to all the frames, including the ones in the future. Instead, $\\mathbf { z }$ will be sampled from assumed prior $p ( \\mathbf { z } )$ . ",
+        "bbox": [
+            174,
+            607,
+            825,
+            690
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            709,
+            326,
+            726
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To evaluate SV2P, we test it on three real-world video datasets by comparing it to the CDNA model (Finn et al., 2016), as a deterministic baseline, as well as a baseline that outputs the last seen frame as the prediction. We compare SV2P with an auto-regressive stochastic model, video pixel networks (VPN) (Kalchbrenner et al., 2017). We use the parallel multi-resolution implementation of VPN from Reed et al. (2017), which is an order of magnitude faster than the original VPN, but still requires more than a minute to generate one second of $6 4 \\times 6 4$ video. In all of these experiments, we plot the results of sampling the latent once per video (SV2P time-invariant latent) and once per frame (SV2P time-variant latent). We strongly encourage readers to view https://goo.gl/iywUHc for videos of the results which are more illustrative than printed frames. ",
+        "bbox": [
+            174,
+            741,
+            825,
+            867
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            883,
+            287,
+            897
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We quantitatively and qualitatively evaluate SV2P on following real-world datasets: ",
+        "bbox": [
+            174,
+            909,
+            720,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• BAIR robot pushing dataset (Ebert et al., 2017): This dataset contains action-conditioned videos collected by a Sawyer robotic arm pushing a variety of objects. All of the videos in this datasets have similar table top settings with static background. Each video also has recorded actions taken by the robotic arm which correspond to the commanded gripper pose. An interesting property of this dataset is the fact that the arm movements are quite unpredictable in the absence of actions (compared to the robot pushing dataset (Finn et al., 2016) which the arm moves to the center of the bin). For this dataset, we train the models to predict the next ten frames given the first two, both in action-conditioned and action-free settings. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Human3.6M (Ionescu et al., 2014): Humans and animals are one of the most interesting sources of stochasticity in natural videos, which behave in complex ways as a consequence of unpredictable intentions. To study human motion prediction, we use the Human3.6M dataset which consists of actors performing various actions in a room. We used the pre-processing and testing format of Finn et al. (2016): a $1 0 \\ : \\mathrm { H z }$ frame rate and 10-frame prediction given the previous ten. The videos from this datasets contains various actions performed by humans (walking, talking on the phone, . . . ). Similar to Finn et al. (2016), we included videos from all the performed actions in training dataset while keeping all the videos from an specific actor out for testing. ",
+        "bbox": [
+            173,
+            222,
+            825,
+            334
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Robotic pushing prediction (Finn et al., 2016): We use the robot pushing prediction dataset to compare SV2P with another stochastic prediction method, video pixel networks (VPNs) (Kalchbrenner et al., 2017). VPNs demonstrated excellent results on this dataset in prior work, and therefore robot pushing dataset provides a strong point of comparison. However, in contrast to our method, VPNs do not include latent stochastic variables that represent random events, and rely on an expensive auto-regressive architecture. In this experiment, the models have been trained to predict the next ten frames, given the first two. Similar to BAIR robot pushing dataset, this dataset also contains actions taken by the robotic arm which are the pose of the commanded gripper. ",
+        "bbox": [
+            174,
+            342,
+            825,
+            453
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 QUANTITATIVE COMPARISON ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            418,
+            487
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In our quantitative evaluation, we aim to understand whether the range of possible futures captured by our stochastic model includes the true future. Models that are more stochastic do not necessarily score better on average standard metrics such as PSNR (Huynh-Thu & Ghanbari, 2008) and SSIM (Wang et al., 2004). However, if we are interested primarily in understanding whether the true outcome is within the set of predictions, we can instead evaluate the score of the best sample from multiple random priors. We argue that this is a better metric for stochastic models, since it allows us to understand if uncertain futures contain the true outcome. Figure 5 illustrates how this metric changes with different numbers of samples. By predicting more possible futures, the probability of predicting the true outcome increases, and therefore it is more likely to get a sample with higher PSNR compared to the ground truth. Of course, as with all video prediction metrics, it is imperfect, and is only suitable for understanding the performance of the model when combined with a visual examination of the qualitative results in Section 5.3. ",
+        "bbox": [
+            174,
+            501,
+            496,
+            833
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/78b52c592b0095218a20984840158d2ab406420ba33d49820e2bca9795b24d83.jpg",
+        "image_caption": [
+            "Figure 5: Stochasticity of SV2P predictions on the action-free BAIR dataset. Each line presents the sample with highest PSNR compared to ground truth, after multiple sampling. The number on the right indicates the number of random samples. As can be seen, SV2P predicts highly stochastic videos and, on average, only three samples is enough to predict outcomes with higher quality compared to Finn et al. (2016). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            516,
+            813,
+            715
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To use this metric, we sample 100 latent values from prior $\\mathbf { z } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ and use them to predict 100 videos and show the result of the sample with ",
+        "bbox": [
+            174,
+            840,
+            496,
+            882
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "highest PSNR. For a fair comparison to VPN, we use the same best out of 100 samples for our stochastic baseline. Since even the fast implementation of VPN is quite slow, we limit the comparison with VPN to only last dataset with 256 test samples. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/82a2595a17bf365c56d24b74d6c233663fec9ac13b964cf26182ff353185ecb9.jpg",
+        "image_caption": [
+            "Figure 6: Quantitative comparison of the prediction methods. The stochastic models have been sampled 100 times and the results with the best PSNR have been displayed. For SV2P, we demonstrate the results of both time-variant and time-invariant latent sampling. Repeat shows the results of the lower bound prediction by repeating the last seen frame as the prediction. In the last column, we compare the results of video pixel networks (VPN). All the models, including Finn et al. (2016), have been trained up to the frame marked by vertical separator and the results beyond this line display their generalization. The plots are the average SSIM and PSNR over the test set and shadow is the $9 5 \\%$ confidence interval. In all of these graphs, higher is better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            99,
+            823,
+            369
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 displays the quantitative comparison of the predictions on all of the datasets. In this graph, the top row is a PSNR comparison and the bottom row is SSIM, while each column represents a different dataset. To evaluate the generalization of the models beyond what they have been trained for, we generate more frames than what the models observed during training time. The length of the training sequences is marked by a vertical separator in all of the graphs, and the results beyond this line represent extrapolation to longer sequences. ",
+        "bbox": [
+            174,
+            521,
+            825,
+            604
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Overall, SV2P with both time-variant and time-invariant latent sampling outperform all of the other baselines, by predicting higher quality videos with higher PSNR and SSIM. Time-varying latent sampling is more stable beyond the time horizon used during training (Figure 6b). One possible explanation for this behaviour is that the time-invariant latent has to include the information required for predicting all the frames and therefore, beyond training time, it collapses. This issue is mitigated by a time-variant latent variable which takes a different value at each time step. However, this stability is not always the case as it is more evident in late frames of Figure 6a. ",
+        "bbox": [
+            174,
+            612,
+            825,
+            710
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One other interesting observation is that the time-invariant model outperforms the time-variant model in the Human3.6M dataset. In this dataset, the most important latent event – the action performed by the actor – is consistent across the whole video which is easier to capture using timeinvariant latent. ",
+        "bbox": [
+            174,
+            717,
+            825,
+            772
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 QUALITATIVE COMPARISON ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            794,
+            408,
+            808
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We can better understand the performance of the proposed model by visual examination of the qualitative results. We highlight some of the most important and observable differences in predictions by different models in Figures 8-11 1. In all of these figures, the $\\mathbf { X }$ -axis is time (i.e., each row is one video). The first row is the ground truth video, and the second row is the result of Finn et al. (2016). The result of sampling the latent from approximated posterior is provided in the third row. For stochastic methods, we show the best (highest PSNR) and worst (lowest PSNR) predictions out of 100 samples (as discussed in Section 5.2), as well as two random predicted videos from our model. ",
+        "bbox": [
+            174,
+            821,
+            825,
+            891
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 8 illustrates two examples from the BAIR robot pushing dataset in the action-free setting. As a consequence of the high stochasticity in the movement of the arm in absence of actions, Finn et al. (2016) only blurs the arm out, while SV2P predicts varied but coherent movements of the arm. Note that, although each predicted movements of the arm is random, it is still in the valid range of possible outcomes (i.e., there is no sudden jump of the arm nor random movement of the objects). The proposed model also learned how to move objects in cases where they have been pushed by the predicted movements of the arm, as can be seen in the zoomed images of both samples. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            236
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the action-conditioned setting (Figure 9), the differences are more subtle: the range of possible outcomes is narrower, but we can still observe stochasticity in the behavior of the pushed objects. Interactions between the arm and objects are uncertain due to ambiguity in depth, friction, and mass, and SV2P is able to capture some of this variation. Since these variations are subtle and occupy a smaller part of the images, we illustrate this with zoomed insets in Figure 9. Some examples of varied object movements can be found in last three rows of right example of Figure 9. SV2P also generates sharper outputs, compared to Finn et al. (2016) as is evident in the left example of Figure 9. ",
+        "bbox": [
+            174,
+            243,
+            496,
+            450
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ef3813f7c9049d9bb6445c5d0f95258bf228837402cb7c08d1cb69e202cc394.jpg",
+        "image_caption": [
+            "Figure 7: Quantitative comparison of the predicted frames on Human3.6M dataset using confidence of object detection as quality metric. The y-axis demonstrates the average confidence of Huang et al. (2016) in detecting humans in predicted frames. Based on this metric, SV2P predicts images with more meaningful semantics compared to to Finn et al. (2016). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            517,
+            261,
+            784,
+            454
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Please note that the approximate posterior $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ is still trained with the evidence lower bound (ELBO), which means that the posterior must compress the information of the future events. Perfect reconstruction of high-quality images from posterior distributions over latent states is an open problem, and the results in our experiments compare favorably to those typically observed even in single-image VAEs (e.g. see Xue et al. (2016)). This is why the model cannot reconstruct all the future frames perfectly, even though when latent values are sampled from $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { 0 : T } )$ . ",
+        "bbox": [
+            174,
+            459,
+            496,
+            612
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            611,
+            544,
+            625
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 displays two examples from the Human3.6M dataset. In absence of actions, Finn et al. (2016) manages to separate the foreground from background, but cannot predict what happens next accurately. This results in distorted or blurred foregrounds. On the other hand, SV2P predicts a variety of different outcomes, and moves the actor accordingly. Note that PSNR and SSIM are measuring reconstruction loss with respect to the ground truth and they may not generally present a better prediction. For some applications, a prediction with lower PSNR/SSIM might have higher quality and be more interesting. A good example is the prediction with the worst PSNR in Figure 10- right, where the model predicts that the actor is spinning in his chair with relatively high quality. However, this output has the lowest PSNR compared to the ground truth. ",
+        "bbox": [
+            174,
+            632,
+            825,
+            757
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "However, pixel-wise metrics such as PSNR and SSIM may not be the best measures for semantic evaluation of predicted frames. Therefore, we use the confidence of an object detector to show the predicted frames contain useful semantic information. For this purpose, we use the open-sourced implementation of Huang et al. (2016) to compare the quality of predicted frames in Human3.6M dataset. As it can be seen in Figure 7, SV2P predicted frames which the human inside can be detected with higher confidence, compared to Finn et al. (2016). ",
+        "bbox": [
+            174,
+            763,
+            825,
+            848
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, Figure 11 demonstrates results on the Google robot pushing dataset. The qualitative and quantitative results in Figure 11 and 6 both indicate that SV2P produces substantially better predictions than VPNs. The quantitative results suggest that our best-of-100 metric is a reasonable measure of performance: the VPN predictions are more noisy, but simply increasing noise is not sufficient to increase the quality of the best sample. The stochasticity in our predictions is more coherent, corresponding to differences in object or arm motion, while much of the stochasticity in the VPN predictions resembles noise in the image, as well as visible artifacts when predicting for substantially longer time horizons. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            179,
+            318,
+            195
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We proposed stochastic variational video prediction (SV2P), an approach for multi-step video prediction based on variational inference. Our primary contributions include an effective stochastic prediction method with latent variables, a network architecture that succeeds on natural videos, and a training procedure that provides for stable optimization. The source code for our method will be released upon acceptance. We evaluated our proposed method on three real-world datasets in actionconditioned and action-free settings, as well as one toy dataset which has been carefully designed to highlight the importance of the stochasticity in video prediction. Both qualitative and quantitative results indicate higher quality predictions compared to other deterministic and stochastic baselines. ",
+        "bbox": [
+            174,
+            210,
+            825,
+            323
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "SV2P can be expanded in numerous ways. First, the current inference network design is fully convolutional, which exposes multiple limitations, such as unmodeled spatial correlations between the latent variables. The model could be improved by incorporating the spatial correlation induced by the convolutions into the prior, using a learned structured prior in place of the standard spherical Gaussian. Time-variant posterior approximation to reflect the new information that is revealed as the video progresses, is another possible SV2P improvement. However, as discussed in Section 3, this requires incentivizing the inference network to incorporate the latent information at training time. This would allow time-variant latent distributions which is more aligned with generative neural models for time-series(Johnson et al., 2016; Gao et al., 2016; Krishnan et al., 2017). ",
+        "bbox": [
+            174,
+            329,
+            825,
+            455
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Another exciting direction for future research would be to study how stochastic predictions can be used to act in the real world, producing model-based reinforcement learning methods that can execute risk-sensitive behaviors from raw image observations. Accounting for risk in this way could be especially important in safety-critical settings, such as robotics. ",
+        "bbox": [
+            176,
+            462,
+            825,
+            517
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            539,
+            357,
+            553
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Matt Johnson for providing feedback on an early draft of the paper, and Alex Lee for fixing bugs in the deterministic version of the model. This material is based upon work supported by the National Science Foundation under award no. 1725729 and was partially done while author was interning at Google Brain. ",
+        "bbox": [
+            174,
+            569,
+            825,
+            626
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            646,
+            285,
+            661
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for largescale machine learning. In OSDI, volume 16, pp. 265–283, 2016. ",
+        "bbox": [
+            176,
+            678,
+            823,
+            719
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Byron Boots, Arunkumar Byravan, and Dieter Fox. Learning predictive models of a depth camera & manipulator from raw execution traces. In International Conference on Robotics and Automation (ICRA), 2014. ",
+        "bbox": [
+            174,
+            728,
+            823,
+            770
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Samuel R. Bowman, Luke Vilnis, Oriol Vinyals, Andrew M. Dai, Rafal Jozefowicz, and Samy ´ Bengio. Generating sentences from a continuous space. In Proceedings of the 20th SIGNLL Conference on Computational Natural Language Learning, CoNLL 2016, Berlin, Germany, August 11-12, 2016, pp. 10–21, 2016. URL http://aclweb.org/anthology/K/K16/ K16-1002.pdf. ",
+        "bbox": [
+            174,
+            779,
+            823,
+            849
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andreja Bubic, D Yves Von Cramon, and Ricarda I Schubotz. Prediction, cognition and the brain. Frontiers in human neuroscience, 4, 2010. ",
+        "bbox": [
+            173,
+            858,
+            820,
+            887
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Baoyang Chen, Wenmin Wang, Jinzhuo Wang, Xiongtao Chen, and Weimian Li. Video imagination from a single image with transformation generation. arXiv preprint arXiv:1706.04124, 2017. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8457f3b2981417099f8ec0f8472dc0520463d5a03763affd471257be50b7a32.jpg",
+        "image_caption": [
+            "Figure 8: Comparing the results of SV2P with Finn et al. (2016) (second row) on action-free BAIR robot pushing dataset. Fourth and fifth rows are the predictions with minimum and maximum PSNR out of 100 random outputs with time-invariant latent sampling. The last two rows are random predicted outcomes. The numbers on top indicate the predicted frame number. In lack of actions and therefore high stochasticity, Finn et al. (2016) only blurs the robotic arm out while the proposed method predicts sharper frames on each sampling. SV2P also predicts the interaction dynamics between random movements of the arm and the objects. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            99,
+            818,
+            424
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/aeffc9ce522fc01f59fc532bfeee8dd2cf2e3e965715578d204755eb3faf00fc.jpg",
+        "image_caption": [
+            "Figure 9: Similar comparison as Figure 8 this time action-conditioned with time-variant latent sampling. SV2P predicts sharper and slightly variant outcomes compared to Finn et al. (2016). This is mostly evident in zoomed in objects which have been pushed by the arm. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            540,
+            818,
+            866
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/01075ef00723db057ec71d5426419f9a4646c302104026d6443451ee9d56ea2f.jpg",
+        "image_caption": [
+            "Figure 10: Prediction results on the action-free Human3.6M dataset. SV2P predicts a different outcome on each sampling given the latent. In the left example, the model predicts walking as well as stopping which result in different outputs in predicted future frames. Similarly, the right example demonstrates various outcomes including spinning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            98,
+            818,
+            424
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/c92dfa1707968d59cea926015c300dfbf4b9777833a4dc8ab341f2d5f1a65bed.jpg",
+        "image_caption": [
+            "Figure 11: Comparing the results of video pixel networks (VPN) (Kalchbrenner et al., 2017; Reed et al., 2017) with SV2P on the robotic pushing dataset. We use the same best PSNR out of 100 random samples for both methods. Besides stochastic movements of the pushed objects, another source of stochasticity is the starting lag in movements of the robotic arm. SV2P generates sharper images compared to Finn et al. (2016) (notice the pushed objects in zoomed images) with less noise compared to Reed et al. (2017) (look at the accumulated noise in later frames). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            500,
+            823,
+            825
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Silvia Chiappa, Sebastien Racani ´ ere, Daan Wierstra, and Shakir Mohamed. Recurrent environment \\` simulators. In Proceedings of the International Conference on Learning Representations (ICLR), 2017. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Bert De Brabandere, Xu Jia, Tinne Tuytelaars, and Luc Van Gool. Dynamic filter networks. In Neural Information Processing Systems (NIPS), 2016. ",
+        "bbox": [
+            173,
+            155,
+            823,
+            185
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Emily Denton and Vighnesh Birodkar. Unsupervised learning of disentangled representations from video. arXiv preprint arXiv:1705.10915, 2017. ",
+        "bbox": [
+            174,
+            194,
+            821,
+            223
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Frederik Ebert, Chelsea Finn, Alex X. Lee, and Sergey Levine. Self-Supervised Visual Planning with Temporal Skip Connections. Conference on Robot Learning (CoRL), 2017. ",
+        "bbox": [
+            173,
+            232,
+            821,
+            262
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Chelsea Finn and Sergey Levine. Deep visual foresight for planning robot motion. In International Conference on Robotics and Automation (ICRA), 2017. ",
+        "bbox": [
+            174,
+            270,
+            823,
+            300
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Chelsea Finn, Ian Goodfellow, and Sergey Levine. Unsupervised learning for physical interaction through video prediction. In Advances in Neural Information Processing Systems, 2016. ",
+        "bbox": [
+            173,
+            309,
+            823,
+            338
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Katerina Fragkiadaki, Jonathan Huang, Alex Alemi, Sudheendra Vijayanarasimhan, Susanna Ricco, and Rahul Sukthankar. Motion prediction under multimodality with conditional stochastic networks. CoRR, abs/1705.02082, 2017. ",
+        "bbox": [
+            173,
+            347,
+            826,
+            390
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Yuanjun Gao, Evan W Archer, Liam Paninski, and John P Cunningham. Linear dynamical neural population models through nonlinear embeddings. In Advances in Neural Information Processing Systems, pp. 163–171, 2016. ",
+        "bbox": [
+            174,
+            400,
+            825,
+            443
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow. Nips 2016 tutorial: Generative adversarial networks. arXiv preprint arXiv:1701.00160, 2016. ",
+        "bbox": [
+            168,
+            452,
+            825,
+            481
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, 2014. ",
+        "bbox": [
+            173,
+            489,
+            823,
+            534
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. International Conference on Learning Representations (ICLR), 2016. ",
+        "bbox": [
+            173,
+            542,
+            825,
+            599
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Huang, Vivek Rathod, Chen Sun, Menglong Zhu, Anoop Korattikara, Alireza Fathi, Ian Fischer, Zbigniew Wojna, Yang Song, Sergio Guadarrama, et al. Speed/accuracy trade-offs for modern convolutional object detectors. arXiv preprint arXiv:1611.10012, 2016. ",
+        "bbox": [
+            174,
+            608,
+            825,
+            652
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Quan Huynh-Thu and Mohammed Ghanbari. Scope of validity of psnr in image/video quality assessment. Electronics letters, 2008. ",
+        "bbox": [
+            173,
+            661,
+            823,
+            690
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu. Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments. IEEE transactions on pattern analysis and machine intelligence, 36(7), 2014. ",
+        "bbox": [
+            174,
+            699,
+            825,
+            742
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Matthew Johnson, David K Duvenaud, Alex Wiltschko, Ryan P Adams, and Sandeep R Datta. Composing graphical models with neural networks for structured representations and fast inference. In Advances in neural information processing systems, pp. 2946–2954, 2016. ",
+        "bbox": [
+            176,
+            751,
+            821,
+            795
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Nal Kalchbrenner, Aaron van den Oord, Karen Simonyan, Ivo Danihelka, Oriol Vinyals, Alex ¨ Graves, and Koray Kavukcuoglu. Video pixel networks. International Conference on Machine Learning (ICML), 2017. ",
+        "bbox": [
+            174,
+            804,
+            823,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Max Welling. Auto-encoding variational bayes. International Conference on Learning Representations (ICLR), 2014. ",
+        "bbox": [
+            173,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Rahul G Krishnan, Uri Shalit, and David Sontag. Structured inference networks for nonlinear state space models. In AAAI, pp. 2101–2109, 2017. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Yitong Li, Martin Renqiang Min, Dinghan Shen, David Carlson, and Lawrence Carin. Video generation from text. arXiv preprint arXiv:1710.00421, 2017. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Ziwei Liu, Raymond Yeh, Xiaoou Tang, Yiming Liu, and Aseem Agarwala. Video frame synthesis using deep voxel flow. International Conference on Computer Vision (ICCV), 2017. ",
+        "bbox": [
+            173,
+            140,
+            823,
+            171
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "William Lotter, Gabriel Kreiman, and David Cox. Deep predictive coding networks for video prediction and unsupervised learning. International Conference on Learning Representations (ICLR), 2017. ",
+        "bbox": [
+            178,
+            178,
+            823,
+            222
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. International Conference on Learning Representations (ICLR), 2016. ",
+        "bbox": [
+            173,
+            229,
+            823,
+            260
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Junhyuk Oh, Xiaoxiao Guo, Honglak Lee, Richard L Lewis, and Satinder Singh. Action-conditional video prediction using deep networks in atari games. In Advances in Neural Information Processing Systems, 2015. ",
+        "bbox": [
+            173,
+            267,
+            823,
+            310
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "MarcAurelio Ranzato, Arthur Szlam, Joan Bruna, Michael Mathieu, Ronan Collobert, and Sumit Chopra. Video (language) modeling: a baseline for generative models of natural videos. arXiv preprint arXiv:1412.6604, 2014. ",
+        "bbox": [
+            173,
+            319,
+            826,
+            363
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Scott E. Reed, Aaron van den Oord, Nal Kalchbrenner, Sergio Gomez Colmenarejo, Ziyu Wang, ¨ Yutian Chen, Dan Belov, and Nando de Freitas. Parallel multiscale autoregressive density estimation. International Conference on Machine Learning (ICML), 2017. ",
+        "bbox": [
+            174,
+            371,
+            825,
+            414
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Rui Shu, James Brofos, Frank Zhang, Hung Hai Bui, Mohammad Ghavamzadeh, and Mykel Kochenderfer. Stochastic video prediction with conditional density estimation. In ECCV Workshop on Action and Anticipation for Visual Learning, 2016. ",
+        "bbox": [
+            173,
+            422,
+            825,
+            465
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Nitish Srivastava, Elman Mansimov, and Ruslan Salakhudinov. Unsupervised learning of video representations using lstms. In International Conference on Machine Learning, 2015. ",
+        "bbox": [
+            174,
+            473,
+            823,
+            503
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Sergey Tulyakov, Ming-Yu Liu, Xiaodong Yang, and Jan Kautz. Mocogan: Decomposing motion and content for video generation. arXiv preprint arXiv:1707.04993, 2017. ",
+        "bbox": [
+            174,
+            511,
+            823,
+            541
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Carl Vondrick and Antonio Torralba. Generating the future with adversarial transformers. In Computer Vision and Pattern Recognition (CVPR), 2017. ",
+        "bbox": [
+            174,
+            549,
+            821,
+            579
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Anticipating the future by watching unlabeled video. arXiv preprint arXiv:1504.08023, 2015. ",
+        "bbox": [
+            173,
+            587,
+            823,
+            617
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Generating videos with scene dynamics. In Advances In Neural Information Processing Systems, 2016. ",
+        "bbox": [
+            173,
+            625,
+            823,
+            655
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Jacob Walker, Carl Doersch, Abhinav Gupta, and Martial Hebert. An uncertain future: Forecasting from static images using variational autoencoders. In European Conference on Computer Vision, pp. 835–851. Springer, 2016. ",
+        "bbox": [
+            173,
+            662,
+            825,
+            705
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 2004. ",
+        "bbox": [
+            168,
+            713,
+            823,
+            744
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "SHI Xingjian, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional lstm network: A machine learning approach for precipitation nowcasting. In Advances in Neural Information Processing Systems, 2015. ",
+        "bbox": [
+            176,
+            752,
+            823,
+            796
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Tianfan Xue, Jiajun Wu, Katherine Bouman, and Bill Freeman. Visual dynamics: Probabilistic future frame synthesis via cross convolutional networks. In Advances in Neural Information Processing Systems, 2016. ",
+        "bbox": [
+            174,
+            804,
+            823,
+            847
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. International Conference on Computer Vision (ICCV), 2017. ",
+        "bbox": [
+            174,
+            856,
+            825,
+            898
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            375,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 contains details of the network architectures used as generative and inference models. In all of the experiments we used the same set of hyper-parameters which can be found in Table 1. ",
+        "bbox": [
+            171,
+            133,
+            826,
+            162
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a855776096651ab9f8362a058eb449c0b39e3195842c9caab343d585a74deb4.jpg",
+        "table_caption": [
+            "Table 1: Hyper-parameters used for experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Generative Network</td></tr><tr><td>model type batch size learning rate scheduled sampling (k) #of masks</td><td>CDNA 16 0.001 900.0 10</td></tr><tr><td># of iterations InferenceNetwork</td><td>200000</td></tr><tr><td>latent minimumo starting β final β # of latent channels # step 1 iterations</td><td>-5.0 0.0 0.001 1 50000</td></tr><tr><td># step 2 iterations # step 3 iterations</td><td>50000 100000</td></tr><tr><td>Optimization</td><td>ADAM</td></tr><tr><td>Method β1</td><td>0.9</td></tr><tr><td>β2 E</td><td>0.999 1e-8</td></tr></table>",
+        "bbox": [
+            374,
+            202,
+            622,
+            483
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the first step of training, we disable the inference network and instead sample latent values from $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ . In step 2, the latent values will be sampled from the approximated posterior $q _ { \\phi } ( { \\bf z } | { \\bf x } _ { 0 : T } ) =$ $\\mathcal { N } \\big ( \\mu ( \\mathbf { x } _ { 0 : T } ) , \\sigma ( \\mathbf { x } _ { 0 : T } ) \\big )$ . Please note that the inference network approximates $\\log ( \\sigma )$ instead of $\\sigma$ for numerical stability. To gradually switch from Step 2 of training procedure to Step 3, we increase $\\beta$ linearly from its starting value to its end value over the length of training. ",
+        "bbox": [
+            173,
+            498,
+            825,
+            570
+        ],
+        "page_idx": 14
+    }
+]